Subgroups — C₆₇×#61007 #30740258

Order 17152.

1704 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3510	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3263	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3262	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3233	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3211	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3202	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#3200	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3199	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#3183	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3168	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3121	C₂	1	4	yes	maximal, complemented
12	C₆₇×#3118	C₂	1	4	yes	maximal, complemented
13	C₆₇×#2922	C₂	1	1	yes	maximal, characteristic, complemented
14	#559×C₆₇	—	2	4	no	—
15	#556×C₆₇	—	2	4	no	—
16	#551×C₆₇	—	2	4	no	—
17	#551×C₆₇	—	2	4	no	—
18	#542×C₆₇	C₂²	1	4	yes	—
19	#542×C₆₇	C₂²	1	4	yes	complemented
20	#536×C₆₇	C₂²	1	2	yes	—
21	#535×C₆₇	C₂²	1	2	yes	—
22	#533×C₆₇	C₂²	1	2	yes	complemented
23	#529×C₆₇	C₂²	1	4	yes	—
24	#529×C₆₇	C₂²	1	4	yes	complemented
25	C₂×#124×C₆₇	C₂²	1	2	yes	complemented
26	#519×C₆₇	C₂²	1	2	yes	complemented
27	#518×C₆₇	C₂²	1	2	yes	complemented
28	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
29	#499×C₆₇	C₂²	1	2	yes	complemented
30	#499×C₆₇	C₂²	1	2	yes	complemented
31	#495×C₆₇	C₂²	1	2	yes	—
32	#495×C₆₇	C₂²	1	2	yes	—
33	#488×C₆₇	C₂²	1	4	yes	complemented
34	#488×C₆₇	C₂²	1	4	yes	complemented
35	#485×C₆₇	C₂²	1	4	yes	complemented
36	#485×C₆₇	C₂²	1	4	yes	complemented
37	#482×C₆₇	C₂²	1	4	yes	—
38	#482×C₆₇	C₂²	1	4	yes	—
39	#477×C₆₇	C₂²	1	4	yes	complemented
40	#476×C₆₇	C₂²	1	4	yes	complemented
41	#475×C₆₇	C₂²	1	4	yes	complemented
42	#474×C₆₇	C₂²	1	4	yes	complemented
43	#470×C₆₇	C₂²	1	2	yes	complemented
44	#469×C₆₇	C₂²	1	2	yes	—
45	#468×C₆₇	C₂²	1	4	yes	complemented
46	#467×C₆₇	C₂²	1	2	yes	complemented
47	#466×C₆₇	C₂²	1	2	yes	—
48	#465×C₆₇	C₂²	1	4	yes	complemented
49	#464×C₆₇	C₂²	1	4	yes	complemented
50	#464×C₆₇	C₂²	1	4	yes	complemented
51	#463×C₆₇	C₂²	1	4	yes	—
52	#463×C₆₇	C₂²	1	4	yes	—
53	#451×C₆₇	C₂²	1	4	yes	complemented
54	#450×C₆₇	C₂²	1	4	yes	complemented
55	#448×C₆₇	C₂²	1	4	yes	—
56	#447×C₆₇	C₂²	1	4	yes	—
57	#442×C₆₇	C₂²	1	2	yes	—
58	#441×C₆₇	C₂²	1	2	yes	complemented
59	#440×C₆₇	C₂²	1	4	yes	complemented
60	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	complemented
61	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
62	C₄×D₈×C₆₇	C₂²	1	2	yes	—
63	#433×C₆₇	C₂²	1	4	yes	complemented
64	#431×C₆₇	C₂²	1	4	yes	complemented
65	#424×C₆₇	C₂²	1	1	yes	characteristic, complemented
66	#424×C₆₇	C₂²	1	1	yes	characteristic, complemented
67	#408×C₆₇	C₂²	1	1	yes	characteristic
