Subgroups — C₆₇×#61020 #30740271

Order 17152.

1668 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3512	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3509	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3261	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3260	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3240	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3236	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3218	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3216	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3201	C₂	1	4	yes	maximal, complemented
10	C₆₇×#3181	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3172	C₂	1	2	yes	maximal, complemented
12	C₆₇×#3127	C₂	1	2	yes	maximal, complemented
13	C₆₇×#3126	C₂	1	2	yes	maximal, complemented
14	C₆₇×#3120	C₂	1	4	yes	maximal, complemented
15	C₆₇×#2923	C₂	1	1	yes	maximal, characteristic, complemented
16	#557×C₆₇	—	2	2	no	—
17	#556×C₆₇	—	2	2	no	—
18	#553×C₆₇	—	2	2	no	—
19	#553×C₆₇	—	2	2	no	—
20	#553×C₆₇	—	2	2	no	—
21	#552×C₆₇	—	2	2	no	—
22	#552×C₆₇	—	2	2	no	—
23	#552×C₆₇	—	2	2	no	—
24	#541×C₆₇	C₂²	1	2	yes	—
25	#541×C₆₇	C₂²	1	1	yes	characteristic
26	#541×C₆₇	C₂²	1	2	yes	—
27	#541×C₆₇	C₂²	1	1	yes	characteristic
28	#540×C₆₇	C₂²	1	1	yes	characteristic
29	#540×C₆₇	C₂²	1	1	yes	characteristic
30	#537×C₆₇	C₂²	1	2	yes	—
31	#536×C₆₇	C₂²	1	2	yes	—
32	#533×C₆₇	C₂²	1	2	yes	—
33	#533×C₆₇	C₂²	1	2	yes	—
34	#533×C₆₇	C₂²	1	1	yes	characteristic
35	#532×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
37	#529×C₆₇	C₂²	1	2	yes	—
38	#529×C₆₇	C₂²	1	2	yes	—
39	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
40	#520×C₆₇	C₂²	1	2	yes	—
41	#519×C₆₇	C₂²	1	2	yes	—
42	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
43	#501×C₆₇	C₂²	1	2	yes	—
44	#501×C₆₇	C₂²	1	2	yes	—
45	#497×C₆₇	C₂²	1	2	yes	—
46	#497×C₆₇	C₂²	1	2	yes	—
47	#491×C₆₇	C₂²	1	2	yes	—
48	#491×C₆₇	C₂²	1	2	yes	—
49	#489×C₆₇	C₂²	1	2	yes	—
50	#489×C₆₇	C₂²	1	2	yes	—
51	#485×C₆₇	C₂²	1	4	yes	—
52	#484×C₆₇	C₂²	1	4	yes	—
53	#483×C₆₇	C₂²	1	4	yes	—
54	#482×C₆₇	C₂²	1	4	yes	—
55	#479×C₆₇	C₂²	1	4	yes	—
56	#479×C₆₇	C₂²	1	4	yes	—
57	#479×C₆₇	C₂²	1	4	yes	—
58	#478×C₆₇	C₂²	1	4	yes	—
59	#470×C₆₇	C₂²	1	4	yes	—
60	#468×C₆₇	C₂²	1	4	yes	—
61	#467×C₆₇	C₂²	1	4	yes	—
62	#465×C₆₇	C₂²	1	4	yes	—
63	#462×C₆₇	C₂²	1	4	yes	—
64	#462×C₆₇	C₂²	1	4	yes	—
65	#461×C₆₇	C₂²	1	4	yes	—
66	#460×C₆₇	C₂²	1	4	yes	—
67	#452×C₆₇	C₂²	1	4	yes	—
68	#452×C₆₇	C₂²	1	4	yes	—
69	#451×C₆₇	C₂²	1	4	yes	—
70	#448×C₆₇	C₂²	1	4	yes	—
71	#441×C₆₇	C₂²	1	4	yes	—
72	#440×C₆₇	C₂²	1	4	yes	—
73	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	—
74	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	—
75	#433×C₆₇	C₂²	1	2	yes	—
76	#433×C₆₇	C₂²	1	2	yes	—
77	#432×C₆₇	C₂²	1	2	yes	—
78	#432×C₆₇	C₂²	1	2	yes	—
79	#424×C₆₇	C₂²	1	2	yes	—
80	#408×C₆₇	C₂²	1	2	yes	—
81	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
82	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
83	C₂×#41×C₆₇	C₂³	1	2	yes	—
84	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
85	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
