Subgroups — C₆₇×#11111 #30690362

Order 17152.

874 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2911	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2896	C₂	1	2	yes	maximal
3	C₂×#408×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#338×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1729	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#117×C₆₇	C₂²	1	2	yes	—
11	#428×C₆₇	—	2	4	no	—
12	#427×C₆₇	—	2	2	no	—
13	#419×C₆₇	—	2	2	no	—
14	#418×C₆₇	—	2	2	no	—
15	#417×C₆₇	—	2	4	no	—
16	#416×C₆₇	—	2	4	no	—
17	#408×C₆₇	—	2	2	no	—
18	#408×C₆₇	C₄	1	4	yes	complemented
19	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
20	#338×C₆₇	—	2	2	no	—
21	#336×C₆₇	—	2	4	no	—
22	C₂×#41×C₆₇	—	2	1	no	—
23	C₂×#41×C₆₇	—	2	1	no	—
24	C₂×#41×C₆₇	—	2	2	no	—
25	C₂×#41×C₆₇	—	2	2	no	—
26	C₂×#41×C₆₇	—	2	2	no	—
27	C₂×#41×C₆₇	—	2	2	no	—
28	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×#41×C₆₇	Q₈	1	2	yes	—
30	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×D₄×C₆₇	—	2	1	no	—
32	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	—	2	1	no	—
34	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	—	2	2	no	—
43	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
45	C₂²×C₈×C₆₇	—	2	2	no	—
46	C₂²×C₈×C₆₇	—	2	1	no	—
47	C₂²×C₈×C₆₇	—	2	1	no	—
48	C₂²×C₈×C₆₇	—	2	2	no	—
49	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₆₇	D₄	1	4	yes	—
51	C₂²×C₈×C₆₇	D₄	1	2	yes	—
52	#117×C₆₇	—	2	2	no	—
53	#117×C₆₇	—	2	2	no	—
54	#117×C₆₇	—	2	2	no	—
55	#117×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
57	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄²×C₆₇	—	2	2	no	—
59	#107×C₆₇	—	2	4	no	—
60	#106×C₆₇	—	2	4	no	—
61	#103×C₆₇	—	2	8	no	—
62	#102×C₆₇	—	2	8	no	—
63	C₂²⋊C₈×C₆₇	—	2	8	no	—
64	C₂⁴×C₆₇	—	2	1	no	—
65	C₂⁴×C₆₇	#41	1	1	yes	characteristic
66	#41×C₆₇	—	2	4	no	—
67	#41×C₆₇	—	2	4	no	—
68	#41×C₆₇	—	2	4	no	—
69	#41×C₆₇	—	2	4	no	—
70	#41×C₆₇	—	4	2	no	—
71	#41×C₆₇	—	4	2	no	—
72	#41×C₆₇	—	4	2	no	—
73	#41×C₆₇	—	4	2	no	—
74	C₂×Q₈×C₆₇	—	2	1	no	—
75	C₂×Q₈×C₆₇	—	2	2	no	—
76	C₂×Q₈×C₆₇	—	2	2	no	—
77	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
78	C₂×D₄×C₆₇	—	2	2	no	—
79	C₂×D₄×C₆₇	—	2	2	no	—
80	C₂×D₄×C₆₇	—	2	2	no	—
81	C₂×D₄×C₆₇	—	2	1	no	—
82	C₂×D₄×C₆₇	—	2	2	no	—
83	C₂×D₄×C₆₇	—	4	2	no	—
84	C₂×D₄×C₆₇	—	4	2	no	—
85	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
86	C₂²×C₄×C₆₇	—	2	1	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	4	no	—
90	C₂²×C₄×C₆₇	—	4	1	no	—
91	C₂²×C₄×C₆₇	—	4	1	no	—
92	C₂²×C₄×C₆₇	—	4	1	no	—
93	C₂²×C₄×C₆₇	—	4	1	no	—
94	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	#41	1	2	yes	—
102	C₂²×C₄×C₆₇	#41	1	2	yes	—
103	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
104	M₁₆×C₆₇	—	2	4	no	—
105	M₁₆×C₆₇	—	2	4	no	—
106	M₁₆×C₆₇	—	4	2	no	—
107	M₁₆×C₆₇	—	4	2	no	—
108	M₁₆×C₆₇	—	4	2	no	—
109	M₁₆×C₆₇	—	4	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	4	no	—
114	C₂×C₈×C₆₇	—	2	4	no	—
115	C₂×C₈×C₆₇	—	2	4	no	—
116	C₂×C₈×C₆₇	—	2	4	no	—
117	C₂×C₈×C₆₇	—	4	2	no	—
