Subgroups — #16854686

Order 3968.

380 conjugacy classes of subgroups.

The subgroup lattice has 246 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	#14521179	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521128	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521120	C₂	1	2	yes	maximal, complemented
6	C₈×D₁₂₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊D₃₁	C₂	1	2	yes	maximal, complemented
8	#14521108	C₂	1	2	yes	maximal, complemented
9	#14521106	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521003	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	#11758507	C₄	1	1	yes	characteristic
16	#11758506	C₄	1	2	yes	—
17	C₄×C₂²⋊D₃₁	C₂²	1	2	yes	—
18	#11758501	C₄	1	1	yes	characteristic
19	C₂×#92469	C₂²	1	2	yes	—
20	C₂×C₈×D₃₁	C₂²	1	2	yes	—
21	#11758453	C₄	1	2	yes	—
22	#11758452	C₄	1	1	yes	characteristic
23	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
24	#11758449	C₂²	1	2	yes	—
25	#11758447	C₄	1	1	yes	characteristic
26	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
32	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
34	#11758403	C₂²	1	2	yes	—
35	Dic₃₁⋊C₈	C₂²	1	2	yes	—
36	Dic₃₁⋊C₈	C₂²	1	2	yes	—
37	C₈×Dic₃₁	C₂²	1	2	yes	—
38	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
39	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
40	#11758391	C₂²	1	2	yes	—
41	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
45	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₃₁	C₂³	1	2	yes	—
47	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×C₃₁	—	2	2	no	—
49	C₂×C₈×C₃₁	—	2	2	no	—
50	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₃₁	C₂³	1	1	yes	characteristic
57	C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
59	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
61	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
62	(C₂×D₃₁)⋊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	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
66	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
67	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
68	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
69	C₄×Dic₃₁	C₂³	1	2	yes	—
70	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
71	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
72	#92469	—	2	4	no	—
73	C₈×D₃₁	—	2	4	no	—
74	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	1	no	—
79	C₂×C₄×C₃₁	—	2	1	no	—
80	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
86	C₂²⋊D₃₁	—	2	4	no	—
87	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
88	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
89	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
90	D₁₂₄	—	2	2	no	—
91	C₄×D₃₁	—	2	2	no	—
92	C₄×D₃₁	—	2	2	no	—
93	Dic₆₂	—	2	2	no	—
94	C₈×C₃₁	—	2	1	no	—
95	C₈×C₃₁	—	2	1	no	—
96	C₈×C₃₁	#41	1	2	yes	—
97	C₈×C₃₁	#41	1	2	yes	—
98	C₃₁⋊C₈	—	2	2	no	—
99	C₃₁⋊C₈	—	2	2	no	—
100	#2966	—	31	1	no	maximal, Sylow
101	C₂²×C₃₁	—	2	1	no	—
102	C₂²×C₃₁	—	2	1	no	—
103	C₂²×C₃₁	#131	1	2	yes	—
104	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
105	C₂×D₃₁	—	2	2	no	—
106	C₂×D₃₁	—	2	2	no	—
107	C₂×D₃₁	—	2	2	no	—
108	C₄×C₃₁	—	2	1	no	—
109	C₄×C₃₁	—	2	1	no	—
110	C₄×C₃₁	—	2	1	no	—
111	C₄×C₃₁	—	2	1	no	—
112	C₄×C₃₁	#131	1	2	yes	—
113	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
114	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
115	Dic₃₁	—	2	2	no	—
116	Dic₃₁	—	2	2	no	—
117	Dic₃₁	—	2	2	no	—
118	#529	—	31	1	no	—
119	#446	—	31	1	no	—
120	C₈×Q₈	—	31	1	no	—
121	#436	—	31	1	no	—
122	#435	—	31	2	no	—
123	C₈×D₄	—	31	2	no	—
124	C₈×D₄	—	31	1	no	—
125	#433	—	31	2	no	—
126	#432	—	31	1	no	—
127	#405	—	31	2	no	—
128	C₂×C₄×C₈	D₃₁	1	1	yes	characteristic, complemented
129	C₂×C₃₁	—	2	2	no	—