68	#408×C₆₇	C₂²	1	1	yes	characteristic
69	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
70	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
71	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
73	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
74	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
75	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
76	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
77	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
78	C₂×D₈×C₆₇	C₂³	1	2	yes	—
79	C₂×D₈×C₆₇	C₂³	1	2	yes	—
80	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₄⋊Q₈×C₆₇	—	2	2	no	—
85	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
86	#127×C₆₇	—	2	2	no	—
87	#126×C₆₇	—	2	4	no	—
88	#126×C₆₇	—	2	4	no	—
89	#126×C₆₇	—	2	4	no	—
90	#126×C₆₇	—	2	4	no	—
91	#125×C₆₇	C₂³	1	4	yes	—
92	#124×C₆₇	—	2	2	no	—
93	#124×C₆₇	—	2	2	no	—
94	#124×C₆₇	—	2	4	no	—
95	#124×C₆₇	—	2	4	no	—
96	#124×C₆₇	—	2	4	no	—
97	#124×C₆₇	C₂³	1	4	yes	—
98	#124×C₆₇	C₂³	1	4	yes	—
99	#123×C₆₇	—	2	2	no	—
100	#123×C₆₇	—	2	2	no	—
101	#123×C₆₇	—	2	4	no	—
102	#123×C₆₇	—	2	4	no	—
103	#123×C₆₇	—	2	4	no	—
104	#123×C₆₇	—	2	4	no	—
105	C₂²⋊Q₈×C₆₇	—	2	4	no	—
106	C₂²⋊Q₈×C₆₇	—	2	4	no	—
107	C₂²⋊Q₈×C₆₇	—	2	4	no	—
108	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
109	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
110	#121×C₆₇	C₂³	1	4	yes	—
111	#121×C₆₇	C₂³	1	4	yes	—
112	#120×C₆₇	—	2	2	no	—
113	#120×C₆₇	—	2	2	no	—
114	C₄×Q₈×C₆₇	—	2	4	no	—
115	C₄×Q₈×C₆₇	—	2	4	no	—
116	C₄×Q₈×C₆₇	D₄	1	2	yes	—
117	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
118	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
119	C₄×D₄×C₆₇	—	2	4	no	—
120	C₄×D₄×C₆₇	—	2	4	no	—
121	C₄×D₄×C₆₇	—	2	4	no	—
122	C₄×D₄×C₆₇	D₄	1	2	yes	—
123	C₄×D₄×C₆₇	D₄	1	4	yes	—
124	C₄×D₄×C₆₇	C₂³	1	4	yes	—
125	C₄×D₄×C₆₇	C₂³	1	4	yes	—
126	#117×C₆₇	—	2	4	no	—
127	#117×C₆₇	—	2	4	no	—
128	#117×C₆₇	—	2	4	no	—
129	#117×C₆₇	C₂³	1	2	yes	—
130	#117×C₆₇	C₂³	1	2	yes	—
131	#117×C₆₇	C₂³	1	2	yes	—
132	#117×C₆₇	C₂³	1	1	yes	characteristic
133	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
134	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
135	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
136	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
137	#107×C₆₇	C₂³	1	2	yes	—
138	#107×C₆₇	C₂³	1	2	yes	—
139	#107×C₆₇	C₂³	1	4	yes	—
140	#106×C₆₇	C₂³	1	4	yes	—
141	#106×C₆₇	C₂³	1	2	yes	—
142	#106×C₆₇	C₂³	1	2	yes	—
143	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
144	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
145	#103×C₆₇	C₂³	1	4	yes	—
146	#103×C₆₇	C₂³	1	4	yes	—
147	#103×C₆₇	C₂³	1	4	yes	—
148	#103×C₆₇	C₂³	1	4	yes	—
149	#102×C₆₇	C₂³	1	4	yes	—
150	#102×C₆₇	C₂³	1	4	yes	—
151	#102×C₆₇	C₂³	1	4	yes	—
152	#102×C₆₇	C₂³	1	4	yes	—
