Subgroups — #30824873

Order 17152.

666 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569327	C₂	1	1	yes	maximal, characteristic
2	#22569322	C₂	1	1	yes	maximal, characteristic
3	C₄×#14524051	C₂	1	1	yes	maximal, characteristic
4	#22558975	C₂	1	2	yes	maximal, complemented
5	#22558973	C₂	1	2	yes	maximal, complemented
6	#22558971	C₂	1	2	yes	maximal, complemented
7	#22558963	C₂	1	2	yes	maximal, complemented
8	#22558961	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558955	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16863146	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1565	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524051	C₄	1	1	yes	characteristic
13	C₂×#14524051	C₄	1	1	yes	characteristic
14	C₄×#11761628	C₂²	1	1	yes	characteristic
15	#16863526	—	2	1	no	—
16	#16863523	—	2	1	no	—
17	#16863475	—	2	1	no	—
18	#16863473	—	2	1	no	—
19	#16863467	—	2	2	no	—
20	#16863464	—	2	2	no	—
21	#16863452	C₂²	1	2	yes	—
22	#16863452	C₂²	1	2	yes	—
23	#16863449	C₂²	1	1	yes	characteristic
24	#16863418	C₂²	1	2	yes	—
25	#16863418	C₂²	1	2	yes	—
26	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
27	#16863354	C₄	1	1	yes	characteristic
28	#16863354	C₄	1	1	yes	characteristic
29	#16863353	C₄	1	2	yes	—
30	#16863353	C₄	1	2	yes	—
31	C₄×#11761611	C₂²	1	2	yes	—
32	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₆₇	C₂²	1	2	yes	—
34	#16863149	C₂²	1	2	yes	—
35	#16863148	C₂²	1	1	yes	characteristic
36	#16863148	C₂²	1	1	yes	characteristic
37	#16863147	C₂²	1	2	yes	—
38	#16863146	C₂²	1	2	yes	—
39	#16863146	C₂²	1	2	yes	—
40	#16863146	C₂²	1	1	yes	characteristic
41	#16863146	C₂²	1	1	yes	characteristic
42	#431×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
45	#335×C₆₇	C₂²	1	2	yes	—
46	#334×C₆₇	C₂²	1	2	yes	—
47	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
48	#14524051	—	2	4	no	—
49	#14524051	—	2	4	no	—
50	C₂×#11761611	—	2	2	no	—
51	C₂×#11761611	—	2	2	no	—
52	C₂×#11761611	C₂×C₄	1	2	yes	—
53	C₂×#11761611	C₂×C₄	1	2	yes	—
54	C₂²×C₆₇⋊C₈	—	2	1	no	—
55	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
58	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
60	C₄×Dic₁₃₄	Q₈	1	1	yes	characteristic
61	C₄×D₂₆₈	D₄	1	1	yes	characteristic
62	C₄×D₂₆₈	Q₈	1	1	yes	characteristic
63	C₄×C₂²⋊D₆₇	D₄	1	2	yes	—
64	C₄×C₂²⋊D₆₇	Q₈	1	2	yes	—
65	#14524002	C₂×C₄	1	1	yes	characteristic
66	#14524000	C₂×C₄	1	2	yes	—
67	C₄²×D₆₇	C₂³	1	2	yes	—
68	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
69	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
70	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
71	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
72	Dic₆₇⋊C₈	—	2	2	no	—
73	Dic₆₇⋊C₈	—	2	2	no	—
74	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
75	(C₂×D₆₇)⋊C₈	—	2	2	no	—
76	(C₂×D₆₇)⋊C₈	—	2	2	no	—
77	#14523953	—	2	2	no	—
78	#14523953	—	2	2	no	—
79	#14523953	C₂×C₄	1	2	yes	—
80	#14523953	C₂×C₄	1	2	yes	—
81	C₄×C₆₇⋊C₈	—	2	1	no	—
82	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
83	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
84	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
85	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
86	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
87	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
88	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
89	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₂²×C₈×C₆₇	—	2	1	no	—
91	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
