Subgroups — #30825799

Order 17152.

720 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570252	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	C₈×#11761615	C₂	1	2	yes	maximal, complemented
6	C₈×#11761613	C₂	1	2	yes	maximal, complemented
7	#22558995	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558651	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558622	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11761612	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2966	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₈×D₆₇	—	2	1	no	—
13	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
14	#16863729	C₄	1	1	yes	characteristic
15	#16863712	C₄	1	1	yes	characteristic
16	#16863693	C₄	1	2	yes	—
17	#16863608	C₄	1	2	yes	—
18	#16863546	C₄	1	1	yes	characteristic
19	#16863529	C₄	1	1	yes	characteristic
20	C₄×#11761615	C₂²	1	2	yes	—
21	C₄×#11761613	C₂²	1	2	yes	—
22	C₈×D₂₆₈	—	2	1	no	—
23	C₈×D₂₆₈	—	2	1	no	—
24	C₈×D₂₆₈	C₂²	1	2	yes	—
25	C₈×D₂₆₈	C₂²	1	2	yes	—
26	#16863449	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	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	C₈×C₆₇⋊C₈	C₂²	1	2	yes	complemented
38	#529×C₆₇	C₂²	1	1	yes	characteristic
39	#446×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
41	#436×C₆₇	C₂²	1	1	yes	characteristic, complemented
42	#435×C₆₇	—	2	1	no	—
43	C₈×D₄×C₆₇	—	2	1	no	—
44	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	#433×C₆₇	—	2	1	no	—
46	#432×C₆₇	C₂²	1	1	yes	characteristic
47	#405×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	C₂×#11761615	C₂×C₄	1	2	yes	—
52	C₂×#11761613	C₂×C₄	1	2	yes	—
53	C₂×#11761612	—	2	1	no	—
54	C₂²×C₆₇⋊C₈	—	2	1	no	—
55	C₂×C₈×D₆₇	—	2	2	no	—
56	C₂×C₈×D₆₇	—	2	2	no	—
57	C₂×C₈×D₆₇	—	4	4	no	—
58	C₄×D₂₆₈	—	2	1	no	—
59	C₄×D₂₆₈	—	2	1	no	—
60	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
61	C₄×D₂₆₈	C₂³	1	2	yes	—
62	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
64	#14523976	C₂×C₄	1	2	yes	—
65	#14523974	C₂×C₄	1	2	yes	—
66	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
67	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
68	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
69	#14523967	C₂×C₄	1	2	yes	—
70	#14523964	C₂×C₄	1	2	yes	—
71	#14523963	C₂×C₄	1	2	yes	—
72	#14523963	C₂×C₄	1	2	yes	—
73	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
75	(C₂×D₆₇)⋊C₈	—	2	2	no	—
76	(C₂×D₆₇)⋊C₈	—	2	2	no	—
77	#14523954	—	2	1	no	—
78	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
79	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
80	C₂×M₁₆×C₆₇	—	2	1	no	—
81	C₂²×C₈×C₆₇	—	2	1	no	—
82	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
83	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
84	#126×C₆₇	—	2	1	no	—
85	#125×C₆₇	C₂×C₄	1	1	yes	characteristic
86	#124×C₆₇	C₂×C₄	1	1	yes	characteristic
87	#123×C₆₇	—	2	1	no	—
88	C₂²⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
