Subgroups — #30825871

Order 17152.

624 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	#22570281	C₂	1	1	yes	maximal, characteristic
2	C₈×#11761628	C₂	1	1	yes	maximal, characteristic
3	#22569143	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559061	C₂	1	2	yes	maximal, complemented
5	#22559047	C₂	1	2	yes	maximal, complemented
6	#22559029	C₂	1	2	yes	maximal, complemented
7	#22559023	C₂	1	2	yes	maximal, complemented
8	#22558998	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558649	C₂	1	1	yes	maximal, characteristic
10	#22558631	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863953	C₂²	1	1	yes	characteristic
13	C₄×#11761628	C₂²	1	1	yes	characteristic
14	#16863738	C₄	1	2	yes	—
15	#16863682	C₄	1	1	yes	characteristic
16	#16863677	C₄	1	1	yes	characteristic
17	#16863607	C₄	1	1	yes	characteristic
18	#16863602	C₄	1	1	yes	characteristic
19	#16863560	C₄	1	2	yes	—
20	#16863526	—	2	1	no	—
21	#16863526	C₂²	1	2	yes	—
22	C₈×Dic₁₃₄	—	2	1	no	—
23	C₈×Dic₁₃₄	C₂²	1	2	yes	—
24	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
25	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₆₈	C₂²	1	2	yes	—
27	C₄×#11761617	C₂²	1	2	yes	—
28	C₈×D₂₆₈	C₂²	1	1	yes	characteristic
29	C₈×D₂₆₈	C₂²	1	1	yes	characteristic
30	C₈×C₂²⋊D₆₇	—	2	1	no	—
31	C₈×C₂²⋊D₆₇	—	2	1	no	—
32	#16863449	C₂²	1	1	yes	characteristic
33	#16863447	—	2	1	no	—
34	#16863422	—	2	1	no	—
35	#16863420	C₂²	1	1	yes	characteristic
36	#16863351	—	2	1	no	—
37	C₄×C₈×D₆₇	—	2	1	no	—
38	#16863149	C₂²	1	2	yes	—
39	#16863147	C₂²	1	2	yes	—
40	#16863128	C₂²	1	2	yes	—
41	#16863128	C₂²	1	2	yes	—
42	#16863128	C₂²	1	2	yes	—
43	#16863125	C₂²	1	2	yes	—
44	#431×C₆₇	C₂²	1	1	yes	characteristic
45	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
47	#322×C₆₇	C₂²	1	2	yes	—
48	#322×C₆₇	C₂²	1	2	yes	—
49	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
50	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
51	C₂×#11761617	C₂×C₄	1	2	yes	—
52	C₂×#11761612	—	2	1	no	—
53	C₂×C₈×D₆₇	—	2	1	no	—
54	C₂×C₈×D₆₇	—	2	1	no	—
55	#14524043	C₂×C₄	1	1	yes	characteristic
56	#14524043	C₂×C₄	1	1	yes	characteristic
57	#14524035	—	2	1	no	—
58	#14524022	—	2	1	no	—
59	#14524022	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₃₄	—	2	1	no	—
61	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
63	C₄×Dic₁₃₄	C₂³	1	2	yes	—
64	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
65	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
66	C₄×C₂²⋊D₆₇	—	2	1	no	—
67	C₄×C₂²⋊D₆₇	—	2	1	no	—
68	#14524002	—	2	1	no	—
69	#14524002	C₂×C₄	1	1	yes	characteristic
70	#14524000	—	2	1	no	—
71	C₄²×D₆₇	—	2	1	no	—
72	#14523977	C₂×C₄	1	2	yes	—
73	#14523975	C₂×C₄	1	2	yes	—
74	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
75	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
