Subgroups — #16854809

Order 3968.

330 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521603	C₂	1	1	yes	maximal, characteristic
2	#14521177	C₂	1	2	yes	maximal, complemented
3	#14521176	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521173	C₂	1	2	yes	maximal, complemented
7	#14521111	C₂	1	2	yes	maximal
8	#14521109	C₂	1	2	yes	maximal
9	#14521101	C₂	1	1	yes	maximal, characteristic
10	#14521100	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	#11758523	C₄	1	1	yes	characteristic
14	#11758521	C₄	1	1	yes	characteristic
15	#11758474	C₄	1	1	yes	characteristic
16	#11758473	C₄	1	2	yes	—
17	#11758466	C₄	1	2	yes	—
18	#11758465	C₄	1	1	yes	characteristic
19	#11758449	C₂²	1	2	yes	—
20	C₄²×D₃₁	C₂²	1	1	yes	characteristic
21	C₄×Dic₆₂	C₂²	1	2	yes	—
22	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
23	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
25	C₃₁×#97	C₂²	1	2	yes	—
26	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
28	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
29	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
30	#11758403	C₂²	1	2	yes	—
31	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
32	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
33	Dic₃₁⋊C₈	C₂²	1	2	yes	—
34	Dic₃₁⋊C₈	C₂²	1	2	yes	—
35	C₈×Dic₃₁	C₂²	1	2	yes	—
36	C₈×Dic₃₁	C₂²	1	2	yes	—
37	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
39	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
41	C₂×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
45	C₂×Dic₆₂	C₂×C₄	1	2	yes	—
46	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
52	C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₄²×C₃₁	C₂³	1	2	yes	—
54	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
58	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
59	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
60	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
62	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
63	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
66	C₄×Dic₃₁	C₂³	1	2	yes	—
67	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
69	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
71	Q₈×C₃₁	—	2	2	no	—
72	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
80	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
81	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₄×D₃₁	—	2	2	no	—
83	C₄×D₃₁	—	2	1	no	—
84	C₄×D₃₁	—	2	1	no	—
85	C₄×D₃₁	—	2	2	no	—
86	Dic₆₂	—	2	2	no	—
87	Dic₆₂	—	2	4	no	—
88	C₈×C₃₁	—	2	1	no	—
89	C₈×C₃₁	—	2	1	no	—
90	C₈×C₃₁	—	2	2	no	—
91	C₃₁⋊C₈	—	2	2	no	—
92	C₃₁⋊C₈	—	2	1	no	—
93	C₃₁⋊C₈	—	2	1	no	—
94	#2986	—	31	1	no	maximal, Sylow
95	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₃₁	—	2	1	no	—
97	C₂×D₃₁	—	2	1	no	—
98	C₂×D₃₁	—	2	1	no	—
99	C₄×C₃₁	—	2	2	no	—
100	C₄×C₃₁	—	2	1	no	—
101	C₄×C₃₁	—	2	2	no	—
102	C₄×C₃₁	#131	1	2	yes	—
103	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
104	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
105	Dic₃₁	—	2	1	no	—
106	Dic₃₁	—	2	2	no	—
107	Dic₃₁	—	2	2	no	—
108	Dic₃₁	—	2	1	no	—
109	Dic₃₁	#131	1	2	yes	—
110	#533	—	31	1	no	—
111	#446	—	31	1	no	—
112	#446	—	31	2	no	—
113	#446	D₃₁	1	1	yes	characteristic, complemented
114	C₈×Q₈	—	31	1	no	—
115	C₈×Q₈	—	31	2	no	—
116	C₈×Q₈	—	31	1	no	—
117	#433	—	31	2	no	—
118	#433	—	31	2	no	—
119	#431	—	31	1	no	—
120	#431	—	31	1	no	—
