Subgroups — #16854662

Order 3968.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₆₂	C₂	1	1	yes	maximal, characteristic
2	#14521178	C₂	1	4	yes	maximal, complemented
3	#14521176	C₂	1	2	yes	maximal, complemented
4	#14521173	C₂	1	2	yes	maximal, complemented
5	#14521072	C₂	1	2	yes	maximal
6	C₄×#92474	C₂	1	1	yes	maximal, characteristic
7	M₁₆×Dic₃₁	C₂	1	2	yes	maximal
8	C₃₁×#423	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₆₂	C₄	1	1	yes	characteristic
10	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₃₁	C₄	1	1	yes	characteristic
13	C₂×Dic₃₁⋊C₄	C₄	1	2	yes	—
14	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
15	C₂×#92474	C₂²	1	1	yes	characteristic
16	C₂×#92474	C₂²	1	1	yes	characteristic
17	C₄×Dic₆₂	C₄	1	1	yes	characteristic
18	C₄×Dic₆₂	C₄	1	2	yes	—
19	C₄×Dic₆₂	C₄	1	1	yes	characteristic
20	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
21	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
22	C₄×Dic₆₂	C₂²	1	2	yes	—
23	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
25	#11758403	C₂²	1	4	yes	—
26	Dic₃₁⋊C₈	C₂²	1	4	yes	—
27	Dic₃₁⋊C₈	C₂²	1	4	yes	—
28	C₈×Dic₃₁	C₂²	1	4	yes	—
29	#11758391	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	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
35	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
36	C₂×Dic₆₂	—	2	4	no	—
37	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₆₂	C₂×C₄	1	2	yes	—
39	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
40	M₁₆×C₃₁	—	2	4	no	—
41	C₂×C₈×C₃₁	C₂³	1	4	yes	—
42	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₄²×C₃₁	C₂³	1	1	yes	characteristic
46	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
48	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
49	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
50	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
51	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
52	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
53	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
54	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
55	C₄×Dic₃₁	C₂³	1	2	yes	—
56	C₄×Dic₃₁	C₂³	1	2	yes	—
57	#92474	—	2	2	no	—
58	#92474	Q₈	1	4	yes	complemented
59	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
60	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
61	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₃₁	—	2	2	no	—
63	C₂×C₄×C₃₁	—	4	1	no	—
64	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
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	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₃₁	#41	1	2	yes	—
71	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₃₁	—	2	4	no	—
73	C₂×Dic₃₁	—	4	2	no	—
74	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
75	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
76	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
77	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
78	Dic₆₂	—	2	2	no	—
79	Dic₆₂	—	2	4	no	—
80	Dic₆₂	M₁₆	1	4	yes	complemented
81	C₈×C₃₁	—	2	4	no	—
82	C₃₁⋊C₈	—	2	2	no	—
83	C₃₁⋊C₈	C₂×Q₈	1	4	yes	complemented
84	Q₈×M₁₆	—	31	1	no	maximal, Sylow
85	C₂²×C₃₁	—	2	1	no	—
86	C₂²×C₃₁	—	2	1	no	—
87	C₂²×C₃₁	C₄×Q₈	1	2	yes	—
88	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
89	C₄×C₃₁	—	2	1	no	—
90	C₄×C₃₁	—	2	1	no	—
91	C₄×C₃₁	—	2	1	no	—
