Subgroups — #30825270

Order 17152.

754 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569320	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569233	C₂	1	1	yes	maximal, characteristic, complemented
3	#22559033	C₂	1	2	yes	maximal, complemented
4	SD₁₆⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal, complemented
5	SD₁₆×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
6	D₈×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
7	#22558961	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558591	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×AGL(1,ℤ/8)×C₆₇	C₂	1	1	yes	maximal, characteristic
12	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	complemented
13	D₄×C₆₇⋊C₈	—	2	1	no	—
14	D₄×C₆₇⋊C₈	—	2	1	no	—
15	D₄×C₆₇⋊C₈	—	2	2	no	—
16	D₄×C₆₇⋊C₈	C₂²	1	2	yes	—
17	D₄×C₆₇⋊C₈	C₂²	1	2	yes	—
18	D₄×C₆₇⋊C₈	C₂²	1	2	yes	complemented
19	#16863449	—	2	1	no	—
20	#16863449	C₂²	1	1	yes	characteristic
21	#16863449	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
23	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
26	#16863149	C₂²	1	2	yes	complemented
27	#16863147	C₂²	1	2	yes	complemented
28	Q₈⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
29	Q₈⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
30	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
31	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
32	D₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
33	D₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
34	#16863117	C₂²	1	2	yes	complemented
35	C₈×C₆₇⋊C₈	C₂²	1	2	yes	complemented
36	C₂×AGL(1,ℤ/8)×C₆₇	C₄	1	1	yes	characteristic
37	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
39	#442×C₆₇	C₄	1	2	yes	—
40	#440×C₆₇	C₄	1	2	yes	—
41	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
42	C₄×D₈×C₆₇	C₂²	1	2	yes	—
43	#428×C₆₇	C₄	1	1	yes	characteristic
44	#418×C₆₇	C₄	1	1	yes	characteristic
45	#417×C₆₇	C₄	1	1	yes	characteristic
46	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
47	C₂²×C₆₇⋊C₈	—	2	2	no	—
48	C₂²×C₆₇⋊C₈	—	2	2	no	—
49	C₂²×C₆₇⋊C₈	—	2	2	no	—
50	C₂²×C₆₇⋊C₈	—	2	1	no	—
51	C₂²×C₆₇⋊C₈	—	2	1	no	—
52	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
53	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
54	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
55	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
56	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
57	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
58	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
59	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
60	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
61	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
62	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
63	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
64	C₄×C₆₇⋊C₈	—	2	2	no	—
65	C₄×C₆₇⋊C₈	—	2	1	no	—
66	C₄×C₆₇⋊C₈	—	2	1	no	—
67	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
68	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
70	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂³×C₄×C₆₇	—	2	1	no	—
73	AGL(1,ℤ/8)×C₆₇	C₈	1	8	yes	complemented
74	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
