Subgroups — #16854678

Order 3968.

412 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521180	C₂	1	2	yes	maximal, complemented
2	#14521127	C₂	1	2	yes	maximal, complemented
3	#14521121	C₂	1	4	yes	maximal, complemented
4	#14521102	C₂	1	2	yes	maximal, complemented
5	#14521102	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#92474	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#41×C₃₁	C₄	1	1	yes	characteristic
10	C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
11	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₄×D₄×C₃₁	C₄	1	2	yes	—
13	C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
14	C₄×D₄×C₃₁	C₂²	1	2	yes	—
15	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#117	C₄	1	2	yes	—
17	C₃₁×#117	C₄	1	1	yes	characteristic
18	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
20	C₂×#92474	C₂²	1	4	yes	—
21	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
22	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
23	C₂²⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
24	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
25	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
26	#11758391	C₂²	1	4	yes	—
27	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
28	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
29	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
30	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
31	#41×C₃₁	—	2	4	no	—
32	C₂×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
41	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
42	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
46	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
47	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
48	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₃₁	C₂×C₄	1	2	yes	—
50	C₄²×C₃₁	C₂×C₄	1	2	yes	—
51	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₄²×C₃₁	C₂³	1	2	yes	—
54	C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	#92474	—	2	8	no	—
56	C₂×C₃₁⋊C₈	—	2	4	no	—
57	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
58	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
59	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
60	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
61	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	Q₈×C₃₁	—	2	2	no	—
63	D₄×C₃₁	—	2	4	no	—
64	D₄×C₃₁	—	2	2	no	—
65	C₂×C₄×C₃₁	—	2	4	no	—
66	C₂×C₄×C₃₁	—	2	2	no	—
67	C₂×C₄×C₃₁	—	2	4	no	—
68	C₂×C₄×C₃₁	—	2	2	no	—
69	C₂×C₄×C₃₁	—	2	4	no	—
70	C₂×C₄×C₃₁	M₁₆	1	4	yes	complemented
71	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
80	C₃₁⋊C₈	—	2	2	no	—
81	C₃₁⋊C₈	—	2	4	no	—
82	C₃₁⋊C₈	#41	1	4	yes	complemented
83	#2965	—	31	1	no	maximal, Sylow
84	C₂²×C₃₁	—	2	1	no	—
85	C₂²×C₃₁	—	2	1	no	—
86	C₂²×C₃₁	—	2	2	no	—
87	C₂²×C₃₁	—	2	2	no	—
88	C₂²×C₃₁	—	2	2	no	—
89	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
90	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
91	C₄×C₃₁	—	2	1	no	—
92	C₄×C₃₁	—	2	2	no	—
93	C₄×C₃₁	—	2	2	no	—
94	C₄×C₃₁	—	2	1	no	—
95	C₄×C₃₁	—	2	2	no	—
96	C₄×C₃₁	C₂×M₁₆	1	2	yes	—
97	C₄×C₃₁	C₂×M₁₆	1	2	yes	—
98	C₄×C₃₁	#131	1	2	yes	—
99	C₄×C₃₁	C₂×#41	1	2	yes	—
100	C₄×#41	D₃₁	1	1	yes	characteristic
101	#446	—	31	2	no	—
102	#436	—	31	2	no	—
103	#435	—	31	4	no	—
104	#431	—	31	2	no	—
105	#431	—	31	1	no	—
106	C₄×M₁₆	—	31	2	no	—
107	C₂×C₄×C₈	—	31	1	no	—
108	C₂×C₃₁	—	2	2	no	—
109	C₂×C₃₁	—	4	2	no	—
110	C₂×C₃₁	C₄×#41	1	1	yes	characteristic
111	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