118	C₂×C₈×C₆₇	—	4	1	no	—
119	C₂×C₈×C₆₇	—	4	1	no	—
120	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
121	C₄⋊C₄×C₆₇	—	2	4	no	—
122	C₄⋊C₄×C₆₇	—	2	4	no	—
123	C₂²⋊C₄×C₆₇	—	4	2	no	—
124	C₂²⋊C₄×C₆₇	—	4	2	no	—
125	C₄²×C₆₇	—	4	2	no	—
126	C₄²×C₆₇	—	4	2	no	—
127	C₂³×C₆₇	—	2	2	no	—
128	C₂³×C₆₇	—	4	2	no	—
129	C₂³×C₆₇	—	4	4	no	—
130	C₂³×C₆₇	—	4	1	no	—
131	C₂³×C₆₇	—	4	1	no	—
132	C₂³×C₆₇	#117	1	1	yes	characteristic
133	C₂³×C₆₇	#121	1	1	yes	characteristic
134	C₂³×C₆₇	#121	1	2	yes	—
135	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂³×C₆₇	#126	1	2	yes	—
137	Q₈×C₆₇	—	2	4	no	—
138	Q₈×C₆₇	—	4	2	no	—
139	D₄×C₆₇	—	2	4	no	—
140	D₄×C₆₇	—	4	4	no	—
141	D₄×C₆₇	—	4	4	no	—
142	D₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	4	1	no	—
145	C₂×C₄×C₆₇	—	4	1	no	—
146	C₂×C₄×C₆₇	—	4	2	no	—
147	C₂×C₄×C₆₇	—	4	2	no	—
148	C₂×C₄×C₆₇	—	4	2	no	—
149	C₂×C₄×C₆₇	—	4	2	no	—
150	C₂×C₄×C₆₇	—	4	8	no	—
151	C₂×C₄×C₆₇	—	4	4	no	—
152	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
155	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
156	C₂×C₄×C₆₇	#120	1	2	yes	—
157	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
158	C₂×C₄×C₆₇	#121	1	2	yes	—
159	C₂×C₄×C₆₇	#121	1	2	yes	—
160	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
162	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
163	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
164	C₂×C₄×C₆₇	#123	1	2	yes	—
165	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
166	C₂×C₄×C₆₇	#123	1	2	yes	—
167	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
168	C₂×C₄×C₆₇	#124	1	2	yes	—
169	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
170	C₈×C₆₇	—	2	4	no	—
171	C₈×C₆₇	—	4	2	no	—
172	C₈×C₆₇	—	4	4	no	—
173	C₂²×C₆₇	—	2	4	no	—
174	C₂²×C₆₇	—	2	2	no	—
175	C₂²×C₆₇	—	4	1	no	—
176	C₂²×C₆₇	—	4	1	no	—
177	C₂²×C₆₇	—	4	2	no	—
178	C₂²×C₆₇	—	4	4	no	—
179	C₂²×C₆₇	—	4	2	no	—
180	C₂²×C₆₇	—	8	2	no	—
181	C₂²×C₆₇	—	8	2	no	—
182	C₂²×C₆₇	#380	1	1	yes	characteristic
183	C₂²×C₆₇	#388	1	2	yes	—
184	C₂²×C₆₇	#392	1	1	yes	characteristic
185	C₂²×C₆₇	#396	1	2	yes	—
186	C₂²×C₆₇	#397	1	1	yes	characteristic
187	C₄×C₆₇	—	4	2	no	—
188	C₄×C₆₇	—	8	4	no	—
189	C₄×C₆₇	#385	1	2	yes	—
190	C₄×C₆₇	#386	1	1	yes	characteristic
191	C₄×C₆₇	#390	1	1	yes	characteristic
192	C₄×C₆₇	#393	1	1	yes	characteristic
193	C₄×C₆₇	#394	1	2	yes	—
194	C₄×C₆₇	#399	1	1	yes	characteristic
195	#11111	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂×C₆₇	—	2	1	no	—
197	C₂×C₆₇	—	2	1	no	—
198	C₂×C₆₇	—	2	2	no	—
199	C₂×C₆₇	—	2	2	no	—
200	C₂×C₆₇	—	8	2	no	—
201	C₂×C₆₇	#1439	1	1	yes	characteristic
202	C₂×C₆₇	#2017	1	2	yes	—
203	#2911	C₂×C₆₇	1	1	yes	characteristic, complemented
204	#2896	C₂×C₆₇	1	2	yes	—
205	C₂×#408	C₂×C₆₇	1	1	yes	characteristic
206	C₂×#338	C₂×C₆₇	1	1	yes	characteristic, complemented
207	#1729	C₂×C₆₇	1	2	yes	complemented
208	C₆₇	#11111	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
210	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
