Subgroups — #30826996

Order 17152.

640 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570269	C₂	1	1	yes	maximal, characteristic
2	#22569334	C₂	1	1	yes	maximal, characteristic, complemented
3	#22569275	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559096	C₂	1	2	yes	maximal, complemented
5	#22559070	C₂	1	2	yes	maximal, complemented
6	#22559069	C₂	1	2	yes	maximal, complemented
7	#22559001	C₂	1	2	yes	maximal, complemented
8	#22558997	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558661	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558492	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1476	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863946	C₂²	1	1	yes	characteristic
13	#16863720	C₄	1	2	yes	—
14	#16863691	C₄	1	1	yes	characteristic
15	#16863686	C₄	1	1	yes	characteristic
16	#16863596	C₄	1	1	yes	characteristic
17	#16863591	C₄	1	1	yes	characteristic
18	#16863541	C₄	1	2	yes	—
19	#16863527	C₂²	1	2	yes	—
20	C₄×#11761615	C₂²	1	2	yes	—
21	C₄×#11761613	C₂²	1	2	yes	—
22	#16863475	—	2	1	no	—
23	#16863473	C₂²	1	2	yes	—
24	#16863468	—	2	2	no	—
25	C₈×D₂₆₈	—	2	1	no	—
26	D₄×C₆₇⋊C₈	C₂²	1	2	yes	—
27	#16863461	C₂²	1	2	yes	—
28	#16863453	—	2	1	no	—
29	#16863449	C₂²	1	1	yes	characteristic
30	#16863449	C₂²	1	1	yes	characteristic
31	#16863420	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
33	#16863361	—	2	2	no	—
34	#16863148	C₂²	1	2	yes	—
35	#16863146	C₂²	1	2	yes	—
36	#16863125	C₂²	1	2	yes	—
37	#16863125	C₂²	1	2	yes	—
38	#16863117	C₂²	1	2	yes	—
39	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
40	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
41	#431×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
44	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
45	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#11761628	—	2	1	no	—
48	C₂×#11761615	C₂×C₄	1	2	yes	—
49	C₂×#11761613	C₂×C₄	1	2	yes	—
50	C₂×#11761612	—	2	1	no	—
51	C₂×#11761611	—	2	2	no	—
52	C₂²×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₈×D₆₇	—	2	2	no	—
54	#14524043	C₂×C₄	1	1	yes	characteristic
55	#14524040	C₂×C₄	1	1	yes	characteristic
56	#14524031	—	2	1	no	—
57	#14524017	—	2	1	no	—
58	#14524017	C₂×C₄	1	2	yes	—
59	C₄×D₂₆₈	—	2	1	no	—
60	C₄×D₂₆₈	C₂³	1	2	yes	—
61	#14523976	C₂×C₄	1	2	yes	—
62	#14523974	C₂×C₄	1	2	yes	—
63	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
64	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
65	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
66	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
67	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
68	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
69	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
70	#14523967	C₂×C₄	1	2	yes	—
71	#14523964	C₂×C₄	1	2	yes	—
72	#14523963	C₂×C₄	1	2	yes	—
73	#14523963	C₂×C₄	1	2	yes	—
74	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
75	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
76	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
77	(C₂×D₆₇)⋊C₈	—	2	2	no	—
78	(C₂×D₆₇)⋊C₈	—	2	2	no	—
79	#14523954	—	2	2	no	—
80	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
81	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
82	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₈×C₆₇	—	2	1	no	—
84	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄×D₄×C₆₇	—	2	1	no	—
87	C₄×D₄×C₆₇	—	2	1	no	—
88	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
89	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
