Subgroups — #30825616

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×#14524069	C₂	1	1	yes	maximal, characteristic
3	#22559121	C₂	1	2	yes	maximal, complemented
4	#22559111	C₂	1	2	yes	maximal, complemented
5	#22559097	C₂	1	2	yes	maximal, complemented
6	#22559092	C₂	1	2	yes	maximal, complemented
7	#22558959	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558600	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558594	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11761612	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524069	C₄	1	1	yes	characteristic
13	C₄×#11761628	C₂²	1	1	yes	characteristic
14	#16863526	C₂²	1	1	yes	characteristic, complemented
15	#16863526	C₂²	1	1	yes	characteristic, complemented
16	#16863506	C₄	1	2	yes	—
17	#16863496	C₄	1	2	yes	—
18	C₄×#11761616	C₂²	1	2	yes	—
19	C₄×#11761613	C₂²	1	2	yes	—
20	#16863475	C₂²	1	1	yes	characteristic, complemented
21	#16863475	C₂²	1	1	yes	characteristic, complemented
22	#16863467	—	2	1	no	—
23	#16863467	—	2	1	no	—
24	#16863449	C₂²	1	1	yes	characteristic
25	#16863449	C₂²	1	1	yes	characteristic
26	#16863447	—	2	1	no	—
27	#16863436	C₄	1	1	yes	characteristic
28	#16863401	C₄	1	1	yes	characteristic
29	#16863395	C₄	1	1	yes	characteristic
30	C₄×#11761612	C₂²	1	1	yes	characteristic
31	C₄×#11761611	—	2	1	no	—
32	#16863148	C₂²	1	2	yes	complemented
33	#16863146	C₂²	1	2	yes	complemented
34	#16863126	C₂²	1	2	yes	complemented
35	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
36	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
37	#16863123	C₂²	1	2	yes	complemented
38	#16863118	C₂²	1	2	yes	complemented
39	C₈×C₆₇⋊C₈	C₂²	1	2	yes	complemented
40	C₂²×M₁₆×C₆₇	—	2	1	no	—
41	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
42	#436×C₆₇	—	2	1	no	—
43	#436×C₆₇	—	2	1	no	—
44	#436×C₆₇	C₂²	1	2	yes	—
45	#436×C₆₇	C₂²	1	2	yes	—
46	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
48	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
50	#14524069	—	2	4	no	—
51	C₂×#11761616	C₂×C₄	1	2	yes	—
52	C₂×#11761613	C₂×C₄	1	2	yes	—
53	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
54	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
55	C₂×#11761611	—	2	1	no	—
56	C₂×#11761611	—	2	1	no	—
57	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
59	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
60	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
61	C₄×C₂²⋊D₆₇	—	2	1	no	—
62	C₄×C₂²⋊D₆₇	—	2	1	no	—
63	#14524002	C₂×C₄	1	1	yes	characteristic
64	#14524000	—	2	1	no	—
65	C₄²×D₆₇	—	2	1	no	—
66	#14523976	C₂×C₄	1	2	yes	—
67	#14523974	C₂×C₄	1	2	yes	—
68	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
69	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
70	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
71	Dic₆₇⋊C₈	—	2	1	no	—
72	Dic₆₇⋊C₈	—	2	1	no	—
73	#14523966	C₂×C₄	1	2	yes	—
74	#14523964	C₂×C₄	1	2	yes	—
75	#14523964	C₂×C₄	1	2	yes	—
76	#14523963	C₂×C₄	1	2	yes	—
77	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
78	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
79	(C₂×D₆₇)⋊C₈	—	2	1	no	—
80	(C₂×D₆₇)⋊C₈	—	2	1	no	—
81	#14523954	C₂×C₄	1	2	yes	—
82	#14523953	—	2	1	no	—
83	#14523953	—	2	1	no	—
84	C₄×C₆₇⋊C₈	—	2	1	no	—
85	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
86	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
87	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₂³×C₄×C₆₇	—	2	1	no	—
89	C₂×M₁₆×C₆₇	—	2	2	no	—
90	C₂×M₁₆×C₆₇	—	2	2	no	—
91	C₂×M₁₆×C₆₇	—	4	4	no	—
92	C₂²×C₈×C₆₇	—	2	1	no	—
93	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
94	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
95	C₄×D₄×C₆₇	—	2	1	no	—
96	C₄×D₄×C₆₇	—	2	1	no	—
97	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
