Subgroups — #31011319

Order 17152.

490 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570723	C₂	1	1	yes	maximal, characteristic
2	#22562489	C₂	1	2	yes	maximal, complemented
3	#22562485	C₂	1	4	yes	maximal, complemented
4	#22562482	C₂	1	2	yes	maximal, complemented
5	#22561991	C₂	1	4	yes	maximal
6	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic
8	#16863953	—	2	1	no	—
9	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	#16863801	C₂²	1	2	yes	—
11	#16863798	—	2	1	no	—
12	#16863798	C₂²	1	2	yes	—
13	#16863794	—	2	1	no	—
14	#16863434	—	2	1	no	—
15	#16863434	C₂²	1	2	yes	—
16	#16863223	C₂²	1	8	yes	—
17	#16863221	C₂²	1	8	yes	—
18	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
20	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
21	#533×C₆₇	—	2	1	no	—
22	#500×C₆₇	C₂²	1	2	yes	—
23	#499×C₆₇	—	2	1	no	—
24	#499×C₆₇	C₂²	1	2	yes	—
25	#498×C₆₇	—	2	1	no	—
26	#427×C₆₇	—	2	1	no	—
27	#427×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	#14524043	—	2	2	no	—
30	#14524035	—	2	2	no	—
31	#14524022	—	4	2	no	—
32	C₄×Dic₁₃₄	—	4	2	no	—
33	#14524002	—	2	2	no	—
34	C₂×C₆₇⋊C₁₆	Q₈	1	8	yes	complemented
35	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
36	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
37	#14523977	—	2	2	no	—
38	#14523977	C₂³	1	4	yes	—
39	#14523975	—	2	2	no	—
40	#14523975	—	2	2	no	—
41	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₆₇	—	2	2	no	—
44	#125×C₆₇	—	2	2	no	—
45	C₂²⋊Q₈×C₆₇	—	4	2	no	—
46	C₄×Q₈×C₆₇	—	4	2	no	—
47	#117×C₆₇	—	2	2	no	—
48	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
49	#107×C₆₇	—	2	2	no	—
50	#107×C₆₇	C₂³	1	4	yes	—
51	#106×C₆₇	—	2	2	no	—
52	#106×C₆₇	—	2	2	no	—
53	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
55	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×Dic₁₃₄	—	4	2	no	—
58	C₄⋊Dic₆₇	—	2	4	no	—
59	C₄⋊Dic₆₇	—	4	2	no	—
60	Dic₆₇⋊C₄	—	4	4	no	—
61	C₂²⋊Dic₆₇	—	4	2	no	—
62	C₄×Dic₆₇	—	4	2	no	—
63	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	—
64	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	—
65	C₂×Q₈×C₆₇	—	4	2	no	—
66	C₂²×C₄×C₆₇	D₈	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
69	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₆₇	—	2	4	no	—
75	C₄⋊C₄×C₆₇	—	4	4	no	—
76	C₄⋊C₄×C₆₇	—	4	2	no	—
77	C₂²⋊C₄×C₆₇	—	4	2	no	—
78	C₄²×C₆₇	—	4	2	no	—
79	C₄²×C₆₇	D₈	1	2	yes	—
80	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
83	Q₈×C₆₇	—	8	4	no	—
84	C₂×C₄×C₆₇	—	2	4	no	—
85	C₂×C₄×C₆₇	—	4	4	no	—
86	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
87	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
91	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₆₇	—	4	4	no	—
93	Dic₁₃₄	—	8	4	no	—
94	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
95	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
96	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
97	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
98	C₂²×C₆₇	—	2	2	no	—
99	C₂²×C₆₇	#500	1	2	yes	—
100	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
101	C₄×C₆₇	—	2	2	no	—
102	C₄×C₆₇	—	8	4	no	—
103	C₄×C₆₇	#500	1	2	yes	—
104	C₄×C₆₇	#508	1	4	yes	—
105	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
106	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
107	Dic₆₇	—	8	4	no	—
108	#33726	—	67	1	no	maximal, Sylow
109	C₂×C₆₇	—	2	2	no	—
110	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
111	C₂×C₆₇	C₂×#508	1	2	yes	—
