Subgroups — #30825856

Order 17152.

624 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570289	C₂	1	1	yes	maximal, characteristic
2	#22569184	C₂	1	1	yes	maximal, characteristic
3	#22559059	C₂	1	2	yes	maximal, complemented
4	#22559052	C₂	1	2	yes	maximal, complemented
5	#22559026	C₂	1	2	yes	maximal, complemented
6	#22559019	C₂	1	2	yes	maximal, complemented
7	#22558992	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558656	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558620	C₂	1	1	yes	maximal, characteristic
10	#22558498	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863930	C₂²	1	1	yes	characteristic
13	#16863726	C₄	1	1	yes	characteristic
14	#16863709	C₄	1	1	yes	characteristic
15	#16863703	C₄	1	2	yes	—
16	#16863618	C₄	1	2	yes	—
17	#16863552	C₄	1	1	yes	characteristic
18	#16863535	C₄	1	1	yes	characteristic
19	#16863526	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
21	C₄×#11761618	C₂²	1	2	yes	—
22	C₄×#11761615	C₂²	1	2	yes	—
23	#16863475	C₂²	1	1	yes	characteristic
24	#16863467	—	2	1	no	—
25	C₈×D₂₆₈	C₂²	1	1	yes	characteristic
26	C₈×C₂²⋊D₆₇	—	2	1	no	—
27	#16863455	—	2	1	no	—
28	#16863453	C₂²	1	1	yes	characteristic
29	#16863449	C₂²	1	1	yes	characteristic
30	#16863350	—	2	1	no	—
31	C₄×#11761612	C₂²	1	1	yes	characteristic
32	#16863148	C₂²	1	2	yes	—
33	#16863146	C₂²	1	2	yes	—
34	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
35	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
36	#16863126	C₂²	1	2	yes	—
37	#16863123	C₂²	1	2	yes	—
38	#16863118	C₂²	1	2	yes	—
39	#16863117	C₂²	1	2	yes	—
40	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₈×Q₈×C₆₇	—	2	1	no	—
42	C₈×Q₈×C₆₇	—	2	1	no	—
43	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
44	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
45	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
46	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×C₄×C₈×C₆₇	—	2	1	no	—
48	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×#11761618	C₂×C₄	1	2	yes	—
50	C₂×#11761615	C₂×C₄	1	2	yes	—
51	C₂×#11761612	—	2	1	no	—
52	C₂×#11761611	—	2	1	no	—
53	C₂×C₈×D₆₇	—	2	1	no	—
54	#14524037	—	2	1	no	—
55	#14524035	C₂×C₄	1	1	yes	characteristic
56	#14524031	C₂×C₄	1	1	yes	characteristic
57	#14524026	—	2	1	no	—
58	#14524022	C₂×C₄	1	1	yes	characteristic
59	#14524017	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
61	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
62	C₄×C₂²⋊D₆₇	—	2	1	no	—
63	#14524000	—	2	1	no	—
64	#14523976	C₂×C₄	1	2	yes	—
65	#14523974	C₂×C₄	1	2	yes	—
66	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
67	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
68	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
69	Dic₆₇⋊C₈	—	2	1	no	—
70	Dic₆₇⋊C₈	—	2	1	no	—
71	#14523967	C₂×C₄	1	2	yes	—
72	#14523967	C₂×C₄	1	2	yes	—
73	#14523966	C₂×C₄	1	2	yes	—
74	#14523963	C₂×C₄	1	2	yes	—
75	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
76	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
77	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
78	(C₂×D₆₇)⋊C₈	—	2	1	no	—
79	(C₂×D₆₇)⋊C₈	—	2	1	no	—
80	#14523954	—	2	1	no	—
81	#14523954	—	2	1	no	—
82	#14523953	—	2	1	no	—
83	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
84	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
85	C₈×Dic₆₇	—	2	1	no	—
86	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₂²×C₈×C₆₇	—	2	1	no	—
88	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