75	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
76	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×D₄×C₆₇	—	2	1	no	—
81	C₄×D₄×C₆₇	—	2	1	no	—
82	C₄×D₄×C₆₇	—	2	1	no	—
83	C₄×D₄×C₆₇	—	2	1	no	—
84	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
85	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₄×D₄×C₆₇	C₂³	1	2	yes	—
87	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
88	#117×C₆₇	—	2	1	no	—
89	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₄²×C₆₇	—	2	1	no	—
93	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
94	#107×C₆₇	C₂×C₄	1	2	yes	—
95	#106×C₆₇	C₂×C₄	1	2	yes	—
96	#103×C₆₇	C₂×C₄	1	2	yes	—
97	#102×C₆₇	C₂×C₄	1	2	yes	—
98	#102×C₆₇	C₂×C₄	1	2	yes	—
99	#102×C₆₇	C₂×C₄	1	2	yes	—
100	#97×C₆₇	C₂×C₄	1	2	yes	—
101	C₄×C₈×C₆₇	C₂³	1	2	yes	—
102	C₂×C₆₇⋊C₈	—	2	1	no	—
103	C₂×C₆₇⋊C₈	—	2	1	no	—
104	C₂×C₆₇⋊C₈	—	2	1	no	—
105	C₂×C₆₇⋊C₈	—	2	2	no	—
106	C₂×C₆₇⋊C₈	—	2	1	no	—
107	C₂×C₆₇⋊C₈	—	4	2	no	—
108	C₂×C₆₇⋊C₈	—	8	2	no	—
109	C₂×C₆₇⋊C₈	—	8	2	no	—
110	C₂×C₆₇⋊C₈	—	8	2	no	—
111	C₂×C₆₇⋊C₈	—	8	1	no	—
112	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
113	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
114	C₂⁴×C₆₇	—	2	1	no	—
115	#41×C₆₇	—	2	2	no	—
116	#41×C₆₇	C₂×C₈	1	4	yes	complemented
117	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×D₄×C₆₇	—	2	1	no	—
119	C₂×D₄×C₆₇	—	2	1	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	C₂×C₈	1	4	yes	complemented
122	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₆₇	—	2	1	no	—
125	C₂²×C₄×C₆₇	—	2	1	no	—
126	C₂²×C₄×C₆₇	—	2	1	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	1	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	8	1	no	—
131	C₂²×C₄×C₆₇	—	8	1	no	—
132	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
135	SD₁₆×C₆₇	C₂×C₈	1	8	yes	complemented
136	D₈×C₆₇	C₂×C₈	1	8	yes	complemented
137	M₁₆×C₆₇	—	2	2	no	—
138	M₁₆×C₆₇	C₂×C₈	1	4	yes	complemented
139	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
140	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
141	C₄⋊C₄×C₆₇	—	2	1	no	—
142	C₄⋊C₄×C₆₇	—	2	1	no	—
143	C₄⋊C₄×C₆₇	—	2	1	no	—
144	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
146	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
147	C₂²⋊C₄×C₆₇	—	2	1	no	—
148	C₂²⋊C₄×C₆₇	—	2	2	no	—
149	C₂²⋊C₄×C₆₇	—	2	1	no	—
150	C₂²⋊C₄×C₆₇	—	2	2	no	—
151	C₂²⋊C₄×C₆₇	—	2	1	no	—
152	C₄²×C₆₇	—	2	1	no	—
153	C₄²×C₆₇	—	2	1	no	—
154	C₄²×C₆₇	—	2	1	no	—
155	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
156	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₆₇	#41	1	1	yes	characteristic
158	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
159	C₂³×C₆₇	—	2	1	no	—
160	C₂³×C₆₇	—	2	2	no	—
161	C₂³×C₆₇	—	2	4	no	—
162	C₂³×C₆₇	—	8	1	no	—
163	C₂³×C₆₇	—	8	1	no	—
164	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
165	Q₈×C₆₇	—	2	1	no	—
166	Q₈×C₆₇	C₂²×C₈	1	2	yes	—
167	D₄×C₆₇	—	2	4	no	—
168	D₄×C₆₇	—	2	2	no	—
169	D₄×C₆₇	—	2	1	no	—
170	D₄×C₆₇	—	2	4	no	—
171	D₄×C₆₇	C₂²×C₈	1	2	yes	—
172	D₄×C₆₇	C₂²×C₈	1	4	yes	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	4	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	4	1	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	—	4	1	no	—
