Subgroups — C₆₇×#61014 #30740265

Order 17152.

1768 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3511	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3262	C₂	1	4	yes	maximal, complemented
3	C₆₇×#3230	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3226	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3210	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3207	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3203	C₂	1	4	yes	maximal, complemented
8	C₆₇×#3182	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#3169	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3124	C₂	1	8	yes	maximal, complemented
12	C₆₇×#2922	C₂	1	1	yes	maximal, characteristic, complemented
13	#556×C₆₇	—	2	2	no	—
14	#555×C₆₇	—	2	2	no	—
15	#554×C₆₇	—	2	2	no	—
16	#553×C₆₇	—	2	2	no	—
17	#553×C₆₇	—	2	2	no	—
18	#548×C₆₇	—	2	2	no	—
19	#548×C₆₇	—	2	2	no	—
20	#547×C₆₇	—	2	2	no	—
21	#542×C₆₇	C₂²	1	4	yes	—
22	#542×C₆₇	C₂²	1	4	yes	—
23	#537×C₆₇	C₂²	1	4	yes	—
24	#533×C₆₇	C₂²	1	2	yes	—
25	#533×C₆₇	C₂²	1	2	yes	—
26	#532×C₆₇	C₂²	1	2	yes	—
27	#532×C₆₇	C₂²	1	2	yes	—
28	#532×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
30	#520×C₆₇	C₂²	1	4	yes	—
31	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
32	#500×C₆₇	C₂²	1	1	yes	characteristic
33	#500×C₆₇	C₂²	1	1	yes	characteristic
34	#500×C₆₇	C₂²	1	2	yes	—
35	#494×C₆₇	C₂²	1	1	yes	characteristic
36	#493×C₆₇	C₂²	1	1	yes	characteristic
37	#492×C₆₇	C₂²	1	2	yes	—
38	#487×C₆₇	C₂²	1	2	yes	—
39	#487×C₆₇	C₂²	1	2	yes	—
40	#486×C₆₇	C₂²	1	2	yes	—
41	#486×C₆₇	C₂²	1	2	yes	—
42	#484×C₆₇	C₂²	1	8	yes	—
43	#480×C₆₇	C₂²	1	8	yes	—
44	#477×C₆₇	C₂²	1	8	yes	—
45	#474×C₆₇	C₂²	1	8	yes	—
46	#468×C₆₇	C₂²	1	8	yes	—
47	#467×C₆₇	C₂²	1	4	yes	—
48	#466×C₆₇	C₂²	1	4	yes	—
49	#464×C₆₇	C₂²	1	8	yes	—
50	#463×C₆₇	C₂²	1	8	yes	—
51	#452×C₆₇	C₂²	1	8	yes	—
52	#449×C₆₇	C₂²	1	8	yes	—
53	#440×C₆₇	C₂²	1	8	yes	—
54	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	—
55	C₄×D₈×C₆₇	C₂²	1	4	yes	—
56	#433×C₆₇	C₂²	1	4	yes	—
57	#431×C₆₇	C₂²	1	2	yes	—
58	#431×C₆₇	C₂²	1	2	yes	—
59	#424×C₆₇	C₂²	1	2	yes	—
60	#408×C₆₇	C₂²	1	2	yes	—
61	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
62	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
63	C₂×#41×C₆₇	—	2	2	no	—
64	C₂×#41×C₆₇	—	2	2	no	—
65	C₂×#41×C₆₇	C₂³	1	4	yes	—
66	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×Q₈×C₆₇	—	2	2	no	—
68	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
69	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
70	C₂×D₈×C₆₇	C₂³	1	4	yes	—
71	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
72	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
73	C₄⋊Q₈×C₆₇	—	2	2	no	—
74	C₄⋊Q₈×C₆₇	—	2	2	no	—
75	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
76	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
77	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
78	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
79	#127×C₆₇	C₂³	1	2	yes	—
80	#126×C₆₇	—	2	4	no	—
81	#126×C₆₇	—	2	4	no	—
82	#125×C₆₇	—	2	2	no	—
83	#125×C₆₇	—	2	2	no	—
84	#124×C₆₇	—	2	4	no	—
85	#124×C₆₇	—	2	4	no	—
86	#124×C₆₇	—	2	2	no	—
87	#124×C₆₇	—	2	2	no	—
88	#124×C₆₇	C₂³	1	4	yes	—
89	#123×C₆₇	—	2	4	no	—
90	#123×C₆₇	—	2	4	no	—
91	C₂²⋊Q₈×C₆₇	—	2	4	no	—
92	C₂²⋊Q₈×C₆₇	—	2	2	no	—