90	#117×C₆₇	—	2	1	no	—
91	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	—	2	1	no	—
93	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₂²⋊C₈×C₆₇	—	2	1	no	—
97	C₄×C₈×C₆₇	—	2	1	no	—
98	C₄×C₈×C₆₇	—	2	1	no	—
99	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
101	#11761628	—	8	1	no	—
102	#11761628	—	8	1	no	—
103	C₂×Dic₁₃₄	—	2	1	no	—
104	C₂×D₂₆₈	—	2	1	no	—
105	C₂×D₂₆₈	—	2	1	no	—
106	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
107	C₂×C₂²⋊D₆₇	—	2	2	no	—
108	C₂×C₄×D₆₇	—	2	2	no	—
109	#11761615	—	4	2	no	—
110	#11761613	—	2	4	no	—
111	#11761612	—	8	1	no	—
112	#11761611	—	8	2	no	—
113	C₂×C₆₇⋊C₈	—	2	2	no	—
114	C₂×C₆₇⋊C₈	—	8	2	no	—
115	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
116	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
117	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
118	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
119	C₈×D₆₇	—	8	2	no	—
120	C₄⋊Dic₆₇	—	2	1	no	—
121	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
122	(C₂×D₆₇)⋊C₄	—	2	2	no	—
123	(C₂×D₆₇)⋊C₄	—	2	2	no	—
124	#41×C₆₇	—	2	2	no	—
125	#41×C₆₇	—	4	1	no	—
126	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×D₄×C₆₇	—	2	1	no	—
128	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	—	2	1	no	—
130	C₂²×C₄×C₆₇	—	2	1	no	—
131	C₂²×C₄×C₆₇	—	2	1	no	—
132	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
135	C₂×C₈×C₆₇	—	2	4	no	—
136	C₂×C₈×C₆₇	—	2	1	no	—
137	C₂×C₈×C₆₇	—	2	1	no	—
138	C₂×C₈×C₆₇	—	2	1	no	—
139	C₂×C₈×C₆₇	—	2	1	no	—
140	C₄⋊C₄×C₆₇	—	2	1	no	—
141	C₄⋊C₄×C₆₇	—	2	1	no	—
142	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₂²⋊C₄×C₆₇	—	2	1	no	—
145	C₂²⋊C₄×C₆₇	—	2	1	no	—
146	C₂²⋊C₄×C₆₇	—	2	1	no	—
147	C₄²×C₆₇	—	2	1	no	—
148	C₄²×C₆₇	—	2	1	no	—
149	C₄²×C₆₇	—	2	1	no	—
150	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₆₇	#41	1	1	yes	characteristic
153	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
154	C₂³×C₆₇	—	2	1	no	—
155	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
156	C₂²×D₆₇	—	2	2	no	—
157	Q₈×C₆₇	—	2	1	no	—
158	Q₈×C₆₇	—	2	1	no	—
159	D₄×C₆₇	—	2	1	no	—
160	D₄×C₆₇	—	4	2	no	—
161	D₄×C₆₇	#131	1	2	yes	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	4	1	no	—
172	C₂×C₄×C₆₇	—	4	1	no	—
173	C₂×C₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
180	C₂²⋊D₆₇	—	8	2	no	—
181	C₂×Dic₆₇	—	2	2	no	—
182	D₂₆₈	—	2	2	no	—
183	D₂₆₈	—	2	2	no	—
184	D₂₆₈	—	4	2	no	—
185	D₂₆₈	—	8	1	no	—
186	C₄×D₆₇	—	4	2	no	—
187	C₄×D₆₇	—	4	2	no	—
188	Dic₁₃₄	—	8	1	no	—
189	C₈×C₆₇	—	2	2	no	—
190	C₈×C₆₇	—	2	2	no	—
191	C₈×C₆₇	—	4	1	no	—
192	C₈×C₆₇	—	4	1	no	—
193	C₆₇⋊C₈	—	2	2	no	—
194	C₆₇⋊C₈	—	2	2	no	—
195	C₆₇⋊C₈	—	2	2	no	—
196	C₆₇⋊C₈	—	2	2	no	—
197	C₆₇⋊C₈	—	4	2	no	—
198	C₂²×C₆₇	—	2	1	no	—
199	C₂²×C₆₇	—	2	2	no	—
200	C₂²×C₆₇	—	2	1	no	—
201	C₂²×C₆₇	—	2	1	no	—
202	C₂²×C₆₇	—	4	1	no	—
203	C₂²×C₆₇	—	4	1	no	—
204	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
205	C₂×D₆₇	—	4	2	no	—
206	C₂×D₆₇	—	4	2	no	—
207	C₂×D₆₇	—	4	2	no	—
208	C₄×C₆₇	—	2	1	no	—
209	C₄×C₆₇	—	2	1	no	—
210	C₄×C₆₇	—	2	1	no	—
211	C₄×C₆₇	—	4	1	no	—
212	C₄×C₆₇	—	4	1	no	—
213	C₄×C₆₇	—	4	1	no	—
214	C₄×C₆₇	#443	1	2	yes	—
215	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
216	C₄×C₆₇	#568	1	1	yes	characteristic
217	C₄×C₆₇	#576	1	1	yes	characteristic
218	C₄×C₆₇	#577	1	1	yes	characteristic
219	Dic₆₇	—	4	2	no	—
220	#18015	—	67	1	no	maximal, Sylow
221	C₂×C₆₇	—	4	1	no	—
222	C₂×C₆₇	—	4	2	no	—
223	C₂×C₆₇	#2933	1	1	yes	characteristic
224	C₂×C₆₇	#2946	1	1	yes	characteristic
