Subgroups — #31888610

Order 18176.

414 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	#22587088	C₂	1	2	yes	maximal
2	#22586623	C₂	1	2	yes	maximal
3	#22586173	C₂	1	2	yes	maximal
4	C₄²×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17986052	C₂²	1	4	yes	—
6	#17985752	C₄	1	4	yes	complemented
7	#17985750	C₄	1	4	yes	complemented
8	#17985720	—	2	2	no	—
9	C₈×C₇₁⋊C₈	—	2	2	no	—
10	C₂×C₄×C₈×C₇₁	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₇₁	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₄³×C₇₁	C₂²	1	1	yes	characteristic
15	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	8	yes	—
16	C₂²⋊(C₇₁⋊C₈)	—	2	4	no	—
17	C₄×C₇₁⋊C₈	—	2	4	no	—
18	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₇₁	—	2	2	no	—
22	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₇₁	—	2	4	no	—
25	C₄×C₈×C₇₁	C₈	1	8	yes	complemented
26	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₁	Q₈	1	2	yes	—
31	C₂×C₇₁⋊C₈	—	2	8	no	—
32	C₂²×C₄×C₇₁	—	2	2	no	—
33	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	#41	1	2	yes	—
35	C₂×C₈×C₇₁	—	2	2	no	—
36	C₂×C₈×C₇₁	C₄²	1	2	yes	—
37	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
42	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
43	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
44	C₄²×C₇₁	—	2	8	no	—
45	C₄²×C₇₁	—	2	2	no	—
46	C₄²×C₇₁	C₂×C₈	1	4	yes	—
47	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
48	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
50	C₂³×C₇₁	#117	1	1	yes	characteristic
51	C₂×C₄×C₇₁	—	2	8	no	—
52	C₂×C₄×C₇₁	—	2	4	no	—
53	C₂×C₄×C₇₁	—	2	4	no	—
54	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
57	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
58	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
59	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
60	C₈×C₇₁	—	2	2	no	—
61	C₈×C₇₁	C₄×C₈	1	2	yes	—
62	C₈×C₇₁	C₄×C₈	1	2	yes	—
63	C₈×C₇₁	C₄⋊C₈	1	4	yes	—
64	C₈×C₇₁	C₄⋊C₈	1	4	yes	—
65	C₇₁⋊C₈	—	4	8	no	—
66	C₂²×C₇₁	—	2	2	no	—
67	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₇₁	#431	1	2	yes	—
69	C₄×C₇₁	—	2	8	no	—
70	C₄×C₇₁	—	2	2	no	—
71	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
72	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
73	C₄×C₇₁	C₂×C₄⋊C₈	1	2	yes	—
74	C₄×C₇₁	#443	1	4	yes	—
75	#8266	—	71	1	no	maximal, Sylow
76	C₂×C₇₁	—	2	2	no	—
77	C₂×C₇₁	C₄×C₄⋊C₈	1	1	yes	characteristic
78	C₂×C₇₁	#1772	1	2	yes	—
79	C₄²×C₈	D₇₁	1	1	yes	characteristic
80	#1564	—	71	2	no	—
81	#1450	—	71	2	no	—
82	#1276	—	71	2	no	—
83	C₇₁	#8266	1	1	yes	characteristic, complemented, Sylow
84	#431	—	71	4	no	—
85	C₂×C₄×C₈	Dic₇₁	1	2	yes	—
86	C₂×C₄×C₈	Dic₇₁	1	2	yes	—
87	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
89	C₄³	C₂×D₇₁	1	1	yes	characteristic
90	#335	—	71	4	no	—
91	#334	—	71	4	no	—
92	#322	—	142	2	no	—
93	C₈²	—	142	2	no	—
94	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
95	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
96	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
97	C₂×C₄²	—	2	2	no	—
98	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
99	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
100	C₄⋊C₈	—	71	8	no	—
101	C₂²⋊C₈	—	142	4	no	—
102	C₄×C₈	—	2	4	no	—
103	C₄×C₈	—	142	4	no	—
104	C₄×C₈	C₇₁⋊C₈	1	8	yes	complemented
105	C₄×C₈	Dic₁₄₂	1	2	yes	—
106	C₄×C₈	C₄×D₇₁	1	2	yes	—
107	C₄×C₈	C₄×D₇₁	1	2	yes	—
108	C₄×C₈	D₂₈₄	1	1	yes	characteristic
109	C₄×C₈	D₂₈₄	1	1	yes	characteristic
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
112	C₂²×C₄	#11765865	1	2	yes	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	142	8	no	—
115	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
116	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
117	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
118	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
119	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
120	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	—
121	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	—
122	C₂×C₈	#11765849	1	4	yes	—
123	C₄²	—	2	8	no	—
124	C₄²	—	2	2	no	—
125	C₄²	C₂×C₇₁⋊C₈	1	4	yes	—
126	C₄²	C₂×C₄×D₇₁	1	2	yes	—
127	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
128	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
129	C₂³	#14556087	1	1	yes	characteristic
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
135	C₂×C₄	C₄×D₂₈₄	1	2	yes	—
136	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
137	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
138	C₂×C₄	C₂×#11765849	1	2	yes	—
139	C₈	—	2	2	no	—
140	C₈	—	284	8	no	—
141	C₈	C₄×C₇₁⋊C₈	1	2	yes	—
142	C₈	C₄×C₇₁⋊C₈	1	2	yes	—
143	C₈	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
144	C₈	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	C₄×C₄⋊Dic₇₁	1	1	yes	characteristic
147	C₂²	#17986052	1	2	yes	—
148	C₄	—	2	8	no	—
149	C₄	—	2	2	no	—
150	C₄	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
151	C₄	C₄×#11765849	1	1	yes	characteristic
152	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	2	yes	—
153	C₄	#17986117	1	4	yes	—
154	C₂	—	2	2	no	—
155	C₂	C₄×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
156	C₂	#22588219	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	D₂₈₄
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄²×C₈×C₇₁	C₂
socle	C₂²×C₇₁	C₄×C₄⋊C₄
prime radical	C₂³×C₇₁	#117
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄²×C₈	D₇₁
hypocenter	C₇₁	#8266
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
6. #17985752	C₄	semidirect
7. #17985750	C₄	semidirect
25. C₄×C₈×C₇₁	C₈	semidirect
83. C₇₁	#8266	semidirect
104. C₄×C₈	C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22587088	2
#22586623	2
#22586173	2
C₄²×C₈×C₇₁	2
#8266	71