86	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
87	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
88	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
89	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
90	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
91	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
92	C₄⋊Q₈×C₆₇	—	2	2	no	—
93	C₄⋊Q₈×C₆₇	—	2	2	no	—
94	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
95	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
97	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
98	#127×C₆₇	C₂³	1	1	yes	characteristic
99	#126×C₆₇	—	2	4	no	—
100	#126×C₆₇	—	2	4	no	—
101	#126×C₆₇	—	2	2	no	—
102	#126×C₆₇	—	2	2	no	—
103	#126×C₆₇	—	2	2	no	—
104	#126×C₆₇	—	2	2	no	—
105	#125×C₆₇	—	2	2	no	—
106	#125×C₆₇	—	2	2	no	—
107	#125×C₆₇	—	2	4	no	—
108	#125×C₆₇	—	2	4	no	—
109	#125×C₆₇	—	2	2	no	—
110	#125×C₆₇	—	2	2	no	—
111	#125×C₆₇	—	2	2	no	—
112	#125×C₆₇	C₂³	1	2	yes	—
113	#125×C₆₇	C₂³	1	1	yes	characteristic
114	#125×C₆₇	C₂³	1	1	yes	characteristic
115	#125×C₆₇	C₂³	1	2	yes	—
116	#124×C₆₇	—	2	2	no	—
117	#124×C₆₇	C₂³	1	2	yes	—
118	#123×C₆₇	—	2	2	no	—
119	#123×C₆₇	—	2	2	no	—
120	#123×C₆₇	—	2	2	no	—
121	#123×C₆₇	—	2	2	no	—
122	#123×C₆₇	—	2	2	no	—
123	#123×C₆₇	—	2	2	no	—
124	C₂²⋊Q₈×C₆₇	—	2	2	no	—
125	C₂²⋊Q₈×C₆₇	—	2	2	no	—
126	C₂²⋊Q₈×C₆₇	—	2	2	no	—
127	C₂²⋊Q₈×C₆₇	—	2	2	no	—
128	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
129	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
130	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
131	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
132	#121×C₆₇	—	2	2	no	—
133	#121×C₆₇	—	2	2	no	—
134	#121×C₆₇	—	2	2	no	—
135	#121×C₆₇	—	2	2	no	—
136	#121×C₆₇	—	2	2	no	—
137	#121×C₆₇	—	2	2	no	—
138	#121×C₆₇	C₂³	1	2	yes	—
139	#121×C₆₇	C₂³	1	2	yes	—
140	C₄×Q₈×C₆₇	—	2	4	no	—
141	C₄×Q₈×C₆₇	—	2	2	no	—
142	C₄×Q₈×C₆₇	—	2	2	no	—
143	C₄×Q₈×C₆₇	—	2	2	no	—
144	C₄×Q₈×C₆₇	D₄	1	2	yes	—
145	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
146	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
147	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
148	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
149	C₄×D₄×C₆₇	—	2	2	no	—
150	C₄×D₄×C₆₇	—	2	2	no	—
151	C₄×D₄×C₆₇	—	2	2	no	—
152	C₄×D₄×C₆₇	—	2	2	no	—
153	C₄×D₄×C₆₇	—	2	2	no	—
154	C₄×D₄×C₆₇	D₄	1	2	yes	—
155	C₄×D₄×C₆₇	D₄	1	2	yes	—
156	C₄×D₄×C₆₇	D₄	1	2	yes	—
157	C₄×D₄×C₆₇	C₂³	1	2	yes	—
158	C₄×D₄×C₆₇	C₂³	1	2	yes	—
159	#117×C₆₇	—	2	2	no	—
160	#117×C₆₇	—	2	2	no	—
161	#117×C₆₇	—	2	2	no	—
162	#117×C₆₇	—	2	4	no	—
163	#117×C₆₇	—	2	2	no	—
164	#117×C₆₇	—	2	2	no	—
165	#117×C₆₇	C₂³	1	2	yes	—
166	#117×C₆₇	C₂³	1	1	yes	characteristic
167	#117×C₆₇	C₂³	1	1	yes	characteristic
168	#117×C₆₇	C₂³	1	1	yes	characteristic
169	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
170	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
171	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
172	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
173	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
174	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
175	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
