Subgroups — #30853858

Order 17152.

1042 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863443	C₂	1	1	yes	maximal, characteristic
2	#22568797	C₂	1	1	yes	maximal, characteristic
3	#22568625	C₂	1	2	yes	maximal
4	C₂×#16863348	C₂	1	1	yes	maximal, characteristic
5	#22560093	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560092	C₂	1	2	yes	maximal, complemented
7	#22560091	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560090	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560089	C₂	1	2	yes	maximal, complemented
10	#339×D₆₇	C₂	1	2	yes	maximal
11	C₆₇×#1773	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#11761611	C₂²	1	1	yes	characteristic
13	C₂²×#11761611	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×D₆₇	C₂²	1	2	yes	—
15	#117×D₆₇	C₂²	1	2	yes	—
16	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
17	#16863443	C₄	1	8	yes	complemented
18	C₂×#14523979	C₂²	1	1	yes	characteristic
19	C₂×#14523978	C₂²	1	2	yes	—
20	#16863437	C₄	1	4	yes	complemented
21	#16863435	C₄	1	4	yes	complemented
22	#16863434	C₂²	1	1	yes	characteristic
23	#16863432	C₂²	1	2	yes	—
24	#16863354	C₂²	1	2	yes	—
25	#16863353	—	2	2	no	—
26	#16863352	C₂²	1	2	yes	—
27	#16863351	C₂²	1	2	yes	—
28	#16863350	—	2	2	no	—
29	#16863348	—	2	2	no	—
30	#16863348	C₄	1	8	yes	complemented
31	C₄×#11761611	—	2	2	no	—
32	C₂×#14523954	C₂²	1	1	yes	characteristic
33	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
35	#16863157	C₂²	1	2	yes	—
36	#16863156	C₂²	1	2	yes	—
37	#16863155	C₂²	1	2	yes	—
38	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
39	#427×C₆₇	C₂²	1	1	yes	characteristic
40	#405×C₆₇	C₂²	1	2	yes	—
41	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
42	#339×C₆₇	C₂²	1	2	yes	—
43	C₂²×C₄×D₆₇	—	2	1	no	—
44	C₂²×C₄×D₆₇	C₂³	1	1	yes	characteristic
45	C₂×#11761612	C₂³	1	2	yes	—
46	C₂×#11761611	—	2	2	no	—
47	C₂×#11761611	C₂×C₄	1	4	yes	—
48	C₂×#11761611	C₂×C₄	1	2	yes	—
49	C₂×#11761611	C₂×C₄	1	2	yes	—
50	C₂×#11761611	C₂×C₄	1	4	yes	—
51	C₂×#11761611	C₂×C₄	1	4	yes	—
52	C₂×#11761611	D₄	1	2	yes	—
53	C₂×#11761611	D₄	1	2	yes	—
54	C₂×#11761611	Q₈	1	4	yes	—
55	M₁₆×D₆₇	—	2	4	no	—
56	M₁₆×D₆₇	C₂×C₄	1	8	yes	—
57	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
58	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
59	C₂×C₈×D₆₇	—	2	2	no	—
60	#14524002	C₂³	1	2	yes	—
61	#14524001	C₂×C₄	1	4	yes	—
62	#14524000	—	2	2	no	—
63	#14523999	—	2	2	no	—
64	C₄⋊C₄×D₆₇	C₂×C₄	1	4	yes	—
65	C₂²⋊C₄×D₆₇	—	2	2	no	—
66	C₂×C₄×Dic₆₇	—	2	1	no	—
67	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
68	C₄²×D₆₇	—	2	2	no	—
69	C₄²×D₆₇	—	2	2	no	—
70	C₄²×D₆₇	C₂×C₄	1	4	yes	—
71	#14523979	C₂×C₄	1	4	yes	complemented
72	#14523978	C₂×C₄	1	8	yes	complemented
73	#14523977	C₂×C₄	1	2	yes	complemented
74	#14523976	C₂×C₄	1	4	yes	complemented
75	#14523975	C₂×C₄	1	2	yes	complemented
