Subgroups — #31009332

Order 17152.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562118	C₂	1	8	yes	maximal, complemented
2	#22561968	C₂	1	2	yes	maximal
3	#22561965	C₂	1	1	yes	maximal, characteristic
4	#22561806	C₂	1	2	yes	maximal
5	C₂×#16863172	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic
7	C₂×#14523983	C₂²	1	2	yes	—
8	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
11	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	8	yes	—
12	#16863172	C₂²	1	4	yes	—
13	#16863172	C₂²	1	4	yes	—
14	C₄×C₆₇⋊C₁₆	C₂²	1	4	yes	—
15	#533×C₆₇	C₄	1	1	yes	characteristic
16	#446×C₆₇	C₄	1	4	yes	—
17	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
18	#424×C₆₇	C₄	1	2	yes	—
19	#423×C₆₇	C₄	1	1	yes	characteristic
20	#405×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#14523983	—	2	4	no	—
23	C₂×C₆₇⋊C₁₆	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
25	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
26	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
27	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
28	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₆₇	C₈	1	2	yes	—
32	#125×C₆₇	C₈	1	2	yes	—
33	C₂²⋊Q₈×C₆₇	C₈	1	4	yes	—
34	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
35	#117×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
38	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
39	#97×C₆₇	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₆₇	C₂³	1	4	yes	—
42	C₄×C₈×C₆₇	C₂³	1	2	yes	—
43	C₆₇⋊C₁₆	—	2	2	no	—
44	C₆₇⋊C₁₆	—	2	4	no	—
45	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	—
46	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	M₁₆×C₆₇	—	2	4	no	—
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₆₇	#41	1	2	yes	—
56	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
58	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
59	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
60	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
61	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
64	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
65	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
66	Q₈×C₆₇	—	4	4	no	—
67	C₂×C₄×C₆₇	—	4	2	no	—
68	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₆₇	C₂²×C₈	1	8	yes	—
72	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	#131	1	2	yes	—
74	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
75	C₈×C₆₇	—	2	2	no	—
76	C₈×C₆₇	—	2	4	no	—
77	C₈×C₆₇	C₄×Q₈	1	2	yes	—
78	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
79	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
80	C₂²×C₆₇	—	2	2	no	—
81	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
82	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₆₇	—	2	2	no	—
84	C₄×C₆₇	—	2	2	no	—
85	C₄×C₆₇	—	2	8	no	—
86	C₄×C₆₇	C₈×Q₈	1	2	yes	—
87	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
89	#33475	—	67	1	no	maximal, Sylow
90	C₂×C₆₇	—	2	2	no	—
91	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
92	C₂×C₆₇	#3407	1	2	yes	—
93	#2960	D₆₇	1	1	yes	characteristic
94	#2183	—	67	8	no	—
95	#2151	—	67	2	no	—
96	#2150	—	67	1	no	—
97	#2108	—	67	2	no	—
98	C₂×#346	—	67	1	no	—
99	C₆₇	#33475	1	1	yes	characteristic, complemented, Sylow
100	#533	Dic₆₇	1	1	yes	characteristic
101	C₂×M₃₂	—	67	2	no	—
102	C₂²×C₁₆	—	67	1	no	—
103	C₂²×C₁₆	—	67	1	no	—
104	#446	Dic₆₇	1	4	yes	—
105	C₈×Q₈	C₂×D₆₇	1	4	yes	—
106	#424	Dic₆₇	1	2	yes	—
107	#423	Dic₆₇	1	1	yes	characteristic
108	#405	C₂×D₆₇	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
110	C₄⋊C₁₆	—	67	4	no	—
111	C₄⋊C₁₆	—	67	8	no	—
112	#346	—	67	4	no	—
113	#346	—	67	4	no	—
114	C₄×C₁₆	—	67	4	no	—
115	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
116	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
118	C₄⋊Q₈	C₆₇⋊C₈	1	2	yes	—
119	#125	C₆₇⋊C₈	1	2	yes	—
120	C₂²⋊Q₈	C₆₇⋊C₈	1	4	yes	—
121	C₄×Q₈	C₂×Dic₆₇	1	4	yes	—
122	#117	C₂×Dic₆₇	1	2	yes	—
123	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
124	M₃₂	—	134	4	no	—
125	C₂×C₁₆	—	67	4	no	—
126	C₂×C₁₆	—	67	2	no	—
127	C₂×C₁₆	—	67	4	no	—
128	C₂×C₁₆	—	67	2	no	—
129	C₂×C₁₆	—	134	2	no	—
130	C₄⋊C₈	C₂×Dic₆₇	1	4	yes	—
131	C₄⋊C₈	C₂×Dic₆₇	1	8	yes	—
132	#97	C₂×Dic₆₇	1	4	yes	—
133	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
134	C₄×C₈	C₂²×D₆₇	1	4	yes	—
135	C₄×C₈	C₂²×D₆₇	1	2	yes	—
136	C₂×Q₈	C₂×C₆₇⋊C₈	1	4	yes	—
137	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
138	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
139	M₁₆	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
142	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
143	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₆₇	1	2	yes	—
145	C₂×C₈	#11761629	1	2	yes	—
146	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
147	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
148	C₄⋊C₄	C₂×C₆₇⋊C₈	1	8	yes	—
149	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
150	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
151	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
152	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
153	C₄²	C₂²×Dic₆₇	1	2	yes	—
154	C₄²	C₂²×Dic₆₇	1	4	yes	—
155	C₁₆	—	67	4	no	—
156	C₁₆	—	134	2	no	—
157	C₁₆	—	134	4	no	—
158	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
159	Q₈	—	4	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
162	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
163	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
164	C₂×C₄	C₂²×C₆₇⋊C₈	1	8	yes	—
165	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
166	C₂×C₄	#14524050	1	2	yes	—
167	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	4	no	—
170	C₈	Q₈×Dic₆₇	1	2	yes	—
171	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
172	C₈	C₂×#11761629	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	Q₈×C₆₇⋊C₈	1	2	yes	—
175	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	8	no	—
179	C₄	Q₈×C₆₇⋊C₈	1	2	yes	—
180	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
181	C₄	C₂×#14524050	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×Q₈×C₆₇⋊C₈	1	1	yes	characteristic
184	C₂	#22573312	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Fitting	C₆₇×#2960	C₂
prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2960	D₆₇
hypocenter	C₆₇	#33475
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #22562118	C₂	semidirect
24. C₂×C₆₇⋊C₁₆	Q₈	semidirect
99. C₆₇	#33475	semidirect

Maximal subgroups

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

subgroup	index
#22562118	2
#22561968	2
#22561965	2
#22561806	2
C₂×#16863172	2
C₆₇×#2960	2
#33475	67