Subgroups — #30825412

Order 17152.

748 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	C₄×#14524068	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22559071	C₂	1	2	yes	maximal, complemented
4	#22559064	C₂	1	2	yes	maximal, complemented
5	#22559043	C₂	1	2	yes	maximal, complemented
6	#22559036	C₂	1	2	yes	maximal, complemented
7	#22558962	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558599	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558596	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558498	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2965	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524068	C₄	1	1	yes	characteristic
13	C₂²×C₈×D₆₇	—	2	1	no	—
14	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
15	#16863506	C₄	1	2	yes	—
16	#16863494	C₄	1	2	yes	—
17	C₄×#11761615	C₂²	1	2	yes	—
18	C₄×#11761613	C₂²	1	2	yes	—
19	C₈×D₂₆₈	—	2	1	no	—
20	C₈×D₂₆₈	—	2	1	no	—
21	C₈×D₂₆₈	C₂²	1	2	yes	—
22	C₈×D₂₆₈	C₂²	1	2	yes	—
23	#16863449	C₂²	1	1	yes	characteristic
24	#16863439	C₄	1	1	yes	characteristic
25	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
26	#16863400	C₄	1	1	yes	characteristic
27	#16863397	C₄	1	1	yes	characteristic
28	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
29	C₄×#11761612	C₂²	1	1	yes	characteristic
30	#16863148	C₂²	1	2	yes	complemented
31	#16863146	C₂²	1	2	yes	complemented
32	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	complemented
33	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	complemented
34	#16863123	C₂²	1	2	yes	—
35	#16863123	C₂²	1	2	yes	—
36	#16863118	C₂²	1	2	yes	complemented
37	#16863117	C₂²	1	2	yes	complemented
38	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
39	#446×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	#446×C₆₇	C₂²	1	1	yes	characteristic, complemented
41	#436×C₆₇	C₂²	1	1	yes	characteristic, complemented
42	#436×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	#435×C₆₇	—	2	1	no	—
44	#435×C₆₇	—	2	1	no	—
45	#431×C₆₇	—	2	1	no	—
46	#431×C₆₇	C₂²	1	1	yes	characteristic
47	C₄×M₁₆×C₆₇	—	2	1	no	—
48	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
49	C₂²×D₂₆₈	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×D₆₇	—	2	1	no	—
51	#14524068	—	2	4	no	—
52	C₂×#11761615	C₂×C₄	1	2	yes	—
53	C₂×#11761613	C₂×C₄	1	2	yes	—
54	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
55	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₆₇⋊C₈	—	2	1	no	—
57	C₂×C₈×D₆₇	—	2	2	no	—
58	C₂×C₈×D₆₇	—	2	2	no	—
59	C₂×C₈×D₆₇	—	4	4	no	—
60	C₄×D₂₆₈	—	2	1	no	—
61	C₄×D₂₆₈	—	2	1	no	—
62	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
63	C₄×D₂₆₈	C₂³	1	2	yes	—
64	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×(C₂×D₆₇)⋊C₄	—	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)	—	2	1	no	—
70	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
71	#14523967	C₂×C₄	1	2	yes	—
72	#14523964	C₂×C₄	1	2	yes	—
73	#14523963	C₂×C₄	1	2	yes	—
74	#14523963	C₂×C₄	1	2	yes	—
75	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
76	(C₂×D₆₇)⋊C₈	—	2	2	no	—
77	(C₂×D₆₇)⋊C₈	—	2	2	no	—
78	#14523954	C₂×C₄	1	2	yes	—
79	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
80	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
81	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂×M₁₆×C₆₇	—	2	1	no	—
83	C₂×M₁₆×C₆₇	—	2	1	no	—
84	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
85	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