76	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
77	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
78	Dic₆₇⋊C₈	—	2	2	no	—
79	Dic₆₇⋊C₈	—	2	2	no	—
80	Dic₆₇⋊C₈	—	2	1	no	—
81	Dic₆₇⋊C₈	—	2	1	no	—
82	#14523968	C₂×C₄	1	2	yes	—
83	#14523968	C₂×C₄	1	2	yes	—
84	#14523968	C₂×C₄	1	2	yes	—
85	#14523965	C₂×C₄	1	2	yes	—
86	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
87	(C₂×D₆₇)⋊C₈	—	2	1	no	—
88	(C₂×D₆₇)⋊C₈	—	2	1	no	—
89	#14523953	—	2	2	no	—
90	C₄×C₆₇⋊C₈	—	2	1	no	—
91	C₈×Dic₆₇	—	2	2	no	—
92	C₈×Dic₆₇	—	2	1	no	—
93	C₈×Dic₆₇	—	2	1	no	—
94	C₂×M₁₆×C₆₇	—	2	1	no	—
95	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
96	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
99	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
101	#97×C₆₇	—	2	1	no	—
102	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
103	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
104	C₄×C₈×C₆₇	C₂³	1	2	yes	—
105	C₄×C₈×C₆₇	C₂³	1	2	yes	—
106	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
107	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
108	#11761628	—	2	2	no	—
109	#11761628	—	2	2	no	—
110	C₂×Dic₁₃₄	—	2	1	no	—
111	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
112	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
113	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₂²⋊D₆₇	—	2	1	no	—
115	C₂×C₄×D₆₇	—	2	1	no	—
116	C₂×C₄×D₆₇	—	2	1	no	—
117	C₂×C₄×D₆₇	—	2	1	no	—
118	Dic₂₆₈	—	2	4	no	—
119	#11761617	—	2	4	no	—
120	#11761612	—	4	2	no	—
121	C₂×C₆₇⋊C₈	—	2	1	no	—
122	C₂×C₆₇⋊C₈	—	2	1	no	—
123	C₂×C₆₇⋊C₈	—	2	2	no	—
124	C₈×D₆₇	—	4	2	no	—
125	C₈×D₆₇	—	4	2	no	—
126	C₄⋊Dic₆₇	—	2	1	no	—
127	C₄⋊Dic₆₇	—	2	1	no	—
128	C₄⋊Dic₆₇	—	2	1	no	—
129	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
130	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
132	Dic₆₇⋊C₄	—	2	1	no	—
133	Dic₆₇⋊C₄	—	2	2	no	—
134	Dic₆₇⋊C₄	—	2	1	no	—
135	Dic₆₇⋊C₄	—	2	2	no	—
136	C₂²⋊Dic₆₇	—	2	2	no	—
137	C₂²⋊Dic₆₇	—	2	1	no	—
138	(C₂×D₆₇)⋊C₄	—	2	1	no	—
139	(C₂×D₆₇)⋊C₄	—	2	1	no	—
140	C₄×Dic₆₇	—	2	1	no	—
141	C₄×Dic₆₇	—	2	1	no	—
142	C₄×Dic₆₇	—	2	1	no	—
143	C₄×Dic₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
147	M₁₆×C₆₇	—	4	2	no	—
148	C₂×C₈×C₆₇	—	2	2	no	—
149	C₂×C₈×C₆₇	—	2	2	no	—
150	C₂×C₈×C₆₇	—	4	1	no	—
151	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
152	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
153	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
156	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
159	C₄²×C₆₇	#41	1	1	yes	characteristic
160	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
161	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
162	C₂²×D₆₇	—	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	1	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
174	C₂²⋊D₆₇	—	2	4	no	—