98	C₄×D₄×C₆₇	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	C₄⋊C₈×C₆₇	—	2	1	no	—
103	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
104	C₂²⋊C₈×C₆₇	—	2	2	no	—
105	C₂²⋊C₈×C₆₇	—	2	2	no	—
106	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
107	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
108	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
109	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
110	#11761628	—	2	2	no	—
111	#11761628	—	2	2	no	—
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	#11761616	—	2	4	no	—
119	#11761613	—	2	4	no	—
120	#11761612	—	2	2	no	—
121	#11761612	M₁₆	1	4	yes	complemented
122	#11761611	—	4	2	no	—
123	#11761611	—	4	2	no	—
124	C₂×C₆₇⋊C₈	—	2	1	no	—
125	C₂×C₆₇⋊C₈	—	2	1	no	—
126	C₂×C₆₇⋊C₈	—	2	2	no	—
127	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
128	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
129	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
131	Dic₆₇⋊C₄	—	2	1	no	—
132	Dic₆₇⋊C₄	—	2	1	no	—
133	C₂²⋊Dic₆₇	—	2	1	no	—
134	(C₂×D₆₇)⋊C₄	—	2	1	no	—
135	(C₂×D₆₇)⋊C₄	—	2	1	no	—
136	C₄×Dic₆₇	—	2	1	no	—
137	C₄×Dic₆₇	—	2	1	no	—
138	C₄×Dic₆₇	—	2	1	no	—
139	C₂⁴×C₆₇	—	2	1	no	—
140	C₂×D₄×C₆₇	—	2	1	no	—
141	C₂×D₄×C₆₇	—	2	2	no	—
142	C₂×D₄×C₆₇	—	2	2	no	—
143	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
144	C₂²×C₄×C₆₇	—	2	1	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	4	2	no	—
148	C₂²×C₄×C₆₇	—	4	2	no	—
149	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
152	M₁₆×C₆₇	—	4	4	no	—
153	M₁₆×C₆₇	—	4	4	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	4	1	no	—
157	C₂×C₈×C₆₇	—	4	2	no	—
158	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
159	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
161	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
162	C₄⋊C₄×C₆₇	—	2	1	no	—
163	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
164	C₂²⋊C₄×C₆₇	—	2	2	no	—
165	C₂²⋊C₄×C₆₇	—	2	2	no	—
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	—	2	2	no	—
171	C₂³×C₆₇	—	4	2	no	—
172	C₂³×C₆₇	—	4	2	no	—
173	C₂³×C₆₇	—	4	2	no	—
174	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
175	C₂²×D₆₇	—	2	1	no	—
176	D₄×C₆₇	—	2	2	no	—
177	D₄×C₆₇	—	2	2	no	—
178	D₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	4	1	no	—
183	C₂×C₄×C₆₇	—	4	4	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
191	C₂×C₄×C₆₇	#131	1	2	yes	—
192	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
195	C₂²⋊D₆₇	—	4	2	no	—
196	C₂×Dic₆₇	—	2	1	no	—
197	C₂×Dic₆₇	—	2	1	no	—
198	C₂×Dic₆₇	—	2	1	no	—
199	D₂₆₈	—	2	1	no	—
200	D₂₆₈	—	2	1	no	—
201	C₄×D₆₇	—	4	2	no	—
202	C₄×D₆₇	—	4	1	no	—
203	C₄×D₆₇	—	4	1	no	—
204	C₄×D₆₇	—	4	2	no	—
205	Dic₁₃₄	—	2	1	no	—
206	Dic₁₃₄	—	2	1	no	—
207	C₈×C₆₇	—	2	1	no	—
208	C₈×C₆₇	—	2	1	no	—
209	C₈×C₆₇	—	2	1	no	—
210	C₈×C₆₇	—	4	2	no	—
211	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	complemented
212	C₆₇⋊C₈	—	2	2	no	—
213	C₆₇⋊C₈	—	4	2	no	—
214	C₆₇⋊C₈	—	4	1	no	—
215	C₆₇⋊C₈	—	4	1	no	—
216	C₆₇⋊C₈	C₂×M₁₆	1	4	yes	complemented
217	C₂²×C₆₇	—	2	1	no	—
218	C₂²×C₆₇	—	2	1	no	—
219	C₂²×C₆₇	—	4	2	no	—
220	C₂²×C₆₇	—	4	2	no	—
221	C₂²×C₆₇	—	4	2	no	—
222	C₂²×C₆₇	—	8	4	no	—
223	C₂²×C₆₇	#436	1	2	yes	—
224	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
225	C₂×D₆₇	—	4	1	no	—
226	C₂×D₆₇	—	4	1	no	—
227	C₂×D₆₇	—	4	1	no	—
228	C₄×C₆₇	—	2	1	no	—
229	C₄×C₆₇	—	2	1	no	—
230	C₄×C₆₇	—	2	1	no	—
231	C₄×C₆₇	—	4	2	no	—
232	C₄×C₆₇	#436	1	2	yes	—
233	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
234	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
235	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
236	C₄×C₆₇	#575	1	1	yes	characteristic
237	Dic₆₇	—	4	1	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	—	8	2	no	—