89	#121×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄×D₄×C₆₇	—	2	1	no	—
92	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
93	#117×C₆₇	—	2	1	no	—
94	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
95	C₄⋊C₈×C₆₇	—	2	1	no	—
96	C₄⋊C₈×C₆₇	—	2	1	no	—
97	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₂²⋊C₈×C₆₇	—	2	1	no	—
100	C₂²⋊C₈×C₆₇	—	2	1	no	—
101	C₂²⋊C₈×C₆₇	—	2	1	no	—
102	#97×C₆₇	—	2	1	no	—
103	#97×C₆₇	—	2	1	no	—
104	C₄×C₈×C₆₇	—	2	1	no	—
105	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
106	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
107	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
108	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
109	C₂³×D₆₇	—	2	1	no	—
110	C₂×D₂₆₈	—	2	1	no	—
111	C₂×D₂₆₈	—	2	4	no	—
112	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
113	C₂²×Dic₆₇	—	2	1	no	—
114	C₂×C₄×D₆₇	—	2	2	no	—
115	C₂×C₄×D₆₇	—	2	2	no	—
116	C₂×C₄×D₆₇	—	4	2	no	—
117	C₂×C₄×D₆₇	—	4	2	no	—
118	#11761615	—	2	4	no	—
119	#11761613	—	2	4	no	—
120	#11761612	—	2	4	no	—
121	C₂×C₆₇⋊C₈	—	2	2	no	—
122	C₂×C₆₇⋊C₈	—	2	2	no	—
123	C₂×C₆₇⋊C₈	—	4	2	no	—
124	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
125	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
126	C₈×D₆₇	—	4	4	no	—
127	C₈×D₆₇	—	4	4	no	—
128	C₄⋊Dic₆₇	—	2	1	no	—
129	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
130	(C₂×D₆₇)⋊C₄	—	2	2	no	—
131	(C₂×D₆₇)⋊C₄	—	2	2	no	—
132	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂×D₄×C₆₇	—	2	1	no	—
134	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
139	M₁₆×C₆₇	—	4	2	no	—
140	C₂×C₈×C₆₇	—	2	2	no	—
141	C₂×C₈×C₆₇	—	2	1	no	—
142	C₂×C₈×C₆₇	—	2	1	no	—
143	C₂×C₈×C₆₇	—	4	1	no	—
144	C₂×C₈×C₆₇	—	4	2	no	—
145	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₄⋊C₄×C₆₇	—	2	1	no	—
150	C₄⋊C₄×C₆₇	—	2	1	no	—
151	C₄⋊C₄×C₆₇	—	2	1	no	—
152	C₄⋊C₄×C₆₇	—	2	1	no	—
153	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
154	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
155	C₂²⋊C₄×C₆₇	—	2	1	no	—
156	C₂²⋊C₄×C₆₇	—	2	1	no	—
157	C₂²⋊C₄×C₆₇	—	2	1	no	—
158	C₂²⋊C₄×C₆₇	—	2	1	no	—
159	C₂²⋊C₄×C₆₇	—	2	1	no	—
160	C₄²×C₆₇	—	2	1	no	—
161	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
162	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₄²×C₆₇	#41	1	1	yes	characteristic
164	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
165	C₂³×C₆₇	—	2	1	no	—
166	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
167	C₂²×D₆₇	—	2	4	no	—
168	C₂²×D₆₇	—	2	2	no	—
169	C₂²×D₆₇	—	4	2	no	—
170	C₂²×D₆₇	—	4	2	no	—
171	Q₈×C₆₇	—	2	2	no	—
172	D₄×C₆₇	—	2	2	no	—
173	D₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	4	1	no	—
182	C₂×C₄×C₆₇	—	4	1	no	—
183	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
189	C₂×Dic₆₇	—	2	2	no	—
190	C₂×Dic₆₇	—	4	2	no	—
191	D₂₆₈	—	2	4	no	—
192	D₂₆₈	—	2	2	no	—
193	D₂₆₈	#131	1	4	yes	—