153	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
154	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
155	#97×C₆₇	C₂³	1	2	yes	—
156	C₄×C₈×C₆₇	C₂³	1	2	yes	—
157	C₄×C₈×C₆₇	C₂³	1	2	yes	—
158	C₄×C₈×C₆₇	C₂³	1	2	yes	—
159	C₂⁴×C₆₇	—	2	2	no	—
160	#41×C₆₇	—	4	1	no	—
161	#41×C₆₇	—	4	1	no	—
162	C₂×Q₈×C₆₇	—	2	2	no	—
163	C₂×Q₈×C₆₇	—	2	4	no	—
164	C₂×Q₈×C₆₇	—	4	2	no	—
165	C₂×Q₈×C₆₇	—	4	2	no	—
166	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
167	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
168	C₂×D₄×C₆₇	—	2	4	no	—
169	C₂×D₄×C₆₇	—	2	4	no	—
170	C₂×D₄×C₆₇	—	2	2	no	—
171	C₂×D₄×C₆₇	—	4	2	no	—
172	C₂×D₄×C₆₇	—	4	2	no	—
173	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
175	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₆₇	—	2	2	no	—
177	C₂²×C₄×C₆₇	—	2	4	no	—
178	C₂²×C₄×C₆₇	—	2	4	no	—
179	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
180	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
183	Q₁₆×C₆₇	—	4	2	no	—
184	Q₁₆×C₆₇	—	4	2	no	—
185	SD₁₆×C₆₇	—	4	4	no	—
186	SD₁₆×C₆₇	—	4	4	no	—
187	D₈×C₆₇	—	4	2	no	—
188	D₈×C₆₇	—	4	2	no	—
189	M₁₆×C₆₇	—	4	1	no	—
190	M₁₆×C₆₇	—	4	1	no	—
191	C₂×C₈×C₆₇	—	4	1	no	—
192	C₂×C₈×C₆₇	—	4	1	no	—
193	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
194	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
195	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
196	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
197	C₄⋊C₄×C₆₇	—	2	4	no	—
198	C₄⋊C₄×C₆₇	—	2	4	no	—
199	C₄⋊C₄×C₆₇	—	2	4	no	—
200	C₄⋊C₄×C₆₇	—	2	4	no	—
201	C₄⋊C₄×C₆₇	—	2	4	no	—
202	C₄⋊C₄×C₆₇	—	2	4	no	—
203	C₄⋊C₄×C₆₇	—	2	4	no	—
204	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
205	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
206	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
207	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
208	C₂²⋊C₄×C₆₇	—	2	2	no	—
209	C₂²⋊C₄×C₆₇	—	2	2	no	—
210	C₂²⋊C₄×C₆₇	—	2	4	no	—
211	C₂²⋊C₄×C₆₇	—	2	4	no	—
212	C₂²⋊C₄×C₆₇	—	2	4	no	—
213	C₂²⋊C₄×C₆₇	—	2	4	no	—
214	C₂²⋊C₄×C₆₇	—	2	4	no	—
215	C₂²⋊C₄×C₆₇	—	2	4	no	—
216	C₂²⋊C₄×C₆₇	—	2	4	no	—
217	C₂²⋊C₄×C₆₇	—	2	4	no	—
218	C₂²⋊C₄×C₆₇	—	2	4	no	—
219	C₂²⋊C₄×C₆₇	—	2	4	no	—
220	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
221	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
222	C₄²×C₆₇	—	2	4	no	—
223	C₄²×C₆₇	—	2	4	no	—
224	C₄²×C₆₇	—	2	2	no	—
225	C₄²×C₆₇	—	2	2	no	—
226	C₄²×C₆₇	C₂×D₄	1	2	yes	—
227	C₄²×C₆₇	C₂×D₄	1	4	yes	—
228	C₄²×C₆₇	C₂⁴	1	2	yes	—
229	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
230	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
231	C₂³×C₆₇	—	2	4	no	—
232	C₂³×C₆₇	—	4	4	no	—
233	C₂³×C₆₇	—	8	4	no	—
234	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
235	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
