Subgroups — #16854677

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	C₄×#92495	C₂	1	1	yes	maximal, characteristic
2	#14521179	C₂	1	2	yes	maximal, complemented
3	#14521128	C₂	1	2	yes	maximal, complemented
4	#14521120	C₂	1	4	yes	maximal, complemented
5	#14521102	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521100	C₂	1	2	yes	maximal, complemented
7	C₄×#92469	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#92495	C₄	1	1	yes	characteristic
10	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₄×C₂²⋊D₃₁	C₄	1	2	yes	—
13	C₄×C₂²⋊D₃₁	C₂²	1	2	yes	—
14	#11758503	C₄	1	1	yes	characteristic
15	C₂×#92469	C₂²	1	4	yes	—
16	C₄×D₁₂₄	C₄	1	1	yes	characteristic
17	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
18	#11758449	C₄	1	2	yes	—
19	C₄²×D₃₁	C₂²	1	2	yes	—
20	C₄×Dic₆₂	C₄	1	1	yes	characteristic
21	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₄×C₈×C₃₁	C₂²	1	2	yes	—
24	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
26	(C₂×D₃₁)⋊C₈	C₂²	1	4	yes	—
27	#11758403	C₂²	1	4	yes	—
28	Dic₃₁⋊C₈	C₂²	1	4	yes	—
29	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
30	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
31	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
35	#92495	—	2	4	no	—
36	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₃₁	C₂³	1	2	yes	—
40	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₃₁	—	2	4	no	—
42	C₂×C₈×C₃₁	C₂³	1	2	yes	—
43	C₂×C₈×C₃₁	C₂³	1	2	yes	—
44	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₃₁	C₂³	1	1	yes	characteristic
47	C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
49	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
50	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
51	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
53	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
55	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
56	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
57	C₄×Dic₃₁	C₂³	1	2	yes	—
58	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
59	#92469	—	2	8	no	—
60	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₃₁	—	2	2	no	—
63	C₂×C₄×C₃₁	—	2	2	no	—
64	C₂×C₄×C₃₁	M₁₆	1	4	yes	complemented
65	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₃₁	—	2	4	no	—
72	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
73	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
74	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
75	D₁₂₄	—	2	2	no	—
76	C₄×D₃₁	—	2	4	no	—
77	C₄×D₃₁	—	2	4	no	—
78	C₄×D₃₁	—	2	4	no	—
79	Dic₆₂	—	2	2	no	—
80	C₈×C₃₁	—	2	2	no	—
81	C₈×C₃₁	#41	1	4	yes	complemented
82	C₃₁⋊C₈	—	2	4	no	—
83	#2965	—	31	1	no	maximal, Sylow
84	C₂²×C₃₁	—	2	1	no	—
85	C₂²×C₃₁	—	2	1	no	—
86	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
87	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
88	C₂×D₃₁	—	2	2	no	—
89	C₂×D₃₁	—	2	2	no	—
90	C₂×D₃₁	—	2	2	no	—
91	C₄×C₃₁	—	2	1	no	—
92	C₄×C₃₁	—	2	1	no	—
93	C₄×C₃₁	C₂×M₁₆	1	2	yes	—
94	C₄×C₃₁	C₂×M₁₆	1	2	yes	—
95	C₄×C₃₁	#131	1	2	yes	—
96	C₄×C₃₁	C₂×#41	1	2	yes	—
97	Dic₃₁	—	2	2	no	—
98	Dic₃₁	—	2	2	no	—
99	Dic₃₁	—	2	2	no	—
100	C₄×#41	—	31	1	no	—
101	#446	—	31	2	no	—
102	#436	—	31	2	no	—
103	#435	—	31	4	no	—
104	#431	—	31	1	no	—
105	#431	—	31	2	no	—
106	C₄×M₁₆	—	31	2	no	—
107	C₂×C₄×C₈	D₃₁	1	1	yes	characteristic, complemented