183	C₂×C₄×C₆₇	—	4	1	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	8	1	no	—
186	C₂×C₄×C₆₇	—	8	2	no	—
187	C₂×C₄×C₆₇	—	8	1	no	—
188	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
192	C₂×C₄×C₆₇	#131	1	2	yes	—
193	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
196	C₈×C₆₇	—	2	2	no	—
197	C₈×C₆₇	C₂²×C₈	1	4	yes	—
198	C₆₇⋊C₈	—	2	2	no	—
199	C₆₇⋊C₈	—	2	2	no	—
200	C₆₇⋊C₈	—	2	2	no	—
201	C₆₇⋊C₈	—	4	1	no	—
202	C₆₇⋊C₈	—	4	1	no	—
203	C₆₇⋊C₈	—	4	2	no	—
204	C₆₇⋊C₈	—	4	1	no	—
205	C₂²×C₆₇	—	2	1	no	—
206	C₂²×C₆₇	—	2	1	no	—
207	C₂²×C₆₇	—	2	2	no	—
208	C₂²×C₆₇	—	2	4	no	—
209	C₂²×C₆₇	—	4	2	no	—
210	C₂²×C₆₇	—	4	1	no	—
211	C₂²×C₆₇	—	4	1	no	—
212	C₂²×C₆₇	—	4	2	no	—
213	C₂²×C₆₇	—	8	4	no	—
214	C₂²×C₆₇	C₈×D₄	1	2	yes	—
215	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
216	C₄×C₆₇	—	2	2	no	—
217	C₄×C₆₇	—	2	1	no	—
218	C₄×C₆₇	—	2	1	no	—
219	C₄×C₆₇	—	2	1	no	—
220	C₄×C₆₇	—	4	1	no	—
221	C₄×C₆₇	—	4	2	no	—
222	C₄×C₆₇	—	4	1	no	—
223	C₄×C₆₇	C₈×D₄	1	2	yes	—
224	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
225	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
226	C₄×C₆₇	#576	1	1	yes	characteristic
227	C₄×C₆₇	#577	1	1	yes	characteristic
228	#17852	—	67	1	no	maximal, Sylow
229	C₂×C₆₇	—	2	2	no	—
230	C₂×C₆₇	—	4	2	no	—
231	C₂×C₆₇	—	4	4	no	—
232	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
233	C₂×C₆₇	#2946	1	1	yes	characteristic
234	C₂×C₆₇	#2956	1	1	yes	characteristic
235	#2988	—	67	1	no	—
236	#2972	—	67	1	no	—
237	C₄×AGL(1,ℤ/8)	D₆₇	1	1	yes	characteristic
238	#1580	—	67	2	no	—
239	SD₁₆⋊C₈	—	67	2	no	—
240	C₈×SD₁₆	—	67	2	no	—
241	C₈×D₈	—	67	2	no	—
242	#1565	—	67	1	no	—
243	#1476	—	67	1	no	—
244	C₂×D₄⋊C₈	—	67	1	no	—
245	C₈×M₁₆	—	67	1	no	—
246	C₆₇	#17852	1	1	yes	characteristic, complemented, Sylow
247	C₂×AGL(1,ℤ/8)	Dic₆₇	1	1	yes	characteristic
248	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
249	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
250	C₈×Q₈	—	67	2	no	—
251	#442	Dic₆₇	1	2	yes	—
252	#440	Dic₆₇	1	2	yes	—
253	C₄×SD₁₆	C₂×D₆₇	1	2	yes	—
254	C₄×D₈	C₂×D₆₇	1	2	yes	—
255	C₈×D₄	—	67	2	no	—
256	C₈×D₄	—	67	2	no	—
257	C₈×D₄	—	67	2	no	—
258	C₈×D₄	—	134	2	no	—
259	C₈×D₄	—	134	1	no	—
260	C₈×D₄	—	134	1	no	—
261	#431	—	67	1	no	—
262	#431	—	67	1	no	—
263	#431	—	134	1	no	—
264	#428	Dic₆₇	1	1	yes	characteristic
265	C₂×C₄⋊C₈	—	67	1	no	—
266	C₂×C₄⋊C₈	—	134	1	no	—
267	#418	Dic₆₇	1	1	yes	characteristic
268	#417	Dic₆₇	1	1	yes	characteristic
269	C₂×C₂²⋊C₈	—	134	2	no	—
270	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
271	C₂×C₄×C₈	—	67	1	no	—
272	#335	—	67	2	no	—
273	#334	—	67	2	no	—
274	Q₈⋊C₈	—	67	1	no	—
275	Q₈⋊C₈	—	67	1	no	—
276	D₄⋊C₈	—	67	1	no	—
277	D₄⋊C₈	—	67	2	no	—
278	D₄⋊C₈	—	67	2	no	—
279	D₄⋊C₈	—	67	1	no	—
280	#322	—	67	2	no	—
281	C₈²	—	67	2	no	—
282	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
283	C₂²×D₄	C₂×Dic₆₇	1	1	yes	characteristic
284	C₂³×C₄	—	2	1	no	—
285	AGL(1,ℤ/8)	C₆₇⋊C₈	1	8	yes	complemented
286	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	—
287	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
288	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