194	C₄×D₆₇	—	4	4	no	—
195	C₄×D₆₇	—	4	2	no	—
196	C₄×D₆₇	—	4	2	no	—
197	C₈×C₆₇	—	2	1	no	—
198	C₈×C₆₇	—	2	1	no	—
199	C₈×C₆₇	—	2	1	no	—
200	C₈×C₆₇	—	4	1	no	—
201	C₈×C₆₇	—	4	1	no	—
202	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
203	C₆₇⋊C₈	—	2	4	no	—
204	C₆₇⋊C₈	—	2	2	no	—
205	C₆₇⋊C₈	—	2	2	no	—
206	C₆₇⋊C₈	—	4	2	no	—
207	C₂²×C₆₇	—	2	1	no	—
208	C₂²×C₆₇	—	2	1	no	—
209	C₂²×C₆₇	—	4	1	no	—
210	C₂²×C₆₇	—	4	1	no	—
211	C₂²×C₆₇	—	4	1	no	—
212	C₂²×C₆₇	#443	1	2	yes	—
213	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
214	C₂×D₆₇	—	2	4	no	—
215	C₂×D₆₇	—	4	8	no	—
216	C₂×D₆₇	—	4	2	no	—
217	C₂×D₆₇	—	4	2	no	—
218	C₄×C₆₇	—	2	1	no	—
219	C₄×C₆₇	—	2	1	no	—
220	C₄×C₆₇	—	2	1	no	—
221	C₄×C₆₇	—	2	1	no	—
222	C₄×C₆₇	—	4	1	no	—
223	C₄×C₆₇	—	4	1	no	—
224	C₄×C₆₇	—	4	1	no	—
225	C₄×C₆₇	C₂×#131	1	2	yes	—
226	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
227	C₄×C₆₇	#575	1	1	yes	characteristic
228	Dic₆₇	—	4	2	no	—
229	#17910	—	67	1	no	maximal, Sylow
230	C₂×C₆₇	—	2	2	no	—
231	C₂×C₆₇	—	8	1	no	—
232	C₂×C₆₇	#2929	1	1	yes	characteristic
233	C₂×C₆₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
235	D₆₇	—	4	4	no	—
236	#3105	—	67	1	no	—
237	#2966	D₆₇	1	1	yes	characteristic, complemented
238	C₂×C₈×D₄	—	67	1	no	—
239	#1586	—	67	2	no	—
240	#1585	—	67	2	no	—
241	C₈×SD₁₆	—	67	2	no	—
242	C₈×D₈	—	67	2	no	—
243	#1573	—	67	1	no	—
244	#1490	—	67	1	no	—
245	#1483	—	67	1	no	—
246	C₈×M₁₆	—	67	1	no	—
247	C₆₇	#17910	1	1	yes	characteristic, complemented, Sylow
248	C₂³×C₈	—	134	1	no	—
249	#529	C₂×D₆₇	1	1	yes	characteristic
250	C₂×C₄×D₄	—	67	1	no	—
251	#484	—	67	1	no	—
252	#481	—	67	1	no	—
253	#478	—	67	2	no	—
254	#463	—	67	2	no	—
255	#450	—	67	1	no	—
256	#447	—	67	1	no	—
257	#446	C₂×D₆₇	1	1	yes	characteristic, complemented
258	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic, complemented
259	C₄×SD₁₆	—	67	2	no	—
260	C₄×D₈	—	67	2	no	—
261	#436	C₂×D₆₇	1	1	yes	characteristic, complemented
262	#435	—	2	1	no	—
263	C₈×D₄	—	2	1	no	—
264	C₈×D₄	—	67	2	no	—
265	C₈×D₄	—	67	2	no	—
266	C₈×D₄	—	134	1	no	—
267	C₈×D₄	—	134	1	no	—
268	C₈×D₄	C₂×D₆₇	1	1	yes	characteristic, complemented
269	#433	—	2	1	no	—
270	#432	C₂×D₆₇	1	1	yes	characteristic
271	#431	—	67	1	no	—
272	C₂×C₄⋊C₈	—	67	1	no	—
273	C₂×C₂²⋊C₈	—	134	1	no	—
274	#405	—	2	1	no	—
275	C₄×M₁₆	—	67	1	no	—
276	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
277	#335	—	67	2	no	—
278	#334	—	67	2	no	—
279	Q₈⋊C₈	—	67	2	no	—
280	D₄⋊C₈	—	67	2	no	—
281	D₄⋊C₈	—	67	2	no	—
282	D₄⋊C₈	—	67	2	no	—
283	#322	—	67	2	no	—
284	C₈²	—	67	2	no	—
285	C₂²×D₄	—	67	1	no	—
286	C₂³×C₄	—	134	1	no	—
287	C₂×SD₁₆	—	67	2	no	—
288	C₂×D₈	—	67	2	no	—
289	C₂×M₁₆	—	2	1	no	—
290	C₂×M₁₆	—	134	1	no	—
291	C₂²×C₈	—	2	1	no	—
292	C₂²×C₈	—	268	4	no	—
293	C₂²×C₈	—	134	1	no	—
294	C₂²×C₈	—	134	2	no	—
295	C₂²×C₈	—	134	2	no	—
296	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