236	Q₈×C₆₇	—	2	4	no	—
237	Q₈×C₆₇	—	2	4	no	—
238	Q₈×C₆₇	—	4	1	no	—
239	Q₈×C₆₇	—	4	4	no	—
240	Q₈×C₆₇	—	8	4	no	—
241	D₄×C₆₇	—	2	4	no	—
242	D₄×C₆₇	—	2	4	no	—
243	D₄×C₆₇	—	4	2	no	—
244	D₄×C₆₇	—	4	1	no	—
245	D₄×C₆₇	—	4	4	no	—
246	D₄×C₆₇	—	8	4	no	—
247	D₄×C₆₇	—	8	4	no	—
248	C₂×C₄×C₆₇	—	2	1	no	—
249	C₂×C₄×C₆₇	—	2	1	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	2	2	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	4	no	—
255	C₂×C₄×C₆₇	—	4	2	no	—
256	C₂×C₄×C₆₇	—	4	4	no	—
257	C₂×C₄×C₆₇	—	4	4	no	—
258	C₂×C₄×C₆₇	—	8	4	no	—
259	C₂×C₄×C₆₇	—	8	4	no	—
260	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
262	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
263	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
266	C₈×C₆₇	—	2	2	no	—
267	C₈×C₆₇	—	2	2	no	—
268	C₈×C₆₇	—	2	2	no	—
269	C₈×C₆₇	—	2	2	no	—
270	C₂²×C₆₇	—	2	2	no	—
271	C₂²×C₆₇	—	2	4	no	—
272	C₂²×C₆₇	—	2	2	no	—
273	C₂²×C₆₇	—	2	1	no	—
274	C₂²×C₆₇	—	4	4	no	—
275	C₂²×C₆₇	—	4	4	no	—
276	C₂²×C₆₇	—	4	4	no	—
277	C₂²×C₆₇	—	16	4	no	—
278	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
279	C₄×C₆₇	—	2	2	no	—
280	C₄×C₆₇	—	2	2	no	—
281	C₄×C₆₇	—	2	4	no	—
282	C₄×C₆₇	—	2	2	no	—
283	C₄×C₆₇	—	2	1	no	—
284	C₄×C₆₇	—	4	4	no	—
285	C₄×C₆₇	—	4	4	no	—
286	C₄×C₆₇	—	4	4	no	—
287	C₄×C₆₇	#585	1	1	yes	characteristic
288	C₄×C₆₇	#585	1	1	yes	characteristic
289	#61007	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
290	C₂×C₆₇	—	4	1	no	—
291	C₂×C₆₇	—	4	2	no	—
292	C₂×C₆₇	—	8	4	no	—
293	C₂×C₆₇	#3449	1	1	yes	characteristic
294	C₂×C₆₇	#3584	1	2	yes	—
295	#3510	C₂×C₆₇	1	2	yes	complemented
296	#3263	C₂×C₆₇	1	2	yes	complemented
297	#3262	C₂×C₆₇	1	2	yes	complemented
298	#3233	C₂×C₆₇	1	4	yes	complemented
299	#3211	C₂×C₆₇	1	4	yes	complemented
300	#3202	C₂×C₆₇	1	1	yes	characteristic, complemented
301	#3200	C₂×C₆₇	1	2	yes	complemented
302	#3199	C₂×C₆₇	1	1	yes	characteristic
303	#3183	C₂×C₆₇	1	2	yes	complemented
304	#3168	C₂×C₆₇	1	2	yes	complemented
305	#3121	C₂×C₆₇	1	4	yes	complemented
306	#3118	C₂×C₆₇	1	4	yes	complemented
307	#2922	C₂×C₆₇	1	1	yes	characteristic, complemented
308	C₆₇	#61007	1	1	yes	characteristic, direct product factor, Sylow
309	#559	—	2	4	no	—
310	#556	—	2	4	no	—
311	#551	—	2	4	no	—
312	#551	—	2	4	no	—
313	#542	C₂²×C₆₇	1	4	yes	—
314	#542	C₂²×C₆₇	1	4	yes	complemented
315	#536	C₂²×C₆₇	1	2	yes	—
316	#535	C₂²×C₆₇	1	2	yes	—
317	#533	C₂²×C₆₇	1	2	yes	complemented
318	#529	C₂²×C₆₇	1	4	yes	—
319	#529	C₂²×C₆₇	1	4	yes	complemented
320	C₂×#124	C₂²×C₆₇	1	2	yes	complemented
321	#519	C₂²×C₆₇	1	2	yes	complemented
322	#518	C₂²×C₆₇	1	2	yes	complemented
323	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
324	#499	C₂²×C₆₇	1	2	yes	complemented
325	#499	C₂²×C₆₇	1	2	yes	complemented
326	#495	C₂²×C₆₇	1	2	yes	—
327	#495	C₂²×C₆₇	1	2	yes	—