76	#14523974	C₂×C₄	1	4	yes	complemented
77	#14523954	—	2	2	no	—
78	#14523954	C₂×C₄	1	4	yes	complemented
79	#14523953	—	2	2	no	—
80	#14523953	—	2	2	no	—
81	C₄×C₆₇⋊C₈	—	2	2	no	—
82	C₈×Dic₆₇	—	2	2	no	—
83	C₈×Dic₆₇	—	2	2	no	—
84	C₈×Dic₆₇	C₂×C₄	1	4	yes	complemented
85	C₈×Dic₆₇	C₂×C₄	1	4	yes	complemented
86	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
87	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	#117×C₆₇	C₂³	1	2	yes	—
90	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
91	#108×C₆₇	C₂×C₄	1	4	yes	complemented
92	#107×C₆₇	C₂×C₄	1	2	yes	complemented
93	#106×C₆₇	C₂×C₄	1	2	yes	complemented
94	#97×C₆₇	—	2	2	no	—
95	#97×C₆₇	C₂×C₄	1	4	yes	complemented
96	C₄×C₈×C₆₇	—	2	2	no	—
97	C₂³×D₆₇	#41	1	1	yes	characteristic
98	C₂²×Dic₆₇	—	2	1	no	—
99	C₂²×Dic₆₇	#41	1	1	yes	characteristic
100	C₂×C₄×D₆₇	—	2	4	no	—
101	C₂×C₄×D₆₇	—	2	4	no	—
102	C₂×C₄×D₆₇	—	4	4	no	—
103	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
104	C₂×C₄×D₆₇	C₂²×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	C₂×C₄×D₆₇	#41	1	2	yes	—
109	#11761612	—	2	2	no	—
110	#11761612	C₂²×C₄	1	4	yes	—
111	#11761611	—	2	8	no	—
112	#11761611	C₄²	1	8	yes	—
113	#11761611	C₄²	1	8	yes	—
114	#11761611	C₄⋊C₄	1	8	yes	—
115	#11761611	C₄⋊C₄	1	8	yes	—
116	C₂×C₆₇⋊C₈	—	2	2	no	—
117	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
118	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
119	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
120	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
121	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
124	C₈×D₆₇	—	2	8	no	—
125	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
126	Dic₆₇⋊C₄	—	2	4	no	—
127	C₂²⋊Dic₆₇	—	2	2	no	—
128	(C₂×D₆₇)⋊C₄	—	2	4	no	—
129	C₄×Dic₆₇	—	2	2	no	—
130	C₄×Dic₆₇	—	2	1	no	—
131	C₄×Dic₆₇	—	2	4	no	—
132	C₄×Dic₆₇	—	2	4	no	—
133	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
136	M₁₆×C₆₇	—	2	2	no	—
137	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
138	C₂×C₈×C₆₇	—	2	2	no	—
139	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
142	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
143	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
144	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
145	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
146	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
147	C₂²⋊C₄×C₆₇	—	2	2	no	—
148	C₄²×C₆₇	—	2	2	no	—
149	C₄²×C₆₇	—	2	1	no	—
150	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
151	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
152	C₂²×D₆₇	—	4	2	no	—
153	C₂²×D₆₇	#117	1	2	yes	—
154	C₂²×D₆₇	C₄×D₄	1	2	yes	—
155	C₂²×D₆₇	C₄×Q₈	1	2	yes	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	4	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
165	C₂×Dic₆₇	—	2	4	no	—
166	C₂×Dic₆₇	—	2	2	no	—
167	C₂×Dic₆₇	—	2	4	no	—