225	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
226	D₆₇	—	8	2	no	—
227	#3106	—	67	1	no	—
228	#2990	—	67	1	no	—
229	#2979	—	67	1	no	—
230	#1590	—	67	2	no	—
231	#1586	—	67	2	no	—
232	#1585	—	67	2	no	—
233	C₈×D₈	—	67	2	no	—
234	#1574	—	67	1	no	—
235	#1492	—	67	1	no	—
236	#1476	D₆₇	1	1	yes	characteristic, complemented
237	#1450	—	67	1	no	—
238	C₆₇	#18015	1	1	yes	characteristic, complemented, Sylow
239	#532	—	67	1	no	—
240	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
241	#482	—	67	2	no	—
242	#477	—	67	1	no	—
243	#476	—	67	1	no	—
244	#460	—	67	1	no	—
245	#459	—	67	1	no	—
246	#449	—	67	2	no	—
247	#446	—	67	2	no	—
248	C₄×SD₁₆	—	67	2	no	—
249	C₄×D₈	—	67	2	no	—
250	#436	—	67	2	no	—
251	#436	—	134	1	no	—
252	#435	—	134	2	no	—
253	C₈×D₄	—	67	2	no	—
254	C₈×D₄	—	67	2	no	—
255	C₈×D₄	—	134	1	no	—
256	#432	—	134	1	no	—
257	#431	—	67	1	no	—
258	#431	—	67	1	no	—
259	#431	C₂×D₆₇	1	1	yes	characteristic
260	#423	—	67	1	no	—
261	C₂×C₄⋊C₈	—	134	1	no	—
262	#408	—	134	2	no	—
263	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
264	#335	—	67	2	no	—
265	#334	—	67	2	no	—
266	Q₈⋊C₈	C₂×D₆₇	1	1	yes	characteristic
267	Q₈⋊C₈	C₂×D₆₇	1	1	yes	characteristic
268	D₄⋊C₈	—	67	2	no	—
269	D₄⋊C₈	—	67	2	no	—
270	D₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
271	D₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
272	#322	—	67	2	no	—
273	C₈²	—	67	2	no	—
274	C₂×#41	—	134	1	no	—
275	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
276	C₂×SD₁₆	—	67	2	no	—
277	C₂×D₈	—	67	2	no	—
278	C₂×M₁₆	—	134	2	no	—
279	C₂×M₁₆	—	134	1	no	—
280	C₂²×C₈	—	2	1	no	—
281	C₂²×C₈	—	134	2	no	—
282	C₂²×C₈	—	134	2	no	—
283	C₄⋊Q₈	—	67	1	no	—
284	#127	—	67	1	no	—
285	#124	—	134	1	no	—
286	#121	—	67	2	no	—
287	#121	—	134	1	no	—
288	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
289	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
290	C₄×D₄	—	2	1	no	—
291	C₄×D₄	—	2	1	no	—
292	C₄×D₄	—	67	2	no	—
293	C₄×D₄	—	134	1	no	—
294	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
295	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
296	#117	—	2	1	no	—
297	#117	C₄×D₆₇	1	1	yes	characteristic
298	C₂×C₄²	—	2	1	no	—
299	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
300	#107	—	67	2	no	—
301	#106	—	67	2	no	—
302	C₄⋊C₈	—	67	1	no	—
303	C₄⋊C₈	—	67	1	no	—
304	C₄⋊C₈	—	67	1	no	—
305	C₄⋊C₈	—	67	1	no	—
306	C₄⋊C₈	—	67	2	no	—
307	C₄⋊C₈	—	134	2	no	—
308	C₄⋊C₈	—	134	1	no	—
309	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
310	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
311	#103	—	67	2	no	—
312	#102	—	67	2	no	—
313	#102	—	67	2	no	—
314	#102	—	67	2	no	—
315	C₂²⋊C₈	—	2	1	no	—
316	C₂²⋊C₈	—	67	2	no	—
317	C₂²⋊C₈	—	67	2	no	—
318	C₂²⋊C₈	—	134	2	no	—
319	C₂²⋊C₈	—	134	2	no	—
320	C₂²⋊C₈	—	134	2	no	—
321	#97	—	134	2	no	—
322	C₄×C₈	—	2	1	no	—
323	C₄×C₈	—	2	1	no	—
324	C₄×C₈	—	67	2	no	—
325	C₄×C₈	—	67	2	no	—
326	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
327	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
328	#41	—	2	2	no	—
329	#41	—	4	1	no	—
330	#41	—	536	1	no	—
331	#41	—	536	1	no	—
332	C₂×Q₈	—	134	1	no	—
333	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
334	C₂×D₄	—	2	1	no	—
335	C₂×D₄	—	67	2	no	—
336	C₂×D₄	—	134	2	no	—
337	C₂×D₄	—	134	1	no	—
338	C₂×D₄	—	134	1	no	—
339	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	134	2	no	—
344	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
345	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
346	C₂²×C₄	#11761627	1	1	yes	characteristic