328	#488	C₂²×C₆₇	1	4	yes	complemented
329	#488	C₂²×C₆₇	1	4	yes	complemented
330	#485	C₂²×C₆₇	1	4	yes	complemented
331	#485	C₂²×C₆₇	1	4	yes	complemented
332	#482	C₂²×C₆₇	1	4	yes	—
333	#482	C₂²×C₆₇	1	4	yes	—
334	#477	C₂²×C₆₇	1	4	yes	complemented
335	#476	C₂²×C₆₇	1	4	yes	complemented
336	#475	C₂²×C₆₇	1	4	yes	complemented
337	#474	C₂²×C₆₇	1	4	yes	complemented
338	#470	C₂²×C₆₇	1	2	yes	complemented
339	#469	C₂²×C₆₇	1	2	yes	—
340	#468	C₂²×C₆₇	1	4	yes	complemented
341	#467	C₂²×C₆₇	1	2	yes	complemented
342	#466	C₂²×C₆₇	1	2	yes	—
343	#465	C₂²×C₆₇	1	4	yes	complemented
344	#464	C₂²×C₆₇	1	4	yes	complemented
345	#464	C₂²×C₆₇	1	4	yes	complemented
346	#463	C₂²×C₆₇	1	4	yes	—
347	#463	C₂²×C₆₇	1	4	yes	—
348	#451	C₂²×C₆₇	1	4	yes	complemented
349	#450	C₂²×C₆₇	1	4	yes	complemented
350	#448	C₂²×C₆₇	1	4	yes	—
351	#447	C₂²×C₆₇	1	4	yes	—
352	#442	C₂²×C₆₇	1	2	yes	—
353	#441	C₂²×C₆₇	1	2	yes	complemented
354	#440	C₂²×C₆₇	1	4	yes	complemented
355	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	complemented
356	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
357	C₄×D₈	C₂²×C₆₇	1	2	yes	—
358	#433	C₂²×C₆₇	1	4	yes	complemented
359	#431	C₂²×C₆₇	1	4	yes	complemented
360	#424	C₂²×C₆₇	1	1	yes	characteristic, complemented
361	#424	C₂²×C₆₇	1	1	yes	characteristic, complemented
362	#408	C₂²×C₆₇	1	1	yes	characteristic
363	#408	C₂²×C₆₇	1	1	yes	characteristic
364	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
365	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
366	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
367	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
368	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
369	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
370	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
371	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
372	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
373	C₂×D₈	C₂³×C₆₇	1	2	yes	—
374	C₂×D₈	C₂³×C₆₇	1	2	yes	—
375	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
376	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
377	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
378	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
379	C₄⋊Q₈	—	2	2	no	—
380	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
381	#127	—	2	2	no	—
382	#126	—	2	4	no	—
383	#126	—	2	4	no	—
384	#126	—	2	4	no	—
385	#126	—	2	4	no	—
386	#125	C₂³×C₆₇	1	4	yes	—
387	#124	—	2	2	no	—
388	#124	—	2	2	no	—
389	#124	—	2	4	no	—
390	#124	—	2	4	no	—
391	#124	—	2	4	no	—
392	#124	C₂³×C₆₇	1	4	yes	—
393	#124	C₂³×C₆₇	1	4	yes	—
394	#123	—	2	2	no	—
395	#123	—	2	2	no	—
396	#123	—	2	4	no	—
397	#123	—	2	4	no	—
398	#123	—	2	4	no	—
399	#123	—	2	4	no	—
400	C₂²⋊Q₈	—	2	4	no	—
401	C₂²⋊Q₈	—	2	4	no	—
402	C₂²⋊Q₈	—	2	4	no	—
403	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
