Subgroups — #16854716

Order 3968.

370 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	#14521180	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521127	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521127	C₂	1	2	yes	maximal, complemented
5	#14521121	C₂	1	2	yes	maximal, complemented
6	D₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521110	C₂	1	2	yes	maximal, complemented
8	#14521102	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521005	C₂	1	2	yes	maximal, complemented
10	C₄×#92474	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#529	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#126	C₄	1	2	yes	—
13	C₃₁×#125	C₄	1	1	yes	characteristic
14	C₃₁×#124	C₄	1	1	yes	characteristic
15	C₃₁×#123	C₄	1	2	yes	—
16	C₃₁×C₂²⋊Q₈	C₄	1	1	yes	characteristic
17	C₃₁×#121	C₄	1	1	yes	characteristic
18	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#117	C₂²	1	2	yes	—
22	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×#92474	C₂²	1	1	yes	characteristic
24	C₂×#92474	C₂²	1	1	yes	characteristic
25	C₂×#92474	C₂²	1	2	yes	—
26	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
27	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
28	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
29	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
30	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
31	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
32	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
33	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
34	#11758391	C₂²	1	1	yes	characteristic
35	#11758391	C₂²	1	2	yes	—
36	#11758391	C₂²	1	1	yes	characteristic
37	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
38	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
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×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
50	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
52	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
54	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
55	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
56	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
57	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
59	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₄²×C₃₁	C₂³	1	2	yes	—
64	#92474	—	2	2	no	—
65	#92474	—	2	2	no	—
66	#92474	—	2	4	no	—
67	C₂×C₃₁⋊C₈	—	2	4	no	—
68	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
69	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
70	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
75	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₃₁	—	2	2	no	—
77	D₄×C₃₁	—	2	2	no	—
78	D₄×C₃₁	—	4	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	1	no	—
83	C₂×C₄×C₃₁	—	2	1	no	—
84	C₂×C₄×C₃₁	—	4	1	no	—
85	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
94	C₃₁⋊C₈	—	2	2	no	—
95	C₃₁⋊C₈	—	2	1	no	—
96	C₃₁⋊C₈	—	2	2	no	—
97	C₃₁⋊C₈	—	2	1	no	—
98	C₃₁⋊C₈	#41	1	2	yes	—
99	C₃₁⋊C₈	#41	1	2	yes	—
100	#2969	—	31	1	no	maximal, Sylow
101	C₂²×C₃₁	—	2	2	no	—
102	C₂²×C₃₁	—	2	1	no	—
103	C₂²×C₃₁	—	2	1	no	—
104	C₂²×C₃₁	—	2	2	no	—
105	C₂²×C₃₁	—	2	1	no	—
106	C₂²×C₃₁	—	2	2	no	—
107	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
108	C₄×C₃₁	—	2	2	no	—
109	C₄×C₃₁	—	2	1	no	—
110	C₄×C₃₁	—	2	1	no	—
111	C₄×C₃₁	—	2	2	no	—
112	C₄×C₃₁	—	2	1	no	—
113	C₄×C₃₁	—	2	1	no	—
114	C₄×C₃₁	—	2	2	no	—
115	C₄×C₃₁	#131	1	2	yes	—
116	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
117	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
118	#529	D₃₁	1	1	yes	characteristic
119	#446	—	31	1	no	—
120	C₈×Q₈	—	31	1	no	—
121	#436	—	31	1	no	—
122	#436	—	31	2	no	—
123	#435	—	31	2	no	—
124	C₈×D₄	—	31	1	no	—
125	#433	—	31	2	no	—
126	#431	—	31	1	no	—
127	#405	—	31	2	no	—
128	C₄×M₁₆	—	31	1	no	—
129	C₂×C₃₁	—	4	1	no	—
130	C₂×C₃₁	—	4	2	no	—
131	C₂×C₃₁	C₄×#41	1	1	yes	characteristic
132	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