93	C₂²⋊Q₈×C₆₇	—	2	2	no	—
94	C₂²⋊Q₈×C₆₇	—	2	4	no	—
95	C₂²⋊Q₈×C₆₇	—	2	4	no	—
96	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
97	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
98	#121×C₆₇	—	2	4	no	—
99	#121×C₆₇	—	2	4	no	—
100	#121×C₆₇	—	2	4	no	—
101	#121×C₆₇	—	2	2	no	—
102	#121×C₆₇	—	2	2	no	—
103	#121×C₆₇	—	2	2	no	—
104	#121×C₆₇	—	2	2	no	—
105	#121×C₆₇	C₂³	1	4	yes	—
106	#121×C₆₇	C₂³	1	4	yes	—
107	C₄×Q₈×C₆₇	—	2	2	no	—
108	C₄×Q₈×C₆₇	—	2	2	no	—
109	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
110	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
111	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
112	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
113	C₄×D₄×C₆₇	—	2	4	no	—
114	C₄×D₄×C₆₇	—	2	4	no	—
115	C₄×D₄×C₆₇	—	2	2	no	—
116	C₄×D₄×C₆₇	—	2	2	no	—
117	C₄×D₄×C₆₇	—	2	2	no	—
118	C₄×D₄×C₆₇	—	2	2	no	—
119	C₄×D₄×C₆₇	D₄	1	2	yes	—
120	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
121	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
122	C₄×D₄×C₆₇	D₄	1	2	yes	—
123	C₄×D₄×C₆₇	C₂³	1	4	yes	—
124	C₄×D₄×C₆₇	C₂³	1	4	yes	—
125	#117×C₆₇	—	2	2	no	—
126	#117×C₆₇	—	2	2	no	—
127	#117×C₆₇	—	2	4	no	—
128	#117×C₆₇	C₂³	1	2	yes	—
129	#117×C₆₇	C₂³	1	1	yes	characteristic
130	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
132	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
133	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
134	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
135	#107×C₆₇	C₂³	1	4	yes	—
136	#107×C₆₇	C₂³	1	4	yes	—
137	#107×C₆₇	C₂³	1	8	yes	—
138	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
139	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
140	#103×C₆₇	C₂³	1	8	yes	—
141	#103×C₆₇	C₂³	1	8	yes	—
142	#102×C₆₇	C₂³	1	8	yes	—
143	#102×C₆₇	C₂³	1	8	yes	—
144	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
145	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
146	#97×C₆₇	C₂³	1	2	yes	—
147	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
148	C₄×C₈×C₆₇	C₂³	1	2	yes	—
149	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
150	C₄×C₈×C₆₇	C₂³	1	2	yes	—
151	#41×C₆₇	—	4	2	no	—
152	#41×C₆₇	—	4	8	no	—
153	#41×C₆₇	—	8	4	no	—
154	#41×C₆₇	—	8	4	no	—
155	C₂×Q₈×C₆₇	—	2	2	no	—
156	C₂×Q₈×C₆₇	—	2	2	no	—
157	C₂×Q₈×C₆₇	—	2	4	no	—
158	C₂×Q₈×C₆₇	—	8	4	no	—
159	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
160	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
161	C₂×D₄×C₆₇	—	2	4	no	—
162	C₂×D₄×C₆₇	—	2	4	no	—
163	C₂×D₄×C₆₇	—	2	4	no	—
164	C₂×D₄×C₆₇	—	2	2	no	—
165	C₂×D₄×C₆₇	—	2	2	no	—
166	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
167	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
168	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₆₇	—	2	4	no	—
170	C₂²×C₄×C₆₇	—	2	4	no	—
171	C₂²×C₄×C₆₇	—	2	4	no	—
172	C₂²×C₄×C₆₇	—	2	4	no	—
173	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
176	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
179	Q₁₆×C₆₇	—	4	4	no	—
180	SD₁₆×C₆₇	—	4	8	no	—
181	D₈×C₆₇	—	4	4	no	—
182	M₁₆×C₆₇	—	4	2	no	—
183	C₂×C₈×C₆₇	—	4	2	no	—
184	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
185	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
186	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
187	C₄⋊C₄×C₆₇	—	2	4	no	—