168	C₂×Dic₆₇	#117	1	2	yes	—
169	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
170	C₂×Dic₆₇	C₄×Q₈	1	2	yes	—
171	C₄×D₆₇	—	4	8	no	—
172	C₄×D₆₇	—	8	4	no	—
173	C₄×D₆₇	C₂×C₄²	1	4	yes	—
174	C₄×D₆₇	C₂×C₄⋊C₄	1	4	yes	—
175	C₄×D₆₇	C₄×D₄	1	4	yes	—
176	C₄×D₆₇	C₄×Q₈	1	4	yes	—
177	C₈×C₆₇	—	2	4	no	—
178	C₈×C₆₇	C₂×C₄²	1	2	yes	—
179	C₈×C₆₇	C₂×C₄²	1	2	yes	—
180	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
181	C₆₇⋊C₈	—	2	4	no	—
182	C₆₇⋊C₈	C₂×C₄²	1	4	yes	—
183	C₆₇⋊C₈	C₂×C₄⋊C₄	1	2	yes	—
184	C₆₇⋊C₈	C₂×C₄⋊C₄	1	2	yes	—
185	C₂²×C₆₇	—	2	2	no	—
186	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
187	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
188	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
189	C₂×D₆₇	—	2	4	no	—
190	C₂×D₆₇	—	4	2	no	—
191	C₂×D₆₇	—	4	2	no	—
192	C₂×D₆₇	C₄×C₄⋊C₄	1	4	yes	—
193	C₄×C₆₇	—	4	2	no	—
194	C₄×C₆₇	—	4	4	no	—
195	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
196	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
197	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
198	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
199	Dic₆₇	—	2	4	no	—
200	Dic₆₇	—	4	4	no	—
201	Dic₆₇	C₄×C₄⋊C₄	1	4	yes	—
202	#20445	—	67	1	no	maximal, Sylow
203	C₂×C₆₇	—	2	1	no	—
204	C₂×C₆₇	—	2	1	no	—
205	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
206	C₂×C₆₇	#2956	1	2	yes	—
207	D₆₇	—	2	4	no	—
208	C₂×#430	—	67	1	no	—
209	#2911	—	67	1	no	—
210	#2873	—	67	2	no	—
211	C₂×C₄×M₁₆	—	67	1	no	—
212	#1773	—	67	1	no	—
213	#1773	—	67	2	no	—
214	#1773	D₆₇	1	1	yes	characteristic, complemented
215	#1772	—	67	1	no	—
216	#1772	—	67	1	no	—
217	#1772	—	67	2	no	—
218	C₂×#339	—	67	2	no	—
219	C₆₇	#20445	1	1	yes	characteristic, complemented, Sylow
220	C₂²×M₁₆	—	67	1	no	—
221	C₂²×M₁₆	—	67	1	no	—
222	C₂²×M₁₆	—	67	2	no	—
223	C₂×#117	—	67	2	no	—
224	C₂²×C₄²	—	67	1	no	—
225	#430	—	67	8	no	—
226	C₂×#108	—	67	2	no	—
227	C₂×#108	—	67	1	no	—
228	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
229	#428	—	67	4	no	—
230	#428	—	67	4	no	—
231	#427	—	67	1	no	—
232	#427	—	67	2	no	—
233	#427	C₂×D₆₇	1	1	yes	characteristic
234	#405	—	67	2	no	—
235	#405	—	67	2	no	—
236	#405	—	67	2	no	—
237	#405	—	134	2	no	—
238	#405	—	134	2	no	—
239	#405	C₂×D₆₇	1	2	yes	—
240	C₄×M₁₆	—	67	8	no	—
241	C₄×M₁₆	—	134	2	no	—
242	C₄×M₁₆	—	134	2	no	—
243	C₂×#97	—	67	1	no	—
244	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
245	C₂×C₄×C₈	—	67	1	no	—
246	C₂×C₄×C₈	—	67	1	no	—
247	#339	—	67	2	no	—
248	#339	—	67	2	no	—
249	#339	—	67	2	no	—
250	#339	C₂×D₆₇	1	2	yes	—
251	C₂³×C₄	—	67	1	no	—
252	C₂³×C₄	—	134	1	no	—
253	C₂×M₁₆	—	67	4	no	—
254	C₂×M₁₆	—	67	4	no	—
255	C₂×M₁₆	—	67	2	no	—
256	C₂×M₁₆	—	67	2	no	—
257	C₂×M₁₆	—	67	4	no	—
258	C₂×M₁₆	—	67	2	no	—
259	C₂×M₁₆	—	67	2	no	—
260	C₂×M₁₆	—	67	4	no	—
261	C₂×M₁₆	—	67	2	no	—
262	C₂×M₁₆	—	67	8	no	—
263	C₂×M₁₆	—	134	2	no	—