89	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
90	C₄×Q₈×C₆₇	—	2	1	no	—
91	C₄×Q₈×C₆₇	—	2	1	no	—
92	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
93	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
95	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄²×C₆₇	—	2	1	no	—
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₆₇	—	2	2	no	—
99	C₄⋊C₈×C₆₇	—	2	2	no	—
100	C₄⋊C₈×C₆₇	—	2	1	no	—
101	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
102	C₄×C₈×C₆₇	—	2	2	no	—
103	C₄×C₈×C₆₇	—	2	2	no	—
104	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
105	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
106	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
107	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
108	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
109	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₂²⋊D₆₇	—	2	1	no	—
111	C₂×C₄×D₆₇	—	2	1	no	—
112	#11761618	—	2	4	no	—
113	#11761615	—	2	4	no	—
114	#11761612	—	4	2	no	—
115	#11761611	—	4	2	no	—
116	C₂×C₆₇⋊C₈	—	2	1	no	—
117	C₂×C₆₇⋊C₈	—	2	1	no	—
118	C₂×C₆₇⋊C₈	—	2	2	no	—
119	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
120	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
121	C₈×D₆₇	—	4	2	no	—
122	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
124	Dic₆₇⋊C₄	—	2	1	no	—
125	Dic₆₇⋊C₄	—	2	1	no	—
126	Dic₆₇⋊C₄	—	2	1	no	—
127	Dic₆₇⋊C₄	—	2	1	no	—
128	C₂²⋊Dic₆₇	—	2	1	no	—
129	(C₂×D₆₇)⋊C₄	—	2	1	no	—
130	(C₂×D₆₇)⋊C₄	—	2	1	no	—
131	(C₂×D₆₇)⋊C₄	—	2	1	no	—
132	(C₂×D₆₇)⋊C₄	—	2	1	no	—
133	C₄×Dic₆₇	—	2	1	no	—
134	C₂×Q₈×C₆₇	—	2	4	no	—
135	C₂×Q₈×C₆₇	—	2	1	no	—
136	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	—	2	1	no	—
139	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
142	C₂×C₈×C₆₇	—	2	2	no	—
143	C₂×C₈×C₆₇	—	2	2	no	—
144	C₂×C₈×C₆₇	—	4	1	no	—
145	C₂×C₈×C₆₇	—	4	2	no	—
146	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₄⋊C₄×C₆₇	—	2	2	no	—
151	C₄⋊C₄×C₆₇	—	2	1	no	—
152	C₄⋊C₄×C₆₇	—	2	2	no	—
153	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
154	C₄²×C₆₇	—	2	2	no	—
155	C₄²×C₆₇	—	2	2	no	—
156	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₄²×C₆₇	#41	1	1	yes	characteristic
159	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
160	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
161	C₂²×D₆₇	—	2	1	no	—
162	Q₈×C₆₇	—	2	4	no	—
163	Q₈×C₆₇	—	2	2	no	—
164	Q₈×C₆₇	#131	1	4	yes	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
179	C₂²⋊D₆₇	—	4	2	no	—
180	C₂×Dic₆₇	—	2	1	no	—
181	C₂×Dic₆₇	—	2	1	no	—
182	C₂×Dic₆₇	—	2	1	no	—
183	D₂₆₈	—	2	2	no	—
184	C₄×D₆₇	—	4	1	no	—
185	C₄×D₆₇	—	4	1	no	—
186	Dic₁₃₄	—	2	2	no	—
187	C₈×C₆₇	—	2	1	no	—
188	C₈×C₆₇	—	2	1	no	—
189	C₈×C₆₇	—	2	1	no	—
190	C₈×C₆₇	—	4	2	no	—
191	C₈×C₆₇	#137	1	2	yes	—
192	C₆₇⋊C₈	—	2	2	no	—
193	C₆₇⋊C₈	—	2	2	no	—
194	C₆₇⋊C₈	—	4	2	no	—
195	C₆₇⋊C₈	—	4	1	no	—
196	C₆₇⋊C₈	—	4	1	no	—
197	C₂²×C₆₇	—	2	1	no	—
198	C₂²×C₆₇	—	2	1	no	—
199	C₂²×C₆₇	#444	1	2	yes	—
200	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
201	C₂×D₆₇	—	4	1	no	—
202	C₂×D₆₇	—	4	1	no	—
203	C₂×D₆₇	—	4	1	no	—
204	C₄×C₆₇	—	2	4	no	—
205	C₄×C₆₇	—	2	1	no	—
206	C₄×C₆₇	—	2	1	no	—
207	C₄×C₆₇	—	2	1	no	—
208	C₄×C₆₇	—	2	1	no	—
209	C₄×C₆₇	—	4	2	no	—
210	C₄×C₆₇	C₂×#131	1	2	yes	—
211	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
212	C₄×C₆₇	#575	1	1	yes	characteristic
213	Dic₆₇	—	4	1	no	—
214	Dic₆₇	—	4	1	no	—
215	Dic₆₇	—	4	1	no	—
216	#17915	—	67	1	no	maximal, Sylow
217	C₂×C₆₇	—	2	2	no	—
218	C₂×C₆₇	#2929	1	1	yes	characteristic