211	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
212	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
213	C₂×#117	C₂²×C₆₇	1	2	yes	—
214	#428	—	2	4	no	—
215	#427	—	2	2	no	—
216	#419	—	2	2	no	—
217	#418	—	2	2	no	—
218	#417	—	2	4	no	—
219	#416	—	2	4	no	—
220	#408	—	2	2	no	—
221	#408	C₄×C₆₇	1	4	yes	complemented
222	C₂×C₂²⋊C₈	—	2	2	no	—
223	#338	—	2	2	no	—
224	#336	—	2	4	no	—
225	C₂×#41	—	2	1	no	—
226	C₂×#41	—	2	1	no	—
227	C₂×#41	—	2	2	no	—
228	C₂×#41	—	2	2	no	—
229	C₂×#41	—	2	2	no	—
230	C₂×#41	—	2	2	no	—
231	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
232	C₂×#41	Q₈×C₆₇	1	2	yes	—
233	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
234	C₂²×D₄	—	2	1	no	—
235	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
236	C₂³×C₄	—	2	1	no	—
237	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
238	C₂×M₁₆	—	2	2	no	—
239	C₂×M₁₆	—	2	2	no	—
240	C₂×M₁₆	—	2	2	no	—
241	C₂×M₁₆	—	2	2	no	—
242	C₂×M₁₆	—	2	2	no	—
243	C₂×M₁₆	—	2	2	no	—
244	C₂×M₁₆	—	2	2	no	—
245	C₂×M₁₆	—	2	2	no	—
246	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
247	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
248	C₂²×C₈	—	2	2	no	—
249	C₂²×C₈	—	2	1	no	—
250	C₂²×C₈	—	2	1	no	—
251	C₂²×C₈	—	2	2	no	—
252	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
253	C₂²×C₈	D₄×C₆₇	1	4	yes	—
254	C₂²×C₈	D₄×C₆₇	1	2	yes	—
255	#117	—	2	2	no	—
256	#117	—	2	2	no	—
257	#117	—	2	2	no	—
258	#117	—	2	2	no	—
259	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
260	C₂×C₂²⋊C₄	—	2	2	no	—
261	C₂×C₄²	—	2	2	no	—
262	#107	—	2	4	no	—
263	#106	—	2	4	no	—
264	#103	—	2	8	no	—
265	#102	—	2	8	no	—
266	C₂²⋊C₈	—	2	8	no	—
267	C₂⁴	—	2	1	no	—
268	C₂⁴	#41×C₆₇	1	1	yes	characteristic
269	#41	—	2	4	no	—
270	#41	—	2	4	no	—
271	#41	—	2	4	no	—
272	#41	—	2	4	no	—
273	#41	—	4	2	no	—
274	#41	—	4	2	no	—
275	#41	—	4	2	no	—
276	#41	—	4	2	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	C₄⋊C₄×C₆₇	1	2	yes	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	4	2	no	—
287	C₂×D₄	—	4	2	no	—
288	C₂×D₄	C₄⋊C₄×C₆₇	1	2	yes	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	4	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	—	4	1	no	—
295	C₂²×C₄	—	4	1	no	—
296	C₂²×C₄	—	4	1	no	—
297	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
298	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
300	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
302	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
303	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
304	C₂²×C₄	#41×C₆₇	1	2	yes	—
305	C₂²×C₄	#41×C₆₇	1	2	yes	—
306	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
307	M₁₆	—	2	4	no	—
308	M₁₆	—	2	4	no	—
309	M₁₆	—	4	2	no	—
310	M₁₆	—	4	2	no	—
311	M₁₆	—	4	2	no	—
312	M₁₆	—	4	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	4	no	—
317	C₂×C₈	—	2	4	no	—
318	C₂×C₈	—	2	4	no	—
319	C₂×C₈	—	2	4	no	—
320	C₂×C₈	—	4	2	no	—
321	C₂×C₈	—	4	1	no	—
322	C₂×C₈	—	4	1	no	—
323	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
324	C₄⋊C₄	—	2	4	no	—