243	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
244	C₂×C₆₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
245	C₂×C₆₇	#2955	1	1	yes	characteristic
246	D₆₇	—	8	1	no	—
247	#2965	—	67	1	no	—
248	C₄×AGL(1,ℤ/8)	—	67	1	no	—
249	C₂×#436	D₆₇	1	1	yes	characteristic, complemented
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	—	67	1	no	—
258	C₆₇	#17891	1	1	yes	characteristic, complemented, Sylow
259	C₂×AGL(1,ℤ/8)	—	67	1	no	—
260	C₂²×M₁₆	—	2	1	no	—
261	C₄×#41	—	67	1	no	—
262	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
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	—	2	1	no	—
270	#436	—	2	1	no	—
271	#436	—	67	1	no	—
272	#436	—	67	1	no	—
273	#436	C₂×D₆₇	1	2	yes	—
274	#436	C₂×D₆₇	1	2	yes	—
275	#435	—	134	1	no	—
276	#435	—	134	1	no	—
277	#431	—	67	1	no	—
278	#431	—	67	1	no	—
279	#431	—	134	1	no	—
280	#428	—	67	1	no	—
281	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
282	#418	—	67	1	no	—
283	#417	—	67	1	no	—
284	C₂×C₂²⋊C₈	—	2	1	no	—
285	C₄×M₁₆	—	67	1	no	—
286	C₄×M₁₆	—	134	1	no	—
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	—	67	2	no	—
295	C₈²	—	67	2	no	—
296	C₂×#41	—	67	1	no	—
297	C₂²×D₄	C₄×D₆₇	1	1	yes	characteristic
298	C₂³×C₄	—	2	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	—	2	2	no	—
303	C₂×M₁₆	—	2	2	no	—
304	C₂×M₁₆	—	4	4	no	—
305	C₂×M₁₆	—	67	1	no	—
306	C₂×M₁₆	—	67	1	no	—
307	C₂×M₁₆	—	134	1	no	—
308	C₂×M₁₆	—	134	1	no	—
309	C₂²×C₈	—	2	1	no	—
310	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
311	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
312	C₄×Q₈	—	67	1	no	—
313	C₄×Q₈	—	67	1	no	—
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	67	1	no	—
317	C₄×D₄	—	67	1	no	—
318	C₄×D₄	—	134	1	no	—
319	C₄×D₄	—	134	1	no	—
320	C₄×D₄	C₄×D₆₇	1	2	yes	—
321	C₄×D₄	C₂²×D₆₇	1	2	yes	—
322	#117	—	67	1	no	—
323	#117	—	134	1	no	—
324	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
325	C₂×C₂²⋊C₄	—	2	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	—	134	1	no	—
334	C₄⋊C₈	—	134	1	no	—
335	C₄⋊C₈	—	134	1	no	—
336	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
337	#103	—	67	2	no	—
338	#102	—	67	2	no	—
339	#102	—	67	2	no	—
340	#102	—	67	2	no	—
341	C₂²⋊C₈	—	2	2	no	—
342	C₂²⋊C₈	—	2	2	no	—
343	C₂²⋊C₈	—	134	1	no	—
344	C₂²⋊C₈	—	134	1	no	—
345	C₂²⋊C₈	—	134	1	no	—
346	C₂²⋊C₈	—	134	1	no	—
347	#97	—	67	2	no	—
348	#97	—	134	1	no	—
349	#97	—	134	1	no	—
350	C₄×C₈	—	67	2	no	—
351	C₄×C₈	—	67	2	no	—
352	C₄×C₈	—	134	1	no	—
353	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
354	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
355	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
356	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
357	C₂⁴	—	2	1	no	—
358	#41	—	134	2	no	—
359	#41	—	134	2	no	—
360	C₂×Q₈	—	67	1	no	—
361	C₂×D₄	—	2	1	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	67	1	no	—
365	C₂×D₄	—	134	1	no	—
366	C₂×D₄	C₂×C₄×D₆₇	1	2	yes	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	4	2	no	—
371	C₂²×C₄	—	4	2	no	—
372	C₂²×C₄	—	134	1	no	—
373	C₂²×C₄	—	134	1	no	—
374	C₂²×C₄	—	134	1	no	—
375	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
376	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
377	C₂²×C₄	#11761628	1	1	yes	characteristic
378	SD₁₆	—	134	4	no	—
379	D₈	—	134	4	no	—
380	M₁₆	—	4	4	no	—
381	M₁₆	—	4	4	no	—
382	M₁₆	—	268	2	no	—
383	M₁₆	—	268	2	no	—
384	M₁₆	—	67	4	no	—
385	M₁₆	—	134	2	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	4	1	no	—
389	C₂×C₈	—	4	2	no	—
390	C₂×C₈	—	67	2	no	—
391	C₂×C₈	—	67	2	no	—
392	C₂×C₈	—	134	1	no	—
393	C₂×C₈	—	134	1	no	—
394	C₂×C₈	—	134	2	no	—
395	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