92	C₄×C₃₁	C₄×Q₈	1	2	yes	—
93	C₄×C₃₁	C₂×M₁₆	1	2	yes	—
94	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
95	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
96	Dic₃₁	—	2	2	no	—
97	Dic₃₁	—	2	2	no	—
98	Dic₃₁	—	2	2	no	—
99	Dic₃₁	C₂×M₁₆	1	4	yes	—
100	C₂×C₄×Q₈	—	31	1	no	—
101	#446	—	31	2	no	—
102	#446	—	31	4	no	—
103	C₈×Q₈	—	31	2	no	—
104	#423	—	31	2	no	—
105	#423	D₃₁	1	1	yes	characteristic
106	C₄×M₁₆	—	31	2	no	—
107	C₄×M₁₆	—	31	1	no	—
108	C₂×C₃₁	—	2	2	no	—
109	C₂×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
110	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
111	C₂×C₃₁	#568	1	1	yes	characteristic
112	C₂²×Q₈	—	31	1	no	—
113	C₂×M₁₆	—	31	1	no	—
114	C₂×M₁₆	—	31	1	no	—
115	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
116	C₄×Q₈	—	31	1	no	—
117	C₄×Q₈	—	31	2	no	—
118	C₄×Q₈	—	31	1	no	—
119	C₄×Q₈	—	31	1	no	—
120	C₄×Q₈	—	31	2	no	—
121	C₄×Q₈	—	31	1	no	—
122	C₂×C₄⋊C₄	—	31	2	no	—
123	C₂×C₄⋊C₄	—	31	1	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	4	no	—
128	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
129	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
130	#97	—	31	4	no	—
131	#97	—	31	2	no	—
132	C₄×C₈	—	31	4	no	—
133	C₄×C₈	—	31	2	no	—
134	C₃₁	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
135	C₂×Q₈	—	31	1	no	—
136	C₂×Q₈	—	31	2	no	—
137	C₂×Q₈	—	31	1	no	—
138	C₂×Q₈	—	62	4	no	—
139	C₂²×C₄	—	31	2	no	—
140	C₂²×C₄	—	31	2	no	—
141	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
142	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
144	M₁₆	—	2	4	no	—
145	M₁₆	—	31	4	no	—
146	M₁₆	—	62	2	no	—
147	C₂×C₈	—	31	2	no	—
148	C₂×C₈	—	31	2	no	—
149	C₂×C₈	C₂²×D₃₁	1	4	yes	—
150	C₄⋊C₄	—	31	2	no	—
151	C₄⋊C₄	—	31	2	no	—
152	C₄⋊C₄	—	31	2	no	—
153	C₄⋊C₄	—	31	2	no	—
154	C₄⋊C₄	—	31	2	no	—
155	C₄⋊C₄	—	31	1	no	—
156	C₄⋊C₄	—	31	1	no	—
157	C₄²	—	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	C₂×C₄×D₃₁	1	1	yes	characteristic
166	Q₈	—	31	4	no	—
167	Q₈	—	62	4	no	—
168	Q₈	—	62	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	31	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	—	124	2	no	—
177	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
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	#92497	1	2	yes	—
183	C₂×C₄	Q₈×D₃₁	1	2	yes	—
184	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	31	4	no	—
187	C₈	—	62	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	#11758464	1	2	yes	—
191	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	31	4	no	—
196	C₄	—	62	2	no	—
197	C₄	—	62	2	no	—
198	C₄	—	62	2	no	—
199	C₄	#11758464	1	2	yes	—
200	C₄	M₁₆×D₃₁	1	2	yes	—
201	C₄	C₂×#92497	1	1	yes	characteristic
202	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₂×#11758464	1	1	yes	characteristic
205	C₂	C₂×M₁₆×D₃₁	1	1	yes	characteristic
206	C₂	#14521815	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#423	C₂
prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#423	D₃₁
hypocenter	C₃₁	Q₈×M₁₆
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
2. #14521178	C₂	semidirect
3. #14521176	C₂	semidirect
4. #14521173	C₂	semidirect
58. #92474	Q₈	semidirect
80. Dic₆₂	M₁₆	semidirect
83. C₃₁⋊C₈	C₂×Q₈	semidirect
134. C₃₁	Q₈×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₆₂	2
#14521178	2
#14521176	2
#14521173	2
#14521072	2
C₄×#92474	2
M₁₆×Dic₃₁	2
C₃₁×#423	2
Q₈×M₁₆	31