188	C₄⋊C₄×C₆₇	—	2	2	no	—
189	C₄⋊C₄×C₆₇	—	2	2	no	—
190	C₄⋊C₄×C₆₇	—	2	4	no	—
191	C₄⋊C₄×C₆₇	—	2	4	no	—
192	C₄⋊C₄×C₆₇	—	2	4	no	—
193	C₄⋊C₄×C₆₇	—	2	2	no	—
194	C₄⋊C₄×C₆₇	—	2	2	no	—
195	C₄⋊C₄×C₆₇	—	2	4	no	—
196	C₄⋊C₄×C₆₇	—	2	4	no	—
197	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
198	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
199	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
200	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
201	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
202	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
203	C₂²⋊C₄×C₆₇	—	2	4	no	—
204	C₂²⋊C₄×C₆₇	—	2	4	no	—
205	C₂²⋊C₄×C₆₇	—	2	4	no	—
206	C₂²⋊C₄×C₆₇	—	2	4	no	—
207	C₂²⋊C₄×C₆₇	—	2	4	no	—
208	C₂²⋊C₄×C₆₇	—	2	4	no	—
209	C₂²⋊C₄×C₆₇	—	2	4	no	—
210	C₂²⋊C₄×C₆₇	—	2	4	no	—
211	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
212	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
213	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
214	C₄²×C₆₇	—	2	4	no	—
215	C₄²×C₆₇	—	2	4	no	—
216	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₆₇	C₂×D₄	1	2	yes	—
218	C₄²×C₆₇	C₂×D₄	1	2	yes	—
219	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₆₇	C₂⁴	1	2	yes	—
221	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
222	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
223	C₂³×C₆₇	—	2	4	no	—
224	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
225	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
226	Q₈×C₆₇	—	2	8	no	—
227	Q₈×C₆₇	—	4	1	no	—
228	Q₈×C₆₇	—	4	4	no	—
229	Q₈×C₆₇	—	4	4	no	—
230	Q₈×C₆₇	—	8	4	no	—
231	D₄×C₆₇	—	2	8	no	—
232	D₄×C₆₇	—	4	2	no	—
233	D₄×C₆₇	—	4	1	no	—
234	D₄×C₆₇	—	4	8	no	—
235	D₄×C₆₇	—	4	4	no	—
236	D₄×C₆₇	—	4	4	no	—
237	D₄×C₆₇	—	8	4	no	—
238	D₄×C₆₇	—	8	4	no	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	4	8	no	—
249	C₂×C₄×C₆₇	—	8	4	no	—
250	C₂×C₄×C₆₇	—	8	4	no	—
251	C₂×C₄×C₆₇	—	8	4	no	—
252	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
261	C₈×C₆₇	—	2	2	no	—
262	C₈×C₆₇	—	2	2	no	—
263	C₈×C₆₇	—	2	4	no	—
264	C₂²×C₆₇	—	2	2	no	—
265	C₂²×C₆₇	—	2	1	no	—
266	C₂²×C₆₇	—	2	2	no	—
267	C₂²×C₆₇	—	2	1	no	—
268	C₂²×C₆₇	—	2	1	no	—
269	C₂²×C₆₇	—	2	2	no	—
270	C₂²×C₆₇	—	4	4	no	—
271	C₂²×C₆₇	—	4	4	no	—
272	C₂²×C₆₇	—	4	4	no	—
273	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
274	C₄×C₆₇	—	2	2	no	—
275	C₄×C₆₇	—	2	1	no	—
276	C₄×C₆₇	—	2	1	no	—
277	C₄×C₆₇	—	2	2	no	—
278	C₄×C₆₇	—	2	1	no	—
279	C₄×C₆₇	—	2	2	no	—
280	C₄×C₆₇	—	4	4	no	—
281	C₄×C₆₇	—	4	4	no	—
282	C₄×C₆₇	—	4	4	no	—
283	C₄×C₆₇	#576	1	2	yes	—
284	C₄×C₆₇	#578	1	2	yes	—
285	C₄×C₆₇	#585	1	2	yes	—
286	#61014	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
287	C₂×C₆₇	—	4	2	no	—
288	C₂×C₆₇	—	4	1	no	—
289	C₂×C₆₇	—	8	4	no	—
290	C₂×C₆₇	#3449	1	1	yes	characteristic
291	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
292	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
293	#3511	C₂×C₆₇	1	2	yes	complemented
294	#3262	C₂×C₆₇	1	4	yes	complemented
295	#3230	C₂×C₆₇	1	2	yes	complemented
296	#3226	C₂×C₆₇	1	2	yes	complemented
297	#3210	C₂×C₆₇	1	2	yes	complemented
298	#3207	C₂×C₆₇	1	2	yes	complemented
299	#3203	C₂×C₆₇	1	4	yes	complemented
300	#3182	C₂×C₆₇	1	2	yes	complemented
301	#3170	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#3169	C₂×C₆₇	1	1	yes	characteristic, complemented