347	SD₁₆	—	268	2	no	—
348	D₈	—	134	4	no	—
349	M₁₆	—	536	2	no	—
350	M₁₆	—	536	1	no	—
351	C₂×C₈	—	2	4	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	536	2	no	—
357	C₂×C₈	—	536	2	no	—
358	C₂×C₈	—	67	2	no	—
359	C₂×C₈	—	67	2	no	—
360	C₂×C₈	—	67	2	no	—
361	C₂×C₈	—	67	2	no	—
362	C₂×C₈	—	134	2	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	67	2	no	—
366	C₄⋊C₄	—	134	1	no	—
367	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
368	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
369	C₂²⋊C₄	—	2	1	no	—
370	C₂²⋊C₄	—	2	1	no	—
371	C₂²⋊C₄	—	2	1	no	—
372	C₂²⋊C₄	—	134	2	no	—
373	C₂²⋊C₄	—	134	2	no	—
374	C₄²	—	2	1	no	—
375	C₄²	—	2	1	no	—
376	C₄²	—	2	1	no	—
377	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
378	C₄²	D₄×D₆₇	1	1	yes	characteristic
379	C₄²	#11761627	1	1	yes	characteristic
380	C₄²	C₂³×D₆₇	1	1	yes	characteristic
381	C₂³	—	2	1	no	—
382	C₂³	—	134	2	no	—
383	C₂³	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
384	Q₈	—	2	1	no	—
385	Q₈	—	2	1	no	—
386	Q₈	—	536	1	no	—
387	D₄	—	2	1	no	—
388	D₄	—	4	2	no	—
389	D₄	—	268	2	no	—
390	D₄	—	536	1	no	—
391	D₄	—	536	2	no	—
392	D₄	—	134	2	no	—
393	D₄	—	134	2	no	—
394	D₄	#14524049	1	2	yes	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	268	2	no	—
408	C₂×C₄	—	268	2	no	—
409	C₂×C₄	—	134	2	no	—
410	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
411	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
412	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
413	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
414	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
415	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
416	C₈	—	2	2	no	—
417	C₈	—	2	2	no	—
418	C₈	—	4	1	no	—
419	C₈	—	4	1	no	—
420	C₈	—	268	2	no	—
421	C₈	—	134	2	no	—
422	C₈	—	134	2	no	—
423	C₈	—	134	2	no	—
424	C₈	—	134	2	no	—
425	C₂²	—	2	1	no	—
426	C₂²	—	2	1	no	—
427	C₂²	—	2	1	no	—
428	C₂²	—	2	2	no	—
429	C₂²	—	4	1	no	—
430	C₂²	—	4	1	no	—
431	C₂²	—	268	2	no	—
432	C₂²	—	268	2	no	—
433	C₂²	—	268	2	no	—
434	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
435	C₄	—	2	1	no	—
436	C₄	—	2	1	no	—
437	C₄	—	2	1	no	—
438	C₄	—	4	1	no	—
439	C₄	—	4	1	no	—
440	C₄	—	4	1	no	—
441	C₄	—	268	2	no	—
442	C₄	#16863510	1	2	yes	—
443	C₄	C₂×#14524049	1	1	yes	characteristic
444	C₄	#16864162	1	1	yes	characteristic
445	C₄	#16864198	1	1	yes	characteristic
446	C₄	#16864203	1	1	yes	characteristic
447	C₂	—	4	1	no	—
448	C₂	—	4	2	no	—
449	C₂	—	536	2	no	—
450	C₂	#22568930	1	1	yes	characteristic
451	C₂	#22569044	1	1	yes	characteristic
452	C₂	C₂×#16863510	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×#1476	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22570269	C₂
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1476	D₆₇
hypocenter	C₆₇	#18015
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1476

Complements

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

N	H ≅ G/N	type
2. #22569334	C₂	semidirect
3. #22569275	C₂	semidirect
4. #22559096	C₂	semidirect
5. #22559070	C₂	semidirect
6. #22559069	C₂	semidirect
7. #22559001	C₂	semidirect
8. #22558997	C₂	semidirect
9. #22558661	C₂	semidirect
10. #22558492	C₂	semidirect
11. C₆₇×#1476	C₂	semidirect
236. #1476	D₆₇	semidirect
238. C₆₇	#18015	semidirect

Maximal subgroups

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

subgroup	index
#22570269	2
#22569334	2
#22569275	2
#22559096	2
#22559070	2
#22559069	2
#22559001	2
#22558997	2
#22558661	2
#22558492	2
C₆₇×#1476	2
#18015	67