219	C₂×C₆₇	C₄×#137	1	1	yes	characteristic
220	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
221	D₆₇	—	8	1	no	—
222	#3108	—	67	1	no	—
223	#2966	—	67	1	no	—
224	C₂×C₈×Q₈	D₆₇	1	1	yes	characteristic, complemented
225	#1584	—	67	2	no	—
226	#1583	—	67	2	no	—
227	#1579	—	67	2	no	—
228	SD₁₆⋊C₈	—	67	2	no	—
229	#1573	—	67	1	no	—
230	#1491	—	67	1	no	—
231	#1483	—	67	1	no	—
232	#1451	—	67	1	no	—
233	C₆₇	#17915	1	1	yes	characteristic, complemented, Sylow
234	#529	—	67	1	no	—
235	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
236	#483	—	67	1	no	—
237	#480	—	67	1	no	—
238	#479	—	67	2	no	—
239	#464	—	67	2	no	—
240	#451	—	67	1	no	—
241	#448	—	67	1	no	—
242	#446	—	67	1	no	—
243	C₈×Q₈	—	2	1	no	—
244	C₈×Q₈	—	2	1	no	—
245	C₈×Q₈	—	67	1	no	—
246	C₈×Q₈	C₂×D₆₇	1	2	yes	—
247	C₈×Q₈	C₂×D₆₇	1	2	yes	—
248	C₄×Q₁₆	—	67	2	no	—
249	C₄×SD₁₆	—	67	2	no	—
250	#436	—	67	1	no	—
251	#435	—	134	1	no	—
252	C₈×D₄	—	67	1	no	—
253	C₈×D₄	—	134	1	no	—
254	#433	—	134	1	no	—
255	#432	—	67	1	no	—
256	#431	—	67	1	no	—
257	C₂×C₄⋊C₈	—	2	1	no	—
258	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
259	#405	—	134	1	no	—
260	C₄×M₁₆	—	67	1	no	—
261	C₂×C₄×C₈	—	2	1	no	—
262	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
263	#335	—	67	2	no	—
264	#334	—	67	2	no	—
265	Q₈⋊C₈	—	67	2	no	—
266	Q₈⋊C₈	—	67	2	no	—
267	Q₈⋊C₈	—	67	2	no	—
268	D₄⋊C₈	—	67	2	no	—
269	#322	—	67	2	no	—
270	#322	—	67	2	no	—
271	C₂²×Q₈	C₄×D₆₇	1	1	yes	characteristic
272	C₂×Q₁₆	—	67	2	no	—
273	C₂×SD₁₆	—	67	2	no	—
274	C₂×M₁₆	—	134	1	no	—
275	C₂×M₁₆	—	134	1	no	—
276	C₂²×C₈	—	2	1	no	—
277	C₂²×C₈	—	134	1	no	—
278	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
279	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
280	#126	—	134	1	no	—
281	#125	—	67	1	no	—
282	#124	—	67	1	no	—
283	#123	—	134	1	no	—
284	C₂²⋊Q₈	—	67	1	no	—
285	#121	—	67	1	no	—
286	C₄×Q₈	—	2	1	no	—
287	C₄×Q₈	—	2	1	no	—
288	C₄×Q₈	—	67	1	no	—
289	C₄×Q₈	C₄×D₆₇	1	2	yes	—
290	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
291	C₄×D₄	—	67	1	no	—
292	C₄×D₄	—	134	1	no	—
293	#117	—	134	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
296	C₂×C₄²	—	2	1	no	—
297	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
298	#107	—	67	2	no	—
299	#106	—	67	2	no	—
300	C₄⋊C₈	—	2	2	no	—
301	C₄⋊C₈	—	2	2	no	—
302	C₄⋊C₈	—	2	1	no	—
303	C₄⋊C₈	—	67	2	no	—
304	C₄⋊C₈	—	67	1	no	—
305	C₄⋊C₈	—	67	1	no	—
306	C₄⋊C₈	—	134	1	no	—
307	C₄⋊C₈	—	134	1	no	—
308	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
309	#103	—	67	2	no	—
310	#103	—	67	2	no	—
311	#103	—	67	2	no	—
312	#102	—	67	2	no	—
313	C₂²⋊C₈	—	67	1	no	—
314	C₂²⋊C₈	—	67	1	no	—
315	C₂²⋊C₈	—	134	1	no	—
316	C₂²⋊C₈	—	134	1	no	—
317	C₂²⋊C₈	—	134	1	no	—
318	#97	—	134	1	no	—
319	#97	—	134	1	no	—
320	#97	—	134	1	no	—
321	C₄×C₈	—	2	2	no	—
322	C₄×C₈	—	2	2	no	—
323	C₄×C₈	—	67	2	no	—
324	C₄×C₈	—	67	2	no	—
325	C₄×C₈	—	134	1	no	—
326	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
327	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
328	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
329	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
330	C₂×Q₈	—	2	4	no	—
331	C₂×Q₈	—	2	1	no	—
332	C₂×Q₈	—	67	1	no	—
333	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
334	C₂×D₄	—	67	1	no	—
335	C₂×D₄	—	134	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	134	1	no	—
339	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
340	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
