Subgroups — #31011331

Order 17152.

482 conjugacy classes of subgroups.

The subgroup lattice has 216 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	#22562496	C₂	1	4	yes	maximal, complemented
3	#22562492	C₂	1	4	yes	maximal, complemented
4	#22561991	C₂	1	4	yes	maximal
5	C₂×#16863172	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic
7	#16863953	—	2	1	no	—
8	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	#16863801	C₂²	1	2	yes	—
10	#16863798	—	2	1	no	—
11	#16863798	C₂²	1	2	yes	—
12	#16863794	—	2	1	no	—
13	#16863434	—	2	1	no	—
14	#16863434	C₂²	1	2	yes	—
15	#16863223	C₂²	1	8	yes	—
16	#16863221	C₂²	1	8	yes	—
17	#16863172	C₂²	1	2	yes	—
18	#16863172	C₂²	1	2	yes	—
19	#533×C₆₇	—	2	1	no	—
20	#500×C₆₇	C₂²	1	2	yes	—
21	#499×C₆₇	—	2	1	no	—
22	#499×C₆₇	C₂²	1	2	yes	—
23	#498×C₆₇	—	2	1	no	—
24	#427×C₆₇	—	2	1	no	—
25	#427×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#14524043	—	2	2	no	—
28	#14524035	—	2	2	no	—
29	#14524022	—	4	2	no	—
30	C₄×Dic₁₃₄	—	4	2	no	—
31	#14524002	—	2	2	no	—
32	C₂×C₆₇⋊C₁₆	Q₈	1	8	yes	complemented
33	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
34	#14523977	—	2	2	no	—
35	#14523977	C₂³	1	4	yes	—
36	#14523975	—	2	2	no	—
37	#14523975	—	2	2	no	—
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₆₇	—	2	2	no	—
41	#125×C₆₇	—	2	2	no	—
42	C₂²⋊Q₈×C₆₇	—	4	2	no	—
43	C₄×Q₈×C₆₇	—	4	2	no	—
44	#117×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
46	#107×C₆₇	—	2	2	no	—
47	#107×C₆₇	C₂³	1	4	yes	—
48	#106×C₆₇	—	2	2	no	—
49	#106×C₆₇	—	2	2	no	—
50	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×Dic₁₃₄	—	4	2	no	—
55	C₄⋊Dic₆₇	—	2	4	no	—
56	C₄⋊Dic₆₇	—	4	2	no	—
57	Dic₆₇⋊C₄	—	4	4	no	—
58	C₂²⋊Dic₆₇	—	4	2	no	—
59	C₄×Dic₆₇	—	4	2	no	—
60	C₆₇⋊C₁₆	C₂×Q₈	1	8	yes	—
61	C₂×Q₈×C₆₇	—	4	2	no	—
62	C₂²×C₄×C₆₇	D₈	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
65	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
69	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₆₇	—	2	4	no	—
71	C₄⋊C₄×C₆₇	—	4	4	no	—
72	C₄⋊C₄×C₆₇	—	4	2	no	—
73	C₂²⋊C₄×C₆₇	—	4	2	no	—
74	C₄²×C₆₇	—	4	2	no	—
75	C₄²×C₆₇	D₈	1	2	yes	—
76	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
79	Q₈×C₆₇	—	8	4	no	—
80	C₂×C₄×C₆₇	—	4	2	no	—
81	C₂×C₄×C₆₇	—	4	4	no	—
82	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
88	C₂×Dic₆₇	—	4	4	no	—
89	Dic₁₃₄	—	8	4	no	—
90	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
91	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
92	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
93	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
94	C₂²×C₆₇	—	2	2	no	—
95	C₂²×C₆₇	#500	1	2	yes	—
96	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₆₇	—	2	2	no	—
98	C₄×C₆₇	—	2	2	no	—
99	C₄×C₆₇	—	8	4	no	—
100	C₄×C₆₇	#500	1	2	yes	—
101	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
103	Dic₆₇	—	8	4	no	—
104	#33728	—	67	1	no	maximal, Sylow
105	C₂×C₆₇	—	2	2	no	—
106	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
107	C₂×C₆₇	#3414	1	2	yes	—
108	#3160	—	67	1	no	—
109	#3160	D₆₇	1	1	yes	characteristic
110	#2255	—	67	4	no	—
111	#2255	—	67	4	no	—
112	#2158	—	67	4	no	—
