Subgroups — #16854878

Order 3968.

346 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521583	C₂	1	1	yes	maximal, characteristic
2	#14521178	C₂	1	2	yes	maximal, complemented
3	#14521125	C₂	1	2	yes	maximal, complemented
4	#14521119	C₂	1	2	yes	maximal, complemented
5	#14521118	C₂	1	2	yes	maximal, complemented
6	#14521109	C₂	1	2	yes	maximal, complemented
7	#14521106	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521074	C₂	1	2	yes	maximal, complemented
9	C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	#11758507	C₄	1	1	yes	characteristic
14	#11758506	C₄	1	2	yes	—
15	C₄×C₂²⋊D₃₁	C₂²	1	2	yes	—
16	#11758501	C₄	1	1	yes	characteristic
17	C₂×#92469	C₂²	1	2	yes	—
18	C₂×C₈×D₃₁	C₂²	1	2	yes	—
19	#11758453	C₄	1	2	yes	—
20	#11758452	C₄	1	1	yes	characteristic
21	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
22	#11758449	C₂²	1	2	yes	—
23	#11758447	C₄	1	1	yes	characteristic
24	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
25	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
26	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
27	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
28	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
29	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
30	#11758403	C₂²	1	2	yes	—
31	Dic₃₁⋊C₈	C₂²	1	2	yes	—
32	Dic₃₁⋊C₈	C₂²	1	2	yes	—
33	C₈×Dic₃₁	C₂²	1	2	yes	—
34	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
35	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
36	#11758391	C₂²	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	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
41	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₃₁	C₂³	1	2	yes	—
43	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
44	M₁₆×C₃₁	—	4	1	no	—
45	M₁₆×C₃₁	—	4	1	no	—
46	C₂×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂×C₈×C₃₁	C₂³	1	2	yes	—
48	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
53	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
54	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
56	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
58	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
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	C₂³	1	2	yes	—
64	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
65	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
66	#92469	—	4	2	no	—
67	C₈×D₃₁	—	4	2	no	—
68	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₃₁	—	2	1	no	—
71	C₂×C₄×C₃₁	—	2	1	no	—
72	C₂×C₄×C₃₁	—	4	1	no	—
73	C₂×C₄×C₃₁	—	4	1	no	—
74	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
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	C₂²⋊D₃₁	—	4	2	no	—
81	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
82	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
83	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
84	D₁₂₄	—	4	1	no	—
85	C₄×D₃₁	—	2	2	no	—
86	C₄×D₃₁	—	2	2	no	—
87	Dic₆₂	—	4	1	no	—
88	C₈×C₃₁	—	2	2	no	—
89	C₈×C₃₁	—	2	2	no	—
90	C₃₁⋊C₈	—	2	2	no	—
91	C₃₁⋊C₈	—	2	2	no	—
92	#2995	—	31	1	no	maximal, Sylow
93	C₂²×C₃₁	—	2	1	no	—
94	C₂²×C₃₁	—	2	1	no	—
95	C₂²×C₃₁	—	2	1	no	—
96	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
97	C₂×D₃₁	—	2	2	no	—
98	C₂×D₃₁	—	2	2	no	—
99	C₂×D₃₁	—	2	2	no	—
100	C₄×C₃₁	—	2	1	no	—
101	C₄×C₃₁	—	2	1	no	—
102	C₄×C₃₁	—	2	1	no	—
103	C₄×C₃₁	—	2	1	no	—
104	C₄×C₃₁	—	2	1	no	—
105	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
106	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
107	Dic₃₁	—	2	2	no	—
