Subgroups — #30825620

Order 17152.

724 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569175	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14524071	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863475	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559124	C₂	1	2	yes	maximal, complemented
5	#22559113	C₂	1	2	yes	maximal, complemented
6	#22559101	C₂	1	2	yes	maximal, complemented
7	#22559095	C₂	1	2	yes	maximal, complemented
8	#22558961	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558601	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558597	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524071	C₄	1	1	yes	characteristic
13	C₂²×#11761611	—	2	1	no	—
14	C₄×#11761628	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
16	#16863526	C₂²	1	1	yes	characteristic, complemented
17	#16863526	C₂²	1	1	yes	characteristic, complemented
18	#16863509	C₄	1	2	yes	—
19	#16863498	C₄	1	2	yes	—
20	C₄×#11761617	C₂²	1	2	yes	—
21	C₄×D₅₃₆	C₂²	1	2	yes	—
22	#16863475	—	2	1	no	—
23	#16863475	—	2	1	no	—
24	#16863475	C₂²	1	2	yes	—
25	#16863475	C₂²	1	2	yes	—
26	#16863475	C₂²	1	1	yes	characteristic, complemented
27	#16863475	C₂²	1	1	yes	characteristic, complemented
28	#16863467	—	2	1	no	—
29	#16863467	—	2	1	no	—
30	#16863449	C₂²	1	1	yes	characteristic
31	#16863447	—	2	1	no	—
32	#16863438	C₄	1	1	yes	characteristic
33	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
34	#16863402	C₄	1	1	yes	characteristic
35	#16863398	C₄	1	1	yes	characteristic
36	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
37	C₄×#11761611	—	2	1	no	—
38	#16863149	C₂²	1	2	yes	complemented
39	#16863147	C₂²	1	2	yes	complemented
40	#16863128	C₂²	1	2	yes	complemented
41	#16863125	C₂²	1	2	yes	—
42	#16863125	C₂²	1	2	yes	—
43	#16863125	C₂²	1	2	yes	complemented
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	complemented
48	C₈²×C₆₇	C₂²	1	2	yes	complemented
49	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₂₆₈	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×D₆₇	—	2	1	no	—
52	#14524071	—	2	4	no	—
53	C₂×#11761617	C₂×C₄	1	2	yes	—
54	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
55	C₂×#11761611	—	2	1	no	—
56	C₂×#11761611	—	2	1	no	—
57	C₂×#11761611	—	2	2	no	—
58	C₂×#11761611	—	2	2	no	—
59	C₂×#11761611	—	4	4	no	—
60	C₂²×C₆₇⋊C₈	—	2	1	no	—
61	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
63	C₄×D₂₆₈	—	2	1	no	—
64	C₄×D₂₆₈	—	2	1	no	—
65	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
66	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
67	C₄×D₂₆₈	C₂³	1	2	yes	—
68	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
69	C₄×C₂²⋊D₆₇	—	2	1	no	—
70	C₄×C₂²⋊D₆₇	—	2	1	no	—
71	#14524002	C₂×C₄	1	1	yes	characteristic
72	#14524000	—	2	1	no	—
73	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
75	C₄²×D₆₇	—	2	1	no	—
76	#14523977	C₂×C₄	1	2	yes	—
77	#14523975	C₂×C₄	1	2	yes	—
78	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
79	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
80	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
81	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
82	Dic₆₇⋊C₈	—	2	1	no	—
83	Dic₆₇⋊C₈	—	2	1	no	—
84	#14523968	C₂×C₄	1	2	yes	—
