Subgroups — #16854643

Order 3968.

364 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521602	C₂	1	1	yes	maximal, characteristic
2	#14521177	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₆₂	C₂	1	4	yes	maximal, complemented
4	#14521077	C₂	1	1	yes	maximal, characteristic
5	#14521076	C₂	1	1	yes	maximal, characteristic
6	#14521072	C₂	1	1	yes	maximal, characteristic
7	#14521004	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#405	C₂	1	1	yes	maximal, characteristic
10	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#117	C₂²	1	1	yes	characteristic
13	#11758503	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
15	C₂×#92474	C₂²	1	1	yes	characteristic
16	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
17	#11758467	C₄	1	2	yes	—
18	#11758465	C₄	1	2	yes	—
19	#11758455	C₄	1	4	yes	—
20	C₄×Dic₆₂	C₂²	1	4	yes	—
21	C₃₁×#97	C₂²	1	2	yes	—
22	C₄×C₈×C₃₁	C₂²	1	2	yes	—
23	#11758403	C₂²	1	2	yes	—
24	Dic₃₁⋊C₈	C₂²	1	4	yes	—
25	Dic₃₁⋊C₈	C₂²	1	4	yes	—
26	C₈×Dic₃₁	C₂²	1	2	yes	—
27	C₈×Dic₃₁	C₂²	1	2	yes	—
28	C₈×Dic₃₁	C₂²	1	2	yes	—
29	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
30	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
32	C₂×Dic₆₂	C₂×C₄	1	4	yes	—
33	M₁₆×C₃₁	—	2	2	no	—
34	C₂×C₈×C₃₁	Q₈	1	4	yes	complemented
35	C₂×C₈×C₃₁	C₂³	1	2	yes	—
36	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
39	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
40	C₄²×C₃₁	C₂³	1	2	yes	—
41	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
43	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
44	Dic₃₁⋊C₄	C₂×C₄	1	4	yes	—
45	Dic₃₁⋊C₄	C₂×C₄	1	4	yes	—
46	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₃₁	C₂³	1	2	yes	—
49	C₄×Dic₃₁	C₂³	1	2	yes	—
50	#92474	—	2	2	no	—
51	C₂×C₃₁⋊C₈	—	2	2	no	—
52	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
53	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
54	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₃₁	#41	1	2	yes	—
59	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₃₁	—	2	4	no	—
61	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
62	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
63	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
64	Dic₆₂	—	2	8	no	—
65	C₈×C₃₁	—	2	2	no	—
66	C₈×C₃₁	C₂×Q₈	1	2	yes	—
67	C₈×C₃₁	C₂×Q₈	1	2	yes	—
68	C₃₁⋊C₈	—	2	2	no	—
69	C₃₁⋊C₈	—	2	2	no	—
70	#2960	—	31	1	no	maximal, Sylow
71	C₂²×C₃₁	—	2	2	no	—
72	C₂²×C₃₁	C₄×Q₈	1	2	yes	—
73	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₃₁	—	2	4	no	—
75	C₄×C₃₁	C₄×Q₈	1	2	yes	—
76	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
77	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
78	Dic₃₁	—	2	4	no	—
79	Dic₃₁	—	2	2	no	—
80	Dic₃₁	#131	1	4	yes	—
81	#533	—	31	1	no	—
82	#446	—	31	4	no	—
83	C₈×Q₈	—	31	4	no	—
84	#424	—	31	1	no	—
85	#424	—	31	1	no	—
86	#423	—	31	1	no	—
87	#405	—	31	1	no	—
88	#405	D₃₁	1	1	yes	characteristic
89	C₂×C₄×C₈	—	31	1	no	—
90	C₂×C₃₁	—	2	2	no	—
91	C₂×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₂×C₃₁	C₂×#131	1	2	yes	—
93	C₂×M₁₆	—	31	1	no	—
94	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
95	C₂²×C₈	—	31	1	no	—
96	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
97	C₄⋊Q₈	—	31	2	no	—
98	#125	—	31	2	no	—
99	C₂²⋊Q₈	—	31	4	no	—
100	C₄×Q₈	—	31	4	no	—
101	#117	—	31	1	no	—
102	#117	C₂×D₃₁	1	1	yes	characteristic
103	C₂×C₄²	—	31	1	no	—
104	C₄⋊C₈	—	31	4	no	—
105	C₄⋊C₈	—	31	4	no	—
106	C₄⋊C₈	—	31	4	no	—
107	#97	—	31	2	no	—
108	#97	C₂×D₃₁	1	2	yes	—
109	C₄×C₈	—	31	2	no	—
110	C₄×C₈	—	31	2	no	—
111	C₄×C₈	—	31	2	no	—
112	C₄×C₈	C₂×D₃₁	1	2	yes	—
113	C₃₁	#2960	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₈	—	31	4	no	—
115	C₂²×C₄	—	31	2	no	—
116	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
117	M₁₆	—	2	2	no	—
118	M₁₆	—	62	2	no	—
119	C₂×C₈	—	31	2	no	—
120	C₂×C₈	—	31	2	no	—
121	C₂×C₈	—	62	2	no	—
122	C₂×C₈	Dic₆₂	1	4	yes	complemented
123	C₂×C₈	C₂²×D₃₁	1	2	yes	—
124	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
125	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
126	C₄⋊C₄	—	31	4	no	—
127	C₄⋊C₄	—	31	4	no	—
128	C₄⋊C₄	—	31	2	no	—
129	C₄⋊C₄	—	31	4	no	—
130	C₄⋊C₄	C₄×D₃₁	1	2	yes	—
131	C₂²⋊C₄	—	31	2	no	—
132	C₂²⋊C₄	C₄×D₃₁	1	2	yes	—
133	C₄²	—	31	1	no	—
134	C₄²	—	31	1	no	—
135	C₄²	—	31	2	no	—
136	C₄²	—	31	2	no	—
137	C₄²	C₂²×D₃₁	1	2	yes	—
138	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
139	Q₈	—	62	8	no	—
140	C₂×C₄	—	31	2	no	—
141	C₂×C₄	—	31	2	no	—
142	C₂×C₄	—	31	4	no	—
143	C₂×C₄	—	62	4	no	—
144	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
146	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
147	C₂×C₄	#92495	1	2	yes	—
148	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	62	2	no	—
151	C₈	—	62	2	no	—
152	C₈	C₂×Dic₆₂	1	2	yes	—
153	C₈	C₂×Dic₆₂	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	C₄×Dic₆₂	1	2	yes	—
156	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	31	4	no	—
159	C₄	—	62	2	no	—
160	C₄	—	62	4	no	—
161	C₄	C₄×Dic₆₂	1	2	yes	—
162	C₄	C₂²×Dic₆₂	1	1	yes	characteristic
163	C₄	C₂×#92495	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₂×C₄×Dic₆₂	1	1	yes	characteristic
166	C₂	#131×D₃₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#405	C₂
prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#405	D₃₁
hypocenter	C₃₁	#2960
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #14521177	C₂	semidirect
3. C₈×Dic₆₂	C₂	semidirect
34. C₂×C₈×C₃₁	Q₈	semidirect
113. C₃₁	#2960	semidirect
122. C₂×C₈	Dic₆₂	semidirect

Maximal subgroups

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

subgroup	index
#14521602	2
#14521177	2
C₈×Dic₆₂	2
#14521077	2
#14521076	2
#14521072	2
#14521004	2
C₂×C₈×Dic₃₁	2
C₃₁×#405	2
#2960	31