108	Dic₃₁	—	2	2	no	—
109	Dic₃₁	—	2	2	no	—
110	#529	—	31	1	no	—
111	#446	—	31	2	no	—
112	#436	—	31	2	no	—
113	#435	—	31	2	no	—
114	#435	—	31	2	no	—
115	#433	—	31	2	no	—
116	#432	—	31	1	no	—
117	#424	—	31	2	no	—
118	#423	D₃₁	1	1	yes	characteristic, complemented
119	C₂×C₃₁	—	4	1	no	—
120	C₂×C₃₁	#520	1	1	yes	characteristic
121	C₂×C₃₁	#568	1	1	yes	characteristic
122	C₂×C₃₁	#568	1	1	yes	characteristic
123	D₃₁	—	4	2	no	—
124	C₂×M₁₆	—	31	2	no	—
125	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
126	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
127	C₂²×C₈	—	31	2	no	—
128	#126	—	31	2	no	—
129	#125	—	31	1	no	—
130	#124	—	31	1	no	—
131	#123	—	31	2	no	—
132	C₂²⋊Q₈	—	31	1	no	—
133	#121	—	31	1	no	—
134	C₄×Q₈	—	31	1	no	—
135	C₄×D₄	—	31	2	no	—
136	C₄×D₄	—	31	1	no	—
137	#117	—	31	2	no	—
138	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
139	C₄⋊C₈	—	31	2	no	—
140	C₄⋊C₈	—	31	2	no	—
141	C₄⋊C₈	—	31	1	no	—
142	C₄⋊C₈	—	31	1	no	—
143	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
144	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
145	C₂²⋊C₈	—	31	2	no	—
146	C₂²⋊C₈	—	31	2	no	—
147	C₂²⋊C₈	—	31	2	no	—
148	#97	—	31	2	no	—
149	#97	—	31	2	no	—
150	C₄×C₈	—	31	2	no	—
151	C₃₁	#2995	1	1	yes	characteristic, complemented, Sylow
152	C₂×Q₈	—	31	1	no	—
153	C₂×D₄	—	31	2	no	—
154	C₂×D₄	—	31	1	no	—
155	C₂²×C₄	—	31	2	no	—
156	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
157	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
159	M₁₆	—	4	1	no	—
160	M₁₆	—	4	1	no	—
161	M₁₆	—	124	2	no	—
162	C₂×C₈	—	31	2	no	—
163	C₂×C₈	—	31	2	no	—
164	C₂×C₈	—	124	2	no	—
165	C₂×C₈	C₂²×D₃₁	1	2	yes	—
166	C₂×C₈	C₂²×D₃₁	1	2	yes	—
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	1	no	—
172	C₄⋊C₄	—	31	1	no	—
173	C₂²⋊C₄	—	31	2	no	—
174	C₂²⋊C₄	—	31	2	no	—
175	C₂²⋊C₄	—	31	2	no	—
176	C₂²⋊C₄	—	31	2	no	—
177	C₂²⋊C₄	—	31	2	no	—
178	C₄²	—	31	2	no	—
179	C₄²	C₄×D₃₁	1	1	yes	characteristic
180	C₄²	C₄×D₃₁	1	1	yes	characteristic
181	C₄²	C₂²×D₃₁	1	1	yes	characteristic
182	C₄²	C₂²×D₃₁	1	1	yes	characteristic
183	C₂³	—	31	2	no	—
184	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
185	Q₈	—	124	1	no	—
186	D₄	—	124	1	no	—
187	D₄	—	124	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	31	2	no	—
193	C₂×C₄	—	31	2	no	—
194	C₂×C₄	—	31	2	no	—
195	C₂×C₄	—	62	2	no	—
196	C₂×C₄	—	62	2	no	—
197	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	62	2	no	—
206	C₈	—	62	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	62	2	no	—
211	C₂²	—	62	2	no	—
212	C₂²	—	62	2	no	—
213	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	62	2	no	—
220	C₄	—	62	2	no	—
221	C₄	—	62	2	no	—
222	C₄	#11758561	1	1	yes	characteristic
223	C₄	#11758564	1	1	yes	characteristic
224	C₂	—	4	1	no	—
225	C₂	—	124	2	no	—
226	C₂	#14521530	1	1	yes	characteristic
227	C₂	#14521815	1	1	yes	characteristic
228	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	#11758507	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#423	D₃₁
hypocenter	C₃₁	#2995
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. #14521125	C₂	semidirect
4. #14521119	C₂	semidirect
5. #14521118	C₂	semidirect
6. #14521109	C₂	semidirect
7. #14521106	C₂	semidirect
8. #14521074	C₂	semidirect
9. C₃₁×#423	C₂	semidirect
118. #423	D₃₁	semidirect
151. C₃₁	#2995	semidirect

Maximal subgroups

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

subgroup	index
#14521583	2
#14521178	2
#14521125	2
#14521119	2
#14521118	2
#14521109	2
#14521106	2
#14521074	2
C₃₁×#423	2
#2995	31