85	#14523965	C₂×C₄	1	2	yes	—
86	#14523965	C₂×C₄	1	2	yes	—
87	#14523965	C₂×C₄	1	2	yes	—
88	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
89	(C₂×D₆₇)⋊C₈	—	2	2	no	—
90	(C₂×D₆₇)⋊C₈	—	2	2	no	—
91	(C₂×D₆₇)⋊C₈	—	2	1	no	—
92	(C₂×D₆₇)⋊C₈	—	2	1	no	—
93	#14523953	—	2	1	no	—
94	#14523953	—	2	1	no	—
95	C₄×C₆₇⋊C₈	—	2	1	no	—
96	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
98	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
99	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
101	C₄⋊C₈×C₆₇	—	2	1	no	—
102	C₂²⋊C₈×C₆₇	—	2	1	no	—
103	#97×C₆₇	C₂×C₄	1	2	yes	—
104	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
105	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
106	C₄×C₈×C₆₇	C₂³	1	2	yes	—
107	C₄×C₈×C₆₇	C₂³	1	2	yes	—
108	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
109	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
110	C₂³×D₆₇	—	2	1	no	—
111	#11761628	—	2	2	no	—
112	#11761628	—	2	2	no	—
113	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
114	C₂×D₂₆₈	—	2	2	no	—
115	C₂×D₂₆₈	—	2	2	no	—
116	C₂×D₂₆₈	—	2	1	no	—
117	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
119	C₂×C₂²⋊D₆₇	—	2	1	no	—
120	C₂²×Dic₆₇	—	2	1	no	—
121	C₂×C₄×D₆₇	—	2	1	no	—
122	C₂×C₄×D₆₇	—	2	1	no	—
123	C₂×C₄×D₆₇	—	2	2	no	—
124	C₂×C₄×D₆₇	—	2	1	no	—
125	C₂×C₄×D₆₇	—	2	2	no	—
126	C₂×C₄×D₆₇	—	4	2	no	—
127	C₂×C₄×D₆₇	—	4	2	no	—
128	#11761617	—	2	4	no	—
129	D₅₃₆	—	2	4	no	—
130	#11761611	—	4	2	no	—
131	#11761611	—	4	4	no	—
132	#11761611	—	4	2	no	—
133	#11761611	—	4	4	no	—
134	C₂×C₆₇⋊C₈	—	2	1	no	—
135	C₂×C₆₇⋊C₈	—	2	1	no	—
136	C₂×C₆₇⋊C₈	—	2	2	no	—
137	C₂×C₆₇⋊C₈	—	4	2	no	—
138	C₄⋊Dic₆₇	—	2	1	no	—
139	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
142	Dic₆₇⋊C₄	—	2	1	no	—
143	Dic₆₇⋊C₄	—	2	1	no	—
144	C₂²⋊Dic₆₇	—	2	1	no	—
145	(C₂×D₆₇)⋊C₄	—	2	2	no	—
146	(C₂×D₆₇)⋊C₄	—	2	1	no	—
147	(C₂×D₆₇)⋊C₄	—	2	2	no	—
148	(C₂×D₆₇)⋊C₄	—	2	1	no	—
149	C₄×Dic₆₇	—	2	1	no	—
150	C₄×Dic₆₇	—	2	1	no	—
151	C₄×Dic₆₇	—	2	1	no	—
152	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
155	M₁₆×C₆₇	—	2	2	no	—
156	M₁₆×C₆₇	M₁₆	1	4	yes	complemented
157	C₂×C₈×C₆₇	—	2	2	no	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	—	4	1	no	—
160	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
161	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
162	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
166	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
167	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₆₇	#41	1	1	yes	characteristic
169	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
170	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
171	C₂²×D₆₇	—	2	1	no	—
172	C₂²×D₆₇	—	2	2	no	—
173	C₂²×D₆₇	—	4	2	no	—
174	C₂²×D₆₇	—	4	2	no	—
175	C₂²×D₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	4	1	no	—
179	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
183	C₂×C₄×C₆₇	#131	1	2	yes	—
184	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
187	C₂²⋊D₆₇	—	4	2	no	—
188	C₂×Dic₆₇	—	2	1	no	—
189	C₂×Dic₆₇	—	2	1	no	—
190	C₂×Dic₆₇	—	2	1	no	—
191	C₂×Dic₆₇	—	2	2	no	—
192	C₂×Dic₆₇	—	4	2	no	—
193	D₂₆₈	—	2	1	no	—
194	D₂₆₈	—	2	2	no	—
195	D₂₆₈	—	2	2	no	—
196	D₂₆₈	—	2	1	no	—
197	D₂₆₈	—	4	2	no	—
198	C₄×D₆₇	—	4	2	no	—