303	#3124	C₂×C₆₇	1	8	yes	complemented
304	#2922	C₂×C₆₇	1	1	yes	characteristic, complemented
305	C₆₇	#61014	1	1	yes	characteristic, direct product factor, Sylow
306	#556	—	2	2	no	—
307	#555	—	2	2	no	—
308	#554	—	2	2	no	—
309	#553	—	2	2	no	—
310	#553	—	2	2	no	—
311	#548	—	2	2	no	—
312	#548	—	2	2	no	—
313	#547	—	2	2	no	—
314	#542	C₂²×C₆₇	1	4	yes	—
315	#542	C₂²×C₆₇	1	4	yes	—
316	#537	C₂²×C₆₇	1	4	yes	—
317	#533	C₂²×C₆₇	1	2	yes	—
318	#533	C₂²×C₆₇	1	2	yes	—
319	#532	C₂²×C₆₇	1	2	yes	—
320	#532	C₂²×C₆₇	1	2	yes	—
321	#532	C₂²×C₆₇	1	2	yes	—
322	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
323	#520	C₂²×C₆₇	1	4	yes	—
324	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
325	#500	C₂²×C₆₇	1	1	yes	characteristic
326	#500	C₂²×C₆₇	1	1	yes	characteristic
327	#500	C₂²×C₆₇	1	2	yes	—
328	#494	C₂²×C₆₇	1	1	yes	characteristic
329	#493	C₂²×C₆₇	1	1	yes	characteristic
330	#492	C₂²×C₆₇	1	2	yes	—
331	#487	C₂²×C₆₇	1	2	yes	—
332	#487	C₂²×C₆₇	1	2	yes	—
333	#486	C₂²×C₆₇	1	2	yes	—
334	#486	C₂²×C₆₇	1	2	yes	—
335	#484	C₂²×C₆₇	1	8	yes	—
336	#480	C₂²×C₆₇	1	8	yes	—
337	#477	C₂²×C₆₇	1	8	yes	—
338	#474	C₂²×C₆₇	1	8	yes	—
339	#468	C₂²×C₆₇	1	8	yes	—
340	#467	C₂²×C₆₇	1	4	yes	—
341	#466	C₂²×C₆₇	1	4	yes	—
342	#464	C₂²×C₆₇	1	8	yes	—
343	#463	C₂²×C₆₇	1	8	yes	—
344	#452	C₂²×C₆₇	1	8	yes	—
345	#449	C₂²×C₆₇	1	8	yes	—
346	#440	C₂²×C₆₇	1	8	yes	—
347	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	—
348	C₄×D₈	C₂²×C₆₇	1	4	yes	—
349	#433	C₂²×C₆₇	1	4	yes	—
350	#431	C₂²×C₆₇	1	2	yes	—
351	#431	C₂²×C₆₇	1	2	yes	—
352	#424	C₂²×C₆₇	1	2	yes	—
353	#408	C₂²×C₆₇	1	2	yes	—
354	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
355	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
356	C₂×#41	—	2	2	no	—
357	C₂×#41	—	2	2	no	—
358	C₂×#41	C₂³×C₆₇	1	4	yes	—
359	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
360	C₂²×Q₈	—	2	2	no	—
361	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
362	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
363	C₂×D₈	C₂³×C₆₇	1	4	yes	—
364	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
365	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
366	C₄⋊Q₈	—	2	2	no	—
367	C₄⋊Q₈	—	2	2	no	—
368	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
369	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
370	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
371	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
372	#127	C₂³×C₆₇	1	2	yes	—
373	#126	—	2	4	no	—
374	#126	—	2	4	no	—
375	#125	—	2	2	no	—
376	#125	—	2	2	no	—
377	#124	—	2	4	no	—
378	#124	—	2	4	no	—
379	#124	—	2	2	no	—
380	#124	—	2	2	no	—
381	#124	C₂³×C₆₇	1	4	yes	—
382	#123	—	2	4	no	—
383	#123	—	2	4	no	—
384	C₂²⋊Q₈	—	2	4	no	—
385	C₂²⋊Q₈	—	2	2	no	—
386	C₂²⋊Q₈	—	2	2	no	—
387	C₂²⋊Q₈	—	2	4	no	—
388	C₂²⋊Q₈	—	2	4	no	—
389	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
390	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
391	#121	—	2	4	no	—
392	#121	—	2	4	no	—
393	#121	—	2	4	no	—
394	#121	—	2	2	no	—
395	#121	—	2	2	no	—
396	#121	—	2	2	no	—
397	#121	—	2	2	no	—
398	#121	C₂³×C₆₇	1	4	yes	—
399	#121	C₂³×C₆₇	1	4	yes	—
400	C₄×Q₈	—	2	2	no	—
401	C₄×Q₈	—	2	2	no	—
402	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