112	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
113	C₂×#41	Dic₃₁	1	1	yes	characteristic
114	C₂×M₁₆	—	31	4	no	—
115	C₂²×C₈	—	31	2	no	—
116	C₄×Q₈	Dic₃₁	1	1	yes	characteristic
117	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
118	C₄×D₄	Dic₃₁	1	2	yes	—
119	C₄×D₄	Dic₃₁	1	1	yes	characteristic
120	C₄×D₄	C₂×D₃₁	1	2	yes	—
121	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
122	#117	Dic₃₁	1	2	yes	—
123	#117	Dic₃₁	1	1	yes	characteristic
124	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
125	C₂×C₄²	C₂×D₃₁	1	2	yes	—
126	C₄⋊C₈	—	31	4	no	—
127	C₄⋊C₈	—	31	2	no	—
128	C₂²⋊C₈	—	31	2	no	—
129	C₂²⋊C₈	—	31	4	no	—
130	#97	—	31	4	no	—
131	C₄×C₈	—	31	2	no	—
132	C₄×C₈	—	31	1	no	—
133	C₄×C₈	—	31	1	no	—
134	C₄×C₈	—	31	2	no	—
135	C₃₁	#2965	1	1	yes	characteristic, complemented, Sylow
136	#41	—	2	4	no	—
137	C₂×Q₈	C₂×Dic₃₁	1	1	yes	characteristic
138	C₂×D₄	C₂×Dic₃₁	1	2	yes	—
139	C₂×D₄	C₂×Dic₃₁	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
141	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
142	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
144	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
146	M₁₆	—	62	8	no	—
147	C₂×C₈	—	31	2	no	—
148	C₂×C₈	—	31	2	no	—
149	C₂×C₈	—	31	4	no	—
150	C₂×C₈	—	62	4	no	—
151	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
152	C₄⋊C₄	C₂×Dic₃₁	1	1	yes	characteristic
153	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
154	C₄⋊C₄	C₂×Dic₃₁	1	1	yes	characteristic
155	C₂²⋊C₄	C₂×Dic₃₁	1	2	yes	—
156	C₂²⋊C₄	C₂×Dic₃₁	1	2	yes	—
157	C₂²⋊C₄	C₂×Dic₃₁	1	2	yes	—
158	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
159	C₄²	C₂×Dic₃₁	1	2	yes	—
160	C₄²	C₂×Dic₃₁	1	2	yes	—
161	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
162	C₄²	C₂²×D₃₁	1	1	yes	characteristic
163	C₄²	C₂²×D₃₁	1	2	yes	—
164	C₄²	C₂²×D₃₁	1	1	yes	characteristic
165	C₂³	C₂²×Dic₃₁	1	2	yes	—
166	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	#92474	1	4	yes	complemented
176	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
178	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
179	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
180	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
181	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
183	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
185	C₈	—	31	4	no	—
186	C₈	—	62	2	no	—
187	C₈	—	62	4	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	C₂×#92474	1	2	yes	—
194	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	2	no	—
200	C₄	C₂×#92474	1	2	yes	—
201	C₄	C₂×#92474	1	2	yes	—
202	C₄	#11758525	1	2	yes	—
203	C₄	#41×D₃₁	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	4	2	no	—
206	C₂	#41×Dic₃₁	1	1	yes	characteristic
207	C₂	C₂²×#92474	1	1	yes	characteristic
208	C₂	C₂×#11758525	1	1	yes	characteristic

	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	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₄×#41	D₃₁
hypocenter	C₃₁	#2965
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. #14521180	C₂	semidirect
2. #14521127	C₂	semidirect
3. #14521121	C₂	semidirect
4. #14521102	C₂	semidirect
5. #14521102	C₂	semidirect
6. C₄×#92474	C₂	semidirect
7. C₂×C₄×C₃₁⋊C₈	C₂	semidirect
70. C₂×C₄×C₃₁	M₁₆	semidirect
82. C₃₁⋊C₈	#41	semidirect
135. C₃₁	#2965	semidirect
175. C₂×C₄	#92474	semidirect

Maximal subgroups

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

subgroup	index
#14521180	2
#14521127	2
#14521121	2
#14521102	2
C₄×#92474	2
C₂×C₄×C₃₁⋊C₈	2
C₄×#41×C₃₁	2
#2965	31