199	C₄×D₆₇	—	4	4	no	—
200	C₄×D₆₇	—	4	2	no	—
201	C₄×D₆₇	—	4	1	no	—
202	C₄×D₆₇	—	4	1	no	—
203	C₄×D₆₇	—	4	2	no	—
204	C₄×D₆₇	—	4	2	no	—
205	Dic₁₃₄	—	2	1	no	—
206	Dic₁₃₄	—	2	1	no	—
207	C₈×C₆₇	—	2	2	no	—
208	C₈×C₆₇	—	2	1	no	—
209	C₈×C₆₇	—	2	1	no	—
210	C₈×C₆₇	—	2	1	no	—
211	C₈×C₆₇	—	4	2	no	—
212	C₈×C₆₇	C₂×M₁₆	1	4	yes	complemented
213	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	complemented
214	C₆₇⋊C₈	—	4	2	no	—
215	C₆₇⋊C₈	—	4	1	no	—
216	C₆₇⋊C₈	—	4	1	no	—
217	C₂²×C₆₇	—	2	1	no	—
218	C₂²×C₆₇	—	2	1	no	—
219	C₂²×C₆₇	#436	1	2	yes	—
220	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
221	C₂×D₆₇	—	4	1	no	—
222	C₂×D₆₇	—	4	2	no	—
223	C₂×D₆₇	—	4	2	no	—
224	C₂×D₆₇	—	4	1	no	—
225	C₂×D₆₇	—	4	1	no	—
226	C₂×D₆₇	—	4	2	no	—
227	C₂×D₆₇	—	8	4	no	—
228	C₄×C₆₇	—	2	1	no	—
229	C₄×C₆₇	—	2	1	no	—
230	C₄×C₆₇	—	2	1	no	—
231	C₄×C₆₇	#436	1	2	yes	—
232	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
233	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
234	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
235	C₄×C₆₇	#575	1	1	yes	characteristic
236	Dic₆₇	—	4	1	no	—
237	Dic₆₇	—	4	2	no	—
238	Dic₆₇	—	4	1	no	—
239	Dic₆₇	—	4	1	no	—
240	#17891	—	67	1	no	maximal, Sylow
241	C₂×C₆₇	—	2	2	no	—
242	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
243	C₂×C₆₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₂×C₆₇	#2955	1	1	yes	characteristic
245	D₆₇	—	8	1	no	—
246	D₆₇	—	8	2	no	—
247	#2965	—	67	1	no	—
248	C₄×AGL(1,ℤ/8)	—	67	1	no	—
249	C₂×#436	—	67	1	no	—
250	#1594	—	67	2	no	—
251	#1592	—	67	2	no	—
252	#1590	—	67	2	no	—
253	#1589	—	67	2	no	—
254	#1565	—	67	1	no	—
255	#1478	—	67	1	no	—
256	#1477	—	67	1	no	—
257	C₈×M₁₆	D₆₇	1	1	yes	characteristic, complemented
258	C₆₇	#17891	1	1	yes	characteristic, complemented, Sylow
259	C₂×AGL(1,ℤ/8)	—	67	1	no	—
260	C₂²×M₁₆	—	134	1	no	—
261	C₄×#41	—	67	1	no	—
262	C₂×C₄×D₄	—	67	1	no	—
263	#446	—	67	1	no	—
264	#446	—	67	1	no	—
265	#442	—	67	2	no	—
266	#440	—	67	2	no	—
267	C₄×SD₁₆	—	67	2	no	—
268	C₄×D₈	—	67	2	no	—
269	#436	—	67	1	no	—
270	#436	—	67	2	no	—
271	#436	—	67	2	no	—
272	#436	—	67	1	no	—
273	#436	—	134	1	no	—
274	#436	—	134	1	no	—
275	#435	—	134	1	no	—
276	#435	—	134	1	no	—
277	#431	—	67	1	no	—
278	#431	—	134	1	no	—
279	#431	C₂×D₆₇	1	1	yes	characteristic
280	#428	—	67	1	no	—
281	C₂×C₄⋊C₈	—	67	1	no	—
282	#418	—	67	1	no	—
283	#417	—	67	1	no	—
284	C₂×C₂²⋊C₈	—	134	1	no	—
285	C₄×M₁₆	—	134	1	no	—
286	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
287	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
288	#335	—	67	2	no	—
289	#334	—	67	2	no	—
290	Q₈⋊C₈	—	67	2	no	—
291	D₄⋊C₈	—	67	2	no	—
292	D₄⋊C₈	—	67	2	no	—
293	D₄⋊C₈	—	67	2	no	—
294	#322	C₂×D₆₇	1	2	yes	complemented
295	C₈²	C₂×D₆₇	1	2	yes	complemented
296	C₂×#41	—	67	1	no	—
297	C₂²×D₄	—	67	1	no	—
298	C₂³×C₄	—	134	1	no	—
299	AGL(1,ℤ/8)	—	134	4	no	—
300	C₂×SD₁₆	—	67	2	no	—
301	C₂×D₈	—	67	2	no	—
302	C₂×M₁₆	—	268	4	no	—
303	C₂×M₁₆	—	134	2	no	—
304	C₂×M₁₆	—	134	2	no	—
305	C₂×M₁₆	—	134	1	no	—
306	C₂×M₁₆	—	134	1	no	—
307	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
308	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
309	C₂²×C₈	—	134	1	no	—
310	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