289	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
290	C₂²×C₈	—	134	1	no	—
291	C₂²×C₈	—	134	2	no	—
292	C₂²×C₈	—	134	2	no	—
293	C₂²×C₈	—	134	2	no	—
294	C₂²×C₈	—	134	1	no	—
295	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
296	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
297	C₄×D₄	—	2	1	no	—
298	C₄×D₄	—	2	1	no	—
299	C₄×D₄	—	2	1	no	—
300	C₄×D₄	—	2	1	no	—
301	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
302	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
303	C₄×D₄	C₂²×D₆₇	1	2	yes	—
304	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
305	#117	—	2	1	no	—
306	#117	C₂×Dic₆₇	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
308	C₂×C₂²⋊C₄	—	2	1	no	—
309	C₂×C₄²	—	2	1	no	—
310	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
311	#107	C₂×Dic₆₇	1	2	yes	—
312	#106	C₂×Dic₆₇	1	2	yes	—
313	C₄⋊C₈	—	67	1	no	—
314	C₄⋊C₈	—	67	1	no	—
315	C₄⋊C₈	—	67	2	no	—
316	C₄⋊C₈	—	67	1	no	—
317	C₄⋊C₈	—	67	1	no	—
318	C₄⋊C₈	—	134	2	no	—
319	C₄⋊C₈	—	134	1	no	—
320	#103	C₂×Dic₆₇	1	2	yes	—
321	#102	C₂×Dic₆₇	1	2	yes	—
322	#102	C₂×Dic₆₇	1	2	yes	—
323	#102	C₂×Dic₆₇	1	2	yes	—
324	C₂²⋊C₈	—	67	2	no	—
325	C₂²⋊C₈	—	134	2	no	—
326	C₂²⋊C₈	—	134	2	no	—
327	C₂²⋊C₈	—	134	2	no	—
328	C₂²⋊C₈	—	134	1	no	—
329	#97	C₂×Dic₆₇	1	2	yes	—
330	C₄×C₈	—	67	2	no	—
331	C₄×C₈	—	67	1	no	—
332	C₄×C₈	—	67	1	no	—
333	C₄×C₈	—	134	2	no	—
334	C₄×C₈	—	134	1	no	—
335	C₄×C₈	—	134	1	no	—
336	C₄×C₈	C₂²×D₆₇	1	2	yes	—
337	C₂⁴	—	2	1	no	—
338	#41	—	2	2	no	—
339	#41	C₂×C₆₇⋊C₈	1	4	yes	complemented
340	C₂×Q₈	C₂²×Dic₆₇	1	1	yes	characteristic
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	C₂×C₆₇⋊C₈	1	4	yes	complemented
345	C₂×D₄	C₂²×Dic₆₇	1	1	yes	characteristic
346	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	8	1	no	—
354	C₂²×C₄	—	8	1	no	—
355	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
356	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
357	C₂²×C₄	#11761627	1	1	yes	characteristic
358	SD₁₆	C₂×C₆₇⋊C₈	1	8	yes	complemented
359	D₈	C₂×C₆₇⋊C₈	1	8	yes	complemented
360	M₁₆	—	2	2	no	—
361	M₁₆	C₂×C₆₇⋊C₈	1	4	yes	complemented
362	C₂×C₈	—	268	2	no	—
363	C₂×C₈	—	536	1	no	—
364	C₂×C₈	—	536	2	no	—
365	C₂×C₈	—	536	2	no	—
366	C₂×C₈	—	536	2	no	—
367	C₂×C₈	—	67	2	no	—
368	C₂×C₈	—	67	2	no	—
369	C₂×C₈	—	134	1	no	—
370	C₂×C₈	—	134	1	no	—
371	C₂×C₈	—	134	2	no	—
372	C₂×C₈	—	134	1	no	—
373	C₂×C₈	—	134	1	no	—
374	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
375	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
380	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
381	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
382	C₂²⋊C₄	—	2	1	no	—
383	C₂²⋊C₄	—	2	2	no	—
384	C₂²⋊C₄	—	2	1	no	—
385	C₂²⋊C₄	—	2	2	no	—
386	C₂²⋊C₄	—	2	1	no	—
387	C₄²	—	2	1	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
391	C₄²	D₄×D₆₇	1	1	yes	characteristic
392	C₄²	#11761627	1	1	yes	characteristic
393	C₄²	C₂³×D₆₇	1	1	yes	characteristic
394	C₂³	—	2	1	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	2	4	no	—
397	C₂³	—	8	1	no	—
398	C₂³	—	8	1	no	—
399	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
400	Q₈	—	2	1	no	—
401	Q₈	C₂²×C₆₇⋊C₈	1	2	yes	—
402	D₄	—	2	4	no	—
403	D₄	—	2	2	no	—
404	D₄	—	2	1	no	—
405	D₄	—	2	4	no	—
