Subgroups — #31009326

Order 17152.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562118	C₂	1	4	yes	maximal, complemented
2	Q₈×C₆₇⋊C₁₆	C₂	1	4	yes	maximal, complemented
3	#22561968	C₂	1	2	yes	maximal
4	#22561965	C₂	1	1	yes	maximal, characteristic
5	#22561806	C₂	1	2	yes	maximal
6	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523983	C₂²	1	2	yes	—
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
12	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	8	yes	—
13	#16863172	C₂²	1	4	yes	—
14	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
15	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
16	C₄×C₆₇⋊C₁₆	C₂²	1	4	yes	—
17	#533×C₆₇	C₄	1	1	yes	characteristic
18	#446×C₆₇	C₄	1	4	yes	—
19	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
20	#424×C₆₇	C₄	1	2	yes	—
21	#423×C₆₇	C₄	1	1	yes	characteristic
22	#405×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#14523983	—	2	4	no	—
25	C₂×C₆₇⋊C₁₆	—	2	2	no	—
26	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
27	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
28	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
29	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
30	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₆₇	C₈	1	2	yes	—
35	#125×C₆₇	C₈	1	2	yes	—
36	C₂²⋊Q₈×C₆₇	C₈	1	4	yes	—
37	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
38	#117×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
41	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
42	#97×C₆₇	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₆₇	C₂³	1	2	yes	—
45	C₄×C₈×C₆₇	C₂³	1	4	yes	—
46	C₆₇⋊C₁₆	—	2	2	no	—
47	C₆₇⋊C₁₆	—	2	4	no	—
48	C₆₇⋊C₁₆	C₂×Q₈	1	2	yes	—
49	C₆₇⋊C₁₆	C₂×Q₈	1	2	yes	—
50	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
51	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	M₁₆×C₆₇	—	2	4	no	—
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₆₇	#41	1	2	yes	—
60	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
62	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
63	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
64	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
65	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
67	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
69	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
70	Q₈×C₆₇	—	2	8	no	—
71	C₂×C₄×C₆₇	—	2	4	no	—
72	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₆₇	C₂²×C₈	1	8	yes	—
75	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₆₇	#131	1	2	yes	—
78	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₆₇	—	2	2	no	—
80	C₈×C₆₇	—	2	4	no	—
81	C₈×C₆₇	C₄×Q₈	1	2	yes	—
82	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
83	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
84	C₂²×C₆₇	—	2	2	no	—
85	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
86	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
87	C₄×C₆₇	—	2	2	no	—
88	C₄×C₆₇	—	2	8	no	—
89	C₄×C₆₇	C₈×Q₈	1	2	yes	—
90	C₄×C₆₇	#504	1	4	yes	—
91	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
93	#33474	—	67	1	no	maximal, Sylow
94	C₂×C₆₇	—	2	2	no	—
95	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
96	C₂×C₆₇	C₂×#504	1	2	yes	—
97	#2960	D₆₇	1	1	yes	characteristic
98	#2183	—	67	4	no	—
99	Q₈×C₁₆	—	67	4	no	—
100	#2151	—	67	2	no	—
101	#2150	—	67	1	no	—
102	#2108	—	67	2	no	—
103	C₂×C₄×C₁₆	—	67	1	no	—
104	C₆₇	#33474	1	1	yes	characteristic, complemented, Sylow
105	#533	Dic₆₇	1	1	yes	characteristic
106	C₂×M₃₂	—	67	2	no	—
107	C₂²×C₁₆	—	67	1	no	—
108	C₂²×C₁₆	—	67	1	no	—
109	#446	Dic₆₇	1	4	yes	—
110	C₈×Q₈	C₂×D₆₇	1	4	yes	—
111	#424	Dic₆₇	1	2	yes	—
112	#423	Dic₆₇	1	1	yes	characteristic
113	#405	C₂×D₆₇	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
115	C₄⋊C₁₆	—	67	4	no	—
116	C₄⋊C₁₆	—	67	8	no	—
117	#346	—	67	4	no	—
118	C₄×C₁₆	—	67	2	no	—
119	C₄×C₁₆	—	67	2	no	—
120	C₄×C₁₆	—	67	4	no	—
121	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
122	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
124	C₄⋊Q₈	C₆₇⋊C₈	1	2	yes	—
125	#125	C₆₇⋊C₈	1	2	yes	—
126	C₂²⋊Q₈	C₆₇⋊C₈	1	4	yes	—
127	C₄×Q₈	C₂×Dic₆₇	1	4	yes	—
128	#117	C₂×Dic₆₇	1	2	yes	—
129	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
130	M₃₂	—	134	4	no	—
131	C₂×C₁₆	—	67	4	no	—
132	C₂×C₁₆	—	67	1	no	—
133	C₂×C₁₆	—	67	1	no	—
134	C₂×C₁₆	—	67	2	no	—
135	C₂×C₁₆	—	67	4	no	—
136	C₂×C₁₆	—	134	2	no	—
137	C₄⋊C₈	C₂×Dic₆₇	1	4	yes	—
138	C₄⋊C₈	C₂×Dic₆₇	1	8	yes	—
139	#97	C₂×Dic₆₇	1	4	yes	—
140	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
141	C₄×C₈	C₂²×D₆₇	1	2	yes	—
142	C₄×C₈	C₂²×D₆₇	1	4	yes	—
143	C₂×Q₈	C₂×C₆₇⋊C₈	1	4	yes	—
144	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
145	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
146	M₁₆	—	2	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
149	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
150	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₆₇	1	2	yes	—
152	C₂×C₈	#11761629	1	2	yes	—
153	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
154	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
155	C₄⋊C₄	C₂×C₆₇⋊C₈	1	8	yes	—
156	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
157	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
158	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
159	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
160	C₄²	C₂²×Dic₆₇	1	2	yes	—
161	C₄²	C₂²×Dic₆₇	1	4	yes	—
162	C₁₆	—	67	2	no	—
163	C₁₆	—	67	2	no	—
164	C₁₆	—	134	2	no	—
165	C₁₆	—	134	4	no	—
166	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
167	Q₈	—	2	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
170	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
171	C₂×C₄	C₂²×C₆₇⋊C₈	1	8	yes	—
172	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
173	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
174	C₂×C₄	#14524050	1	2	yes	—
175	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	4	no	—
178	C₈	Q₈×Dic₆₇	1	2	yes	—
179	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
180	C₈	C₂×#11761629	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	Q₈×C₆₇⋊C₈	1	2	yes	—
183	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	8	no	—
186	C₄	Q₈×C₆₇⋊C₈	1	2	yes	—
187	C₄	#16863813	1	4	yes	—
188	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
189	C₄	C₂×#14524050	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×Q₈×C₆₇⋊C₈	1	1	yes	characteristic
192	C₂	C₂×#16863813	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	#33474
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
2. Q₈×C₆₇⋊C₁₆	C₂	semidirect
26. C₂×C₆₇⋊C₁₆	Q₈	semidirect
104. C₆₇	#33474	semidirect

Maximal subgroups

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

subgroup	index
#22562118	2
Q₈×C₆₇⋊C₁₆	2
#22561968	2
#22561965	2
#22561806	2
C₂×C₄×C₆₇⋊C₁₆	2
C₆₇×#2960	2
#33474	67