311	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
312	C₄×Q₈	—	67	1	no	—
313	C₄×Q₈	—	67	1	no	—
314	C₄×D₄	—	67	2	no	—
315	C₄×D₄	—	67	1	no	—
316	C₄×D₄	—	67	2	no	—
317	C₄×D₄	—	67	1	no	—
318	C₄×D₄	—	134	1	no	—
319	C₄×D₄	—	134	1	no	—
320	C₄×D₄	—	134	1	no	—
321	C₄×D₄	—	134	1	no	—
322	#117	—	67	1	no	—
323	#117	—	134	1	no	—
324	C₂×C₄⋊C₄	—	67	1	no	—
325	C₂×C₂²⋊C₄	—	134	1	no	—
326	C₂×C₄²	—	134	1	no	—
327	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
328	#107	—	67	2	no	—
329	#106	—	67	2	no	—
330	C₄⋊C₈	—	2	1	no	—
331	C₄⋊C₈	—	67	1	no	—
332	C₄⋊C₈	—	67	1	no	—
333	C₄⋊C₈	—	67	2	no	—
334	C₄⋊C₈	—	134	1	no	—
335	C₄⋊C₈	—	134	1	no	—
336	C₄⋊C₈	—	134	1	no	—
337	#103	—	67	2	no	—
338	#102	—	67	2	no	—
339	#102	—	67	2	no	—
340	#102	—	67	2	no	—
341	C₂²⋊C₈	—	2	1	no	—
342	C₂²⋊C₈	—	134	1	no	—
343	C₂²⋊C₈	—	134	2	no	—
344	C₂²⋊C₈	—	134	2	no	—
345	C₂²⋊C₈	—	134	1	no	—
346	C₂²⋊C₈	—	134	1	no	—
347	#97	—	134	1	no	—
348	#97	—	134	1	no	—
349	#97	C₄×D₆₇	1	2	yes	—
350	C₄×C₈	—	134	1	no	—
351	C₄×C₈	D₂₆₈	1	1	yes	characteristic
352	C₄×C₈	D₂₆₈	1	1	yes	characteristic
353	C₄×C₈	C₂²×D₆₇	1	2	yes	—
354	C₄×C₈	C₂²×D₆₇	1	2	yes	—
355	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
356	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
357	C₂⁴	—	134	1	no	—
358	#41	—	134	2	no	—
359	#41	—	134	2	no	—
360	C₂×Q₈	—	67	1	no	—
361	C₂×D₄	—	67	1	no	—
362	C₂×D₄	—	67	2	no	—
363	C₂×D₄	—	134	1	no	—
364	C₂×D₄	—	134	1	no	—
365	C₂×D₄	—	134	2	no	—
366	C₂×D₄	—	134	2	no	—
367	C₂²×C₄	—	268	2	no	—
368	C₂²×C₄	—	268	2	no	—
369	C₂²×C₄	—	134	1	no	—
370	C₂²×C₄	—	134	1	no	—
371	C₂²×C₄	—	134	1	no	—
372	C₂²×C₄	—	134	2	no	—
373	C₂²×C₄	—	134	1	no	—
374	C₂²×C₄	—	134	2	no	—
375	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
377	C₂²×C₄	#11761628	1	1	yes	characteristic
378	SD₁₆	—	134	4	no	—
379	D₈	—	134	4	no	—
380	M₁₆	—	2	2	no	—
381	M₁₆	—	268	4	no	—
382	M₁₆	—	268	2	no	—
383	M₁₆	—	268	4	no	—
384	M₁₆	—	268	2	no	—
385	M₁₆	#11761611	1	4	yes	complemented
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	4	1	no	—
389	C₂×C₈	—	268	2	no	—
390	C₂×C₈	—	134	1	no	—
391	C₂×C₈	—	134	1	no	—
392	C₂×C₈	—	134	2	no	—
393	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
394	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
395	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
396	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
397	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
398	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
399	C₄⋊C₄	—	67	1	no	—
400	C₄⋊C₄	—	67	1	no	—
401	C₄⋊C₄	—	67	2	no	—
402	C₄⋊C₄	—	134	1	no	—
403	C₄⋊C₄	—	134	1	no	—
404	C₄⋊C₄	—	134	1	no	—
405	C₂²⋊C₄	—	134	1	no	—
406	C₂²⋊C₄	—	134	2	no	—
407	C₂²⋊C₄	—	134	1	no	—
408	C₂²⋊C₄	—	134	2	no	—
409	C₂²⋊C₄	—	134	1	no	—
410	C₄²	—	134	1	no	—
411	C₄²	—	134	1	no	—
412	C₄²	—	134	1	no	—
413	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
414	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
415	C₄²	#11761628	1	1	yes	characteristic
416	C₄²	C₂³×D₆₇	1	1	yes	characteristic
417	C₂³	—	268	2	no	—
418	C₂³	—	268	2	no	—
419	C₂³	—	268	2	no	—
420	C₂³	—	134	1	no	—
421	C₂³	—	134	2	no	—
422	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
423	Q₈	—	134	1	no	—
424	Q₈	—	134	1	no	—