406	D₄	C₂²×C₆₇⋊C₈	1	2	yes	—
407	D₄	C₂²×C₆₇⋊C₈	1	4	yes	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	8	2	no	—
421	C₂×C₄	—	8	1	no	—
422	C₂×C₄	—	8	1	no	—
423	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
424	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
425	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
426	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
427	C₂×C₄	#14524050	1	2	yes	—
428	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
429	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
430	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
431	C₈	—	2	2	no	—
432	C₈	—	268	1	no	—
433	C₈	—	268	2	no	—
434	C₈	—	268	1	no	—
435	C₈	—	268	1	no	—
436	C₈	—	134	2	no	—
437	C₈	—	134	2	no	—
438	C₈	—	134	2	no	—
439	C₈	C₂²×C₆₇⋊C₈	1	4	yes	—
440	C₂²	—	2	1	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	2	2	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	4	2	no	—
445	C₂²	—	4	1	no	—
446	C₂²	—	4	1	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	8	4	no	—
449	C₂²	D₄×C₆₇⋊C₈	1	2	yes	—
450	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
451	C₄	—	2	1	no	—
452	C₄	—	2	1	no	—
453	C₄	—	2	1	no	—
454	C₄	—	2	2	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	1	no	—
457	C₄	—	4	1	no	—
458	C₄	D₄×C₆₇⋊C₈	1	2	yes	—
459	C₄	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
460	C₄	C₂×#14524050	1	1	yes	characteristic
461	C₄	#16864200	1	1	yes	characteristic
462	C₄	#16864206	1	1	yes	characteristic
463	C₂	—	2	2	no	—
464	C₂	—	4	4	no	—
465	C₂	—	4	2	no	—
466	C₂	C₂×D₄×C₆₇⋊C₈	1	1	yes	characteristic
467	C₂	#22569042	1	1	yes	characteristic
468	C₂	#22569119	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂³×C₈
Frattini	C₄²	C₂³×D₆₇
Fitting	C₄×AGL(1,ℤ/8)×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×AGL(1,ℤ/8)×C₆₇	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₄×AGL(1,ℤ/8)	D₆₇
hypocenter	C₆₇	#17852
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #22569320	C₂	semidirect
2. #22569233	C₂	semidirect
3. #22559033	C₂	semidirect
4. SD₁₆⋊(C₆₇⋊C₈)	C₂	semidirect
5. SD₁₆×C₆₇⋊C₈	C₂	semidirect
6. D₈×C₆₇⋊C₈	C₂	semidirect
7. #22558961	C₂	semidirect
8. #22558591	C₂	semidirect
9. C₂×D₄⋊(C₆₇⋊C₈)	C₂	semidirect
10. M₁₆×C₆₇⋊C₈	C₂	semidirect
12. Q₈×C₆₇⋊C₈	C₂²	semidirect
18. D₄×C₆₇⋊C₈	C₂²	semidirect
26. #16863149	C₂²	semidirect
27. #16863147	C₂²	semidirect
28. Q₈⋊(C₆₇⋊C₈)	C₂²	semidirect
29. Q₈⋊(C₆₇⋊C₈)	C₂²	semidirect
32. D₄⋊(C₆₇⋊C₈)	C₂²	semidirect
33. D₄⋊(C₆₇⋊C₈)	C₂²	semidirect
34. #16863117	C₂²	semidirect
35. C₈×C₆₇⋊C₈	C₂²	semidirect
73. AGL(1,ℤ/8)×C₆₇	C₈	semidirect
116. #41×C₆₇	C₂×C₈	semidirect
121. C₂×D₄×C₆₇	C₂×C₈	semidirect
135. SD₁₆×C₆₇	C₂×C₈	semidirect
136. D₈×C₆₇	C₂×C₈	semidirect
138. M₁₆×C₆₇	C₂×C₈	semidirect
246. C₆₇	#17852	semidirect
285. AGL(1,ℤ/8)	C₆₇⋊C₈	semidirect
339. #41	C₂×C₆₇⋊C₈	semidirect
344. C₂×D₄	C₂×C₆₇⋊C₈	semidirect
358. SD₁₆	C₂×C₆₇⋊C₈	semidirect
359. D₈	C₂×C₆₇⋊C₈	semidirect
361. M₁₆	C₂×C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22569320	2
#22569233	2
#22559033	2
SD₁₆⋊(C₆₇⋊C₈)	2
SD₁₆×C₆₇⋊C₈	2
D₈×C₆₇⋊C₈	2
#22558961	2
#22558591	2
C₂×D₄⋊(C₆₇⋊C₈)	2
M₁₆×C₆₇⋊C₈	2
C₄×AGL(1,ℤ/8)×C₆₇	2
#17852	67