396	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
397	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
398	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	67	1	no	—
401	C₄⋊C₄	—	67	1	no	—
402	C₄⋊C₄	—	134	1	no	—
403	C₄⋊C₄	—	134	1	no	—
404	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
405	C₂²⋊C₄	—	2	2	no	—
406	C₂²⋊C₄	—	2	2	no	—
407	C₂²⋊C₄	—	134	1	no	—
408	C₂²⋊C₄	—	134	1	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×C₂²⋊D₆₇	1	1	yes	characteristic
415	C₄²	#11761628	1	1	yes	characteristic
416	C₄²	C₂³×D₆₇	1	1	yes	characteristic
417	C₂³	—	2	2	no	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	134	1	no	—
422	C₂³	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
423	Q₈	—	134	1	no	—
424	Q₈	—	134	1	no	—
425	D₄	—	2	2	no	—
426	D₄	—	2	2	no	—
427	D₄	—	4	2	no	—
428	D₄	—	268	2	no	—
429	D₄	—	134	1	no	—
430	D₄	—	134	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	4	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	268	2	no	—
440	C₂×C₄	—	268	1	no	—
441	C₂×C₄	—	268	1	no	—
442	C₂×C₄	—	268	2	no	—
443	C₂×C₄	—	134	1	no	—
444	C₂×C₄	—	134	1	no	—
445	C₂×C₄	—	134	1	no	—
446	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
447	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
448	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
449	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
450	C₂×C₄	#14524051	1	2	yes	—
451	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
452	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
453	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
454	C₈	—	2	1	no	—
455	C₈	—	2	1	no	—
456	C₈	—	2	1	no	—
457	C₈	—	4	2	no	—
458	C₈	—	268	2	no	—
459	C₈	—	268	1	no	—
460	C₈	—	268	1	no	—
461	C₈	—	67	4	no	—
462	C₈	—	134	2	no	—
463	C₈	#14524069	1	2	yes	complemented
464	C₂²	—	2	1	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	4	2	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	8	4	no	—
470	C₂²	—	268	1	no	—
471	C₂²	—	268	1	no	—
472	C₂²	—	268	1	no	—
473	C₂²	#16863472	1	2	yes	—
474	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	4	2	no	—
479	C₄	—	268	1	no	—
480	C₄	—	268	1	no	—
481	C₄	—	268	1	no	—
482	C₄	#16863472	1	2	yes	—
483	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
484	C₄	C₂×#14524051	1	1	yes	characteristic
485	C₄	C₂×#14524069	1	1	yes	characteristic
486	C₄	#16864194	1	1	yes	characteristic
487	C₂	—	2	2	no	—
488	C₂	—	8	2	no	—
489	C₂	—	536	1	no	—
490	C₂	C₂×#16863472	1	1	yes	characteristic
491	C₂	C₄×#14524069	1	1	yes	characteristic
492	C₂	#22569110	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×#436×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524069	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#436	D₆₇
hypocenter	C₆₇	#17891
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #22569175	C₂	semidirect
3. #22559121	C₂	semidirect
4. #22559111	C₂	semidirect
5. #22559097	C₂	semidirect
6. #22559092	C₂	semidirect
7. #22558959	C₂	semidirect
8. #22558600	C₂	semidirect
9. #22558594	C₂	semidirect
10. C₈×#11761612	C₂	semidirect
11. C₂×#436×C₆₇	C₂	semidirect
14. #16863526	C₂²	semidirect
15. #16863526	C₂²	semidirect
20. #16863475	C₂²	semidirect
21. #16863475	C₂²	semidirect
32. #16863148	C₂²	semidirect
33. #16863146	C₂²	semidirect
34. #16863126	C₂²	semidirect
37. #16863123	C₂²	semidirect
38. #16863118	C₂²	semidirect
39. C₈×C₆₇⋊C₈	C₂²	semidirect
121. #11761612	M₁₆	semidirect
211. C₈×C₆₇	AGL(1,ℤ/8)	semidirect
216. C₆₇⋊C₈	C₂×M₁₆	semidirect
249. C₂×#436	D₆₇	semidirect
258. C₆₇	#17891	semidirect
463. C₈	#14524069	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×#14524069	2
#22559121	2
#22559111	2
#22559097	2
#22559092	2
#22558959	2
#22558600	2
#22558594	2
C₈×#11761612	2
C₂×#436×C₆₇	2
#17891	67