86	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₄×D₄×C₆₇	—	2	1	no	—
89	C₄×D₄×C₆₇	—	2	1	no	—
90	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
92	#117×C₆₇	—	2	1	no	—
93	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₆₇	—	2	1	no	—
95	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
96	C₄⋊C₈×C₆₇	—	2	1	no	—
97	C₄⋊C₈×C₆₇	—	2	1	no	—
98	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₂²⋊C₈×C₆₇	—	2	1	no	—
101	C₂²⋊C₈×C₆₇	—	2	1	no	—
102	C₂²⋊C₈×C₆₇	—	2	1	no	—
103	#97×C₆₇	—	2	1	no	—
104	#97×C₆₇	—	2	1	no	—
105	C₄×C₈×C₆₇	—	2	1	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	C₂³×D₆₇	—	2	1	no	—
111	C₂×D₂₆₈	—	2	1	no	—
112	C₂×D₂₆₈	—	2	2	no	—
113	C₂×D₂₆₈	M₁₆	1	4	yes	complemented
114	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
115	C₂²×Dic₆₇	—	2	1	no	—
116	C₂×C₄×D₆₇	—	2	2	no	—
117	C₂×C₄×D₆₇	—	2	2	no	—
118	C₂×C₄×D₆₇	—	4	2	no	—
119	C₂×C₄×D₆₇	—	4	2	no	—
120	#11761615	—	2	4	no	—
121	#11761613	—	2	4	no	—
122	#11761612	—	2	2	no	—
123	#11761612	—	2	2	no	—
124	C₂×C₆₇⋊C₈	—	2	2	no	—
125	C₂×C₆₇⋊C₈	—	2	2	no	—
126	C₂×C₆₇⋊C₈	—	4	2	no	—
127	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
128	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
129	C₈×D₆₇	—	4	4	no	—
130	C₈×D₆₇	—	4	4	no	—
131	C₄⋊Dic₆₇	—	2	1	no	—
132	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
133	(C₂×D₆₇)⋊C₄	—	2	2	no	—
134	(C₂×D₆₇)⋊C₄	—	2	2	no	—
135	#41×C₆₇	—	2	2	no	—
136	#41×C₆₇	—	2	2	no	—
137	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂×D₄×C₆₇	—	2	1	no	—
139	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	2	1	no	—
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
146	M₁₆×C₆₇	—	4	2	no	—
147	M₁₆×C₆₇	—	4	2	no	—
148	C₂×C₈×C₆₇	—	2	2	no	—
149	C₂×C₈×C₆₇	—	2	1	no	—
150	C₂×C₈×C₆₇	—	2	1	no	—
151	C₂×C₈×C₆₇	—	4	1	no	—
152	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₆₇	—	2	1	no	—
157	C₄⋊C₄×C₆₇	—	2	1	no	—
158	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₂²⋊C₄×C₆₇	—	2	1	no	—
161	C₂²⋊C₄×C₆₇	—	2	1	no	—
162	C₂²⋊C₄×C₆₇	—	2	1	no	—
163	C₄²×C₆₇	—	2	1	no	—
164	C₄²×C₆₇	—	2	1	no	—
165	C₄²×C₆₇	—	2	1	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	1	no	—
171	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
172	C₂²×D₆₇	—	2	2	no	—
173	C₂²×D₆₇	—	2	4	no	—
174	C₂²×D₆₇	—	4	2	no	—
175	C₂²×D₆₇	—	4	2	no	—
176	Q₈×C₆₇	—	2	1	no	—
177	Q₈×C₆₇	—	2	1	no	—
178	D₄×C₆₇	—	2	1	no	—
179	D₄×C₆₇	—	2	1	no	—
180	D₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	4	1	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	1	no	—
189	C₂×C₄×C₆₇	—	4	1	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
195	C₂×C₄×C₆₇	#131	1	2	yes	—
196	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
199	C₂×Dic₆₇	—	2	2	no	—
200	C₂×Dic₆₇	—	4	2	no	—
201	D₂₆₈	—	2	2	no	—
202	D₂₆₈	—	2	4	no	—
203	D₂₆₈	C₂×M₁₆	1	4	yes	—
204	C₄×D₆₇	—	4	4	no	—
205	C₄×D₆₇	—	4	2	no	—
206	C₄×D₆₇	—	4	2	no	—
207	C₈×C₆₇	—	2	1	no	—
208	C₈×C₆₇	—	2	1	no	—
209	C₈×C₆₇	—	2	1	no	—
210	C₈×C₆₇	—	4	1	no	—
211	C₈×C₆₇	—	4	1	no	—
212	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	complemented
213	C₆₇⋊C₈	—	2	2	no	—
214	C₆₇⋊C₈	—	2	2	no	—
215	C₆₇⋊C₈	—	4	2	no	—
216	C₆₇⋊C₈	—	4	2	no	—
217	C₂²×C₆₇	—	2	1	no	—
218	C₂²×C₆₇	—	2	1	no	—
219	C₂²×C₆₇	—	4	1	no	—
220	C₂²×C₆₇	—	4	1	no	—
221	C₂²×C₆₇	—	4	1	no	—
222	C₂²×C₆₇	#435	1	2	yes	—
223	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
224	C₂×D₆₇	—	2	4	no	—
225	C₂×D₆₇	—	4	8	no	—
226	C₂×D₆₇	—	4	2	no	—
227	C₂×D₆₇	—	4	2	no	—
228	C₄×C₆₇	—	2	1	no	—
229	C₄×C₆₇	—	2	1	no	—
230	C₄×C₆₇	—	2	1	no	—
