Subgroups — #16848109

Order 3968.

568 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14520917	C₂	1	2	yes	maximal, complemented
2	#14520915	C₂	1	2	yes	maximal, complemented
3	#14520912	C₂	1	2	yes	maximal
4	#14520909	C₂	1	2	yes	maximal
5	C₄×C₄⋊Dic₃₁	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊Dic₃₁	C₂	1	2	yes	maximal
7	#14520895	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₃₁	C₄	1	8	yes	complemented
13	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊Dic₃₁	C₂²	1	4	yes	—
18	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
19	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
20	C₃₁⋊#95	C₂²	1	4	yes	—
21	C₃₁⋊#95	C₂²	1	4	yes	—
22	C₃₁⋊#95	C₂²	1	4	yes	—
23	C₃₁⋊#95	C₂²	1	4	yes	—
24	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	—	2	4	no	—
26	C₂²×C₄×C₃₁	—	2	4	no	—
27	C₂²×C₄×C₃₁	—	2	4	no	—
28	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
37	C₂²×Dic₃₁	C₂³	1	8	yes	—
38	C₄²×C₃₁	C₂×C₄	1	4	yes	—
39	C₄²×C₃₁	C₂×C₄	1	4	yes	—
40	C₄²×C₃₁	C₂×C₄	1	4	yes	—
41	C₄²×C₃₁	C₂×C₄	1	4	yes	—
42	C₂²⋊Dic₃₁	—	2	8	no	—
43	C₄⋊Dic₃₁	—	2	8	no	—
44	C₄×Dic₃₁	—	2	8	no	—
45	C₂³×C₃₁	—	2	2	no	—
46	C₂³×C₃₁	—	2	2	no	—
47	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
48	C₂³×C₃₁	#41	1	4	yes	—
49	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₃₁	—	2	8	no	—
51	C₂×C₄×C₃₁	—	2	2	no	—
52	C₂×C₄×C₃₁	—	2	2	no	—
53	C₂×C₄×C₃₁	—	2	2	no	—
54	C₂×C₄×C₃₁	—	2	2	no	—
55	C₂×C₄×C₃₁	—	2	8	no	—
56	C₂×C₄×C₃₁	—	2	8	no	—
57	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₃₁	#41	1	4	yes	—
64	C₂×C₄×C₃₁	#41	1	4	yes	—
65	C₂×Dic₃₁	—	2	8	no	—
66	C₂×Dic₃₁	—	2	8	no	—
67	C₂×Dic₃₁	—	2	8	no	—
68	#2283	—	31	1	no	maximal, Sylow
69	C₂²×C₃₁	—	2	2	no	—
70	C₂²×C₃₁	—	2	2	no	—
71	C₂²×C₃₁	—	2	8	no	—
72	C₂²×C₃₁	#117	1	4	yes	—
73	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
74	C₂²×C₃₁	C₂×#41	1	2	yes	—
75	C₂²×C₃₁	C₂×#41	1	2	yes	—
76	C₂²×C₃₁	C₂×#41	1	2	yes	—
77	C₄×C₃₁	—	2	2	no	—
78	C₄×C₃₁	—	2	2	no	—
79	C₄×C₃₁	—	2	2	no	—
80	C₄×C₃₁	—	2	2	no	—
81	C₄×C₃₁	#117	1	4	yes	—
82	C₄×C₃₁	#117	1	4	yes	—
83	Dic₃₁	—	4	8	no	—
84	C₂²×C₄²	D₃₁	1	1	yes	characteristic
85	#383	—	31	2	no	—
86	#382	—	31	2	no	—
87	#381	—	31	2	no	—
88	#380	—	31	2	no	—
89	C₄×C₄⋊C₄	—	31	2	no	—
90	C₄×C₂²⋊C₄	—	31	2	no	—
91	#376	—	31	2	no	—
92	C₂×C₃₁	—	2	4	no	—
93	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
94	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
95	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
96	C₂×C₃₁	#529	1	4	yes	—
97	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
98	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
99	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	31	4	no	—
101	C₂×C₂²⋊C₄	—	31	4	no	—
102	C₂×C₄²	—	31	4	no	—
103	C₂×C₄²	Dic₃₁	1	8	yes	complemented
104	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
105	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
106	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
107	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
108	#95	—	31	4	no	—
109	#95	—	31	4	no	—
110	#95	—	31	4	no	—
111	#95	—	31	4	no	—
112	C₃₁	#2283	1	1	yes	characteristic, complemented, Sylow
113	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
114	C₂²×C₄	—	2	4	no	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	—	31	8	no	—
118	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
119	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
120	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
121	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
127	C₄⋊C₄	—	62	8	no	—
128	C₂²⋊C₄	—	62	8	no	—
129	C₄²	—	62	8	no	—
130	C₄²	C₂×Dic₃₁	1	4	yes	—
131	C₄²	C₂×Dic₃₁	1	4	yes	—
132	C₄²	C₂×Dic₃₁	1	4	yes	—
133	C₄²	C₂×Dic₃₁	1	4	yes	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	2	no	—
136	C₂³	#92495	1	4	yes	—
137	C₂³	C₂²×Dic₃₁	1	2	yes	—
138	C₂³	C₂³×D₃₁	1	1	yes	characteristic
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	62	8	no	—
147	C₂×C₄	—	62	8	no	—
148	C₂×C₄	—	62	8	no	—
149	C₂×C₄	#92495	1	4	yes	—
150	C₂×C₄	#92495	1	4	yes	—
151	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
152	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
153	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
154	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
155	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
156	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	8	no	—
160	C₂²	#11758503	1	4	yes	—
161	C₂²	C₂×#92495	1	2	yes	—
162	C₂²	C₂×#92495	1	2	yes	—
163	C₂²	C₂×#92495	1	2	yes	—
164	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	124	8	no	—
170	C₄	#11758503	1	4	yes	—
171	C₄	#11758503	1	4	yes	—
172	C₂	—	2	4	no	—
173	C₂	C₂×#11758503	1	1	yes	characteristic
174	C₂	C₂×#11758503	1	1	yes	characteristic
175	C₂	C₂×#11758503	1	1	yes	characteristic
176	C₂	#14521583	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Frattini	C₂³	C₂³×D₃₁
Fitting	C₂²×C₄²×C₃₁	C₂
socle, cosocle	C₂³×C₃₁	C₂⁴
prime radical	C₂⁴×C₃₁	C₂³
hypercenter	C₂²×C₄²	D₃₁
hypocenter	C₃₁	#2283
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #14520917	C₂	semidirect
2. #14520915	C₂	semidirect
5. C₄×C₄⋊Dic₃₁	C₂	semidirect
7. #14520895	C₂	semidirect
12. C₂×C₄²×C₃₁	C₄	semidirect
103. C₂×C₄²	Dic₃₁	semidirect
112. C₃₁	#2283	semidirect

Maximal subgroups

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

subgroup	index
#14520917	2
#14520915	2
#14520912	2
#14520909	2
C₄×C₄⋊Dic₃₁	2
C₄×C₂²⋊Dic₃₁	2
#14520895	2
C₂²×C₄²×C₃₁	2
#2283	31