121	C₂×C₃₁	#519	1	1	yes	characteristic
122	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
123	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
124	D₃₁	—	4	1	no	—
125	C₄⋊Q₈	—	31	1	no	—
126	C₄⋊Q₈	—	31	1	no	—
127	#125	—	31	2	no	—
128	C₂²⋊Q₈	—	31	1	no	—
129	C₂²⋊Q₈	—	31	2	no	—
130	C₂²⋊Q₈	—	31	1	no	—
131	C₄×Q₈	—	31	2	no	—
132	C₄×Q₈	—	31	1	no	—
133	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
134	#117	—	31	2	no	—
135	C₂×C₄²	—	31	1	no	—
136	C₄⋊C₈	—	31	1	no	—
137	C₄⋊C₈	—	31	1	no	—
138	C₄⋊C₈	—	31	2	no	—
139	C₄⋊C₈	—	31	2	no	—
140	C₄⋊C₈	—	31	1	no	—
141	C₄⋊C₈	—	31	2	no	—
142	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
143	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
144	C₂²⋊C₈	—	31	2	no	—
145	C₂²⋊C₈	—	31	1	no	—
146	C₂²⋊C₈	—	31	1	no	—
147	#97	—	31	2	no	—
148	#97	C₂×D₃₁	1	2	yes	—
149	C₄×C₈	—	31	2	no	—
150	C₄×C₈	—	31	2	no	—
151	C₄×C₈	—	31	1	no	—
152	C₄×C₈	—	31	2	no	—
153	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
154	C₃₁	#2986	1	1	yes	characteristic, complemented, Sylow
155	C₂×Q₈	—	31	2	no	—
156	C₂×Q₈	—	31	1	no	—
157	C₂×Q₈	C₄×D₃₁	1	1	yes	characteristic
158	C₂²×C₄	—	31	1	no	—
159	C₂²×C₄	—	31	1	no	—
160	C₂²×C₄	—	31	1	no	—
161	C₂×C₈	—	31	2	no	—
162	C₂×C₈	—	31	1	no	—
163	C₂×C₈	—	31	1	no	—
164	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
165	C₂×C₈	C₂²×D₃₁	1	2	yes	—
166	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
167	C₄⋊C₄	—	31	2	no	—
168	C₄⋊C₄	—	31	2	no	—
169	C₄⋊C₄	—	31	2	no	—
170	C₄⋊C₄	—	31	2	no	—
171	C₄⋊C₄	—	31	2	no	—
172	C₄⋊C₄	—	31	2	no	—
173	C₄⋊C₄	—	31	1	no	—
174	C₄⋊C₄	C₄×D₃₁	1	2	yes	—
175	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
176	C₂²⋊C₄	—	31	2	no	—
177	C₂²⋊C₄	—	31	2	no	—
178	C₄²	—	31	1	no	—
179	C₄²	—	31	1	no	—
180	C₄²	—	31	1	no	—
181	C₄²	—	31	2	no	—
182	C₄²	C₂²×D₃₁	1	1	yes	characteristic
183	C₄²	C₂²×D₃₁	1	2	yes	—
184	C₂³	—	31	1	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	62	4	no	—
187	Q₈	—	62	2	no	—
188	C₂×C₄	—	31	1	no	—
189	C₂×C₄	—	31	1	no	—
190	C₂×C₄	—	31	1	no	—
191	C₂×C₄	—	31	2	no	—
192	C₂×C₄	—	31	2	no	—
193	C₂×C₄	—	62	2	no	—
194	C₂×C₄	—	62	2	no	—
195	C₂×C₄	—	62	1	no	—
196	C₂×C₄	—	62	1	no	—
197	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	2	2	no	—
205	C₈	—	62	2	no	—
206	C₈	—	62	1	no	—
207	C₈	—	62	1	no	—
208	C₂²	—	62	1	no	—
209	C₂²	—	62	1	no	—
210	C₂²	—	62	1	no	—
211	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	31	2	no	—
216	C₄	—	62	1	no	—
217	C₄	—	62	2	no	—
218	C₄	—	62	1	no	—
219	C₄	—	62	2	no	—
220	C₄	#11758484	1	2	yes	—
221	C₄	#11758564	1	1	yes	characteristic
222	C₄	#11758567	1	1	yes	characteristic
223	C₂	—	124	1	no	—
224	C₂	#14521525	1	1	yes	characteristic
225	C₂	#131×D₃₁	1	1	yes	characteristic
226	C₂	C₂×#11758484	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#446	C₂
prime radical	C₂²×D₃₁	C₂²×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#446	D₃₁
hypocenter	C₃₁	#2986
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #446

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. Q₈×C₃₁⋊C₈	C₂	semidirect
5. C₈×Dic₆₂	C₂	semidirect
6. #14521173	C₂	semidirect
11. C₃₁×#446	C₂	semidirect
113. #446	D₃₁	semidirect
154. C₃₁	#2986	semidirect

Maximal subgroups

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

subgroup	index
#14521603	2
#14521177	2
#14521176	2
Q₈×C₃₁⋊C₈	2
C₈×Dic₆₂	2
#14521173	2
#14521111	2
#14521109	2
#14521101	2
#14521100	2
C₃₁×#446	2
#2986	31