403	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
404	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
405	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
406	C₄×D₄	—	2	4	no	—
407	C₄×D₄	—	2	4	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	2	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	D₄×C₆₇	1	2	yes	—
413	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
414	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
415	C₄×D₄	D₄×C₆₇	1	2	yes	—
416	C₄×D₄	C₂³×C₆₇	1	4	yes	—
417	C₄×D₄	C₂³×C₆₇	1	4	yes	—
418	#117	—	2	2	no	—
419	#117	—	2	2	no	—
420	#117	—	2	4	no	—
421	#117	C₂³×C₆₇	1	2	yes	—
422	#117	C₂³×C₆₇	1	1	yes	characteristic
423	C₂×C₄⋊C₄	—	2	2	no	—
424	C₂×C₄⋊C₄	—	2	4	no	—
425	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
426	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
427	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
428	#107	C₂³×C₆₇	1	4	yes	—
429	#107	C₂³×C₆₇	1	4	yes	—
430	#107	C₂³×C₆₇	1	8	yes	—
431	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
432	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
433	#103	C₂³×C₆₇	1	8	yes	—
434	#103	C₂³×C₆₇	1	8	yes	—
435	#102	C₂³×C₆₇	1	8	yes	—
436	#102	C₂³×C₆₇	1	8	yes	—
437	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
438	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
439	#97	C₂³×C₆₇	1	2	yes	—
440	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
441	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
442	C₄×C₈	C₂³×C₆₇	1	2	yes	—
443	C₄×C₈	C₂³×C₆₇	1	2	yes	—
444	#41	—	4	2	no	—
445	#41	—	4	8	no	—
446	#41	—	8	4	no	—
447	#41	—	8	4	no	—
448	C₂×Q₈	—	2	2	no	—
449	C₂×Q₈	—	2	2	no	—
450	C₂×Q₈	—	2	4	no	—
451	C₂×Q₈	—	8	4	no	—
452	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
453	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
454	C₂×D₄	—	2	4	no	—
455	C₂×D₄	—	2	4	no	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
460	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
461	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
467	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
469	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
470	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
471	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
472	Q₁₆	—	4	4	no	—
473	SD₁₆	—	4	8	no	—
474	D₈	—	4	4	no	—
475	M₁₆	—	4	2	no	—
476	C₂×C₈	—	4	2	no	—
477	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
478	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
479	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
480	C₄⋊C₄	—	2	4	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	4	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
491	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
492	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
493	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
494	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
495	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	2	4	no	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	2	4	no	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
505	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
506	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
507	C₄²	—	2	4	no	—
508	C₄²	—	2	4	no	—
