Subgroups — C₃₁×#21372 #21478131

Order 7936.

714 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2921	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#433	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2873	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1950	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1855	C₂	1	4	yes	maximal
6	C₃₁×#1778	C₂	1	2	yes	maximal, complemented
7	C₄×C₃₁×#108	C₂	1	2	yes	maximal, complemented
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
10	C₃₁×#433	—	2	4	no	—
11	C₃₁×#433	C₄	1	8	yes	—
12	C₂×C₃₁×#108	C₂²	1	4	yes	—
13	C₂×C₃₁×#108	C₂²	1	4	yes	—
14	C₃₁×#423	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₃₁×#407	C₂²	1	2	yes	—
17	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₃₁×#405	—	2	4	no	—
19	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
20	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
21	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
22	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#341	C₂²	1	4	yes	—
25	C₃₁×#341	C₂²	1	4	yes	—
26	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂×M₁₆×C₃₁	—	2	8	no	—
29	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
30	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
31	C₂²×C₈×C₃₁	—	2	2	no	—
32	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
34	C₃₁×#117	C₂×C₄	1	4	yes	—
35	C₃₁×#117	D₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₃₁	Q₈	1	2	yes	—
37	C₂×C₂²⋊C₄×C₃₁	Q₈	1	2	yes	—
38	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×#108	—	2	8	no	—
41	C₃₁×#108	—	2	8	no	—
42	C₃₁×C₄⋊C₈	—	2	4	no	—
43	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
44	C₃₁×C₂²⋊C₈	—	2	4	no	—
45	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
46	C₃₁×#97	—	2	2	no	—
47	C₃₁×#97	—	2	2	no	—
48	C₃₁×#97	—	2	2	no	—
49	C₃₁×#97	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₃₁	—	2	2	no	—
51	C₄×C₈×C₃₁	—	2	2	no	—
52	C₄×C₈×C₃₁	—	2	2	no	—
53	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
54	C₂⁴×C₃₁	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	—	2	4	no	—
56	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
62	M₁₆×C₃₁	—	2	8	no	—
63	M₁₆×C₃₁	—	2	4	no	—
64	M₁₆×C₃₁	—	2	4	no	—
65	M₁₆×C₃₁	—	2	4	no	—
66	M₁₆×C₃₁	#41	1	8	yes	—
67	C₂×C₈×C₃₁	—	2	8	no	—
68	C₂×C₈×C₃₁	—	2	4	no	—
69	C₂×C₈×C₃₁	—	2	4	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
74	C₄⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
75	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
76	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
77	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
78	C₄²×C₃₁	C₂×D₄	1	2	yes	—
79	C₄²×C₃₁	C₂×D₄	1	2	yes	—
80	#21372	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₃₁	—	2	2	no	—
82	C₂³×C₃₁	—	4	2	no	—
83	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
84	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	4	no	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	4	2	no	—
90	C₂×C₄×C₃₁	—	4	2	no	—
91	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
97	C₈×C₃₁	—	2	4	no	—
98	C₈×C₃₁	—	2	4	no	—
99	C₈×C₃₁	—	2	8	no	—
100	C₈×C₃₁	—	4	4	no	—
101	#2921	C₂×C₃₁	1	1	yes	characteristic
102	C₂×#433	C₂×C₃₁	1	1	yes	characteristic
103	#2873	C₂×C₃₁	1	1	yes	characteristic
104	#1950	C₂×C₃₁	1	4	yes	complemented
105	#1855	C₂×C₃₁	1	4	yes	—
106	#1778	C₂×C₃₁	1	2	yes	complemented
107	C₄×#108	C₂×C₃₁	1	2	yes	complemented
108	C₂²×C₃₁	—	2	2	no	—
109	C₂²×C₃₁	—	2	4	no	—
110	C₂²×C₃₁	—	2	2	no	—
111	C₂²×C₃₁	—	8	2	no	—
112	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₃₁	#533	1	2	yes	—
114	C₄×C₃₁	—	2	4	no	—
115	C₄×C₃₁	—	2	4	no	—
116	C₄×C₃₁	—	2	2	no	—
117	C₄×C₃₁	C₄×#41	1	2	yes	—
118	C₄×C₃₁	#525	1	2	yes	—
119	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
120	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
121	#433	—	2	4	no	—