425	D₄	—	268	2	no	—
426	D₄	—	268	2	no	—
427	D₄	—	134	1	no	—
428	D₄	—	134	2	no	—
429	D₄	—	134	2	no	—
430	D₄	—	134	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	268	4	no	—
435	C₂×C₄	—	268	2	no	—
436	C₂×C₄	—	268	2	no	—
437	C₂×C₄	—	268	2	no	—
438	C₂×C₄	—	268	1	no	—
439	C₂×C₄	—	268	1	no	—
440	C₂×C₄	—	268	2	no	—
441	C₂×C₄	—	268	2	no	—
442	C₂×C₄	—	134	1	no	—
443	C₂×C₄	—	134	1	no	—
444	C₂×C₄	—	134	1	no	—
445	C₂×C₄	—	134	2	no	—
446	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
447	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
448	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
449	C₂×C₄	C₂×#11761611	1	2	yes	—
450	C₂×C₄	#14524049	1	2	yes	—
451	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
452	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
453	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
454	C₈	—	2	2	no	—
455	C₈	—	2	1	no	—
456	C₈	—	2	1	no	—
457	C₈	—	2	1	no	—
458	C₈	—	4	2	no	—
459	C₈	—	268	2	no	—
460	C₈	—	268	1	no	—
461	C₈	—	268	1	no	—
462	C₈	C₂×#11761611	1	4	yes	complemented
463	C₈	#14524071	1	2	yes	complemented
464	C₂²	—	2	1	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	268	2	no	—
467	C₂²	—	268	1	no	—
468	C₂²	—	268	1	no	—
469	C₂²	—	268	2	no	—
470	C₂²	—	268	1	no	—
471	C₂²	—	268	2	no	—
472	C₂²	—	536	4	no	—
473	C₂²	#16863475	1	2	yes	—
474	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	268	2	no	—
479	C₄	—	268	1	no	—
480	C₄	—	268	1	no	—
481	C₄	—	268	1	no	—
482	C₄	#16863475	1	2	yes	—
483	C₄	C₂²×#11761611	1	1	yes	characteristic
484	C₄	C₂×#14524049	1	1	yes	characteristic
485	C₄	C₂×#14524071	1	1	yes	characteristic
486	C₄	#16864196	1	1	yes	characteristic
487	C₂	—	2	2	no	—
488	C₂	—	536	2	no	—
489	C₂	—	536	1	no	—
490	C₂	C₂×#16863475	1	1	yes	characteristic
491	C₂	C₄×#14524071	1	1	yes	characteristic
492	C₂	#22569113	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₈×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524071	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₈×M₁₆	D₆₇
hypocenter	C₆₇	#17891
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #22569175	C₂	semidirect
3. C₂×#16863475	C₂	semidirect
4. #22559124	C₂	semidirect
5. #22559113	C₂	semidirect
6. #22559101	C₂	semidirect
7. #22559095	C₂	semidirect
8. #22558961	C₂	semidirect
9. #22558601	C₂	semidirect
10. #22558597	C₂	semidirect
11. C₈×M₁₆×C₆₇	C₂	semidirect
16. #16863526	C₂²	semidirect
17. #16863526	C₂²	semidirect
26. #16863475	C₂²	semidirect
27. #16863475	C₂²	semidirect
38. #16863149	C₂²	semidirect
39. #16863147	C₂²	semidirect
40. #16863128	C₂²	semidirect
43. #16863125	C₂²	semidirect
47. #322×C₆₇	C₂²	semidirect
48. C₈²×C₆₇	C₂²	semidirect
156. M₁₆×C₆₇	M₁₆	semidirect
212. C₈×C₆₇	C₂×M₁₆	semidirect
213. C₈×C₆₇	AGL(1,ℤ/8)	semidirect
257. C₈×M₁₆	D₆₇	semidirect
258. C₆₇	#17891	semidirect
294. #322	C₂×D₆₇	semidirect
295. C₈²	C₂×D₆₇	semidirect
385. M₁₆	#11761611	semidirect
462. C₈	C₂×#11761611	semidirect
463. C₈	#14524071	semidirect

Maximal subgroups

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

subgroup	index
#22569175	2
C₄×#14524071	2
C₂×#16863475	2
#22559124	2
#22559113	2
#22559101	2
#22559095	2
#22558961	2
#22558601	2
#22558597	2
C₈×M₁₆×C₆₇	2
#17891	67