176	#107×C₆₇	C₂³	1	4	yes	—
177	#107×C₆₇	C₂³	1	4	yes	—
178	#106×C₆₇	C₂³	1	4	yes	—
179	#106×C₆₇	C₂³	1	4	yes	—
180	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
181	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
182	#103×C₆₇	C₂³	1	4	yes	—
183	#103×C₆₇	C₂³	1	4	yes	—
184	#103×C₆₇	C₂³	1	4	yes	—
185	#103×C₆₇	C₂³	1	4	yes	—
186	#103×C₆₇	C₂³	1	4	yes	—
187	#103×C₆₇	C₂³	1	4	yes	—
188	#102×C₆₇	C₂³	1	4	yes	—
189	#102×C₆₇	C₂³	1	4	yes	—
190	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
191	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
192	#97×C₆₇	C₂³	1	2	yes	—
193	#97×C₆₇	C₂³	1	2	yes	—
194	#97×C₆₇	C₂³	1	2	yes	—
195	C₄×C₈×C₆₇	C₂³	1	2	yes	—
196	#41×C₆₇	—	4	2	no	—
197	#41×C₆₇	—	4	4	no	—
198	C₂×Q₈×C₆₇	—	2	2	no	—
199	C₂×Q₈×C₆₇	—	2	2	no	—
200	C₂×Q₈×C₆₇	—	4	4	no	—
201	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
202	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
203	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
204	C₂×D₄×C₆₇	—	2	2	no	—
205	C₂×D₄×C₆₇	—	2	2	no	—
206	C₂×D₄×C₆₇	—	2	2	no	—
207	C₂×D₄×C₆₇	—	2	2	no	—
208	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
209	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
210	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
211	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
212	C₂²×C₄×C₆₇	—	2	2	no	—
213	C₂²×C₄×C₆₇	—	2	2	no	—
214	C₂²×C₄×C₆₇	—	2	2	no	—
215	C₂²×C₄×C₆₇	—	2	2	no	—
216	C₂²×C₄×C₆₇	—	2	2	no	—
217	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
218	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
219	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
220	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
221	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
222	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
223	Q₁₆×C₆₇	—	4	4	no	—
224	Q₁₆×C₆₇	—	4	4	no	—
225	SD₁₆×C₆₇	—	4	4	no	—
226	SD₁₆×C₆₇	—	4	4	no	—
227	M₁₆×C₆₇	—	4	2	no	—
228	C₂×C₈×C₆₇	—	4	2	no	—
229	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
230	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
231	C₄⋊C₄×C₆₇	—	2	2	no	—
232	C₄⋊C₄×C₆₇	—	2	2	no	—
233	C₄⋊C₄×C₆₇	—	2	2	no	—
234	C₄⋊C₄×C₆₇	—	2	2	no	—
235	C₄⋊C₄×C₆₇	—	2	2	no	—
236	C₄⋊C₄×C₆₇	—	2	2	no	—
237	C₄⋊C₄×C₆₇	—	2	2	no	—
238	C₄⋊C₄×C₆₇	—	2	2	no	—
239	C₄⋊C₄×C₆₇	—	2	2	no	—
240	C₄⋊C₄×C₆₇	—	2	4	no	—
241	C₄⋊C₄×C₆₇	—	2	2	no	—
242	C₄⋊C₄×C₆₇	—	2	4	no	—
243	C₄⋊C₄×C₆₇	—	2	4	no	—
244	C₄⋊C₄×C₆₇	—	2	2	no	—
245	C₄⋊C₄×C₆₇	—	2	2	no	—
246	C₄⋊C₄×C₆₇	—	2	2	no	—
247	C₄⋊C₄×C₆₇	—	2	4	no	—
248	C₄⋊C₄×C₆₇	—	2	4	no	—
249	C₄⋊C₄×C₆₇	—	2	4	no	—
250	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
251	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
252	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
253	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
254	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
255	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
256	C₂²⋊C₄×C₆₇	—	2	2	no	—
257	C₂²⋊C₄×C₆₇	—	2	4	no	—
258	C₂²⋊C₄×C₆₇	—	2	4	no	—
259	C₂²⋊C₄×C₆₇	—	2	2	no	—
260	C₂²⋊C₄×C₆₇	—	2	2	no	—
261	C₂²⋊C₄×C₆₇	—	2	2	no	—
262	C₂²⋊C₄×C₆₇	—	2	2	no	—
263	C₂²⋊C₄×C₆₇	—	2	2	no	—