404	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
405	#121	C₂³×C₆₇	1	4	yes	—
406	#121	C₂³×C₆₇	1	4	yes	—
407	#120	—	2	2	no	—
408	#120	—	2	2	no	—
409	C₄×Q₈	—	2	4	no	—
410	C₄×Q₈	—	2	4	no	—
411	C₄×Q₈	D₄×C₆₇	1	2	yes	—
412	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
413	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
414	C₄×D₄	—	2	4	no	—
415	C₄×D₄	—	2	4	no	—
416	C₄×D₄	—	2	4	no	—
417	C₄×D₄	D₄×C₆₇	1	2	yes	—
418	C₄×D₄	D₄×C₆₇	1	4	yes	—
419	C₄×D₄	C₂³×C₆₇	1	4	yes	—
420	C₄×D₄	C₂³×C₆₇	1	4	yes	—
421	#117	—	2	4	no	—
422	#117	—	2	4	no	—
423	#117	—	2	4	no	—
424	#117	C₂³×C₆₇	1	2	yes	—
425	#117	C₂³×C₆₇	1	2	yes	—
426	#117	C₂³×C₆₇	1	2	yes	—
427	#117	C₂³×C₆₇	1	1	yes	characteristic
428	C₂×C₂²⋊C₄	—	2	4	no	—
429	C₂×C₂²⋊C₄	—	2	4	no	—
430	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
431	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
432	#107	C₂³×C₆₇	1	2	yes	—
433	#107	C₂³×C₆₇	1	2	yes	—
434	#107	C₂³×C₆₇	1	4	yes	—
435	#106	C₂³×C₆₇	1	4	yes	—
436	#106	C₂³×C₆₇	1	2	yes	—
437	#106	C₂³×C₆₇	1	2	yes	—
438	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
439	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
440	#103	C₂³×C₆₇	1	4	yes	—
441	#103	C₂³×C₆₇	1	4	yes	—
442	#103	C₂³×C₆₇	1	4	yes	—
443	#103	C₂³×C₆₇	1	4	yes	—
444	#102	C₂³×C₆₇	1	4	yes	—
445	#102	C₂³×C₆₇	1	4	yes	—
446	#102	C₂³×C₆₇	1	4	yes	—
447	#102	C₂³×C₆₇	1	4	yes	—
448	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
449	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
450	#97	C₂³×C₆₇	1	2	yes	—
451	C₄×C₈	C₂³×C₆₇	1	2	yes	—
452	C₄×C₈	C₂³×C₆₇	1	2	yes	—
453	C₄×C₈	C₂³×C₆₇	1	2	yes	—
454	C₂⁴	—	2	2	no	—
455	#41	—	4	1	no	—
456	#41	—	4	1	no	—
457	C₂×Q₈	—	2	2	no	—
458	C₂×Q₈	—	2	4	no	—
459	C₂×Q₈	—	4	2	no	—
460	C₂×Q₈	—	4	2	no	—
461	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
462	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
463	C₂×D₄	—	2	4	no	—
464	C₂×D₄	—	2	4	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	4	2	no	—
467	C₂×D₄	—	4	2	no	—
468	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
469	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
470	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	4	no	—
473	C₂²×C₄	—	2	4	no	—
474	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
475	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
476	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
477	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
478	Q₁₆	—	4	2	no	—
479	Q₁₆	—	4	2	no	—
480	SD₁₆	—	4	4	no	—
481	SD₁₆	—	4	4	no	—
482	D₈	—	4	2	no	—
483	D₈	—	4	2	no	—
484	M₁₆	—	4	1	no	—
485	M₁₆	—	4	1	no	—
486	C₂×C₈	—	4	1	no	—
487	C₂×C₈	—	4	1	no	—
488	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
489	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