130	C₂×C₃₁	C₄×#41	1	1	yes	characteristic
131	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
132	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
133	D₃₁	—	4	2	no	—
134	C₂×M₁₆	—	31	2	no	—
135	C₂²×C₈	—	31	2	no	—
136	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
137	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
138	#126	—	31	2	no	—
139	#125	—	31	1	no	—
140	#124	—	31	1	no	—
141	#123	—	31	2	no	—
142	C₂²⋊Q₈	—	31	1	no	—
143	#121	—	31	1	no	—
144	C₄×Q₈	—	31	1	no	—
145	C₄×D₄	—	31	2	no	—
146	C₄×D₄	—	31	1	no	—
147	#117	—	31	2	no	—
148	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
149	C₄⋊C₈	—	31	2	no	—
150	C₄⋊C₈	—	31	2	no	—
151	C₄⋊C₈	—	31	1	no	—
152	C₄⋊C₈	—	31	1	no	—
153	C₂²⋊C₈	—	31	2	no	—
154	C₂²⋊C₈	—	31	2	no	—
155	C₂²⋊C₈	—	31	2	no	—
156	#97	—	31	2	no	—
157	#97	—	31	2	no	—
158	C₄×C₈	—	31	2	no	—
159	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
160	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
161	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
162	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
163	C₃₁	#2966	1	1	yes	characteristic, complemented, Sylow
164	C₂×Q₈	—	31	1	no	—
165	C₂×D₄	—	31	2	no	—
166	C₂×D₄	—	31	1	no	—
167	C₂²×C₄	—	31	2	no	—
168	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
171	M₁₆	—	62	4	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	31	2	no	—
175	C₂×C₈	—	31	2	no	—
176	C₂×C₈	—	62	4	no	—
177	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
179	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
180	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
181	C₄⋊C₄	—	31	2	no	—
182	C₄⋊C₄	—	31	2	no	—
183	C₄⋊C₄	—	31	2	no	—
184	C₄⋊C₄	—	31	2	no	—
185	C₄⋊C₄	—	31	1	no	—
186	C₄⋊C₄	—	31	1	no	—
187	C₂²⋊C₄	—	31	2	no	—
188	C₂²⋊C₄	—	31	2	no	—
189	C₂²⋊C₄	—	31	2	no	—
190	C₂²⋊C₄	—	31	2	no	—
191	C₂²⋊C₄	—	31	2	no	—
192	C₄²	—	31	2	no	—
193	C₄²	C₄×D₃₁	1	1	yes	characteristic
194	C₄²	C₄×D₃₁	1	1	yes	characteristic
195	C₄²	C₂²×D₃₁	1	1	yes	characteristic
196	C₄²	C₂²×D₃₁	1	1	yes	characteristic
197	C₂³	—	31	2	no	—
198	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
199	Q₈	—	62	2	no	—
200	D₄	—	62	4	no	—
201	D₄	—	62	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	31	2	no	—
207	C₂×C₄	—	31	2	no	—
208	C₂×C₄	—	31	2	no	—
209	C₂×C₄	—	62	2	no	—
210	C₂×C₄	—	62	2	no	—
211	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	62	2	no	—
220	C₈	—	62	2	no	—
221	C₈	#92495	1	2	yes	—
222	C₈	#92495	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	62	2	no	—
226	C₂²	—	62	2	no	—
227	C₂²	—	62	2	no	—
228	C₂²	#11758478	1	2	yes	—
229	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	62	2	no	—
235	C₄	—	62	2	no	—
236	C₄	—	62	2	no	—
237	C₄	#11758478	1	2	yes	—
238	C₄	C₂×#92495	1	1	yes	characteristic
239	C₄	C₂×#92495	1	1	yes	characteristic
240	C₂	—	2	2	no	—
241	C₂	—	124	2	no	—
242	C₂	C₄×#92495	1	1	yes	characteristic
243	C₂	C₂×#11758478	1	1	yes	characteristic
244	C₂	C₂×#11758478	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₄×C₈×C₃₁	C₂
prime radical	#11758507	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×C₄×C₈	D₃₁
hypocenter	C₃₁	#2966
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #14521179	C₂	semidirect
3. C₈×Dic₆₂	C₂	semidirect
4. #14521128	C₂	semidirect
5. #14521120	C₂	semidirect
6. C₈×D₁₂₄	C₂	semidirect
7. C₈×C₂²⋊D₃₁	C₂	semidirect
8. #14521108	C₂	semidirect
9. #14521106	C₂	semidirect
10. #14521003	C₂	semidirect
11. C₂×C₄×C₈×C₃₁	C₂	semidirect
128. C₂×C₄×C₈	D₃₁	semidirect
163. C₃₁	#2966	semidirect

Maximal subgroups

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

subgroup	index
#14521583	2
#14521179	2
C₈×Dic₆₂	2
#14521128	2
#14521120	2
C₈×D₁₂₄	2
C₈×C₂²⋊D₃₁	2
#14521108	2
#14521106	2
#14521003	2
C₂×C₄×C₈×C₃₁	2
#2966	31