Subgroups — #30826511

Order 17152.

560 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569304	C₂	1	2	yes	maximal
2	#22559118	C₂	1	2	yes	maximal, complemented
3	#22559106	C₂	1	2	yes	maximal, complemented
4	#22559028	C₂	1	2	yes	maximal, complemented
5	Q₁₆×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
6	#22558972	C₂	1	1	yes	maximal, characteristic
7	#22558682	C₂	1	2	yes	maximal
8	#22558492	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#3170	C₂	1	1	yes	maximal, characteristic
10	#16863527	—	2	4	no	—
11	#16863527	C₂²	1	4	yes	—
12	Q₈×C₆₇⋊C₈	C₂²	1	4	yes	—
13	#16863457	—	2	4	no	—
14	#16863453	C₂²	1	2	yes	—
15	#16863449	C₂²	1	1	yes	characteristic
16	#16863449	C₂²	1	1	yes	characteristic
17	#16863149	C₂²	1	2	yes	—
18	#16863147	C₂²	1	2	yes	—
19	Q₈⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
20	Q₈⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
21	#16863117	C₂²	1	1	yes	characteristic
22	#16863117	C₂²	1	1	yes	characteristic
23	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
24	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
25	#533×C₆₇	C₂²	1	2	yes	—
26	#500×C₆₇	C₄	1	1	yes	characteristic
27	#494×C₆₇	C₄	1	1	yes	characteristic
28	#487×C₆₇	C₄	1	2	yes	—
29	#467×C₆₇	C₄	1	2	yes	—
30	#467×C₆₇	C₄	1	2	yes	—
31	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
32	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
37	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
38	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
39	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
40	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
41	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
42	#14523954	—	2	4	no	—
43	#14523954	—	2	2	no	—
44	C₄×C₆₇⋊C₈	—	2	4	no	—
45	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
49	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
54	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
55	#125×C₆₇	—	2	2	no	—
56	C₂²⋊Q₈×C₆₇	—	2	2	no	—
57	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
58	C₄×Q₈×C₆₇	—	2	2	no	—
59	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
60	#117×C₆₇	—	2	2	no	—
61	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
62	#107×C₆₇	C₂×C₄	1	2	yes	—
63	#107×C₆₇	C₂×C₄	1	2	yes	—
64	#103×C₆₇	C₂×C₄	1	4	yes	—
65	#103×C₆₇	C₂×C₄	1	4	yes	—
66	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₆₇⋊C₈	—	2	4	no	—
71	C₂×C₆₇⋊C₈	—	2	1	no	—
72	C₂×C₆₇⋊C₈	—	2	1	no	—
73	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
74	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
75	C₂×Q₈×C₆₇	—	2	2	no	—
76	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
80	Q₁₆×C₆₇	—	2	4	no	—
81	Q₁₆×C₆₇	—	2	4	no	—
82	C₂×C₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₄⋊C₄×C₆₇	—	2	2	no	—
91	C₄⋊C₄×C₆₇	—	2	4	no	—
92	C₄⋊C₄×C₆₇	—	2	4	no	—
93	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
94	C₂²⋊C₄×C₆₇	—	2	4	no	—
95	C₄²×C₆₇	—	2	4	no	—
96	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₆₇	#41	1	1	yes	characteristic
99	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
101	Q₈×C₆₇	—	2	4	no	—
102	Q₈×C₆₇	—	2	4	no	—
103	Q₈×C₆₇	—	8	2	no	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	4	1	no	—
110	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
116	C₈×C₆₇	—	2	1	no	—
117	C₈×C₆₇	—	2	1	no	—
118	C₈×C₆₇	#131	1	2	yes	—
119	C₈×C₆₇	#131	1	2	yes	—
120	C₆₇⋊C₈	—	2	2	no	—
121	C₆₇⋊C₈	—	2	2	no	—
122	C₆₇⋊C₈	—	4	1	no	—
123	C₆₇⋊C₈	—	4	1	no	—
124	C₆₇⋊C₈	—	4	4	no	—
125	C₂²×C₆₇	—	2	1	no	—
126	C₂²×C₆₇	—	2	1	no	—
127	C₂²×C₆₇	—	2	1	no	—
128	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
129	C₄×C₆₇	—	2	1	no	—
130	C₄×C₆₇	—	2	1	no	—
131	C₄×C₆₇	—	2	1	no	—
132	C₄×C₆₇	—	2	1	no	—
133	C₄×C₆₇	—	4	4	no	—
134	C₄×C₆₇	—	4	4	no	—
135	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
136	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
137	C₄×C₆₇	#578	1	2	yes	—
138	#17972	—	67	1	no	maximal, Sylow
139	C₂×C₆₇	—	4	1	no	—
140	C₂×C₆₇	#2931	1	1	yes	characteristic
141	C₂×C₆₇	#2950	1	1	yes	characteristic
142	C₂×C₆₇	#2957	1	1	yes	characteristic
143	#3170	D₆₇	1	1	yes	characteristic
144	#2985	—	67	2	no	—
