Subgroups — #16854801

Order 3968.

364 conjugacy classes of subgroups.

The subgroup lattice has 198 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	#14521177	C₂	1	4	yes	maximal, complemented
3	#14521176	C₂	1	2	yes	maximal, complemented
4	#14521173	C₂	1	2	yes	maximal, complemented
5	#14521103	C₂	1	2	yes	maximal
6	#14521102	C₂	1	1	yes	maximal, characteristic
7	#14521099	C₂	1	2	yes	maximal
8	C₃₁×#432	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₆₂	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃₁	C₄	1	1	yes	characteristic
12	C₂×Dic₃₁⋊C₄	C₄	1	2	yes	—
13	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
14	C₄×Dic₆₂	C₄	1	1	yes	characteristic
15	C₄×Dic₆₂	C₄	1	2	yes	—
16	C₄×Dic₆₂	C₄	1	1	yes	characteristic
17	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
18	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
19	C₄×Dic₆₂	C₂²	1	2	yes	—
20	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₃₁×#97	C₂²	1	2	yes	—
23	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
24	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
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	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	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	C₂×C₈×C₃₁	C₂³	1	4	yes	—
41	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₃₁	C₂³	1	1	yes	characteristic
44	C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
47	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
48	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
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	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
53	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
54	C₄×Dic₃₁	C₂³	1	2	yes	—
55	C₄×Dic₃₁	C₂³	1	2	yes	—
56	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
57	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
58	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₃₁	—	2	2	no	—
60	C₂×C₄×C₃₁	—	2	2	no	—
61	C₂×C₄×C₃₁	—	2	2	no	—
62	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₃₁	—	2	4	no	—
69	C₂×Dic₃₁	—	2	4	no	—
70	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
71	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
72	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
73	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
74	Dic₆₂	—	2	2	no	—
75	Dic₆₂	—	2	4	no	—
76	Dic₆₂	M₁₆	1	4	yes	—
77	C₈×C₃₁	—	2	4	no	—
78	C₃₁⋊C₈	—	2	2	no	—
79	C₃₁⋊C₈	—	2	2	no	—
80	#2984	—	31	1	no	maximal, Sylow
81	C₂²×C₃₁	—	2	1	no	—
82	C₂²×C₃₁	—	2	1	no	—
83	C₂²×C₃₁	#131	1	2	yes	—
84	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
85	C₄×C₃₁	—	2	1	no	—
86	C₄×C₃₁	—	2	1	no	—
87	C₄×C₃₁	—	2	1	no	—
88	C₄×C₃₁	—	2	1	no	—
89	C₄×C₃₁	C₂×M₁₆	1	2	yes	—
90	C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
91	Dic₃₁	—	2	2	no	—
92	Dic₃₁	—	2	2	no	—
93	Dic₃₁	—	2	2	no	—
94	Dic₃₁	C₂×M₁₆	1	4	yes	—
95	C₂×C₄×Q₈	—	31	1	no	—
96	#446	—	31	2	no	—
97	#446	—	31	4	no	—
98	C₈×Q₈	—	31	2	no	—
99	#432	—	31	2	no	—
100	#432	D₃₁	1	1	yes	characteristic
101	#431	—	31	1	no	—
102	#431	—	31	2	no	—
103	C₂×C₃₁	—	2	2	no	—
104	C₂×C₃₁	#519	1	1	yes	characteristic
105	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
106	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
107	C₂²×Q₈	—	31	1	no	—
108	C₄×Q₈	—	31	1	no	—
109	C₄×Q₈	—	31	2	no	—
110	C₄×Q₈	—	31	1	no	—
111	C₄×Q₈	—	31	1	no	—
112	C₄×Q₈	—	31	2	no	—
113	C₄×Q₈	—	31	1	no	—
114	C₂×C₄⋊C₄	—	31	2	no	—
115	C₂×C₄⋊C₄	—	31	1	no	—
116	C₂×C₄²	—	31	2	no	—
117	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
118	C₄⋊C₈	—	31	4	no	—
119	C₄⋊C₈	—	31	4	no	—
120	C₄⋊C₈	—	31	2	no	—
121	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
122	C₂²⋊C₈	—	31	1	no	—
123	C₂²⋊C₈	—	31	1	no	—
124	C₂²⋊C₈	C₂×D₃₁	1	2	yes	—
125	#97	—	31	4	no	—
126	#97	C₂×D₃₁	1	2	yes	—
127	C₄×C₈	—	31	4	no	—
128	C₄×C₈	—	31	2	no	—
129	C₃₁	#2984	1	1	yes	characteristic, complemented, Sylow
130	C₂×Q₈	—	31	1	no	—
131	C₂×Q₈	—	31	2	no	—
132	C₂×Q₈	—	31	1	no	—
133	C₂×Q₈	—	62	4	no	—
134	C₂²×C₄	—	31	2	no	—
135	C₂²×C₄	—	31	2	no	—
136	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
137	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
138	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
139	C₂×C₈	—	31	2	no	—
140	C₂×C₈	—	31	2	no	—
141	C₂×C₈	C₂²×D₃₁	1	4	yes	—
142	C₄⋊C₄	—	31	2	no	—
143	C₄⋊C₄	—	31	2	no	—
144	C₄⋊C₄	—	31	2	no	—
145	C₄⋊C₄	—	31	2	no	—
146	C₄⋊C₄	—	31	2	no	—
147	C₄⋊C₄	—	31	1	no	—
148	C₄⋊C₄	—	31	1	no	—
149	C₄²	—	31	2	no	—
150	C₄²	—	31	2	no	—
151	C₄²	—	31	2	no	—
152	C₄²	—	31	2	no	—
153	C₄²	C₄×D₃₁	1	1	yes	characteristic
154	C₄²	C₄×D₃₁	1	1	yes	characteristic
155	C₄²	C₂²×D₃₁	1	1	yes	characteristic
156	C₄²	C₂²×D₃₁	1	1	yes	characteristic
157	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
158	Q₈	—	31	4	no	—
159	Q₈	—	62	4	no	—
160	Q₈	—	62	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	31	2	no	—
165	C₂×C₄	—	31	2	no	—
166	C₂×C₄	—	31	2	no	—
167	C₂×C₄	—	31	2	no	—
168	C₂×C₄	—	62	4	no	—
169	C₂×C₄	—	62	4	no	—
170	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
175	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
176	C₈	—	2	4	no	—
177	C₈	—	62	2	no	—
178	C₈	—	62	2	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	#11758484	1	2	yes	—
182	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	31	4	no	—
188	C₄	—	62	2	no	—
189	C₄	—	62	2	no	—
190	C₄	—	62	2	no	—
191	C₄	M₁₆×D₃₁	1	2	yes	—
192	C₄	#11758567	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	#14521526	1	1	yes	characteristic
195	C₂	C₂×M₁₆×D₃₁	1	1	yes	characteristic
196	C₂	C₂×#11758484	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#432	C₂
prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#432	D₃₁
hypocenter	C₃₁	#2984
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #14521177	C₂	semidirect
3. #14521176	C₂	semidirect
4. #14521173	C₂	semidirect
129. C₃₁	#2984	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
#14521177	2
#14521176	2
#14521173	2
#14521103	2
#14521102	2
#14521099	2
C₃₁×#432	2
#2984	31