92	C₄⋊C₈×C₆₇	—	2	1	no	—
93	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
95	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₂²⋊C₈×C₆₇	—	2	1	no	—
97	#97×C₆₇	C₂×C₄	1	2	yes	—
98	C₄×C₈×C₆₇	—	2	1	no	—
99	C₄×C₈×C₆₇	C₂³	1	2	yes	—
100	#11761628	—	2	2	no	—
101	#11761628	—	2	2	no	—
102	C₂×Dic₁₃₄	C₄⋊C₄	1	1	yes	characteristic
103	C₂×D₂₆₈	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₂²⋊D₆₇	C₄⋊C₄	1	2	yes	—
105	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
106	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
107	C₂×C₄×D₆₇	C₂×Q₈	1	2	yes	—
108	#11761611	—	2	4	no	—
109	#11761611	—	2	4	no	—
110	#11761611	—	8	2	no	—
111	#11761611	—	8	2	no	—
112	C₂×C₆₇⋊C₈	—	2	2	no	—
113	C₂×C₆₇⋊C₈	—	2	1	no	—
114	C₂×C₆₇⋊C₈	—	2	1	no	—
115	C₂×C₆₇⋊C₈	—	2	2	no	—
116	C₂×C₆₇⋊C₈	—	8	1	no	—
117	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	complemented
118	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
122	C₈×D₆₇	—	2	4	no	—
123	C₈×D₆₇	—	2	4	no	—
124	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
125	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
126	Dic₆₇⋊C₄	C₄⋊C₄	1	2	yes	—
127	Dic₆₇⋊C₄	C₄⋊C₄	1	2	yes	—
128	C₂²⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
129	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	2	yes	—
130	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	2	yes	—
131	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
132	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
133	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
134	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
137	M₁₆×C₆₇	—	2	2	no	—
138	M₁₆×C₆₇	M₁₆	1	4	yes	complemented
139	C₂×C₈×C₆₇	—	2	2	no	—
140	C₂×C₈×C₆₇	—	2	1	no	—
141	C₂×C₈×C₆₇	—	2	1	no	—
142	C₂×C₈×C₆₇	—	8	1	no	—
143	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
144	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
145	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
146	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
148	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
149	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×D₆₇	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×C₄×C₆₇	—	2	1	no	—
152	C₂×C₄×C₆₇	—	2	1	no	—
153	C₂×C₄×C₆₇	—	4	1	no	—
154	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
158	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
159	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
162	C₂²⋊D₆₇	—	2	4	no	—
163	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
165	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
166	D₂₆₈	—	2	2	no	—
167	C₄×D₆₇	—	2	2	no	—
168	C₄×D₆₇	—	2	2	no	—
169	C₄×D₆₇	—	2	2	no	—
170	C₄×D₆₇	—	2	2	no	—
171	C₄×D₆₇	—	4	2	no	—
172	Dic₁₃₄	—	2	2	no	—
173	C₈×C₆₇	—	2	2	no	—
174	C₈×C₆₇	—	4	1	no	—
175	C₈×C₆₇	—	4	2	no	—
176	C₈×C₆₇	—	4	1	no	—
177	C₈×C₆₇	C₂×M₁₆	1	4	yes	—
178	C₆₇⋊C₈	—	2	1	no	—
179	C₆₇⋊C₈	—	2	1	no	—
180	C₆₇⋊C₈	—	4	1	no	—
181	C₆₇⋊C₈	—	4	2	no	—
182	C₆₇⋊C₈	—	4	1	no	—
183	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
184	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
185	C₂²×C₆₇	—	2	1	no	—
186	C₂²×C₆₇	—	2	1	no	—
187	C₂²×C₆₇	#423	1	2	yes	—
188	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
189	C₂×D₆₇	—	2	2	no	—
190	C₂×D₆₇	—	2	2	no	—
191	C₂×D₆₇	—	2	2	no	—
192	C₄×C₆₇	—	2	1	no	—
193	C₄×C₆₇	—	2	1	no	—
194	C₄×C₆₇	—	2	1	no	—
195	C₄×C₆₇	#423	1	2	yes	—
196	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
197	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