264	C₂×M₁₆	—	134	4	no	—
265	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
266	C₂²×C₈	—	67	1	no	—
267	C₂²×C₈	—	67	1	no	—
268	C₂²×C₈	—	134	2	no	—
269	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
270	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
271	#117	—	67	2	no	—
272	#117	—	67	4	no	—
273	#117	—	134	2	no	—
274	#117	—	134	2	no	—
275	#117	C₂²×D₆₇	1	2	yes	—
276	C₂×C₄⋊C₄	—	67	4	no	—
277	C₂×C₂²⋊C₄	—	134	2	no	—
278	C₂×C₄²	—	67	1	no	—
279	C₂×C₄²	—	67	4	no	—
280	C₂×C₄²	—	134	2	no	—
281	C₂×C₄²	—	134	2	no	—
282	C₂×C₄²	—	134	1	no	—
283	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
284	#108	—	67	8	no	—
285	#108	—	67	4	no	—
286	#108	C₄×D₆₇	1	4	yes	complemented
287	#107	—	67	2	no	—
288	#107	—	67	4	no	—
289	#107	C₄×D₆₇	1	2	yes	complemented
290	#106	—	67	2	no	—
291	#106	—	67	4	no	—
292	#106	C₄×D₆₇	1	2	yes	complemented
293	#97	—	2	2	no	—
294	#97	—	67	4	no	—
295	#97	—	134	2	no	—
296	#97	—	134	2	no	—
297	#97	—	134	2	no	—
298	#97	C₄×D₆₇	1	4	yes	complemented
299	C₄×C₈	—	2	2	no	—
300	C₄×C₈	—	67	4	no	—
301	C₄×C₈	—	67	4	no	—
302	C₄×C₈	—	134	2	no	—
303	C₄×C₈	—	134	2	no	—
304	C₄×C₈	—	134	2	no	—
305	C₂⁴	—	67	1	no	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	268	4	no	—
308	C₂²×C₄	—	67	1	no	—
309	C₂²×C₄	—	67	2	no	—
310	C₂²×C₄	—	67	2	no	—
311	C₂²×C₄	—	67	2	no	—
312	C₂²×C₄	—	67	2	no	—
313	C₂²×C₄	—	67	2	no	—
314	C₂²×C₄	—	67	2	no	—
315	C₂²×C₄	—	134	4	no	—
316	C₂²×C₄	—	134	4	no	—
317	C₂²×C₄	—	134	1	no	—
318	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
319	M₁₆	—	2	2	no	—
320	M₁₆	—	67	8	no	—
321	M₁₆	—	67	8	no	—
322	M₁₆	—	67	8	no	—
323	M₁₆	—	67	8	no	—
324	M₁₆	—	67	4	no	—
325	M₁₆	—	134	8	no	—
326	M₁₆	—	134	2	no	—
327	M₁₆	C₂×C₄×D₆₇	1	4	yes	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	67	2	no	—
330	C₂×C₈	—	67	2	no	—
331	C₂×C₈	—	67	2	no	—
332	C₂×C₈	—	67	2	no	—
333	C₂×C₈	—	67	1	no	—
334	C₂×C₈	—	67	1	no	—
335	C₂×C₈	—	67	2	no	—
336	C₂×C₈	—	134	8	no	—
337	C₂×C₈	—	134	2	no	—
338	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
339	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
340	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
341	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
342	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
343	C₂×C₈	D₄×D₆₇	1	2	yes	—
344	C₂×C₈	Q₈×D₆₇	1	2	yes	—
345	C₄⋊C₄	—	67	4	no	—
346	C₄⋊C₄	—	134	4	no	—
347	C₄⋊C₄	C₂×C₄×D₆₇	1	4	yes	—
348	C₂²⋊C₄	—	2	2	no	—
349	C₂²⋊C₄	—	134	4	no	—
350	C₂²⋊C₄	—	134	2	no	—
351	C₄²	—	2	2	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	67	2	no	—
354	C₄²	—	134	4	no	—
355	C₄²	—	134	4	no	—
356	C₄²	—	134	2	no	—
357	C₄²	—	134	1	no	—
358	C₄²	C₂×C₄×D₆₇	1	2	yes	—
359	C₂³	—	268	2	no	—
360	C₂³	—	67	2	no	—
361	C₂³	—	67	2	no	—
362	C₂³	—	67	2	no	—
363	C₂³	#41×D₆₇	1	1	yes	characteristic
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	268	8	no	—