490	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
491	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	4	no	—
499	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
500	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
501	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
502	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	4	no	—
511	C₂²⋊C₄	—	2	4	no	—
512	C₂²⋊C₄	—	2	4	no	—
513	C₂²⋊C₄	—	2	4	no	—
514	C₂²⋊C₄	—	2	4	no	—
515	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
516	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
517	C₄²	—	2	4	no	—
518	C₄²	—	2	4	no	—
519	C₄²	—	2	2	no	—
520	C₄²	—	2	2	no	—
521	C₄²	C₂×D₄×C₆₇	1	2	yes	—
522	C₄²	C₂×D₄×C₆₇	1	4	yes	—
523	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
524	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
525	C₄²	C₂⁴×C₆₇	1	2	yes	—
526	C₂³	—	2	4	no	—
527	C₂³	—	4	4	no	—
528	C₂³	—	8	4	no	—
529	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
530	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
531	Q₈	—	2	4	no	—
532	Q₈	—	2	4	no	—
533	Q₈	—	4	1	no	—
534	Q₈	—	4	4	no	—
535	Q₈	—	8	4	no	—
536	D₄	—	2	4	no	—
537	D₄	—	2	4	no	—
538	D₄	—	4	2	no	—
539	D₄	—	4	1	no	—
540	D₄	—	4	4	no	—
541	D₄	—	8	4	no	—
542	D₄	—	8	4	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	4	4	no	—
552	C₂×C₄	—	4	4	no	—
553	C₂×C₄	—	8	4	no	—
554	C₂×C₄	—	8	4	no	—
555	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
556	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
557	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
558	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
559	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
560	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
561	C₈	—	2	2	no	—
562	C₈	—	2	2	no	—
563	C₈	—	2	2	no	—
564	C₈	—	2	2	no	—
565	C₂²	—	2	2	no	—
566	C₂²	—	2	4	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	4	4	no	—
572	C₂²	—	16	4	no	—
573	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
574	C₄	—	2	2	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	4	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	1	no	—
579	C₄	—	4	4	no	—
580	C₄	—	4	4	no	—
581	C₄	—	4	4	no	—
582	C₄	#585×C₆₇	1	1	yes	characteristic
583	C₄	#585×C₆₇	1	1	yes	characteristic
584	C₂	—	4	2	no	—
585	C₂	—	4	1	no	—
586	C₂	—	8	4	no	—
587	C₂	C₆₇×#3449	1	1	yes	characteristic
588	C₂	C₆₇×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₆₇×#3199	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
1. C₆₇×#3510	C₂	semidirect
2. C₆₇×#3263	C₂	semidirect
3. C₆₇×#3262	C₂	semidirect
4. C₆₇×#3233	C₂	semidirect
5. C₆₇×#3211	C₂	semidirect
6. C₆₇×#3202	C₂	semidirect
7. C₆₇×#3200	C₂	semidirect
9. C₆₇×#3183	C₂	semidirect
10. C₆₇×#3168	C₂	semidirect
11. C₆₇×#3121	C₂	semidirect
