Subgroups — C₃₁×#3095 #16843028

Order 3968.

410 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#485	C₂	1	4	yes	maximal, complemented
4	C₃₁×#484	C₂	1	2	yes	maximal, complemented
5	C₃₁×#483	C₂	1	2	yes	maximal, complemented
6	C₃₁×#428	C₂	1	2	yes	maximal, complemented
7	C₃₁×#419	C₂	1	2	yes	maximal, complemented
8	C₃₁×#407	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
12	C₃₁×C₂²⋊Q₈	—	2	1	no	—
13	C₃₁×C₂²⋊Q₈	—	2	1	no	—
14	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
15	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
16	C₃₁×#117	—	2	1	no	—
17	C₃₁×#117	—	2	1	no	—
18	C₃₁×#117	C₂²	1	2	yes	—
19	C₃₁×#117	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
21	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
25	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
26	C₂×C₄²×C₃₁	—	2	1	no	—
27	C₃₁×#107	C₂²	1	4	yes	complemented
28	C₃₁×#106	C₂²	1	4	yes	complemented
29	C₃₁×#103	C₂²	1	4	yes	complemented
30	C₃₁×#103	C₂²	1	4	yes	complemented
31	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
34	C₂×Q₈×C₃₁	—	2	1	no	—
35	C₂×Q₈×C₃₁	—	2	4	no	—
36	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₃₁	—	2	4	no	—
38	C₂²×C₄×C₃₁	—	2	1	no	—
39	C₂²×C₄×C₃₁	—	2	1	no	—
40	C₂²×C₄×C₃₁	—	2	1	no	—
41	C₂²×C₄×C₃₁	—	2	1	no	—
42	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	M₁₆×C₃₁	—	2	4	no	—
49	C₂×C₈×C₃₁	C₂³	1	4	yes	—
50	C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₄⋊C₄×C₃₁	—	2	1	no	—
53	C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
56	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
57	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
58	C₂²⋊C₄×C₃₁	—	2	2	no	—
59	C₂²⋊C₄×C₃₁	—	2	2	no	—
60	C₂²⋊C₄×C₃₁	—	2	2	no	—
61	C₂²⋊C₄×C₃₁	—	2	2	no	—
62	C₄²×C₃₁	—	2	2	no	—
63	C₄²×C₃₁	—	2	2	no	—
64	C₂³×C₃₁	—	2	2	no	—
65	C₂³×C₃₁	—	2	2	no	—
66	C₂³×C₃₁	—	4	1	no	—
67	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
70	Q₈×C₃₁	—	2	4	no	—
71	Q₈×C₃₁	—	4	2	no	—
72	C₂×C₄×C₃₁	—	2	2	no	—
73	C₂×C₄×C₃₁	—	2	2	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	2	no	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	4	2	no	—
82	C₂×C₄×C₃₁	—	4	2	no	—
83	C₂×C₄×C₃₁	—	4	2	no	—
84	C₂×C₄×C₃₁	—	4	2	no	—
85	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	#41	1	4	yes	—
89	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
90	C₈×C₃₁	—	2	4	no	—
91	#3095	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂²×C₃₁	—	2	1	no	—
93	C₂²×C₃₁	—	2	1	no	—
94	C₂²×C₃₁	—	2	1	no	—
95	C₂²×C₃₁	—	2	1	no	—
96	C₂²×C₃₁	—	2	1	no	—
97	C₂²×C₃₁	—	2	1	no	—
98	C₂²×C₃₁	—	2	1	no	—
99	C₂²×C₃₁	—	4	4	no	—
100	C₂²×C₃₁	#123	1	2	yes	—
101	C₂²×C₃₁	#137	1	2	yes	—
102	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₃₁	—	2	1	no	—
104	C₄×C₃₁	—	2	1	no	—
105	C₄×C₃₁	—	4	2	no	—
106	C₄×C₃₁	—	4	2	no	—
107	C₄×C₃₁	—	4	2	no	—
108	C₄×C₃₁	—	4	2	no	—
109	C₄×C₃₁	#123	1	2	yes	—
110	C₄×C₃₁	C₂×#41	1	2	yes	—
111	C₂×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
112	C₂×#117	C₂×C₃₁	1	1	yes	characteristic
113	#485	C₂×C₃₁	1	4	yes	complemented
114	#484	C₂×C₃₁	1	2	yes	complemented
115	#483	C₂×C₃₁	1	2	yes	complemented
116	#428	C₂×C₃₁	1	2	yes	complemented
117	#419	C₂×C₃₁	1	2	yes	complemented
118	#407	C₂×C₃₁	1	1	yes	characteristic
119	C₂×C₃₁	—	2	2	no	—
120	C₂×C₃₁	—	2	2	no	—
121	C₂×C₃₁	—	4	1	no	—