198	C₄×C₆₇	#577	1	1	yes	characteristic
199	C₄×C₆₇	#577	1	1	yes	characteristic
200	Dic₆₇	—	2	2	no	—
201	Dic₆₇	—	2	2	no	—
202	Dic₆₇	—	2	2	no	—
203	#17798	—	67	1	no	maximal, Sylow
204	C₂×C₆₇	—	2	2	no	—
205	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
206	C₂×C₆₇	#2912	1	1	yes	characteristic
207	C₂×C₆₇	#2916	1	1	yes	characteristic
208	D₆₇	—	4	2	no	—
209	#2989	—	67	1	no	—
210	#2989	—	67	1	no	—
211	C₄×#131	—	67	1	no	—
212	#1569	—	67	2	no	—
213	#1569	—	67	2	no	—
214	#1568	—	67	2	no	—
215	#1566	—	67	2	no	—
216	#1565	—	67	1	no	—
217	#1565	D₆₇	1	1	yes	characteristic, complemented
218	#1564	—	67	1	no	—
219	C₂×#334	—	67	1	no	—
220	C₆₇	#17798	1	1	yes	characteristic, complemented, Sylow
221	C₂×#131	—	67	1	no	—
222	C₂×#131	—	67	1	no	—
223	C₄×#41	—	67	1	no	—
224	#446	—	134	1	no	—
225	#446	—	134	1	no	—
226	#436	—	134	1	no	—
227	#436	—	134	1	no	—
228	#435	—	134	2	no	—
229	#435	—	134	2	no	—
230	#432	—	67	2	no	—
231	#432	—	67	2	no	—
232	#431	—	67	1	no	—
233	#431	C₂×D₆₇	1	1	yes	characteristic
234	#423	—	67	2	no	—
235	#423	—	67	2	no	—
236	C₂×C₄⋊C₈	—	67	1	no	—
237	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
238	#405	—	67	1	no	—
239	#405	—	67	1	no	—
240	#405	—	67	2	no	—
241	#405	—	67	2	no	—
242	C₄×M₁₆	—	67	2	no	—
243	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
244	C₂×C₄×C₈	—	67	1	no	—
245	C₂×C₄×C₈	—	67	2	no	—
246	#335	—	67	1	no	—
247	#335	—	67	1	no	—
248	#335	—	67	2	no	—
249	#335	C₂×D₆₇	1	2	yes	—
250	#334	—	67	2	no	—
251	#334	—	67	2	no	—
252	#334	—	67	1	no	—
253	#334	—	67	1	no	—
254	#334	—	67	2	no	—
255	#334	C₂×D₆₇	1	2	yes	—
256	C₂×#41	—	67	1	no	—
257	#131	—	134	4	no	—
258	#131	—	134	4	no	—
259	C₂×M₁₆	—	67	2	no	—
260	C₂×M₁₆	—	67	2	no	—
261	C₂×M₁₆	—	134	2	no	—
262	C₂×M₁₆	—	134	2	no	—
263	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
264	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
265	C₂²×C₈	—	2	1	no	—
266	C₂²×C₈	—	67	1	no	—
267	C₂²×C₈	—	67	1	no	—
268	C₂²×C₈	—	67	2	no	—
269	C₂²×C₈	—	67	2	no	—
270	C₂²×C₈	—	134	1	no	—
271	C₄×Q₈	—	67	1	no	—
272	C₄×Q₈	—	67	1	no	—
273	C₄×D₄	—	67	2	no	—
274	C₄×D₄	—	67	2	no	—
275	C₄×D₄	—	67	1	no	—
276	C₄×D₄	—	67	1	no	—
277	#117	—	67	1	no	—
278	#117	—	67	2	no	—
279	C₂×C₄²	—	67	2	no	—
280	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	67	2	no	—
283	C₄⋊C₈	—	67	1	no	—
284	C₄⋊C₈	—	67	1	no	—
285	C₄⋊C₈	—	134	2	no	—
286	C₄⋊C₈	—	134	2	no	—
287	C₄⋊C₈	—	134	1	no	—
288	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
289	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
290	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
291	C₂²⋊C₈	—	2	1	no	—
292	C₂²⋊C₈	—	134	2	no	—
293	C₂²⋊C₈	—	134	2	no	—
294	C₂²⋊C₈	—	134	1	no	—
295	#97	—	67	2	no	—
296	#97	—	67	2	no	—
297	#97	—	134	2	no	—
298	#97	—	134	2	no	—
299	#97	C₄×D₆₇	1	2	yes	—
300	C₄×C₈	—	2	1	no	—
301	C₄×C₈	—	67	2	no	—
302	C₄×C₈	—	67	2	no	—
303	C₄×C₈	—	67	1	no	—
304	C₄×C₈	—	67	1	no	—
305	C₄×C₈	—	67	1	no	—
306	C₄×C₈	—	67	1	no	—
307	C₄×C₈	—	134	1	no	—
308	C₄×C₈	C₂²×D₆₇	1	2	yes	—
309	#41	—	134	2	no	—
310	#41	—	134	2	no	—
311	C₂×Q₈	—	67	1	no	—
312	C₂×D₄	—	67	2	no	—
313	C₂×D₄	—	67	1	no	—
314	C₂²×C₄	—	67	2	no	—
315	C₂²×C₄	—	67	2	no	—
316	C₂²×C₄	—	67	2	no	—
317	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
318	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
319	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
320	M₁₆	—	2	2	no	—
321	M₁₆	—	536	2	no	—
322	M₁₆	—	536	2	no	—
323	M₁₆	—	134	4	no	—
324	M₁₆	—	134	4	no	—
325	M₁₆	#11761611	1	4	yes	complemented
