Subgroups — #16858813

Order 3968.

360 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758478	C₂	1	1	yes	maximal, characteristic
2	#14521467	C₂	1	4	yes	maximal, complemented
3	#14521465	C₂	1	4	yes	maximal, complemented
4	C₂×#11758400	C₂	1	1	yes	maximal, characteristic, complemented
5	M₃₂×D₃₁	C₂	1	4	yes	maximal, complemented
6	C₂×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#92495	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×#92474	C₄	1	1	yes	characteristic
10	#11758478	C₄	1	2	yes	—
11	#11758478	C₄	1	2	yes	—
12	#11758478	C₂²	1	2	yes	—
13	#11758478	C₂²	1	2	yes	—
14	C₂×#92469	C₄	1	2	yes	—
15	C₂×C₈×D₃₁	C₂²	1	2	yes	—
16	C₃₁×M₃₂	C₂²	1	2	yes	—
17	C₃₁×M₃₂	C₂²	1	2	yes	complemented
18	C₂×C₁₆×C₃₁	C₂²	1	2	yes	complemented
19	#11758400	C₂²	1	2	yes	—
20	#11758400	C₂²	1	2	yes	complemented
21	C₂×C₃₁⋊C₁₆	C₂²	1	2	yes	complemented
22	#11758386	C₂²	1	8	yes	complemented
23	C₁₆×D₃₁	C₂²	1	8	yes	complemented
24	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₂²⋊D₃₁	C₈	1	2	yes	—
26	#92495	C₈	1	2	yes	—
27	#92495	C₈	1	2	yes	—
28	#92495	C₂×C₄	1	2	yes	—
29	#92495	C₂×C₄	1	2	yes	—
30	C₂×D₁₂₄	C₈	1	1	yes	characteristic
31	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
32	C₂×Dic₆₂	C₈	1	1	yes	characteristic
33	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	#92474	C₂×C₄	1	2	yes	—
40	#92474	C₂×C₄	1	2	yes	—
41	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
42	#92469	C₂×C₄	1	4	yes	—
43	#92469	C₂×C₄	1	4	yes	—
44	C₈×D₃₁	C₂×C₄	1	4	yes	—
45	C₈×D₃₁	C₂³	1	4	yes	—
46	C₁₆×C₃₁	C₂³	1	4	yes	—
47	C₃₁⋊C₁₆	C₂³	1	4	yes	—
48	C₂³×C₃₁	C₂×C₈	1	1	yes	characteristic
49	C₂²×D₃₁	C₂×C₈	1	2	yes	—
50	C₂×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
51	C₂×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
52	C₂×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²⋊D₃₁	C₂×C₈	1	4	yes	—
57	C₂²⋊D₃₁	C₂×C₈	1	4	yes	—
58	C₂×Dic₃₁	C₂×C₈	1	2	yes	—
59	D₁₂₄	C₂×C₈	1	2	yes	—
60	D₁₂₄	C₂×C₈	1	2	yes	—
61	C₄×D₃₁	C₂×C₈	1	4	yes	—
62	C₄×D₃₁	C₂²×C₄	1	4	yes	—
63	Dic₆₂	C₂×C₈	1	2	yes	—
64	Dic₆₂	C₂×C₈	1	2	yes	—
65	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₈×C₃₁	C₂⁴	1	1	yes	characteristic
69	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
70	#3407	—	31	1	no	maximal, Sylow
71	C₂²×C₃₁	—	4	1	no	—
72	C₂²×C₃₁	C₂²×C₈	1	1	yes	characteristic
73	C₂²×C₃₁	C₂²×C₈	1	1	yes	characteristic
74	C₂²×C₃₁	C₂²×C₈	1	1	yes	characteristic
75	C₂×D₃₁	—	4	2	no	—
76	C₂×D₃₁	C₂²×C₈	1	4	yes	—
77	C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
78	C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
79	C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
80	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
81	Dic₃₁	C₂²×C₈	1	4	yes	—
82	C₂×#131	—	31	1	no	—
83	#504	—	31	4	no	—
84	#504	—	31	4	no	—
85	C₂×M₃₂	—	31	1	no	—
86	C₂×M₃₂	—	31	4	no	—
87	C₂×M₃₂	D₃₁	1	1	yes	characteristic, complemented
88	C₂×C₃₁	—	2	1	no	—
89	C₂×C₃₁	—	2	1	no	—
90	C₂×C₃₁	—	2	1	no	—
91	C₂×C₃₁	C₂³×C₈	1	1	yes	characteristic
92	D₃₁	—	2	4	no	—
93	C₂×#41	—	31	1	no	—
94	#131	—	31	2	no	—
95	#131	—	31	2	no	—