264	C₂²⋊C₄×C₆₇	—	2	2	no	—
265	C₂²⋊C₄×C₆₇	—	2	2	no	—
266	C₂²⋊C₄×C₆₇	—	2	2	no	—
267	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
268	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
269	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
270	C₄²×C₆₇	—	2	2	no	—
271	C₄²×C₆₇	—	2	2	no	—
272	C₄²×C₆₇	—	2	4	no	—
273	C₄²×C₆₇	—	2	4	no	—
274	C₄²×C₆₇	—	2	2	no	—
275	C₄²×C₆₇	C₂×D₄	1	2	yes	—
276	C₄²×C₆₇	C₂×D₄	1	2	yes	—
277	C₄²×C₆₇	C₂×D₄	1	2	yes	—
278	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
279	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
280	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
281	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
282	C₂³×C₆₇	—	2	2	no	—
283	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
284	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
285	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
286	Q₈×C₆₇	—	2	4	no	—
287	Q₈×C₆₇	—	2	4	no	—
288	Q₈×C₆₇	—	2	4	no	—
289	Q₈×C₆₇	—	4	1	no	—
290	Q₈×C₆₇	—	4	4	no	—
291	Q₈×C₆₇	—	4	4	no	—
292	D₄×C₆₇	—	2	4	no	—
293	D₄×C₆₇	—	4	1	no	—
294	D₄×C₆₇	—	4	1	no	—
295	D₄×C₆₇	—	4	1	no	—
296	D₄×C₆₇	—	4	4	no	—
297	D₄×C₆₇	—	8	2	no	—
298	D₄×C₆₇	—	8	2	no	—
299	D₄×C₆₇	—	8	2	no	—
300	C₂×C₄×C₆₇	—	2	2	no	—
301	C₂×C₄×C₆₇	—	2	2	no	—
302	C₂×C₄×C₆₇	—	2	2	no	—
303	C₂×C₄×C₆₇	—	2	4	no	—
304	C₂×C₄×C₆₇	—	2	2	no	—
305	C₂×C₄×C₆₇	—	2	4	no	—
306	C₂×C₄×C₆₇	—	2	2	no	—
307	C₂×C₄×C₆₇	—	2	2	no	—
308	C₂×C₄×C₆₇	—	4	2	no	—
309	C₂×C₄×C₆₇	—	4	2	no	—
310	C₂×C₄×C₆₇	—	4	2	no	—
311	C₂×C₄×C₆₇	—	4	4	no	—
312	C₂×C₄×C₆₇	—	4	4	no	—
313	C₂×C₄×C₆₇	—	8	2	no	—
314	C₂×C₄×C₆₇	—	8	2	no	—
315	C₂×C₄×C₆₇	—	8	2	no	—
316	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
317	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
318	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
319	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
320	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
321	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
322	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
323	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
324	C₈×C₆₇	—	2	4	no	—
325	C₈×C₆₇	—	2	4	no	—
326	C₂²×C₆₇	—	2	2	no	—
327	C₂²×C₆₇	—	2	2	no	—
328	C₂²×C₆₇	—	2	2	no	—
329	C₂²×C₆₇	—	2	1	no	—
330	C₂²×C₆₇	—	2	1	no	—
331	C₂²×C₆₇	—	2	1	no	—
332	C₂²×C₆₇	—	4	2	no	—
333	C₂²×C₆₇	—	4	2	no	—
334	C₂²×C₆₇	—	4	2	no	—
335	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
336	C₄×C₆₇	—	2	2	no	—
337	C₄×C₆₇	—	2	2	no	—
338	C₄×C₆₇	—	2	2	no	—
339	C₄×C₆₇	—	2	2	no	—
340	C₄×C₆₇	—	2	1	no	—
341	C₄×C₆₇	—	2	1	no	—
342	C₄×C₆₇	—	2	1	no	—
343	C₄×C₆₇	—	4	2	no	—
344	C₄×C₆₇	—	4	2	no	—
345	C₄×C₆₇	—	4	4	no	—
346	C₄×C₆₇	—	4	2	no	—
347	C₄×C₆₇	—	4	4	no	—
348	C₄×C₆₇	#585	1	2	yes	—
349	#61020	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
350	C₂×C₆₇	—	4	1	no	—
351	C₂×C₆₇	—	4	1	no	—
352	C₂×C₆₇	—	4	1	no	—
353	C₂×C₆₇	—	8	2	no	—
354	C₂×C₆₇	#3449	1	1	yes	characteristic
355	C₂×C₆₇	#3586	1	2	yes	—
356	#3512	C₂×C₆₇	1	1	yes	characteristic
357	#3509	C₂×C₆₇	1	1	yes	characteristic, complemented
358	#3261	C₂×C₆₇	1	2	yes	complemented
359	#3260	C₂×C₆₇	1	2	yes	complemented
360	#3240	C₂×C₆₇	1	2	yes	complemented