297	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
298	#126	—	2	1	no	—
299	#125	C₄×D₆₇	1	1	yes	characteristic
300	#124	C₄×D₆₇	1	1	yes	characteristic
301	#123	—	2	1	no	—
302	C₂²⋊Q₈	C₄×D₆₇	1	1	yes	characteristic
303	#121	C₄×D₆₇	1	1	yes	characteristic
304	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
305	C₄×D₄	—	2	1	no	—
306	C₄×D₄	—	67	2	no	—
307	C₄×D₄	—	67	2	no	—
308	C₄×D₄	—	134	1	no	—
309	C₄×D₄	—	134	1	no	—
310	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
311	#117	—	2	1	no	—
312	C₂×C₄⋊C₄	—	67	1	no	—
313	C₂×C₂²⋊C₄	—	134	1	no	—
314	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
315	#107	—	67	2	no	—
316	#106	—	67	2	no	—
317	C₄⋊C₈	—	2	1	no	—
318	C₄⋊C₈	—	2	1	no	—
319	C₄⋊C₈	—	67	2	no	—
320	C₄⋊C₈	—	67	2	no	—
321	C₄⋊C₈	—	134	1	no	—
322	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
323	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
324	#103	—	67	2	no	—
325	#102	—	67	2	no	—
326	#102	—	67	2	no	—
327	#102	—	67	2	no	—
328	C₂²⋊C₈	—	2	1	no	—
329	C₂²⋊C₈	—	2	1	no	—
330	C₂²⋊C₈	—	2	1	no	—
331	C₂²⋊C₈	—	67	1	no	—
332	C₂²⋊C₈	—	67	1	no	—
333	C₂²⋊C₈	—	134	2	no	—
334	C₂²⋊C₈	—	134	2	no	—
335	#97	—	2	1	no	—
336	#97	—	2	1	no	—
337	#97	—	134	1	no	—
338	C₄×C₈	—	2	1	no	—
339	C₄×C₈	—	67	2	no	—
340	C₄×C₈	—	67	2	no	—
341	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
342	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
343	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
344	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
345	C₂⁴	—	134	1	no	—
346	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	67	2	no	—
349	C₂×D₄	—	134	4	no	—
350	C₂×D₄	—	134	1	no	—
351	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	268	2	no	—
354	C₂²×C₄	—	268	2	no	—
355	C₂²×C₄	—	134	1	no	—
356	C₂²×C₄	—	134	2	no	—
357	C₂²×C₄	—	134	2	no	—
358	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
360	C₂²×C₄	#11761628	1	1	yes	characteristic
361	SD₁₆	—	134	4	no	—
362	D₈	—	134	4	no	—
363	M₁₆	—	4	2	no	—
364	M₁₆	—	134	4	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	4	1	no	—
369	C₂×C₈	—	4	2	no	—
370	C₂×C₈	—	268	4	no	—
371	C₂×C₈	—	268	2	no	—
372	C₂×C₈	—	268	4	no	—
373	C₂×C₈	—	67	2	no	—
374	C₂×C₈	—	67	2	no	—
375	C₂×C₈	—	134	2	no	—
376	C₂×C₈	—	134	2	no	—
377	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
378	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
379	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
380	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	67	2	no	—
386	C₄⋊C₄	—	134	1	no	—
387	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
388	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
389	C₂²⋊C₄	—	2	1	no	—
390	C₂²⋊C₄	—	2	1	no	—
391	C₂²⋊C₄	—	2	1	no	—
392	C₂²⋊C₄	—	2	1	no	—
393	C₂²⋊C₄	—	2	1	no	—
394	C₂²⋊C₄	—	134	2	no	—
395	C₂²⋊C₄	—	134	2	no	—
396	C₄²	—	2	1	no	—
397	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