108	C₂×C₃₁	—	2	2	no	—
109	C₂×C₃₁	C₄×#41	1	1	yes	characteristic
110	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
111	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
112	D₃₁	—	4	2	no	—
113	C₂×#41	—	31	1	no	—
114	C₂×M₁₆	—	31	4	no	—
115	C₂²×C₈	C₂×D₃₁	1	2	yes	—
116	C₄×Q₈	—	31	1	no	—
117	C₄×Q₈	—	31	1	no	—
118	C₄×D₄	—	31	2	no	—
119	C₄×D₄	—	31	2	no	—
120	C₄×D₄	—	31	1	no	—
121	C₄×D₄	—	31	1	no	—
122	#117	—	31	1	no	—
123	#117	—	31	2	no	—
124	C₂×C₄²	—	31	2	no	—
125	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
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	C₂×D₃₁	1	2	yes	—
133	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
134	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
135	C₃₁	#2965	1	1	yes	characteristic, complemented, Sylow
136	#41	—	62	4	no	—
137	C₂×Q₈	—	31	1	no	—
138	C₂×D₄	—	31	2	no	—
139	C₂×D₄	—	31	1	no	—
140	C₂²×C₄	—	31	2	no	—
141	C₂²×C₄	—	31	2	no	—
142	C₂²×C₄	—	31	2	no	—
143	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
146	M₁₆	—	62	8	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	31	4	no	—
149	C₂×C₈	C₂²×D₃₁	1	2	yes	—
150	C₂×C₈	C₂²×D₃₁	1	2	yes	—
151	C₄⋊C₄	—	31	2	no	—
152	C₄⋊C₄	—	31	2	no	—
153	C₄⋊C₄	—	31	1	no	—
154	C₄⋊C₄	—	31	1	no	—
155	C₂²⋊C₄	—	31	2	no	—
156	C₂²⋊C₄	—	31	2	no	—
157	C₂²⋊C₄	—	31	2	no	—
158	C₄²	—	31	2	no	—
159	C₄²	—	31	2	no	—
160	C₄²	—	31	2	no	—
161	C₄²	C₄×D₃₁	1	1	yes	characteristic
162	C₄²	C₄×D₃₁	1	1	yes	characteristic
163	C₄²	C₂²×D₃₁	1	1	yes	characteristic
164	C₄²	C₂²×D₃₁	1	1	yes	characteristic
165	C₂³	—	31	2	no	—
166	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
167	Q₈	—	62	2	no	—
168	D₄	—	62	4	no	—
169	D₄	—	62	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	31	2	no	—
173	C₂×C₄	—	31	2	no	—
174	C₂×C₄	—	31	2	no	—
175	C₂×C₄	—	62	4	no	—
176	C₂×C₄	—	62	4	no	—
177	C₂×C₄	—	62	4	no	—
178	C₂×C₄	#92469	1	4	yes	complemented
179	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	62	4	no	—
187	C₈	#92495	1	4	yes	complemented
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	62	2	no	—
191	C₂²	—	62	2	no	—
192	C₂²	—	62	2	no	—
193	C₂²	C₂×#92469	1	2	yes	—
194	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	62	2	no	—
198	C₄	—	62	2	no	—
199	C₄	—	62	2	no	—
200	C₄	C₂×#92469	1	2	yes	—
201	C₄	C₂×#92469	1	2	yes	—
202	C₄	#11758478	1	2	yes	—
203	C₄	C₂×#92495	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	124	2	no	—
206	C₂	C₄×#92495	1	1	yes	characteristic
207	C₂	C₂²×#92469	1	1	yes	characteristic
208	C₂	C₂×#11758478	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₄×C₈×C₃₁	C₂
prime radical	C₂×#92495	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×C₄×C₈	D₃₁
hypocenter	C₃₁	#2965
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #14521179	C₂	semidirect
3. #14521128	C₂	semidirect
4. #14521120	C₂	semidirect
5. #14521102	C₂	semidirect
6. #14521100	C₂	semidirect
7. C₄×#92469	C₂	semidirect
8. C₂×C₄×C₈×C₃₁	C₂	semidirect
64. C₂×C₄×C₃₁	M₁₆	semidirect
81. C₈×C₃₁	#41	semidirect
107. C₂×C₄×C₈	D₃₁	semidirect
135. C₃₁	#2965	semidirect
178. C₂×C₄	#92469	semidirect
187. C₈	#92495	semidirect

Maximal subgroups

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

subgroup	index
C₄×#92495	2
#14521179	2
#14521128	2
#14521120	2
#14521102	2
#14521100	2
C₄×#92469	2
C₂×C₄×C₈×C₃₁	2
#2965	31