133	C₂×C₃₁	#568	1	1	yes	characteristic
134	C₂×M₁₆	—	31	1	no	—
135	C₂×M₁₆	—	31	1	no	—
136	C₂×M₁₆	—	31	2	no	—
137	C₂²×C₈	—	31	2	no	—
138	#126	Dic₃₁	1	2	yes	—
139	#125	Dic₃₁	1	1	yes	characteristic
140	#124	Dic₃₁	1	1	yes	characteristic
141	#123	Dic₃₁	1	2	yes	—
142	C₂²⋊Q₈	Dic₃₁	1	1	yes	characteristic
143	#121	Dic₃₁	1	1	yes	characteristic
144	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
145	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
146	C₄×D₄	C₂×D₃₁	1	2	yes	—
147	#117	C₂×D₃₁	1	2	yes	—
148	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
149	C₄⋊C₈	—	31	1	no	—
150	C₄⋊C₈	—	31	1	no	—
151	C₄⋊C₈	—	31	2	no	—
152	C₄⋊C₈	—	31	2	no	—
153	C₂²⋊C₈	—	31	2	no	—
154	C₂²⋊C₈	—	31	2	no	—
155	C₂²⋊C₈	—	31	2	no	—
156	#97	—	31	1	no	—
157	#97	—	31	1	no	—
158	#97	—	31	2	no	—
159	C₄×C₈	—	31	1	no	—
160	C₄×C₈	—	31	1	no	—
161	C₄×C₈	—	31	2	no	—
162	C₄×C₈	—	31	2	no	—
163	C₃₁	#2969	1	1	yes	characteristic, complemented, Sylow
164	C₂×Q₈	C₂×Dic₃₁	1	1	yes	characteristic
165	C₂×D₄	C₂×Dic₃₁	1	1	yes	characteristic
166	C₂×D₄	C₂×Dic₃₁	1	2	yes	—
167	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
168	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
171	M₁₆	—	62	2	no	—
172	M₁₆	—	62	2	no	—
173	M₁₆	—	62	4	no	—
174	C₂×C₈	—	31	1	no	—
175	C₂×C₈	—	31	1	no	—
176	C₂×C₈	—	31	2	no	—
177	C₂×C₈	—	31	1	no	—
178	C₂×C₈	—	31	1	no	—
179	C₂×C₈	—	31	2	no	—
180	C₂×C₈	—	62	4	no	—
181	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
182	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
183	C₄⋊C₄	C₂×Dic₃₁	1	1	yes	characteristic
184	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
185	C₄⋊C₄	C₂×Dic₃₁	1	1	yes	characteristic
186	C₄⋊C₄	C₂×Dic₃₁	1	2	yes	—
187	C₂²⋊C₄	C₂×Dic₃₁	1	2	yes	—
188	C₂²⋊C₄	C₂×Dic₃₁	1	2	yes	—
189	C₂²⋊C₄	C₂×Dic₃₁	1	2	yes	—
190	C₂²⋊C₄	C₂×Dic₃₁	1	2	yes	—
191	C₂²⋊C₄	C₂×Dic₃₁	1	2	yes	—
192	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
193	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
194	C₄²	C₂²×D₃₁	1	1	yes	characteristic
195	C₄²	C₂²×D₃₁	1	1	yes	characteristic
196	C₄²	C₂²×D₃₁	1	2	yes	—
197	C₂³	C₂²×Dic₃₁	1	2	yes	—
198	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
199	Q₈	—	2	2	no	—
200	D₄	—	2	2	no	—
201	D₄	—	4	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	—	2	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
209	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
211	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
212	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
213	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
214	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
215	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
217	C₈	—	31	2	no	—
218	C₈	—	31	2	no	—
219	C₈	—	62	2	no	—
220	C₈	—	62	1	no	—
221	C₈	—	62	1	no	—
222	C₈	—	62	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	2	no	—
229	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	2	no	—
237	C₄	#11758525	1	2	yes	—
238	C₄	#41×D₃₁	1	1	yes	characteristic
239	C₄	#41×D₃₁	1	1	yes	characteristic
240	C₂	—	4	1	no	—
241	C₂	—	4	2	no	—
242	C₂	#41×Dic₃₁	1	1	yes	characteristic
243	C₂	C₂×#11758525	1	1	yes	characteristic
244	C₂	#14521816	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#529	C₂
prime radical	C₃₁×#121	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#529	D₃₁
hypocenter	C₃₁	#2969
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #14521180	C₂	semidirect
2. Q₈×C₃₁⋊C₈	C₂	semidirect
3. #14521127	C₂	semidirect
4. #14521127	C₂	semidirect
5. #14521121	C₂	semidirect
6. D₄×C₃₁⋊C₈	C₂	semidirect
7. #14521110	C₂	semidirect
8. #14521102	C₂	semidirect
9. #14521005	C₂	semidirect
10. C₄×#92474	C₂	semidirect
163. C₃₁	#2969	semidirect

Maximal subgroups

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

subgroup	index
#14521180	2
Q₈×C₃₁⋊C₈	2
#14521127	2
#14521121	2
D₄×C₃₁⋊C₈	2
#14521110	2
#14521102	2
#14521005	2
C₄×#92474	2
C₃₁×#529	2
#2969	31