325	C₄⋊C₄	—	2	4	no	—
326	C₂²⋊C₄	—	4	2	no	—
327	C₂²⋊C₄	—	4	2	no	—
328	C₄²	—	4	2	no	—
329	C₄²	—	4	2	no	—
330	C₂³	—	2	2	no	—
331	C₂³	—	4	2	no	—
332	C₂³	—	4	4	no	—
333	C₂³	—	4	1	no	—
334	C₂³	—	4	1	no	—
335	C₂³	#117×C₆₇	1	1	yes	characteristic
336	C₂³	#121×C₆₇	1	1	yes	characteristic
337	C₂³	#121×C₆₇	1	2	yes	—
338	C₂³	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
339	C₂³	#126×C₆₇	1	2	yes	—
340	Q₈	—	2	4	no	—
341	Q₈	—	4	2	no	—
342	D₄	—	2	4	no	—
343	D₄	—	4	4	no	—
344	D₄	—	4	4	no	—
345	D₄	—	4	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	8	no	—
354	C₂×C₄	—	4	4	no	—
355	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
356	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
357	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
358	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
359	C₂×C₄	#120×C₆₇	1	2	yes	—
360	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
361	C₂×C₄	#121×C₆₇	1	2	yes	—
362	C₂×C₄	#121×C₆₇	1	2	yes	—
363	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
364	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
365	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
366	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
367	C₂×C₄	#123×C₆₇	1	2	yes	—
368	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
369	C₂×C₄	#123×C₆₇	1	2	yes	—
370	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
371	C₂×C₄	#124×C₆₇	1	2	yes	—
372	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
373	C₈	—	2	4	no	—
374	C₈	—	4	2	no	—
375	C₈	—	4	4	no	—
376	C₂²	—	2	4	no	—
377	C₂²	—	2	2	no	—
378	C₂²	—	4	1	no	—
379	C₂²	—	4	1	no	—
380	C₂²	—	4	2	no	—
381	C₂²	—	4	4	no	—
382	C₂²	—	4	2	no	—
383	C₂²	—	8	2	no	—
384	C₂²	—	8	2	no	—
385	C₂²	#380×C₆₇	1	1	yes	characteristic
386	C₂²	#388×C₆₇	1	2	yes	—
387	C₂²	#392×C₆₇	1	1	yes	characteristic
388	C₂²	#396×C₆₇	1	2	yes	—
389	C₂²	#397×C₆₇	1	1	yes	characteristic
390	C₄	—	4	2	no	—
391	C₄	—	8	4	no	—
392	C₄	#385×C₆₇	1	2	yes	—
393	C₄	#386×C₆₇	1	1	yes	characteristic
394	C₄	#390×C₆₇	1	1	yes	characteristic
395	C₄	#393×C₆₇	1	1	yes	characteristic
396	C₄	#394×C₆₇	1	2	yes	—
397	C₄	#399×C₆₇	1	1	yes	characteristic
398	C₂	—	2	1	no	—
399	C₂	—	2	1	no	—
400	C₂	—	2	2	no	—
401	C₂	—	2	2	no	—
402	C₂	—	8	2	no	—
403	C₂	C₆₇×#1439	1	1	yes	characteristic
404	C₂	C₆₇×#2017	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	#127
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
socle	C₂²×C₆₇	#392
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2911	C₂	semidirect
4. C₂×#338×C₆₇	C₂	semidirect
5. C₆₇×#1729	C₂	semidirect
18. #408×C₆₇	C₄	semidirect
195. #11111	C₆₇	direct (N × H)
203. #2911	C₂×C₆₇	semidirect
206. C₂×#338	C₂×C₆₇	semidirect
207. #1729	C₂×C₆₇	semidirect
208. C₆₇	#11111	direct (N × H)
221. #408	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2911	2
C₆₇×#2896	2
C₂×#408×C₆₇	2
C₂×#338×C₆₇	2
C₆₇×#1729	2
#11111	67

Subgroups — C67×#11111 #30690362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11111 #30690362