145	#1593	—	67	2	no	—
146	#1591	—	67	2	no	—
147	#1579	—	67	2	no	—
148	C₈×Q₁₆	—	67	2	no	—
149	#1568	—	67	1	no	—
150	#1497	—	67	2	no	—
151	#1450	—	67	1	no	—
152	C₆₇	#17972	1	1	yes	characteristic, complemented, Sylow
153	#533	C₂×D₆₇	1	2	yes	—
154	#500	Dic₆₇	1	1	yes	characteristic
155	#494	Dic₆₇	1	1	yes	characteristic
156	#487	Dic₆₇	1	2	yes	—
157	#467	Dic₆₇	1	2	yes	—
158	#467	Dic₆₇	1	2	yes	—
159	#446	—	67	4	no	—
160	#446	—	134	4	no	—
161	C₈×Q₈	—	67	4	no	—
162	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
163	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
164	#433	—	134	4	no	—
165	#432	—	67	2	no	—
166	#431	—	67	1	no	—
167	#431	—	67	1	no	—
168	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
169	#335	—	67	2	no	—
170	#334	—	67	2	no	—
171	Q₈⋊C₈	—	67	4	no	—
172	Q₈⋊C₈	—	67	4	no	—
173	#322	—	67	1	no	—
174	#322	—	67	1	no	—
175	C₈²	—	67	1	no	—
176	C₈²	—	67	1	no	—
177	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
178	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
179	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
180	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
181	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
182	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
183	#125	—	2	2	no	—
184	C₂²⋊Q₈	—	2	2	no	—
185	C₂²⋊Q₈	C₂×Dic₆₇	1	4	yes	—
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	C₂²×D₆₇	1	4	yes	—
188	#117	—	2	2	no	—
189	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
190	#107	C₂×Dic₆₇	1	2	yes	—
191	#107	C₂×Dic₆₇	1	2	yes	—
192	C₄⋊C₈	—	67	2	no	—
193	C₄⋊C₈	—	67	4	no	—
194	C₄⋊C₈	—	134	4	no	—
195	C₄⋊C₈	—	134	4	no	—
196	C₄⋊C₈	—	134	1	no	—
197	#103	C₂×Dic₆₇	1	4	yes	—
198	#103	C₂×Dic₆₇	1	4	yes	—
199	C₂²⋊C₈	—	67	2	no	—
200	C₂²⋊C₈	—	134	2	no	—
201	C₂²⋊C₈	—	134	1	no	—
202	#97	—	134	4	no	—
203	#97	—	134	2	no	—
204	C₄×C₈	—	67	1	no	—
205	C₄×C₈	—	67	1	no	—
206	C₄×C₈	—	67	1	no	—
207	C₄×C₈	—	67	1	no	—
208	C₄×C₈	—	134	4	no	—
209	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
210	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
211	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
215	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
217	C₂²×C₄	#11761627	1	1	yes	characteristic
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	67	2	no	—
223	C₂×C₈	—	67	2	no	—
224	C₂×C₈	—	134	4	no	—
225	C₂×C₈	—	134	1	no	—
226	C₂×C₈	—	134	1	no	—
227	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
228	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
229	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
230	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
240	C₄²	D₄×D₆₇	1	1	yes	characteristic
241	C₄²	#11761627	1	1	yes	characteristic
242	C₄²	C₂³×D₆₇	1	1	yes	characteristic
243	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
244	Q₈	—	2	4	no	—
245	Q₈	—	2	4	no	—
246	Q₈	—	8	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
254	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
255	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
256	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
258	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	268	1	no	—
262	C₈	—	268	1	no	—
263	C₈	—	268	4	no	—
264	C₈	—	134	2	no	—
265	C₈	—	134	2	no	—
266	C₈	#14524050	1	2	yes	—
267	C₈	#14524050	1	2	yes	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	4	no	—
278	C₄	C₂×#14524050	1	1	yes	characteristic
279	C₄	C₂×#14524050	1	1	yes	characteristic
280	C₄	#16864209	1	2	yes	—
281	C₂	—	4	1	no	—
282	C₂	#22568921	1	1	yes	characteristic
283	C₂	#22569077	1	1	yes	characteristic
284	C₂	#22569126	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#3170	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#3170	D₆₇
hypocenter	C₆₇	#17972
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #22559118	C₂	semidirect
3. #22559106	C₂	semidirect
4. #22559028	C₂	semidirect
5. Q₁₆×C₆₇⋊C₈	C₂	semidirect
152. C₆₇	#17972	semidirect

Maximal subgroups

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

subgroup	index
#22569304	2
#22559118	2
#22559106	2
#22559028	2
Q₁₆×C₆₇⋊C₈	2
#22558972	2
#22558682	2
#22558492	2
C₆₇×#3170	2
#17972	67