122	#433	C₄×C₃₁	1	8	yes	—
123	C₂×#108	C₂²×C₃₁	1	4	yes	—
124	C₂×#108	C₂²×C₃₁	1	4	yes	—
125	#423	C₂²×C₃₁	1	2	yes	—
126	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
127	#407	C₂²×C₃₁	1	2	yes	—
128	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
129	#405	—	2	4	no	—
130	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
131	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
132	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
133	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
135	#341	C₂²×C₃₁	1	4	yes	—
136	#341	C₂²×C₃₁	1	4	yes	—
137	C₂×C₃₁	—	2	2	no	—
138	C₂×C₃₁	—	4	2	no	—
139	C₂×C₃₁	#2296	1	1	yes	characteristic
140	C₂×C₃₁	#2916	1	2	yes	—
141	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	—	2	8	no	—
144	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
145	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
148	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
149	#117	C₂×C₄×C₃₁	1	4	yes	—
150	#117	D₄×C₃₁	1	4	yes	—
151	C₂×C₄⋊C₄	Q₈×C₃₁	1	2	yes	—
152	C₂×C₂²⋊C₄	Q₈×C₃₁	1	2	yes	—
153	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
154	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
155	#108	—	2	8	no	—
156	#108	—	2	8	no	—
157	C₄⋊C₈	—	2	4	no	—
158	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
159	C₂²⋊C₈	—	2	4	no	—
160	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
161	#97	—	2	2	no	—
162	#97	—	2	2	no	—
163	#97	—	2	2	no	—
164	#97	C₂×C₄×C₃₁	1	4	yes	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
169	C₃₁	#21372	1	1	yes	characteristic, direct product factor, Sylow
170	C₂⁴	C₂×Q₈×C₃₁	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
173	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
174	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
175	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
176	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
177	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
178	M₁₆	—	2	8	no	—
179	M₁₆	—	2	4	no	—
180	M₁₆	—	2	4	no	—
181	M₁₆	—	2	4	no	—
182	M₁₆	#41×C₃₁	1	8	yes	—
183	C₂×C₈	—	2	8	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
189	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
190	C₄⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
191	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
192	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
193	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
194	C₄²	C₂×D₄×C₃₁	1	2	yes	—
195	C₄²	C₂×D₄×C₃₁	1	2	yes	—
196	C₂³	—	2	2	no	—
197	C₂³	—	4	2	no	—
198	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
199	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
208	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
209	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
211	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
212	C₈	—	2	4	no	—
213	C₈	—	2	4	no	—
214	C₈	—	2	8	no	—
215	C₈	—	4	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	2	no	—
220	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
221	C₂²	C₃₁×#533	1	2	yes	—
222	C₄	—	2	4	no	—
223	C₄	—	2	4	no	—
224	C₄	—	2	2	no	—
225	C₄	C₄×#41×C₃₁	1	2	yes	—
226	C₄	C₃₁×#525	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	—	4	2	no	—
229	C₂	C₃₁×#2296	1	1	yes	characteristic
230	C₂	C₃₁×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₃₁	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1950	C₂	semidirect
6. C₃₁×#1778	C₂	semidirect
7. C₄×C₃₁×#108	C₂	semidirect
80. #21372	C₃₁	direct (N × H)
104. #1950	C₂×C₃₁	semidirect
106. #1778	C₂×C₃₁	semidirect
107. C₄×#108	C₂×C₃₁	semidirect
169. C₃₁	#21372	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2921	2
C₂×C₃₁×#433	2
C₃₁×#2873	2
C₃₁×#1950	2
C₃₁×#1855	2
C₃₁×#1778	2
C₄×C₃₁×#108	2
#21372	31

Subgroups — C31×#21372 #21478131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21372 #21478131