398	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
399	C₄²	#11761628	1	1	yes	characteristic
400	C₄²	C₂³×D₆₇	1	1	yes	characteristic
401	C₂³	—	2	1	no	—
402	C₂³	—	268	2	no	—
403	C₂³	—	268	2	no	—
404	C₂³	—	134	4	no	—
405	C₂³	—	134	2	no	—
406	C₂³	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
407	Q₈	—	2	2	no	—
408	D₄	—	2	2	no	—
409	D₄	—	4	2	no	—
410	D₄	—	67	4	no	—
411	D₄	—	134	4	no	—
412	D₄	—	134	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	268	4	no	—
423	C₂×C₄	—	268	2	no	—
424	C₂×C₄	—	268	2	no	—
425	C₂×C₄	—	268	2	no	—
426	C₂×C₄	—	134	2	no	—
427	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
428	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
429	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
430	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
431	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
432	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
433	C₈	—	2	1	no	—
434	C₈	—	2	1	no	—
435	C₈	—	2	1	no	—
436	C₈	—	4	1	no	—
437	C₈	—	4	1	no	—
438	C₈	—	268	2	no	—
439	C₈	—	134	4	no	—
440	C₈	—	134	2	no	—
441	C₈	—	134	2	no	—
442	C₈	#14524068	1	2	yes	—
443	C₂²	—	2	1	no	—
444	C₂²	—	2	1	no	—
445	C₂²	—	4	1	no	—
446	C₂²	—	4	1	no	—
447	C₂²	—	4	1	no	—
448	C₂²	—	268	8	no	—
449	C₂²	—	268	2	no	—
450	C₂²	—	268	2	no	—
451	C₂²	—	134	4	no	—
452	C₂²	#16863511	1	2	yes	—
453	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
454	C₄	—	2	1	no	—
455	C₄	—	2	1	no	—
456	C₄	—	2	1	no	—
457	C₄	—	2	1	no	—
458	C₄	—	4	1	no	—
459	C₄	—	4	1	no	—
460	C₄	—	4	1	no	—
461	C₄	—	268	2	no	—
462	C₄	#131×D₆₇	1	2	yes	—
463	C₄	C₂×#14524068	1	1	yes	characteristic
464	C₄	#16864194	1	1	yes	characteristic
465	C₂	—	2	2	no	—
466	C₂	—	8	1	no	—
467	C₂	—	268	4	no	—
468	C₂	#22568901	1	1	yes	characteristic
469	C₂	C₄×#14524068	1	1	yes	characteristic
470	C₂	C₂×#16863511	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#2966	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863529	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#2966	D₆₇
hypocenter	C₆₇	#17910
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2966

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. C₈×#11761615	C₂	semidirect
6. C₈×#11761613	C₂	semidirect
7. #22558995	C₂	semidirect
8. #22558651	C₂	semidirect
9. #22558622	C₂	semidirect
10. C₈×#11761612	C₂	semidirect
11. C₆₇×#2966	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. C₈×C₆₇⋊C₈	C₂²	semidirect
39. #446×C₆₇	C₂²	semidirect
40. C₈×Q₈×C₆₇	C₂²	semidirect
41. #436×C₆₇	C₂²	semidirect
44. C₈×D₄×C₆₇	C₂²	semidirect
237. #2966	D₆₇	semidirect
247. C₆₇	#17910	semidirect
257. #446	C₂×D₆₇	semidirect
258. C₈×Q₈	C₂×D₆₇	semidirect
261. #436	C₂×D₆₇	semidirect
268. C₈×D₄	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22570252	2
C₂×C₈×D₂₆₈	2
#22559071	2
#22559064	2
C₈×#11761615	2
C₈×#11761613	2
#22558995	2
#22558651	2
#22558622	2
C₈×#11761612	2
C₆₇×#2966	2
#17910	67