113	C₂×#346	—	67	1	no	—
114	C₆₇	#33728	1	1	yes	characteristic, complemented, Sylow
115	#533	—	2	1	no	—
116	#533	—	134	1	no	—
117	C₂²×C₁₆	—	67	2	no	—
118	#500	—	67	2	no	—
119	#500	C₂×D₆₇	1	2	yes	—
120	#499	—	2	1	no	—
121	#499	—	67	2	no	—
122	#499	—	134	1	no	—
123	#499	C₂×D₆₇	1	2	yes	—
124	#498	—	2	1	no	—
125	#498	—	134	1	no	—
126	#427	—	2	1	no	—
127	#427	—	67	2	no	—
128	#427	—	134	1	no	—
129	#427	C₂×D₆₇	1	2	yes	—
130	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
131	#367	—	67	8	no	—
132	#366	—	67	8	no	—
133	#346	—	67	2	no	—
134	#346	—	67	2	no	—
135	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	134	2	no	—
139	#125	—	2	2	no	—
140	#125	—	134	2	no	—
141	C₂²⋊Q₈	—	4	2	no	—
142	C₂²⋊Q₈	—	268	2	no	—
143	C₄×Q₈	—	4	2	no	—
144	C₄×Q₈	—	268	2	no	—
145	#117	—	2	2	no	—
146	#117	—	134	2	no	—
147	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
148	C₂×C₁₆	—	67	8	no	—
149	C₂×C₁₆	—	67	4	no	—
150	#107	—	2	2	no	—
151	#107	—	67	4	no	—
152	#107	—	134	2	no	—
153	#107	C₂²×D₆₇	1	4	yes	—
154	#106	—	2	2	no	—
155	#106	—	2	2	no	—
156	#106	—	134	2	no	—
157	#106	—	134	2	no	—
158	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
161	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	268	2	no	—
164	C₂²×C₄	#11761613	1	2	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
166	C₂×C₈	#11761618	1	4	yes	—
167	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
168	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
170	C₂×C₈	Q₈×D₆₇	1	4	yes	—
171	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	268	4	no	—
176	C₄⋊C₄	—	268	2	no	—
177	C₄⋊C₄	—	134	4	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	268	2	no	—
180	C₄²	—	4	2	no	—
181	C₄²	—	268	2	no	—
182	C₄²	#11761613	1	2	yes	—
183	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
184	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
185	C₁₆	—	67	8	no	—
186	C₂³	C₂×#11761613	1	1	yes	characteristic
187	Q₈	—	8	4	no	—
188	Q₈	—	536	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	268	4	no	—
192	C₂×C₄	#14524041	1	2	yes	—
193	C₂×C₄	C₂×#11761613	1	2	yes	—
194	C₂×C₄	C₂×#11761613	1	2	yes	—
195	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
196	C₂×C₄	C₂×#11761618	1	2	yes	—
197	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
198	C₈	#14524041	1	2	yes	—
199	C₈	C₂×#11761618	1	2	yes	—
200	C₈	C₂×#11761618	1	2	yes	—
201	C₈	C₂×Q₈×D₆₇	1	2	yes	—
202	C₂²	—	2	2	no	—
203	C₂²	#16863799	1	2	yes	—
204	C₂²	C₂²×#11761613	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	8	4	no	—
208	C₄	—	536	4	no	—
209	C₄	#16863799	1	2	yes	—
210	C₄	C₂×#14524041	1	1	yes	characteristic
211	C₄	C₂²×#11761618	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	C₂×#16863799	1	1	yes	characteristic
214	C₂	#22573343	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	#33728
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. #22562496	C₂	semidirect
3. #22562492	C₂	semidirect
32. C₂×C₆₇⋊C₁₆	Q₈	semidirect
114. C₆₇	#33728	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
#22562496	2
#22562492	2
#22561991	2
C₂×#16863172	2
C₆₇×#3160	2
#33728	67