341	C₂²×C₄	#11761628	1	1	yes	characteristic
342	Q₁₆	—	134	4	no	—
343	SD₁₆	—	134	4	no	—
344	M₁₆	—	268	2	no	—
345	M₁₆	—	268	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	4	1	no	—
349	C₂×C₈	—	4	2	no	—
350	C₂×C₈	—	268	2	no	—
351	C₂×C₈	—	67	2	no	—
352	C₂×C₈	—	67	2	no	—
353	C₂×C₈	—	134	1	no	—
354	C₂×C₈	—	134	1	no	—
355	C₂×C₈	—	134	2	no	—
356	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
357	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
358	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
359	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	67	1	no	—
364	C₄⋊C₄	—	67	1	no	—
365	C₄⋊C₄	—	134	1	no	—
366	C₄⋊C₄	—	134	1	no	—
367	C₄⋊C₄	—	134	1	no	—
368	C₄⋊C₄	—	134	1	no	—
369	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
370	C₂²⋊C₄	—	134	1	no	—
371	C₂²⋊C₄	—	134	1	no	—
372	C₂²⋊C₄	—	134	1	no	—
373	C₂²⋊C₄	—	134	1	no	—
374	C₂²⋊C₄	—	134	1	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	134	1	no	—
378	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
379	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
380	C₄²	#11761628	1	1	yes	characteristic
381	C₄²	C₂³×D₆₇	1	1	yes	characteristic
382	C₂³	—	134	1	no	—
383	C₂³	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
384	Q₈	—	2	4	no	—
385	Q₈	—	2	2	no	—
386	Q₈	—	134	2	no	—
387	Q₈	#14524049	1	4	yes	—
388	D₄	—	268	2	no	—
389	D₄	—	134	2	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	268	1	no	—
399	C₂×C₄	—	268	1	no	—
400	C₂×C₄	—	134	1	no	—
401	C₂×C₄	—	134	1	no	—
402	C₂×C₄	—	134	1	no	—
403	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
404	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
405	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
406	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
407	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
408	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
409	C₈	—	2	1	no	—
410	C₈	—	2	1	no	—
411	C₈	—	2	1	no	—
412	C₈	—	4	2	no	—
413	C₈	—	268	2	no	—
414	C₈	—	268	1	no	—
415	C₈	—	268	1	no	—
416	C₈	—	134	2	no	—
417	C₈	—	134	2	no	—
418	C₈	#14524074	1	2	yes	—
419	C₂²	—	2	1	no	—
420	C₂²	—	2	1	no	—
421	C₂²	—	268	1	no	—
422	C₂²	—	268	1	no	—
423	C₂²	—	268	1	no	—
424	C₂²	#16863516	1	2	yes	—
425	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
426	C₄	—	2	1	no	—
427	C₄	—	2	1	no	—
428	C₄	—	2	1	no	—
429	C₄	—	2	4	no	—
430	C₄	—	2	1	no	—
431	C₄	—	4	2	no	—
432	C₄	—	268	1	no	—
433	C₄	—	268	1	no	—
434	C₄	—	268	1	no	—
435	C₄	C₂×#14524049	1	2	yes	—
436	C₄	C₂×#14524074	1	1	yes	characteristic
437	C₄	#16864194	1	1	yes	characteristic
438	C₂	—	2	2	no	—
439	C₂	—	536	1	no	—
440	C₂	#22568903	1	1	yes	characteristic
441	C₂	C₄×#14524074	1	1	yes	characteristic
442	C₂	C₂×#16863516	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×C₈×Q₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#14524017	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₆₇
hypocenter	C₆₇	#17915
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #22559059	C₂	semidirect
4. #22559052	C₂	semidirect
5. #22559026	C₂	semidirect
6. #22559019	C₂	semidirect
7. #22558992	C₂	semidirect
8. #22558656	C₂	semidirect
10. #22558498	C₂	semidirect
11. C₂×C₈×Q₈×C₆₇	C₂	semidirect
224. C₂×C₈×Q₈	D₆₇	semidirect
233. C₆₇	#17915	semidirect

Maximal subgroups

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

subgroup	index
#22570289	2
#22569184	2
#22559059	2
#22559052	2
#22559026	2
#22559019	2
#22558992	2
#22558656	2
#22558620	2
#22558498	2
C₂×C₈×Q₈×C₆₇	2
#17915	67