96	#131	—	31	2	no	—
97	#131	—	31	2	no	—
98	C₂×M₁₆	—	31	1	no	—
99	C₂×M₁₆	—	31	2	no	—
100	C₂²×C₈	—	31	2	no	—
101	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
102	M₃₂	—	31	2	no	—
103	M₃₂	—	31	2	no	—
104	M₃₂	—	31	8	no	—
105	M₃₂	C₂×D₃₁	1	2	yes	—
106	M₃₂	C₂×D₃₁	1	2	yes	complemented
107	C₂×C₁₆	—	31	2	no	—
108	C₂×C₁₆	—	31	8	no	—
109	C₂×C₁₆	C₂×D₃₁	1	2	yes	complemented
110	C₃₁	#3407	1	1	yes	characteristic, complemented, Sylow
111	#41	—	31	2	no	—
112	#41	—	31	2	no	—
113	#41	—	31	2	no	—
114	#41	—	31	2	no	—
115	C₂×Q₈	—	31	1	no	—
116	C₂×D₄	—	31	2	no	—
117	C₂×D₄	—	31	1	no	—
118	C₂²×C₄	—	31	2	no	—
119	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
120	M₁₆	—	31	2	no	—
121	M₁₆	—	31	2	no	—
122	M₁₆	—	31	4	no	—
123	M₁₆	—	31	4	no	—
124	C₂×C₈	—	31	2	no	—
125	C₂×C₈	—	31	4	no	—
126	C₂×C₈	—	31	4	no	—
127	C₂×C₈	C₄×D₃₁	1	1	yes	characteristic
128	C₂×C₈	C₄×D₃₁	1	1	yes	characteristic
129	C₂×C₈	C₄×D₃₁	1	1	yes	characteristic
130	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
131	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
132	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
133	C₁₆	—	31	4	no	—
134	C₁₆	C₂²×D₃₁	1	4	yes	—
135	C₂³	—	31	2	no	—
136	C₂³	C₈×D₃₁	1	1	yes	characteristic
137	Q₈	—	31	2	no	—
138	Q₈	—	31	2	no	—
139	D₄	—	31	4	no	—
140	D₄	—	31	4	no	—
141	D₄	—	31	2	no	—
142	D₄	—	31	2	no	—
143	C₂×C₄	—	31	2	no	—
144	C₂×C₄	—	31	4	no	—
145	C₂×C₄	—	31	4	no	—
146	C₂×C₄	C₈×D₃₁	1	1	yes	characteristic
147	C₂×C₄	C₈×D₃₁	1	1	yes	characteristic
148	C₂×C₄	C₈×D₃₁	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
152	C₈	—	31	4	no	—
153	C₈	C₂×C₄×D₃₁	1	1	yes	characteristic
154	C₈	C₂×C₄×D₃₁	1	1	yes	characteristic
155	C₈	C₂×C₄×D₃₁	1	1	yes	characteristic
156	C₈	C₂³×D₃₁	1	1	yes	characteristic
157	C₂²	—	4	1	no	—
158	C₂²	—	31	4	no	—
159	C₂²	—	124	2	no	—
160	C₂²	C₂×C₈×D₃₁	1	1	yes	characteristic
161	C₂²	C₂×C₈×D₃₁	1	1	yes	characteristic
162	C₂²	C₂×C₈×D₃₁	1	1	yes	characteristic
163	C₄	—	31	4	no	—
164	C₄	C₂×C₈×D₃₁	1	1	yes	characteristic
165	C₄	C₂×C₈×D₃₁	1	1	yes	characteristic
166	C₄	C₂×C₈×D₃₁	1	1	yes	characteristic
167	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
168	C₂	—	2	1	no	—
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	—	62	4	no	—
172	C₂	C₂²×C₈×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×D₃₁
commutator, socle	C₂×C₃₁	C₂³×C₈
Fitting	C₂×C₃₁×M₃₂	C₂
prime radical	C₂×#92495	C₄
cosocle	C₈×C₃₁	C₂⁴
hypercenter	C₂×M₃₂	D₃₁
hypocenter	C₃₁	#3407
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. #14521467	C₂	semidirect
3. #14521465	C₂	semidirect
4. C₂×#11758400	C₂	semidirect
5. M₃₂×D₃₁	C₂	semidirect
6. C₂×C₃₁×M₃₂	C₂	semidirect
17. C₃₁×M₃₂	C₂²	semidirect
18. C₂×C₁₆×C₃₁	C₂²	semidirect
20. #11758400	C₂²	semidirect
21. C₂×C₃₁⋊C₁₆	C₂²	semidirect
22. #11758386	C₂²	semidirect
23. C₁₆×D₃₁	C₂²	semidirect
87. C₂×M₃₂	D₃₁	semidirect
106. M₃₂	C₂×D₃₁	semidirect
109. C₂×C₁₆	C₂×D₃₁	semidirect
110. C₃₁	#3407	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11758478	2
#14521467	2
#14521465	2
C₂×#11758400	2
M₃₂×D₃₁	2
C₂×C₃₁×M₃₂	2
#3407	31