112	#3160	—	67	1	no	—
113	#3160	D₆₇	1	1	yes	characteristic
114	#2254	—	67	2	no	—
115	#2253	—	67	4	no	—
116	#2252	—	67	2	no	—
117	#2158	—	67	4	no	—
118	C₂×C₄×C₁₆	—	67	1	no	—
119	C₆₇	#33726	1	1	yes	characteristic, complemented, Sylow
120	#533	—	2	1	no	—
121	#533	—	134	1	no	—
122	C₂²×C₁₆	—	67	2	no	—
123	#500	—	67	2	no	—
124	#500	C₂×D₆₇	1	2	yes	—
125	#499	—	2	1	no	—
126	#499	—	67	2	no	—
127	#499	—	134	1	no	—
128	#499	C₂×D₆₇	1	2	yes	—
129	#498	—	2	1	no	—
130	#498	—	134	1	no	—
131	#427	—	2	1	no	—
132	#427	—	67	2	no	—
133	#427	—	134	1	no	—
134	#427	C₂×D₆₇	1	2	yes	—
135	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
136	#367	—	67	8	no	—
137	#366	—	67	8	no	—
138	C₄×C₁₆	—	67	1	no	—
139	C₄×C₁₆	—	67	2	no	—
140	C₄×C₁₆	—	67	1	no	—
141	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	134	2	no	—
145	#125	—	2	2	no	—
146	#125	—	134	2	no	—
147	C₂²⋊Q₈	—	4	2	no	—
148	C₂²⋊Q₈	—	268	2	no	—
149	C₄×Q₈	—	4	2	no	—
150	C₄×Q₈	—	268	2	no	—
151	#117	—	2	2	no	—
152	#117	—	134	2	no	—
153	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
154	C₂×C₁₆	—	67	8	no	—
155	C₂×C₁₆	—	67	2	no	—
156	C₂×C₁₆	—	67	2	no	—
157	#107	—	2	2	no	—
158	#107	—	67	4	no	—
159	#107	—	134	2	no	—
160	#107	C₂²×D₆₇	1	4	yes	—
161	#106	—	2	2	no	—
162	#106	—	2	2	no	—
163	#106	—	134	2	no	—
164	#106	—	134	2	no	—
165	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
166	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	268	2	no	—
171	C₂²×C₄	#11761613	1	2	yes	—
172	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
173	C₂×C₈	#11761618	1	4	yes	—
174	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
177	C₂×C₈	Q₈×D₆₇	1	4	yes	—
178	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	268	4	no	—
183	C₄⋊C₄	—	268	2	no	—
184	C₄⋊C₄	—	134	4	no	—
185	C₂²⋊C₄	—	4	2	no	—
186	C₂²⋊C₄	—	268	2	no	—
187	C₄²	—	4	2	no	—
188	C₄²	—	268	2	no	—
189	C₄²	#11761613	1	2	yes	—
190	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
191	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
192	C₁₆	—	67	4	no	—
193	C₁₆	—	67	4	no	—
194	C₂³	C₂×#11761613	1	1	yes	characteristic
195	Q₈	—	8	4	no	—
196	Q₈	—	536	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	268	4	no	—
200	C₂×C₄	#14524041	1	2	yes	—
201	C₂×C₄	C₂×#11761613	1	2	yes	—
202	C₂×C₄	C₂×#11761613	1	2	yes	—
203	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
204	C₂×C₄	C₂×#11761618	1	2	yes	—
205	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
206	C₈	#14524041	1	2	yes	—
207	C₈	C₂×#11761618	1	2	yes	—
208	C₈	C₂×#11761618	1	2	yes	—
209	C₈	C₂×Q₈×D₆₇	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	#16863799	1	2	yes	—
212	C₂²	C₂²×#11761613	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	8	4	no	—
215	C₄	—	536	4	no	—
216	C₄	#16863799	1	2	yes	—
217	C₄	#16863826	1	4	yes	—
218	C₄	C₂×#14524041	1	1	yes	characteristic
219	C₄	C₂²×#11761618	1	1	yes	characteristic
220	C₂	—	2	2	no	—
221	C₂	C₂×#16863799	1	1	yes	characteristic
222	C₂	C₂×#16863826	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#3160	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	#3160	D₆₇
hypocenter	C₆₇	#33726
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #22562489	C₂	semidirect
3. #22562485	C₂	semidirect
4. #22562482	C₂	semidirect
34. C₂×C₆₇⋊C₁₆	Q₈	semidirect
119. C₆₇	#33726	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
#22562489	2
#22562485	2
#22562482	2
#22561991	2
C₂×C₄×C₆₇⋊C₁₆	2
C₆₇×#3160	2
#33726	67