361	#3236	C₂×C₆₇	1	2	yes	complemented
362	#3218	C₂×C₆₇	1	2	yes	complemented
363	#3216	C₂×C₆₇	1	2	yes	complemented
364	#3201	C₂×C₆₇	1	4	yes	complemented
365	#3181	C₂×C₆₇	1	2	yes	complemented
366	#3172	C₂×C₆₇	1	2	yes	complemented
367	#3127	C₂×C₆₇	1	2	yes	complemented
368	#3126	C₂×C₆₇	1	2	yes	complemented
369	#3120	C₂×C₆₇	1	4	yes	complemented
370	#2923	C₂×C₆₇	1	1	yes	characteristic, complemented
371	C₆₇	#61020	1	1	yes	characteristic, direct product factor, Sylow
372	#557	—	2	2	no	—
373	#556	—	2	2	no	—
374	#553	—	2	2	no	—
375	#553	—	2	2	no	—
376	#553	—	2	2	no	—
377	#552	—	2	2	no	—
378	#552	—	2	2	no	—
379	#552	—	2	2	no	—
380	#541	C₂²×C₆₇	1	2	yes	—
381	#541	C₂²×C₆₇	1	1	yes	characteristic
382	#541	C₂²×C₆₇	1	2	yes	—
383	#541	C₂²×C₆₇	1	1	yes	characteristic
384	#540	C₂²×C₆₇	1	1	yes	characteristic
385	#540	C₂²×C₆₇	1	1	yes	characteristic
386	#537	C₂²×C₆₇	1	2	yes	—
387	#536	C₂²×C₆₇	1	2	yes	—
388	#533	C₂²×C₆₇	1	2	yes	—
389	#533	C₂²×C₆₇	1	2	yes	—
390	#533	C₂²×C₆₇	1	1	yes	characteristic
391	#532	C₂²×C₆₇	1	1	yes	characteristic
392	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
393	#529	C₂²×C₆₇	1	2	yes	—
394	#529	C₂²×C₆₇	1	2	yes	—
395	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
396	#520	C₂²×C₆₇	1	2	yes	—
397	#519	C₂²×C₆₇	1	2	yes	—
398	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
399	#501	C₂²×C₆₇	1	2	yes	—
400	#501	C₂²×C₆₇	1	2	yes	—
401	#497	C₂²×C₆₇	1	2	yes	—
402	#497	C₂²×C₆₇	1	2	yes	—
403	#491	C₂²×C₆₇	1	2	yes	—
404	#491	C₂²×C₆₇	1	2	yes	—
405	#489	C₂²×C₆₇	1	2	yes	—
406	#489	C₂²×C₆₇	1	2	yes	—
407	#485	C₂²×C₆₇	1	4	yes	—
408	#484	C₂²×C₆₇	1	4	yes	—
409	#483	C₂²×C₆₇	1	4	yes	—
410	#482	C₂²×C₆₇	1	4	yes	—
411	#479	C₂²×C₆₇	1	4	yes	—
412	#479	C₂²×C₆₇	1	4	yes	—
413	#479	C₂²×C₆₇	1	4	yes	—
414	#478	C₂²×C₆₇	1	4	yes	—
415	#470	C₂²×C₆₇	1	4	yes	—
416	#468	C₂²×C₆₇	1	4	yes	—
417	#467	C₂²×C₆₇	1	4	yes	—
418	#465	C₂²×C₆₇	1	4	yes	—
419	#462	C₂²×C₆₇	1	4	yes	—
420	#462	C₂²×C₆₇	1	4	yes	—
421	#461	C₂²×C₆₇	1	4	yes	—
422	#460	C₂²×C₆₇	1	4	yes	—
423	#452	C₂²×C₆₇	1	4	yes	—
424	#452	C₂²×C₆₇	1	4	yes	—
425	#451	C₂²×C₆₇	1	4	yes	—
426	#448	C₂²×C₆₇	1	4	yes	—
427	#441	C₂²×C₆₇	1	4	yes	—
428	#440	C₂²×C₆₇	1	4	yes	—
429	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	—
430	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	—
431	#433	C₂²×C₆₇	1	2	yes	—
432	#433	C₂²×C₆₇	1	2	yes	—
433	#432	C₂²×C₆₇	1	2	yes	—
434	#432	C₂²×C₆₇	1	2	yes	—
435	#424	C₂²×C₆₇	1	2	yes	—
436	#408	C₂²×C₆₇	1	2	yes	—
437	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
438	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
439	C₂×#41	C₂³×C₆₇	1	2	yes	—
440	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
441	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
442	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
443	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
444	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
445	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
446	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
447	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
448	C₄⋊Q₈	—	2	2	no	—
449	C₄⋊Q₈	—	2	2	no	—
