Subgroups — #30772272

Order 17152.

342 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863224	C₂	1	2	yes	maximal
2	C₂×#16863222	C₂	1	2	yes	maximal
3	#22560373	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523977	C₂²	1	4	yes	—
6	C₂×#14523975	—	2	2	no	—
7	#16863262	—	2	2	no	—
8	#16863224	—	2	4	no	—
9	#16863222	—	2	4	no	—
10	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₆₇	C₄	1	4	yes	complemented
14	C₂×C₄⋊Dic₆₇	—	2	4	no	—
15	#14523977	—	2	4	no	—
16	#14523977	—	2	4	no	—
17	#14523975	—	4	4	no	—
18	C₆₇⋊#95	—	4	4	no	—
19	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
25	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
26	C₂×C₁₆×C₆₇	Q₈	1	2	yes	—
27	C₂×C₁₆×C₆₇	Q₈	1	2	yes	—
28	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₆₇	Q₈	1	2	yes	—
30	C₂²×Dic₆₇	—	4	4	no	—
31	C₄⋊Dic₆₇	—	4	4	no	—
32	C₄⋊Dic₆₇	—	4	4	no	—
33	C₂²×C₄×C₆₇	D₈	1	2	yes	—
34	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₆₇	D₈	1	2	yes	—
37	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
38	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
45	C₄²×C₆₇	Q₁₆	1	2	yes	—
46	C₁₆×C₆₇	C₄⋊C₄	1	4	yes	—
47	C₁₆×C₆₇	C₄⋊C₄	1	4	yes	—
48	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₆₇	—	2	4	no	—
50	C₂×C₄×C₆₇	#107	1	4	yes	—
51	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×Dic₆₇	—	8	4	no	—
58	C₂×Dic₆₇	—	8	8	no	—
59	C₈×C₆₇	#107	1	4	yes	—
60	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
63	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
65	C₂²×C₆₇	#493	1	1	yes	characteristic
66	C₂²×C₆₇	#500	1	1	yes	characteristic
67	C₂²×C₆₇	#508	1	4	yes	—
68	C₄×C₆₇	—	2	4	no	—
69	C₄×C₆₇	#384	1	1	yes	characteristic
70	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
71	C₄×C₆₇	#494	1	1	yes	characteristic
72	C₄×C₆₇	#500	1	1	yes	characteristic
73	Dic₆₇	—	16	4	no	—
74	#12907	—	67	1	no	maximal, Sylow
75	C₂×C₆₇	#1845	1	1	yes	characteristic
76	C₂×C₆₇	#2158	1	2	yes	—
77	C₂×C₆₇	#2249	1	2	yes	—
78	C₂×C₆₇	#2253	1	2	yes	—
79	C₂×#367	—	67	2	no	—
80	C₂×#366	—	67	2	no	—
81	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic
82	#1846	—	67	2	no	—
83	C₆₇	#12907	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
85	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
86	C₂×#107	—	67	4	no	—
87	C₂×#106	—	134	2	no	—
88	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
89	#382	—	134	2	no	—
90	#367	—	134	4	no	—
91	#366	—	134	4	no	—
92	C₄×C₁₆	Dic₆₇	1	4	yes	complemented
93	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
94	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
95	C₂×C₄⋊C₄	—	134	4	no	—
96	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
97	C₂×C₁₆	Dic₁₃₄	1	2	yes	—
98	C₂×C₁₆	Dic₁₃₄	1	2	yes	—
99	C₂×C₁₆	D₂₆₈	1	2	yes	—
100	C₂×C₁₆	D₂₆₈	1	2	yes	—
101	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
102	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
103	#107	—	134	4	no	—
104	#107	—	134	4	no	—
105	#106	—	268	4	no	—
106	C₄×C₈	Dic₁₃₄	1	2	yes	—
107	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
108	#95	—	268	4	no	—
109	C₂²×C₄	—	268	4	no	—
110	C₂²×C₄	D₅₃₆	1	2	yes	—
111	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
112	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
113	C₂×C₈	D₅₃₆	1	2	yes	—
114	C₂×C₈	Dic₂₆₈	1	2	yes	—
115	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
116	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
117	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
118	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
119	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
120	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
121	C₄⋊C₄	—	268	4	no	—
122	C₄⋊C₄	—	268	4	no	—
123	C₄²	C₄⋊Dic₆₇	1	2	yes	—
124	C₄²	Dic₂₆₈	1	2	yes	—
125	C₁₆	C₄⋊Dic₆₇	1	4	yes	—
126	C₁₆	C₄⋊Dic₆₇	1	4	yes	—
127	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	536	4	no	—
130	C₂×C₄	—	536	8	no	—
131	C₂×C₄	#14523977	1	4	yes	—
132	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
133	C₂×C₄	#14524040	1	1	yes	characteristic
134	C₂×C₄	#14524043	1	1	yes	characteristic
135	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
136	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
137	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
138	C₈	#14523977	1	4	yes	—
139	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
140	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
141	C₈	#14524043	1	1	yes	characteristic
142	C₈	#14524043	1	1	yes	characteristic
143	C₂²	C₂×#14523977	1	1	yes	characteristic
144	C₂²	#16863773	1	1	yes	characteristic
145	C₂²	#16863801	1	1	yes	characteristic
146	C₂²	#16863828	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	1072	4	no	—
149	C₄	#16863266	1	1	yes	characteristic
150	C₄	C₂×#14523977	1	1	yes	characteristic
151	C₄	#16863776	1	1	yes	characteristic
152	C₄	#16863801	1	1	yes	characteristic
153	C₂	#22560369	1	1	yes	characteristic
154	C₂	#22561993	1	2	yes	—
155	C₂	#22562470	1	2	yes	—
156	C₂	#22562488	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	#12907
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
13. C₄×C₁₆×C₆₇	C₄	semidirect
83. C₆₇	#12907	semidirect
92. C₄×C₁₆	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863224	2
C₂×#16863222	2
#22560373	2
C₂×C₄×C₁₆×C₆₇	2
#12907	67