175	C₂×Dic₆₇	—	2	1	no	—
176	C₂×Dic₆₇	—	2	1	no	—
177	C₂×Dic₆₇	—	2	1	no	—
178	C₂×Dic₆₇	—	2	2	no	—
179	C₂×Dic₆₇	—	2	2	no	—
180	D₂₆₈	—	2	1	no	—
181	D₂₆₈	#131	1	2	yes	—
182	C₄×D₆₇	—	2	4	no	—
183	C₄×D₆₇	—	4	1	no	—
184	C₄×D₆₇	—	4	1	no	—
185	C₄×D₆₇	—	4	2	no	—
186	Dic₁₃₄	—	2	1	no	—
187	Dic₁₃₄	—	2	2	no	—
188	Dic₁₃₄	—	2	2	no	—
189	Dic₁₃₄	—	4	2	no	—
190	Dic₁₃₄	#131	1	2	yes	—
191	C₈×C₆₇	—	2	2	no	—
192	C₈×C₆₇	—	2	2	no	—
193	C₈×C₆₇	—	2	1	no	—
194	C₈×C₆₇	—	2	1	no	—
195	C₈×C₆₇	—	2	1	no	—
196	C₈×C₆₇	—	4	2	no	—
197	C₈×C₆₇	#137	1	2	yes	—
198	C₆₇⋊C₈	—	4	1	no	—
199	C₆₇⋊C₈	—	4	1	no	—
200	C₆₇⋊C₈	—	4	2	no	—
201	C₂²×C₆₇	—	2	1	no	—
202	C₂²×C₆₇	—	2	1	no	—
203	C₂²×C₆₇	—	2	1	no	—
204	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
205	C₂×D₆₇	—	2	2	no	—
206	C₂×D₆₇	—	4	1	no	—
207	C₂×D₆₇	—	4	1	no	—
208	C₄×C₆₇	—	2	1	no	—
209	C₄×C₆₇	—	2	1	no	—
210	C₄×C₆₇	—	2	1	no	—
211	C₄×C₆₇	#444	1	2	yes	—
212	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
213	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
214	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
215	C₄×C₆₇	#575	1	1	yes	characteristic
216	Dic₆₇	—	2	2	no	—
217	Dic₆₇	—	4	2	no	—
218	Dic₆₇	—	4	1	no	—
219	Dic₆₇	—	4	2	no	—
220	Dic₆₇	—	4	1	no	—
221	#17916	—	67	1	no	maximal, Sylow
222	C₂×C₆₇	—	4	1	no	—
223	C₂×C₆₇	#2929	1	1	yes	characteristic
224	C₂×C₆₇	C₄×#137	1	1	yes	characteristic
225	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
226	D₆₇	—	4	2	no	—
227	#3107	—	67	1	no	—
228	C₈×#41	—	67	1	no	—
229	#2960	—	67	1	no	—
230	#1584	—	67	2	no	—
231	#1582	—	67	2	no	—
232	#1579	—	67	2	no	—
233	SD₁₆⋊C₈	—	67	2	no	—
234	#1574	—	67	1	no	—
235	#1489	—	67	1	no	—
236	#1485	—	67	1	no	—
237	#1451	D₆₇	1	1	yes	characteristic, complemented
238	C₆₇	#17916	1	1	yes	characteristic, complemented, Sylow
239	#533	—	67	1	no	—
240	C₄×#41	—	67	1	no	—
241	#485	—	67	2	no	—
242	#475	—	67	1	no	—
243	#474	—	67	1	no	—
244	#462	—	67	1	no	—
245	#461	—	67	1	no	—
246	#452	—	67	2	no	—
247	#446	—	67	2	no	—
248	#446	—	134	1	no	—
249	C₈×Q₈	—	67	2	no	—
250	C₈×Q₈	—	67	1	no	—
251	C₈×Q₈	—	67	1	no	—
252	C₈×Q₈	—	134	1	no	—
253	C₄×Q₁₆	—	67	2	no	—
254	C₄×SD₁₆	—	67	2	no	—
255	C₈×D₄	—	67	1	no	—
256	C₈×D₄	—	67	1	no	—
257	C₈×D₄	—	134	1	no	—
258	C₈×D₄	—	134	1	no	—
259	#431	—	67	1	no	—
260	#431	—	134	1	no	—
261	#431	C₂×D₆₇	1	1	yes	characteristic
262	#424	—	134	1	no	—
263	#423	—	67	1	no	—
264	#405	—	134	1	no	—
265	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
266	C₂×C₄×C₈	—	134	1	no	—
267	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
268	#335	—	67	2	no	—
269	#334	—	67	2	no	—
270	Q₈⋊C₈	—	67	2	no	—
271	Q₈⋊C₈	—	67	2	no	—
272	Q₈⋊C₈	—	67	2	no	—
273	D₄⋊C₈	—	67	2	no	—
274	#322	C₂×D₆₇	1	2	yes	—
275	#322	C₂×D₆₇	1	2	yes	—
276	C₂×#41	—	67	1	no	—
277	C₂×Q₁₆	—	67	2	no	—
278	C₂×SD₁₆	—	67	2	no	—