450	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
451	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
452	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
453	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
454	#127	C₂³×C₆₇	1	1	yes	characteristic
455	#126	—	2	4	no	—
456	#126	—	2	4	no	—
457	#126	—	2	2	no	—
458	#126	—	2	2	no	—
459	#126	—	2	2	no	—
460	#126	—	2	2	no	—
461	#125	—	2	2	no	—
462	#125	—	2	2	no	—
463	#125	—	2	4	no	—
464	#125	—	2	4	no	—
465	#125	—	2	2	no	—
466	#125	—	2	2	no	—
467	#125	—	2	2	no	—
468	#125	C₂³×C₆₇	1	2	yes	—
469	#125	C₂³×C₆₇	1	1	yes	characteristic
470	#125	C₂³×C₆₇	1	1	yes	characteristic
471	#125	C₂³×C₆₇	1	2	yes	—
472	#124	—	2	2	no	—
473	#124	C₂³×C₆₇	1	2	yes	—
474	#123	—	2	2	no	—
475	#123	—	2	2	no	—
476	#123	—	2	2	no	—
477	#123	—	2	2	no	—
478	#123	—	2	2	no	—
479	#123	—	2	2	no	—
480	C₂²⋊Q₈	—	2	2	no	—
481	C₂²⋊Q₈	—	2	2	no	—
482	C₂²⋊Q₈	—	2	2	no	—
483	C₂²⋊Q₈	—	2	2	no	—
484	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
485	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
486	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
487	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
488	#121	—	2	2	no	—
489	#121	—	2	2	no	—
490	#121	—	2	2	no	—
491	#121	—	2	2	no	—
492	#121	—	2	2	no	—
493	#121	—	2	2	no	—
494	#121	C₂³×C₆₇	1	2	yes	—
495	#121	C₂³×C₆₇	1	2	yes	—
496	C₄×Q₈	—	2	4	no	—
497	C₄×Q₈	—	2	2	no	—
498	C₄×Q₈	—	2	2	no	—
499	C₄×Q₈	—	2	2	no	—
500	C₄×Q₈	D₄×C₆₇	1	2	yes	—
501	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
502	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
503	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
504	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
505	C₄×D₄	—	2	2	no	—
506	C₄×D₄	—	2	2	no	—
507	C₄×D₄	—	2	2	no	—
508	C₄×D₄	—	2	2	no	—
509	C₄×D₄	—	2	2	no	—
510	C₄×D₄	D₄×C₆₇	1	2	yes	—
511	C₄×D₄	D₄×C₆₇	1	2	yes	—
512	C₄×D₄	D₄×C₆₇	1	2	yes	—
513	C₄×D₄	C₂³×C₆₇	1	2	yes	—
514	C₄×D₄	C₂³×C₆₇	1	2	yes	—
515	#117	—	2	2	no	—
516	#117	—	2	2	no	—
517	#117	—	2	2	no	—
518	#117	—	2	4	no	—
519	#117	—	2	2	no	—
520	#117	—	2	2	no	—
521	#117	C₂³×C₆₇	1	2	yes	—
522	#117	C₂³×C₆₇	1	1	yes	characteristic
523	#117	C₂³×C₆₇	1	1	yes	characteristic
524	#117	C₂³×C₆₇	1	1	yes	characteristic
525	C₂×C₄⋊C₄	—	2	2	no	—
526	C₂×C₄⋊C₄	—	2	2	no	—
527	C₂×C₄⋊C₄	—	2	2	no	—
528	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
529	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
530	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
531	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
532	#107	C₂³×C₆₇	1	4	yes	—
533	#107	C₂³×C₆₇	1	4	yes	—
534	#106	C₂³×C₆₇	1	4	yes	—
535	#106	C₂³×C₆₇	1	4	yes	—
536	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
537	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
538	#103	C₂³×C₆₇	1	4	yes	—
539	#103	C₂³×C₆₇	1	4	yes	—
540	#103	C₂³×C₆₇	1	4	yes	—
541	#103	C₂³×C₆₇	1	4	yes	—
542	#103	C₂³×C₆₇	1	4	yes	—
543	#103	C₂³×C₆₇	1	4	yes	—
544	#102	C₂³×C₆₇	1	4	yes	—
545	#102	C₂³×C₆₇	1	4	yes	—
546	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
547	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
548	#97	C₂³×C₆₇	1	2	yes	—
