Subgroups — #16854355

Order 3968.

400 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521110	C₂	1	4	yes	maximal, complemented
2	#14521102	C₂	1	4	yes	maximal, complemented
3	#14521076	C₂	1	2	yes	maximal, complemented
4	#14521016	C₂	1	2	yes	maximal
5	C₄×#92474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₃₁	C₄	1	2	yes	—
10	C₄×D₄×C₃₁	C₄	1	4	yes	—
11	C₄×D₄×C₃₁	C₄	1	2	yes	—
12	C₃₁×#117	C₂²	1	2	yes	—
13	C₃₁×#117	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×#92474	C₂²	1	2	yes	—
17	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
18	C₂²⋊(C₃₁⋊C₈)	C₂²	1	8	yes	—
19	C₄⋊(C₃₁⋊C₈)	C₂²	1	8	yes	—
20	#11758391	C₂²	1	2	yes	—
21	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
22	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
23	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
24	#41×C₃₁	—	2	4	no	—
25	C₂×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
31	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
33	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
34	C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
35	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
36	C₄²×C₃₁	C₂×C₄	1	4	yes	—
37	C₄²×C₃₁	C₂×C₄	1	2	yes	—
38	C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₄²×C₃₁	C₂³	1	2	yes	—
40	C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	#92474	—	2	4	no	—
42	C₂×C₃₁⋊C₈	—	2	4	no	—
43	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
44	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
45	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
46	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₃₁	—	2	2	no	—
48	D₄×C₃₁	—	2	4	no	—
49	D₄×C₃₁	—	2	2	no	—
50	C₂×C₄×C₃₁	—	2	4	no	—
51	C₂×C₄×C₃₁	—	2	8	no	—
52	C₂×C₄×C₃₁	—	2	4	no	—
53	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₃₁	#41	1	4	yes	—
59	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
60	C₃₁⋊C₈	—	2	4	no	—
61	C₃₁⋊C₈	—	2	4	no	—
62	#2922	—	31	1	no	maximal, Sylow
63	C₂²×C₃₁	—	2	2	no	—
64	C₂²×C₃₁	—	2	2	no	—
65	C₂²×C₃₁	—	2	4	no	—
66	C₂²×C₃₁	#117	1	2	yes	—
67	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
68	C₄×C₃₁	—	2	2	no	—
69	C₄×C₃₁	—	2	2	no	—
70	C₄×C₃₁	—	2	4	no	—
71	C₄×C₃₁	#117	1	2	yes	—
72	C₄×C₃₁	#131	1	4	yes	—
73	C₄×C₃₁	C₂×#41	1	2	yes	—
74	C₄×#41	D₃₁	1	1	yes	characteristic
75	#433	—	31	4	no	—
76	#431	—	31	4	no	—
77	#424	—	31	2	no	—
78	#408	—	31	2	no	—
79	C₄×M₁₆	—	31	1	no	—
80	C₂×C₄×C₈	—	31	1	no	—
81	C₂×C₃₁	—	2	2	no	—
82	C₂×C₃₁	—	4	2	no	—
83	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
84	C₂×C₃₁	C₂×#131	1	2	yes	—
85	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
86	C₂×M₁₆	—	31	2	no	—
87	C₂²×C₈	—	31	2	no	—
88	C₄×Q₈	Dic₃₁	1	2	yes	—
89	C₄×D₄	Dic₃₁	1	4	yes	—
90	C₄×D₄	Dic₃₁	1	2	yes	—
91	#117	C₂×D₃₁	1	1	yes	characteristic
92	#117	C₂×D₃₁	1	2	yes	—
93	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
94	C₂×C₄²	C₂×D₃₁	1	2	yes	—
95	C₄⋊C₈	—	31	8	no	—
96	C₂²⋊C₈	—	31	8	no	—
97	#97	—	31	2	no	—
98	C₄×C₈	—	31	2	no	—
99	C₄×C₈	—	31	2	no	—
100	C₄×C₈	—	31	2	no	—
101	C₃₁	#2922	1	1	yes	characteristic, complemented, Sylow
102	#41	—	2	4	no	—
103	C₂×Q₈	C₂×Dic₃₁	1	1	yes	characteristic
104	C₂×D₄	C₂×Dic₃₁	1	2	yes	—
105	C₂×D₄	C₂×Dic₃₁	1	1	yes	characteristic
106	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
107	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
108	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
109	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
110	M₁₆	—	62	4	no	—
111	C₂×C₈	—	31	4	no	—
112	C₂×C₈	—	31	4	no	—
113	C₂×C₈	—	62	4	no	—
114	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
115	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
116	C₂²⋊C₄	C₂×Dic₃₁	1	4	yes	—
117	C₂²⋊C₄	C₂×Dic₃₁	1	2	yes	—
118	C₄²	C₂×Dic₃₁	1	4	yes	—
119	C₄²	C₂×Dic₃₁	1	2	yes	—
120	C₄²	C₂²×D₃₁	1	1	yes	characteristic
121	C₄²	C₂²×D₃₁	1	2	yes	—
122	C₄²	C₂²×D₃₁	1	1	yes	characteristic
123	C₂³	C₂²×Dic₃₁	1	2	yes	—
124	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	8	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	#92495	1	4	yes	—
132	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
133	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
134	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
135	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
136	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
137	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
138	C₈	—	62	4	no	—
139	C₈	—	62	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	4	no	—
143	C₂²	#11758503	1	2	yes	—
144	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	4	no	—
148	C₄	#11758503	1	2	yes	—
149	C₄	#11758525	1	4	yes	—
150	C₄	C₂×#92495	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	4	2	no	—
153	C₂	C₂×#11758503	1	1	yes	characteristic
154	C₂	C₂×#11758525	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₄×#41×C₃₁	C₂
prime radical	C₂×#41×C₃₁	C₂²
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₄×#41	D₃₁
hypocenter	C₃₁	#2922
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #14521110	C₂	semidirect
2. #14521102	C₂	semidirect
3. #14521076	C₂	semidirect
5. C₄×#92474	C₂	semidirect
6. C₂×C₄×C₃₁⋊C₈	C₂	semidirect
101. C₃₁	#2922	semidirect

Maximal subgroups

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

subgroup	index
#14521110	2
#14521102	2
#14521076	2
#14521016	2
C₄×#92474	2
C₂×C₄×C₃₁⋊C₈	2
C₄×#41×C₃₁	2
#2922	31