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	8	1	no	—
330	C₂×C₈	—	536	1	no	—
331	C₂×C₈	—	67	4	no	—
332	C₂×C₈	—	67	1	no	—
333	C₂×C₈	—	67	1	no	—
334	C₂×C₈	—	67	1	no	—
335	C₂×C₈	—	67	1	no	—
336	C₂×C₈	—	134	4	no	—
337	C₂×C₈	—	134	1	no	—
338	C₂×C₈	—	134	1	no	—
339	C₂×C₈	—	134	2	no	—
340	C₂×C₈	—	134	2	no	—
341	C₂×C₈	—	134	4	no	—
342	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
343	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
344	C₄⋊C₄	—	67	2	no	—
345	C₄⋊C₄	—	67	2	no	—
346	C₄⋊C₄	—	67	1	no	—
347	C₄⋊C₄	—	67	1	no	—
348	C₂²⋊C₄	—	67	2	no	—
349	C₂²⋊C₄	—	67	2	no	—
350	C₂²⋊C₄	—	67	2	no	—
351	C₄²	—	67	2	no	—
352	C₄²	—	67	2	no	—
353	C₄²	—	67	2	no	—
354	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
355	C₄²	D₄×D₆₇	1	1	yes	characteristic
356	C₄²	Q₈×D₆₇	1	1	yes	characteristic
357	C₄²	C₂³×D₆₇	1	1	yes	characteristic
358	C₂³	—	67	2	no	—
359	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
360	Q₈	—	134	2	no	—
361	D₄	—	134	4	no	—
362	D₄	—	134	2	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	268	2	no	—
367	C₂×C₄	—	67	2	no	—
368	C₂×C₄	—	67	2	no	—
369	C₂×C₄	—	67	2	no	—
370	C₂×C₄	—	134	2	no	—
371	C₂×C₄	—	134	2	no	—
372	C₂×C₄	—	134	2	no	—
373	C₂×C₄	—	134	2	no	—
374	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
375	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
376	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
377	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
378	C₂×C₄	C₂×#11761611	1	2	yes	—
379	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
380	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
381	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
382	C₈	—	2	2	no	—
383	C₈	—	4	1	no	—
384	C₈	—	4	2	no	—
385	C₈	—	4	1	no	—
386	C₈	—	268	2	no	—
387	C₈	—	268	1	no	—
388	C₈	—	268	1	no	—
389	C₈	—	67	2	no	—
390	C₈	—	67	2	no	—
391	C₈	—	134	1	no	—
392	C₈	—	134	1	no	—
393	C₈	C₂×#11761611	1	4	yes	—
394	C₂²	—	2	1	no	—
395	C₂²	—	2	1	no	—
396	C₂²	—	134	2	no	—
397	C₂²	—	134	2	no	—
398	C₂²	—	134	2	no	—
399	C₂²	#16863418	1	2	yes	—
400	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
401	C₄	—	2	1	no	—
402	C₄	—	2	1	no	—
403	C₄	—	2	1	no	—
404	C₄	—	134	2	no	—
405	C₄	—	134	2	no	—
406	C₄	—	134	2	no	—
407	C₄	#16863418	1	2	yes	—
408	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
409	C₄	C₂²×#11761611	1	1	yes	characteristic
410	C₄	#16864205	1	1	yes	characteristic
411	C₄	#16864206	1	1	yes	characteristic
412	C₂	—	2	2	no	—
413	C₂	—	268	2	no	—
414	C₂	C₂×#16863418	1	1	yes	characteristic
415	C₂	#22568803	1	1	yes	characteristic
416	C₂	#22568819	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1565	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1565	D₆₇
hypocenter	C₆₇	#17798
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1565

Complements

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

N	H ≅ G/N	type
4. #22558975	C₂	semidirect
5. #22558973	C₂	semidirect
6. #22558971	C₂	semidirect
7. #22558963	C₂	semidirect
8. #22558961	C₂	semidirect
9. #22558955	C₂	semidirect
10. C₂×#16863146	C₂	semidirect
11. C₆₇×#1565	C₂	semidirect
117. C₂×C₆₇⋊C₈	M₁₆	semidirect
138. M₁₆×C₆₇	M₁₆	semidirect
217. #1565	D₆₇	semidirect
220. C₆₇	#17798	semidirect
325. M₁₆	#11761611	semidirect

Maximal subgroups

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

subgroup	index
#22569327	2
#22569322	2
C₄×#14524051	2
#22558975	2
#22558973	2
#22558971	2
#22558963	2
#22558961	2
#22558955	2
C₂×#16863146	2
C₆₇×#1565	2
#17798	67