549	#97	C₂³×C₆₇	1	2	yes	—
550	#97	C₂³×C₆₇	1	2	yes	—
551	C₄×C₈	C₂³×C₆₇	1	2	yes	—
552	#41	—	4	2	no	—
553	#41	—	4	4	no	—
554	C₂×Q₈	—	2	2	no	—
555	C₂×Q₈	—	2	2	no	—
556	C₂×Q₈	—	4	4	no	—
557	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
558	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
559	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
560	C₂×D₄	—	2	2	no	—
561	C₂×D₄	—	2	2	no	—
562	C₂×D₄	—	2	2	no	—
563	C₂×D₄	—	2	2	no	—
564	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
565	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
566	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
567	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
568	C₂²×C₄	—	2	2	no	—
569	C₂²×C₄	—	2	2	no	—
570	C₂²×C₄	—	2	2	no	—
571	C₂²×C₄	—	2	2	no	—
572	C₂²×C₄	—	2	2	no	—
573	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
574	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
575	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
576	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
577	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
578	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
579	Q₁₆	—	4	4	no	—
580	Q₁₆	—	4	4	no	—
581	SD₁₆	—	4	4	no	—
582	SD₁₆	—	4	4	no	—
583	M₁₆	—	4	2	no	—
584	C₂×C₈	—	4	2	no	—
585	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
586	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	—	2	2	no	—
589	C₄⋊C₄	—	2	2	no	—
590	C₄⋊C₄	—	2	2	no	—
591	C₄⋊C₄	—	2	2	no	—
592	C₄⋊C₄	—	2	2	no	—
593	C₄⋊C₄	—	2	2	no	—
594	C₄⋊C₄	—	2	2	no	—
595	C₄⋊C₄	—	2	2	no	—
596	C₄⋊C₄	—	2	4	no	—
597	C₄⋊C₄	—	2	2	no	—
598	C₄⋊C₄	—	2	4	no	—
599	C₄⋊C₄	—	2	4	no	—
600	C₄⋊C₄	—	2	2	no	—
601	C₄⋊C₄	—	2	2	no	—
602	C₄⋊C₄	—	2	2	no	—
603	C₄⋊C₄	—	2	4	no	—
604	C₄⋊C₄	—	2	4	no	—
605	C₄⋊C₄	—	2	4	no	—
606	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
607	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
608	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
609	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
610	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
611	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
612	C₂²⋊C₄	—	2	2	no	—
613	C₂²⋊C₄	—	2	4	no	—
614	C₂²⋊C₄	—	2	4	no	—
615	C₂²⋊C₄	—	2	2	no	—
616	C₂²⋊C₄	—	2	2	no	—
617	C₂²⋊C₄	—	2	2	no	—
618	C₂²⋊C₄	—	2	2	no	—
619	C₂²⋊C₄	—	2	2	no	—
620	C₂²⋊C₄	—	2	2	no	—
621	C₂²⋊C₄	—	2	2	no	—
622	C₂²⋊C₄	—	2	2	no	—
623	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
624	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
625	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
626	C₄²	—	2	2	no	—
627	C₄²	—	2	2	no	—
628	C₄²	—	2	4	no	—
629	C₄²	—	2	4	no	—
630	C₄²	—	2	2	no	—
631	C₄²	C₂×D₄×C₆₇	1	2	yes	—
632	C₄²	C₂×D₄×C₆₇	1	2	yes	—
633	C₄²	C₂×D₄×C₆₇	1	2	yes	—
634	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
635	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
636	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
637	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
638	C₂³	—	2	2	no	—
639	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
640	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
641	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
