Subgroups — #30772248

Order 17152.

382 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863217	C₂	1	4	yes	maximal
2	#22561965	C₂	1	2	yes	maximal
3	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523983	C₂²	1	4	yes	—
5	#16863217	—	2	8	no	—
6	C₄⋊(C₆₇⋊C₁₆)	—	2	4	no	—
7	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₄³×C₆₇	C₄	1	1	yes	characteristic
12	#14523983	—	2	8	no	—
13	C₂×C₆₇⋊C₁₆	—	2	4	no	—
14	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₆₇	—	2	2	no	—
18	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₆₇	—	2	4	no	—
21	C₄×C₈×C₆₇	C₈	1	2	yes	—
22	C₄×C₈×C₆₇	C₈	1	2	yes	—
23	C₄×C₈×C₆₇	D₄	1	2	yes	—
24	C₄×C₈×C₆₇	D₄	1	4	yes	—
25	C₄×C₈×C₆₇	Q₈	1	4	yes	—
26	C₄×C₈×C₆₇	Q₈	1	2	yes	—
27	C₆₇⋊C₁₆	—	4	4	no	—
28	C₂²×C₄×C₆₇	—	2	2	no	—
29	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
30	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₈×C₆₇	—	2	2	no	—
32	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
33	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
34	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
35	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
36	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
37	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
39	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
40	C₄²×C₆₇	—	2	8	no	—
41	C₄²×C₆₇	—	2	2	no	—
42	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
43	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
44	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
46	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
47	C₂×C₄×C₆₇	—	2	8	no	—
48	C₂×C₄×C₆₇	—	2	4	no	—
49	C₂×C₄×C₆₇	—	2	4	no	—
50	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
51	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
52	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
53	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
56	C₈×C₆₇	—	2	2	no	—
57	C₈×C₆₇	C₄⋊C₈	1	8	yes	—
58	C₈×C₆₇	#127	1	1	yes	characteristic
59	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
60	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
61	C₂²×C₆₇	—	2	2	no	—
62	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₂²×C₆₇	#423	1	2	yes	—
64	C₄×C₆₇	—	2	8	no	—
65	C₄×C₆₇	—	2	2	no	—
66	C₄×C₆₇	#364	1	4	yes	complemented
67	C₄×C₆₇	#384	1	1	yes	characteristic
68	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
69	C₄×C₆₇	#423	1	1	yes	characteristic
70	#12900	—	67	1	no	maximal, Sylow
71	C₂×C₆₇	—	2	2	no	—
72	C₂×C₆₇	#1842	1	1	yes	characteristic
73	C₂×C₆₇	C₂×#364	1	2	yes	—
74	C₂×#364	—	67	4	no	—
75	#2150	—	67	2	no	—
76	C₄²×C₈	D₆₇	1	1	yes	characteristic
77	C₆₇	#12900	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₃₂	—	67	4	no	—
79	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
80	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
82	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
83	C₄³	Dic₆₇	1	1	yes	characteristic
84	#364	—	134	8	no	—
85	C₄⋊C₁₆	—	134	4	no	—
86	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
87	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
88	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
89	C₂×C₄²	—	2	2	no	—
90	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
91	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
92	M₃₂	—	134	8	no	—
93	C₂×C₁₆	—	134	4	no	—
94	C₄×C₈	—	2	4	no	—
95	C₄×C₈	C₆₇⋊C₈	1	2	yes	—
96	C₄×C₈	C₆₇⋊C₈	1	2	yes	—
97	C₄×C₈	Dic₁₃₄	1	4	yes	—
98	C₄×C₈	Dic₁₃₄	1	2	yes	—
99	C₄×C₈	D₂₆₈	1	4	yes	—
100	C₄×C₈	D₂₆₈	1	2	yes	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	#11761612	1	2	yes	—
103	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
106	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
107	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
108	C₂×C₈	#11761612	1	4	yes	—
109	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
110	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
111	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
112	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
113	C₄²	—	2	8	no	—
114	C₄²	—	2	2	no	—
115	C₄²	C₄⋊Dic₆₇	1	4	yes	—
116	C₄²	C₄⋊Dic₆₇	1	2	yes	—
117	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
118	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
119	C₁₆	—	268	4	no	—
120	C₂³	C₂×#11761612	1	1	yes	characteristic
121	C₂×C₄	—	2	8	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
125	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
126	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
127	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
128	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
129	C₂×C₄	C₂×#11761612	1	2	yes	—
130	C₈	—	2	2	no	—
131	C₈	C₄⋊(C₆₇⋊C₈)	1	8	yes	—
132	C₈	#14524040	1	1	yes	characteristic
133	C₈	#14524043	1	2	yes	—
134	C₈	#14524043	1	1	yes	characteristic
135	C₂²	—	2	2	no	—
136	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
137	C₂²	#16863420	1	2	yes	—
138	C₄	—	2	8	no	—
139	C₄	—	2	2	no	—
140	C₄	#16863217	1	4	yes	complemented
141	C₄	#16863266	1	1	yes	characteristic
142	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	2	yes	—
143	C₄	#16863420	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	#22560361	1	1	yes	characteristic
146	C₂	C₂×#16863217	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₄²×C₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₈
prime radical	C₂³×C₆₇	C₂×M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	#12900
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
66. C₄×C₆₇	#364	semidirect
77. C₆₇	#12900	semidirect
140. C₄	#16863217	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863217	2
#22561965	2
C₄²×C₈×C₆₇	2
#12900	67