509	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
510	C₄²	C₂×D₄×C₆₇	1	2	yes	—
511	C₄²	C₂×D₄×C₆₇	1	2	yes	—
512	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
513	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
514	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
515	C₄²	C₂⁴×C₆₇	1	2	yes	—
516	C₂³	—	2	4	no	—
517	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
518	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
519	Q₈	—	2	8	no	—
520	Q₈	—	4	1	no	—
521	Q₈	—	4	4	no	—
522	Q₈	—	4	4	no	—
523	Q₈	—	8	4	no	—
524	D₄	—	2	8	no	—
525	D₄	—	4	2	no	—
526	D₄	—	4	1	no	—
527	D₄	—	4	8	no	—
528	D₄	—	4	4	no	—
529	D₄	—	4	4	no	—
530	D₄	—	8	4	no	—
531	D₄	—	8	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	4	8	no	—
542	C₂×C₄	—	8	4	no	—
543	C₂×C₄	—	8	4	no	—
544	C₂×C₄	—	8	4	no	—
545	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
546	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
547	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
548	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
549	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
550	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
551	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
552	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
553	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
554	C₈	—	2	2	no	—
555	C₈	—	2	2	no	—
556	C₈	—	2	4	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	1	no	—
559	C₂²	—	2	1	no	—
560	C₂²	—	2	2	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	1	no	—
563	C₂²	—	4	4	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	4	4	no	—
566	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
567	C₄	—	2	1	no	—
568	C₄	—	2	2	no	—
569	C₄	—	2	1	no	—
570	C₄	—	2	2	no	—
571	C₄	—	2	2	no	—
572	C₄	—	2	1	no	—
573	C₄	—	4	4	no	—
574	C₄	—	4	4	no	—
575	C₄	—	4	4	no	—
576	C₄	#576×C₆₇	1	2	yes	—
577	C₄	#578×C₆₇	1	2	yes	—
578	C₄	#585×C₆₇	1	2	yes	—
579	C₂	—	4	2	no	—
580	C₂	—	4	1	no	—
581	C₂	—	8	4	no	—
582	C₂	C₆₇×#3449	1	1	yes	characteristic
583	C₂	C₂×#576×C₆₇	1	1	yes	characteristic
584	C₂	C₂×#578×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
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×#3511	C₂	semidirect
2. C₆₇×#3262	C₂	semidirect
3. C₆₇×#3230	C₂	semidirect
4. C₆₇×#3226	C₂	semidirect
5. C₆₇×#3210	C₂	semidirect
6. C₆₇×#3207	C₂	semidirect
7. C₆₇×#3203	C₂	semidirect
8. C₆₇×#3182	C₂	semidirect
9. C₆₇×#3170	C₂	semidirect
10. C₆₇×#3169	C₂	semidirect
11. C₆₇×#3124	C₂	semidirect
12. C₆₇×#2922	C₂	semidirect
286. #61014	C₆₇	direct (N × H)
293. #3511	C₂×C₆₇	semidirect
294. #3262	C₂×C₆₇	semidirect
295. #3230	C₂×C₆₇	semidirect
296. #3226	C₂×C₆₇	semidirect
297. #3210	C₂×C₆₇	semidirect
298. #3207	C₂×C₆₇	semidirect
299. #3203	C₂×C₆₇	semidirect
300. #3182	C₂×C₆₇	semidirect
301. #3170	C₂×C₆₇	semidirect
302. #3169	C₂×C₆₇	semidirect
303. #3124	C₂×C₆₇	semidirect
304. #2922	C₂×C₆₇	semidirect
305. C₆₇	#61014	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×#3511	2
C₆₇×#3262	2
C₆₇×#3230	2
C₆₇×#3226	2
C₆₇×#3210	2
C₆₇×#3207	2
C₆₇×#3203	2
C₆₇×#3182	2
C₆₇×#3170	2
C₆₇×#3169	2
C₆₇×#3124	2
C₆₇×#2922	2
#61014	67

Subgroups — C67×#61014 #30740265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61014 #30740265