279	C₂×M₁₆	—	2	1	no	—
280	C₂×M₁₆	—	134	1	no	—
281	C₂²×C₈	—	134	1	no	—
282	C₂²×C₈	—	134	1	no	—
283	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
284	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
285	C₄⋊Q₈	—	67	1	no	—
286	C₄⋊Q₈	—	67	1	no	—
287	#125	—	134	1	no	—
288	C₂²⋊Q₈	—	67	2	no	—
289	C₂²⋊Q₈	—	134	1	no	—
290	C₄×Q₈	—	67	1	no	—
291	C₄×Q₈	—	67	1	no	—
292	C₄×Q₈	—	67	2	no	—
293	C₄×Q₈	—	134	1	no	—
294	C₄×D₄	—	67	1	no	—
295	C₄×D₄	—	67	1	no	—
296	C₄×D₄	—	134	1	no	—
297	C₄×D₄	—	134	1	no	—
298	#117	—	67	1	no	—
299	#117	—	134	1	no	—
300	#117	—	134	1	no	—
301	C₂×C₄²	—	134	1	no	—
302	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
303	#107	—	67	2	no	—
304	#106	—	67	2	no	—
305	C₄⋊C₈	—	67	2	no	—
306	C₄⋊C₈	—	67	1	no	—
307	C₄⋊C₈	—	67	1	no	—
308	C₄⋊C₈	—	134	2	no	—
309	C₄⋊C₈	—	134	2	no	—
310	C₄⋊C₈	—	134	1	no	—
311	C₄⋊C₈	—	134	1	no	—
312	C₄⋊C₈	—	134	1	no	—
313	C₄⋊C₈	C₄×D₆₇	1	1	yes	characteristic
314	C₄⋊C₈	C₄×D₆₇	1	1	yes	characteristic
315	#103	—	67	2	no	—
316	#103	—	67	2	no	—
317	#103	—	67	2	no	—
318	#102	—	67	2	no	—
319	C₂²⋊C₈	—	134	1	no	—
320	C₂²⋊C₈	—	134	1	no	—
321	C₂²⋊C₈	—	134	1	no	—
322	C₂²⋊C₈	C₄×D₆₇	1	2	yes	—
323	#97	—	2	1	no	—
324	#97	—	134	2	no	—
325	C₄×C₈	—	134	1	no	—
326	C₄×C₈	—	134	1	no	—
327	C₄×C₈	—	134	2	no	—
328	C₄×C₈	—	134	1	no	—
329	C₄×C₈	D₂₆₈	1	1	yes	characteristic
330	C₄×C₈	D₂₆₈	1	1	yes	characteristic
331	C₄×C₈	C₂²×D₆₇	1	2	yes	—
332	C₄×C₈	C₂²×D₆₇	1	2	yes	—
333	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
334	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
335	#41	—	134	2	no	—
336	#41	—	134	2	no	—
337	C₂×Q₈	—	67	1	no	—
338	C₂×Q₈	—	67	2	no	—
339	C₂×Q₈	—	134	1	no	—
340	C₂×D₄	—	67	1	no	—
341	C₂×D₄	—	134	1	no	—
342	C₂²×C₄	—	134	1	no	—
343	C₂²×C₄	—	134	1	no	—
344	C₂²×C₄	—	134	1	no	—
345	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
346	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
347	C₂²×C₄	#11761628	1	1	yes	characteristic
348	Q₁₆	—	134	4	no	—
349	SD₁₆	—	134	4	no	—
350	M₁₆	—	4	2	no	—
351	M₁₆	—	268	2	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	4	1	no	—
355	C₂×C₈	—	268	2	no	—
356	C₂×C₈	—	268	2	no	—
357	C₂×C₈	—	134	1	no	—
358	C₂×C₈	—	134	1	no	—
359	C₂×C₈	—	134	2	no	—
360	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
361	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
362	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
363	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
364	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
365	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
366	C₄⋊C₄	—	67	2	no	—
367	C₄⋊C₄	—	67	1	no	—
368	C₄⋊C₄	—	67	1	no	—
369	C₄⋊C₄	—	134	1	no	—
370	C₄⋊C₄	—	134	1	no	—
371	C₄⋊C₄	—	134	1	no	—
372	C₄⋊C₄	—	134	2	no	—
373	C₄⋊C₄	—	134	1	no	—
374	C₄⋊C₄	—	134	1	no	—
375	C₄⋊C₄	—	134	2	no	—