231	C₄×C₆₇	—	4	1	no	—
232	C₄×C₆₇	—	4	1	no	—
233	C₄×C₆₇	—	4	1	no	—
234	C₄×C₆₇	#435	1	2	yes	—
235	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
236	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
237	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
238	C₄×C₆₇	#575	1	1	yes	characteristic
239	Dic₆₇	—	4	2	no	—
240	#17869	—	67	1	no	maximal, Sylow
241	C₂×C₆₇	—	2	2	no	—
242	C₂×C₆₇	—	8	1	no	—
243	C₂×C₆₇	C₂×#435	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	—	4	4	no	—
247	#2965	D₆₇	1	1	yes	characteristic, complemented
248	C₄×AGL(1,ℤ/8)	—	67	1	no	—
249	C₂×C₈×D₄	—	67	1	no	—
250	#1586	—	67	2	no	—
251	#1585	—	67	2	no	—
252	#1582	—	67	2	no	—
253	#1581	—	67	2	no	—
254	#1565	—	67	1	no	—
255	#1478	—	67	1	no	—
256	#1477	—	67	1	no	—
257	#1451	—	67	1	no	—
258	C₆₇	#17869	1	1	yes	characteristic, complemented, Sylow
259	C₂×AGL(1,ℤ/8)	—	67	1	no	—
260	C₂³×C₈	—	134	1	no	—
261	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
262	C₂×C₄×D₄	—	67	1	no	—
263	#446	C₂×D₆₇	1	1	yes	characteristic, complemented
264	#446	C₂×D₆₇	1	1	yes	characteristic, complemented
265	#442	—	67	2	no	—
266	#440	—	67	2	no	—
267	C₄×SD₁₆	—	67	2	no	—
268	C₄×D₈	—	67	2	no	—
269	#436	C₂×D₆₇	1	1	yes	characteristic, complemented
270	#436	C₂×D₆₇	1	1	yes	characteristic, complemented
271	#435	—	2	1	no	—
272	#435	—	2	1	no	—
273	C₈×D₄	—	67	2	no	—
274	C₈×D₄	—	67	2	no	—
275	C₈×D₄	—	134	1	no	—
276	C₈×D₄	—	134	1	no	—
277	#431	—	2	1	no	—
278	#431	—	67	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	—	2	1	no	—
286	C₄×M₁₆	—	67	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	#322	—	67	2	no	—
296	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
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	—	2	1	no	—
303	C₂×M₁₆	—	2	1	no	—
304	C₂×M₁₆	—	67	1	no	—
305	C₂×M₁₆	—	67	1	no	—
306	C₂²×C₈	—	268	4	no	—
307	C₂²×C₈	—	134	1	no	—
308	C₂²×C₈	—	134	2	no	—
309	C₂²×C₈	—	134	2	no	—
310	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
311	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
312	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
313	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	67	2	no	—
317	C₄×D₄	—	67	2	no	—
318	C₄×D₄	—	134	1	no	—
319	C₄×D₄	—	134	1	no	—
320	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
321	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
322	#117	—	2	1	no	—
323	#117	C₄×D₆₇	1	1	yes	characteristic
324	C₂×C₄⋊C₄	—	67	1	no	—
325	C₂×C₂²⋊C₄	—	134	1	no	—
326	C₂×C₄²	—	2	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	—	2	1	no	—
332	C₄⋊C₈	—	67	2	no	—
333	C₄⋊C₈	—	134	1	no	—
334	C₄⋊C₈	—	134	1	no	—
335	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
336	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
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	—	2	1	no	—
343	C₂²⋊C₈	—	2	1	no	—
344	C₂²⋊C₈	—	134	1	no	—
345	C₂²⋊C₈	—	134	2	no	—
346	C₂²⋊C₈	—	134	2	no	—
347	#97	—	2	1	no	—
348	#97	—	2	1	no	—
349	#97	—	67	2	no	—
350	C₄×C₈	—	2	1	no	—
351	C₄×C₈	—	67	2	no	—
352	C₄×C₈	—	67	2	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	—	134	1	no	—
358	#41	—	2	2	no	—
359	#41	—	2	2	no	—
360	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
361	C₂×D₄	—	2	1	no	—
362	C₂×D₄	—	67	4	no	—
363	C₂×D₄	—	67	2	no	—
364	C₂×D₄	—	134	1	no	—
365	C₂×D₄	—	134	2	no	—
366	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	268	2	no	—
371	C₂²×C₄	—	268	2	no	—
372	C₂²×C₄	—	134	1	no	—
373	C₂²×C₄	—	134	2	no	—