368	C₂×C₄	—	536	4	no	—
369	C₂×C₄	—	67	2	no	—
370	C₂×C₄	—	67	2	no	—
371	C₂×C₄	—	67	2	no	—
372	C₂×C₄	—	67	4	no	—
373	C₂×C₄	—	67	4	no	—
374	C₂×C₄	—	67	4	no	—
375	C₂×C₄	—	67	4	no	—
376	C₂×C₄	—	134	4	no	—
377	C₂×C₄	—	134	4	no	—
378	C₂×C₄	—	134	2	no	—
379	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
380	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
381	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
382	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
383	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
384	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
385	C₈	—	2	4	no	—
386	C₈	—	67	4	no	—
387	C₈	—	67	2	no	—
388	C₈	—	67	2	no	—
389	C₈	—	134	4	no	—
390	C₈	C₄²×D₆₇	1	2	yes	—
391	C₈	C₄²×D₆₇	1	2	yes	—
392	C₈	C₄⋊C₄×D₆₇	1	4	yes	—
393	C₂²	—	2	2	no	—
394	C₂²	—	268	2	no	—
395	C₂²	—	268	2	no	—
396	C₂²	—	67	4	no	—
397	C₂²	—	134	4	no	—
398	C₂²	#117×D₆₇	1	1	yes	characteristic
399	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
400	C₂²	C₄×Q₈×D₆₇	1	1	yes	characteristic
401	C₄	—	4	2	no	—
402	C₄	—	4	4	no	—
403	C₄	—	268	4	no	—
404	C₄	—	67	4	no	—
405	C₄	—	134	4	no	—
406	C₄	C₂×C₄²×D₆₇	1	1	yes	characteristic
407	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
408	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
409	C₄	C₄×Q₈×D₆₇	1	1	yes	characteristic
410	C₂	—	2	1	no	—
411	C₂	—	2	1	no	—
412	C₂	—	134	4	no	—
413	C₂	C₄×C₄⋊C₄×D₆₇	1	1	yes	characteristic
414	C₂	#22569117	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1773	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×D₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1773	D₆₇
hypocenter	C₆₇	#20445
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1773

Complements

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

N	H ≅ G/N	type
5. #22560093	C₂	semidirect
6. #22560092	C₂	semidirect
7. #22560091	C₂	semidirect
8. #22560090	C₂	semidirect
9. #22560089	C₂	semidirect
11. C₆₇×#1773	C₂	semidirect
17. #16863443	C₄	semidirect
20. #16863437	C₄	semidirect
21. #16863435	C₄	semidirect
30. #16863348	C₄	semidirect
71. #14523979	C₂×C₄	semidirect
72. #14523978	C₂×C₄	semidirect
73. #14523977	C₂×C₄	semidirect
74. #14523976	C₂×C₄	semidirect
75. #14523975	C₂×C₄	semidirect
76. #14523974	C₂×C₄	semidirect
78. #14523954	C₂×C₄	semidirect
84. C₈×Dic₆₇	C₂×C₄	semidirect
85. C₈×Dic₆₇	C₂×C₄	semidirect
91. #108×C₆₇	C₂×C₄	semidirect
92. #107×C₆₇	C₂×C₄	semidirect
93. #106×C₆₇	C₂×C₄	semidirect
95. #97×C₆₇	C₂×C₄	semidirect
214. #1773	D₆₇	semidirect
219. C₆₇	#20445	semidirect
286. #108	C₄×D₆₇	semidirect
289. #107	C₄×D₆₇	semidirect
292. #106	C₄×D₆₇	semidirect
298. #97	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863443	2
#22568797	2
#22568625	2
C₂×#16863348	2
#22560093	2
#22560092	2
#22560091	2
#22560090	2
#22560089	2
#339×D₆₇	2
C₆₇×#1773	2
#20445	67