122	C₂×C₃₁	C₂×#123	1	1	yes	characteristic
123	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
124	C₂×C₃₁	#575	1	1	yes	characteristic
125	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
126	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
127	C₂×M₁₆	C₂²×C₃₁	1	2	yes	—
128	C₂²⋊Q₈	—	2	1	no	—
129	C₂²⋊Q₈	—	2	1	no	—
130	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
131	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
132	#117	—	2	1	no	—
133	#117	—	2	1	no	—
134	#117	C₂²×C₃₁	1	2	yes	—
135	#117	C₂²×C₃₁	1	2	yes	—
136	C₂×C₄⋊C₄	—	2	1	no	—
137	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
140	C₂×C₂²⋊C₄	—	2	1	no	—
141	C₂×C₂²⋊C₄	—	2	1	no	—
142	C₂×C₄²	—	2	1	no	—
143	#107	C₂²×C₃₁	1	4	yes	complemented
144	#106	C₂²×C₃₁	1	4	yes	complemented
145	#103	C₂²×C₃₁	1	4	yes	complemented
146	#103	C₂²×C₃₁	1	4	yes	complemented
147	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
148	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
149	C₃₁	#3095	1	1	yes	characteristic, direct product factor, Sylow
150	C₂⁴	D₄×C₃₁	1	1	yes	characteristic
151	C₂×Q₈	—	2	1	no	—
152	C₂×Q₈	—	2	4	no	—
153	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
160	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
161	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
162	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
165	M₁₆	—	2	4	no	—
166	C₂×C₈	C₂³×C₃₁	1	4	yes	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
173	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
174	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	4	1	no	—
184	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
185	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
186	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
187	Q₈	—	2	4	no	—
188	Q₈	—	4	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
204	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₂×C₄	#41×C₃₁	1	4	yes	—
206	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
207	C₈	—	2	4	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	4	4	no	—
216	C₂²	C₃₁×#123	1	2	yes	—
217	C₂²	C₃₁×#137	1	2	yes	—
218	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₄	C₃₁×#123	1	2	yes	—
226	C₄	C₂×#41×C₃₁	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	—	2	2	no	—
229	C₂	—	4	1	no	—
230	C₂	C₂×C₃₁×#123	1	1	yes	characteristic
231	C₂	C₂×C₃₁×#137	1	1	yes	characteristic
232	C₂	C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#485	C₂	semidirect
4. C₃₁×#484	C₂	semidirect
5. C₃₁×#483	C₂	semidirect
6. C₃₁×#428	C₂	semidirect
7. C₃₁×#419	C₂	semidirect
27. C₃₁×#107	C₂²	semidirect
28. C₃₁×#106	C₂²	semidirect
29. C₃₁×#103	C₂²	semidirect
30. C₃₁×#103	C₂²	semidirect
91. #3095	C₃₁	direct (N × H)
113. #485	C₂×C₃₁	semidirect
114. #484	C₂×C₃₁	semidirect
115. #483	C₂×C₃₁	semidirect
116. #428	C₂×C₃₁	semidirect
117. #419	C₂×C₃₁	semidirect
143. #107	C₂²×C₃₁	semidirect
144. #106	C₂²×C₃₁	semidirect
145. #103	C₂²×C₃₁	semidirect
146. #103	C₂²×C₃₁	semidirect
149. C₃₁	#3095	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×C₂²⋊Q₈	2
C₂×C₃₁×#117	2
C₃₁×#485	2
C₃₁×#484	2
C₃₁×#483	2
C₃₁×#428	2
C₃₁×#419	2
C₃₁×#407	2
#3095	31

Subgroups — C31×#3095 #16843028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3095 #16843028