12. C₆₇×#3118	C₂	semidirect
13. C₆₇×#2922	C₂	semidirect
19. #542×C₆₇	C₂²	semidirect
22. #533×C₆₇	C₂²	semidirect
24. #529×C₆₇	C₂²	semidirect
25. C₂×#124×C₆₇	C₂²	semidirect
26. #519×C₆₇	C₂²	semidirect
27. #518×C₆₇	C₂²	semidirect
29. #499×C₆₇	C₂²	semidirect
30. #499×C₆₇	C₂²	semidirect
33. #488×C₆₇	C₂²	semidirect
34. #488×C₆₇	C₂²	semidirect
35. #485×C₆₇	C₂²	semidirect
36. #485×C₆₇	C₂²	semidirect
39. #477×C₆₇	C₂²	semidirect
40. #476×C₆₇	C₂²	semidirect
41. #475×C₆₇	C₂²	semidirect
42. #474×C₆₇	C₂²	semidirect
43. #470×C₆₇	C₂²	semidirect
45. #468×C₆₇	C₂²	semidirect
46. #467×C₆₇	C₂²	semidirect
48. #465×C₆₇	C₂²	semidirect
49. #464×C₆₇	C₂²	semidirect
50. #464×C₆₇	C₂²	semidirect
53. #451×C₆₇	C₂²	semidirect
54. #450×C₆₇	C₂²	semidirect
58. #441×C₆₇	C₂²	semidirect
59. #440×C₆₇	C₂²	semidirect
60. C₄×Q₁₆×C₆₇	C₂²	semidirect
61. C₄×SD₁₆×C₆₇	C₂²	semidirect
63. #433×C₆₇	C₂²	semidirect
64. #431×C₆₇	C₂²	semidirect
65. #424×C₆₇	C₂²	semidirect
66. #424×C₆₇	C₂²	semidirect
69. C₄×M₁₆×C₆₇	C₂²	semidirect
70. C₂×C₄×C₈×C₆₇	C₂²	semidirect
289. #61007	C₆₇	direct (N × H)
295. #3510	C₂×C₆₇	semidirect
296. #3263	C₂×C₆₇	semidirect
297. #3262	C₂×C₆₇	semidirect
298. #3233	C₂×C₆₇	semidirect
299. #3211	C₂×C₆₇	semidirect
300. #3202	C₂×C₆₇	semidirect
301. #3200	C₂×C₆₇	semidirect
303. #3183	C₂×C₆₇	semidirect
304. #3168	C₂×C₆₇	semidirect
305. #3121	C₂×C₆₇	semidirect
306. #3118	C₂×C₆₇	semidirect
307. #2922	C₂×C₆₇	semidirect
308. C₆₇	#61007	direct (N × H)
314. #542	C₂²×C₆₇	semidirect
317. #533	C₂²×C₆₇	semidirect
319. #529	C₂²×C₆₇	semidirect
320. C₂×#124	C₂²×C₆₇	semidirect
321. #519	C₂²×C₆₇	semidirect
322. #518	C₂²×C₆₇	semidirect
324. #499	C₂²×C₆₇	semidirect
325. #499	C₂²×C₆₇	semidirect
328. #488	C₂²×C₆₇	semidirect
329. #488	C₂²×C₆₇	semidirect
330. #485	C₂²×C₆₇	semidirect
331. #485	C₂²×C₆₇	semidirect
334. #477	C₂²×C₆₇	semidirect
335. #476	C₂²×C₆₇	semidirect
336. #475	C₂²×C₆₇	semidirect
337. #474	C₂²×C₆₇	semidirect
338. #470	C₂²×C₆₇	semidirect
340. #468	C₂²×C₆₇	semidirect
341. #467	C₂²×C₆₇	semidirect
343. #465	C₂²×C₆₇	semidirect
344. #464	C₂²×C₆₇	semidirect
345. #464	C₂²×C₆₇	semidirect
348. #451	C₂²×C₆₇	semidirect
349. #450	C₂²×C₆₇	semidirect
353. #441	C₂²×C₆₇	semidirect
354. #440	C₂²×C₆₇	semidirect
355. C₄×Q₁₆	C₂²×C₆₇	semidirect
356. C₄×SD₁₆	C₂²×C₆₇	semidirect
358. #433	C₂²×C₆₇	semidirect
359. #431	C₂²×C₆₇	semidirect
360. #424	C₂²×C₆₇	semidirect
361. #424	C₂²×C₆₇	semidirect
364. C₄×M₁₆	C₂²×C₆₇	semidirect
365. C₂×C₄×C₈	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3510	2
C₆₇×#3263	2
C₆₇×#3262	2
C₆₇×#3233	2
C₆₇×#3211	2
C₆₇×#3202	2
C₆₇×#3200	2
C₆₇×#3199	2
C₆₇×#3183	2
C₆₇×#3168	2
C₆₇×#3121	2
C₆₇×#3118	2
C₆₇×#2922	2
#61007	67

Subgroups — C67×#61007 #30740258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61007 #30740258