376	C₂²⋊C₄	—	134	1	no	—
377	C₂²⋊C₄	—	134	2	no	—
378	C₂²⋊C₄	—	134	1	no	—
379	C₂²⋊C₄	—	134	1	no	—
380	C₄²	—	134	1	no	—
381	C₄²	—	134	1	no	—
382	C₄²	—	134	1	no	—
383	C₄²	—	134	2	no	—
384	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
385	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
386	C₄²	#11761628	1	1	yes	characteristic
387	C₄²	C₂³×D₆₇	1	1	yes	characteristic
388	C₂³	—	134	1	no	—
389	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
390	Q₈	—	268	2	no	—
391	Q₈	—	67	2	no	—
392	Q₈	—	134	2	no	—
393	Q₈	—	134	2	no	—
394	Q₈	—	134	1	no	—
395	D₄	—	67	2	no	—
396	D₄	—	134	4	no	—
397	D₄	—	134	1	no	—
398	C₂×C₄	—	2	2	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	—	268	2	no	—
404	C₂×C₄	—	268	1	no	—
405	C₂×C₄	—	268	1	no	—
406	C₂×C₄	—	134	4	no	—
407	C₂×C₄	—	134	1	no	—
408	C₂×C₄	—	134	1	no	—
409	C₂×C₄	—	134	2	no	—
410	C₂×C₄	—	134	2	no	—
411	C₂×C₄	—	134	1	no	—
412	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
413	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
414	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
415	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
417	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
418	C₈	—	2	2	no	—
419	C₈	—	2	2	no	—
420	C₈	—	2	1	no	—
421	C₈	—	2	1	no	—
422	C₈	—	2	1	no	—
423	C₈	—	4	2	no	—
424	C₈	—	268	1	no	—
425	C₈	—	268	1	no	—
426	C₈	—	268	2	no	—
427	C₈	#14524076	1	2	yes	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	1	no	—
430	C₂²	—	2	1	no	—
431	C₂²	—	268	1	no	—
432	C₂²	—	268	1	no	—
433	C₂²	—	134	2	no	—
434	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
435	C₄	—	2	1	no	—
436	C₄	—	2	1	no	—
437	C₄	—	2	1	no	—
438	C₄	—	268	1	no	—
439	C₄	—	268	1	no	—
440	C₄	—	268	2	no	—
441	C₄	—	268	2	no	—
442	C₄	—	134	2	no	—
443	C₄	#16863519	1	2	yes	—
444	C₄	#131×D₆₇	1	1	yes	characteristic
445	C₄	#131×D₆₇	1	1	yes	characteristic
446	C₄	C₂×#14524076	1	1	yes	characteristic
447	C₄	#16864196	1	1	yes	characteristic
448	C₂	—	4	1	no	—
449	C₂	—	268	2	no	—
450	C₂	#22568905	1	1	yes	characteristic
451	C₂	C₄×#14524076	1	1	yes	characteristic
452	C₂	C₂×#16863519	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1451	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1451	D₆₇
hypocenter	C₆₇	#17916
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #22569143	C₂	semidirect
4. #22559061	C₂	semidirect
5. #22559047	C₂	semidirect
6. #22559029	C₂	semidirect
7. #22559023	C₂	semidirect
8. #22558998	C₂	semidirect
10. #22558631	C₂	semidirect
11. C₆₇×#1451	C₂	semidirect
237. #1451	D₆₇	semidirect
238. C₆₇	#17916	semidirect

Maximal subgroups

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

subgroup	index
#22570281	2
C₈×#11761628	2
#22569143	2
#22559061	2
#22559047	2
#22559029	2
#22559023	2
#22558998	2
#22558649	2
#22558631	2
C₆₇×#1451	2
#17916	67