642	Q₈	—	2	4	no	—
643	Q₈	—	2	4	no	—
644	Q₈	—	2	4	no	—
645	Q₈	—	4	1	no	—
646	Q₈	—	4	4	no	—
647	Q₈	—	4	4	no	—
648	D₄	—	2	4	no	—
649	D₄	—	4	1	no	—
650	D₄	—	4	1	no	—
651	D₄	—	4	1	no	—
652	D₄	—	4	4	no	—
653	D₄	—	8	2	no	—
654	D₄	—	8	2	no	—
655	D₄	—	8	2	no	—
656	C₂×C₄	—	2	2	no	—
657	C₂×C₄	—	2	2	no	—
658	C₂×C₄	—	2	2	no	—
659	C₂×C₄	—	2	4	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	2	4	no	—
662	C₂×C₄	—	2	2	no	—
663	C₂×C₄	—	2	2	no	—
664	C₂×C₄	—	4	2	no	—
665	C₂×C₄	—	4	2	no	—
666	C₂×C₄	—	4	2	no	—
667	C₂×C₄	—	4	4	no	—
668	C₂×C₄	—	4	4	no	—
669	C₂×C₄	—	8	2	no	—
670	C₂×C₄	—	8	2	no	—
671	C₂×C₄	—	8	2	no	—
672	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
673	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
674	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
675	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
676	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
677	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
678	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
679	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
680	C₈	—	2	4	no	—
681	C₈	—	2	4	no	—
682	C₂²	—	2	2	no	—
683	C₂²	—	2	2	no	—
684	C₂²	—	2	2	no	—
685	C₂²	—	2	1	no	—
686	C₂²	—	2	1	no	—
687	C₂²	—	2	1	no	—
688	C₂²	—	4	2	no	—
689	C₂²	—	4	2	no	—
690	C₂²	—	4	2	no	—
691	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
692	C₄	—	2	2	no	—
693	C₄	—	2	2	no	—
694	C₄	—	2	2	no	—
695	C₄	—	2	2	no	—
696	C₄	—	2	1	no	—
697	C₄	—	2	1	no	—
698	C₄	—	2	1	no	—
699	C₄	—	4	2	no	—
700	C₄	—	4	2	no	—
701	C₄	—	4	4	no	—
702	C₄	—	4	2	no	—
703	C₄	—	4	4	no	—
704	C₄	#585×C₆₇	1	2	yes	—
705	C₂	—	4	1	no	—
706	C₂	—	4	1	no	—
707	C₂	—	4	1	no	—
708	C₂	—	8	2	no	—
709	C₂	C₆₇×#3449	1	1	yes	characteristic
710	C₂	C₆₇×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₆₇×#3512	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×#41×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#3509	C₂	semidirect
3. C₆₇×#3261	C₂	semidirect
4. C₆₇×#3260	C₂	semidirect
5. C₆₇×#3240	C₂	semidirect
6. C₆₇×#3236	C₂	semidirect
7. C₆₇×#3218	C₂	semidirect
8. C₆₇×#3216	C₂	semidirect
9. C₆₇×#3201	C₂	semidirect
10. C₆₇×#3181	C₂	semidirect
11. C₆₇×#3172	C₂	semidirect
12. C₆₇×#3127	C₂	semidirect
13. C₆₇×#3126	C₂	semidirect
14. C₆₇×#3120	C₂	semidirect
15. C₆₇×#2923	C₂	semidirect
349. #61020	C₆₇	direct (N × H)
357. #3509	C₂×C₆₇	semidirect
358. #3261	C₂×C₆₇	semidirect
359. #3260	C₂×C₆₇	semidirect
360. #3240	C₂×C₆₇	semidirect
361. #3236	C₂×C₆₇	semidirect
362. #3218	C₂×C₆₇	semidirect
363. #3216	C₂×C₆₇	semidirect
364. #3201	C₂×C₆₇	semidirect
365. #3181	C₂×C₆₇	semidirect
366. #3172	C₂×C₆₇	semidirect
367. #3127	C₂×C₆₇	semidirect
368. #3126	C₂×C₆₇	semidirect
369. #3120	C₂×C₆₇	semidirect
370. #2923	C₂×C₆₇	semidirect
371. C₆₇	#61020	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3512	2
C₆₇×#3509	2
C₆₇×#3261	2
C₆₇×#3260	2
C₆₇×#3240	2
C₆₇×#3236	2
C₆₇×#3218	2
C₆₇×#3216	2
C₆₇×#3201	2
C₆₇×#3181	2
C₆₇×#3172	2
C₆₇×#3127	2
C₆₇×#3126	2
C₆₇×#3120	2
C₆₇×#2923	2
#61020	67

Subgroups — C67×#61020 #30740271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61020 #30740271