374	C₂²×C₄	—	134	2	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	2	no	—
381	M₁₆	—	4	2	no	—
382	M₁₆	—	134	2	no	—
383	M₁₆	—	134	2	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	4	1	no	—
388	C₂×C₈	—	268	4	no	—
389	C₂×C₈	—	268	2	no	—
390	C₂×C₈	—	268	4	no	—
391	C₂×C₈	—	67	2	no	—
392	C₂×C₈	—	67	2	no	—
393	C₂×C₈	—	134	2	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	—	2	1	no	—
401	C₄⋊C₄	—	67	2	no	—
402	C₄⋊C₄	—	134	1	no	—
403	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
404	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
405	C₂²⋊C₄	—	2	1	no	—
406	C₂²⋊C₄	—	2	1	no	—
407	C₂²⋊C₄	—	2	1	no	—
408	C₂²⋊C₄	—	134	2	no	—
409	C₂²⋊C₄	—	134	2	no	—
410	C₄²	—	2	1	no	—
411	C₄²	—	2	1	no	—
412	C₄²	—	2	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	1	no	—
418	C₂³	—	268	2	no	—
419	C₂³	—	268	2	no	—
420	C₂³	—	134	2	no	—
421	C₂³	—	134	4	no	—
422	C₂³	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
423	Q₈	—	2	1	no	—
424	Q₈	—	2	1	no	—
425	D₄	—	2	1	no	—
426	D₄	—	2	1	no	—
427	D₄	—	4	2	no	—
428	D₄	—	67	4	no	—
429	D₄	—	134	2	no	—
430	D₄	—	134	4	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	268	4	no	—
442	C₂×C₄	—	268	2	no	—
443	C₂×C₄	—	268	2	no	—
444	C₂×C₄	—	268	2	no	—
445	C₂×C₄	—	134	2	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	1	no	—
458	C₈	—	4	1	no	—
459	C₈	—	268	2	no	—
460	C₈	—	268	2	no	—
461	C₈	—	134	2	no	—
462	C₈	—	134	2	no	—
463	C₈	#14524068	1	2	yes	complemented
464	C₂²	—	2	1	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	4	1	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	1	no	—
469	C₂²	—	268	8	no	—
470	C₂²	—	268	2	no	—
471	C₂²	—	268	2	no	—
472	C₂²	—	134	4	no	—
473	C₂²	#16863465	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	1	no	—
479	C₄	—	4	1	no	—
480	C₄	—	4	1	no	—
481	C₄	—	268	2	no	—
482	C₄	#16863465	1	2	yes	—
483	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
484	C₄	C₂×#14524051	1	1	yes	characteristic
485	C₄	C₂×#14524068	1	1	yes	characteristic
486	C₄	#16864194	1	1	yes	characteristic
487	C₂	—	2	2	no	—
488	C₂	—	8	1	no	—
489	C₂	—	268	4	no	—
490	C₂	C₂×#16863465	1	1	yes	characteristic
491	C₂	C₄×#14524068	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	C₆₇×#2965	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524068	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#2965	D₆₇
hypocenter	C₆₇	#17869
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2965

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×D₂₆₈	C₂	semidirect
3. #22559071	C₂	semidirect
4. #22559064	C₂	semidirect
5. #22559043	C₂	semidirect
6. #22559036	C₂	semidirect
7. #22558962	C₂	semidirect
8. #22558599	C₂	semidirect
9. #22558596	C₂	semidirect
10. #22558498	C₂	semidirect
11. C₆₇×#2965	C₂	semidirect
30. #16863148	C₂²	semidirect
31. #16863146	C₂²	semidirect
32. Q₈⋊(C₆₇⋊C₈)	C₂²	semidirect
33. D₄⋊(C₆₇⋊C₈)	C₂²	semidirect
36. #16863118	C₂²	semidirect
37. #16863117	C₂²	semidirect
39. #446×C₆₇	C₂²	semidirect
40. #446×C₆₇	C₂²	semidirect
41. #436×C₆₇	C₂²	semidirect
42. #436×C₆₇	C₂²	semidirect
113. C₂×D₂₆₈	M₁₆	semidirect
212. C₈×C₆₇	AGL(1,ℤ/8)	semidirect
247. #2965	D₆₇	semidirect
258. C₆₇	#17869	semidirect
263. #446	C₂×D₆₇	semidirect
264. #446	C₂×D₆₇	semidirect
269. #436	C₂×D₆₇	semidirect
270. #436	C₂×D₆₇	semidirect
463. C₈	#14524068	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14524068	2
C₂×C₈×D₂₆₈	2
#22559071	2
#22559064	2
#22559043	2
#22559036	2
#22558962	2
#22558599	2
#22558596	2
#22558498	2
C₆₇×#2965	2
#17869	67