Subgroups — C₃₁×#33226 #21489985

Order 7936.

970 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#488	C₂	1	4	yes	maximal, complemented
2	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2858	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1958	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1957	C₂	1	2	yes	maximal
6	C₃₁×#1956	C₂	1	2	yes	maximal, complemented
7	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#125	C₂²	1	2	yes	—
11	C₂×C₃₁×#124	C₂²	1	2	yes	—
12	C₂×C₃₁×#124	C₂²	1	2	yes	—
13	C₃₁×#488	—	2	16	no	—
14	C₂×C₃₁×#103	C₂²	1	8	yes	—
15	C₂×C₃₁×#102	C₂²	1	8	yes	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#390	—	2	4	no	—
20	C₃₁×#389	—	2	4	no	—
21	C₃₁×#384	C₂²	1	2	yes	—
22	C₃₁×#382	C₂²	1	2	yes	—
23	C₄³×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
25	C₂²×D₄×C₃₁	—	2	2	no	—
26	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
27	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
28	C₃₁×C₄⋊Q₈	—	2	4	no	—
29	C₃₁×#127	—	2	4	no	—
30	C₃₁×#125	—	2	8	no	—
31	C₃₁×#124	—	2	8	no	—
32	C₃₁×#124	—	2	8	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
37	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
38	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
39	C₂×C₄²×C₃₁	D₄	1	4	yes	—
40	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#103	—	2	16	no	—
44	C₃₁×#102	—	2	16	no	—
45	C₄×C₈×C₃₁	—	2	4	no	—
46	C₄×C₈×C₃₁	—	2	4	no	—
47	C₄×C₈×C₃₁	D₄	1	8	yes	complemented
48	C₃₁×#95	—	2	4	no	—
49	C₂⁴×C₃₁	—	2	2	no	—
50	C₂×Q₈×C₃₁	—	2	4	no	—
51	C₂×Q₈×C₃₁	—	2	4	no	—
52	C₂×Q₈×C₃₁	—	2	4	no	—
53	C₂×D₄×C₃₁	—	2	4	no	—
54	C₂×D₄×C₃₁	—	2	4	no	—
55	C₂×D₄×C₃₁	—	2	4	no	—
56	C₂×D₄×C₃₁	—	4	4	no	—
57	C₂×D₄×C₃₁	—	4	8	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₃₁	—	2	4	no	—
65	C₂×C₈×C₃₁	—	2	4	no	—
66	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
67	C₄⋊C₄×C₃₁	—	2	8	no	—
68	C₄⋊C₄×C₃₁	—	4	4	no	—
69	C₄⋊C₄×C₃₁	—	4	8	no	—
70	C₄⋊C₄×C₃₁	—	4	8	no	—
71	C₂²⋊C₄×C₃₁	—	4	8	no	—
72	C₂²⋊C₄×C₃₁	—	4	8	no	—
73	C₄²×C₃₁	—	2	8	no	—
74	C₄²×C₃₁	C₂×D₄	1	4	yes	—
75	C₄²×C₃₁	#41	1	4	yes	—
76	C₄²×C₃₁	#41	1	4	yes	—
77	#33226	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₃₁	—	4	2	no	—
79	C₂³×C₃₁	—	4	2	no	—
80	C₂³×C₃₁	—	4	2	no	—
81	C₂³×C₃₁	—	4	8	no	—
82	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
83	Q₈×C₃₁	—	4	8	no	—
84	D₄×C₃₁	—	4	8	no	—
85	D₄×C₃₁	—	8	8	no	—
86	C₂×C₄×C₃₁	—	2	4	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	4	2	no	—
89	C₂×C₄×C₃₁	—	4	2	no	—
90	C₂×C₄×C₃₁	—	4	2	no	—
91	C₂×C₄×C₃₁	—	4	4	no	—
92	C₂×C₄×C₃₁	—	4	8	no	—
93	C₂×C₄×C₃₁	#124	1	4	yes	—
94	C₂×C₄×C₃₁	#124	1	4	yes	—
95	C₂×C₄×C₃₁	#127	1	4	yes	—
96	C₂×C₄×C₃₁	#135	1	8	yes	—
97	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
98	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
99	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
100	C₈×C₃₁	—	2	8	no	—
101	C₂×#488	C₂×C₃₁	1	4	yes	complemented
102	#2859	C₂×C₃₁	1	1	yes	characteristic, complemented
103	#2858	C₂×C₃₁	1	1	yes	characteristic
104	#1958	C₂×C₃₁	1	4	yes	complemented
105	#1957	C₂×C₃₁	1	2	yes	—
106	#1956	C₂×C₃₁	1	2	yes	complemented
107	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
108	C₂²×C₃₁	—	8	2	no	—
109	C₂²×C₃₁	—	8	4	no	—
110	C₂²×C₃₁	—	8	4	no	—
111	C₂²×C₃₁	—	8	4	no	—
112	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
113	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
114	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
115	C₂²×C₃₁	C₂×#135	1	4	yes	—
116	C₄×C₃₁	—	2	4	no	—
117	C₄×C₃₁	—	2	4	no	—
118	C₄×C₃₁	—	8	2	no	—
119	C₄×C₃₁	—	8	4	no	—
120	C₄×C₃₁	#488	1	8	yes	complemented
121	C₄×C₃₁	#532	1	4	yes	—
122	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
123	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
124	C₂×#125	C₂²×C₃₁	1	2	yes	—
125	C₂×#124	C₂²×C₃₁	1	2	yes	—
126	C₂×#124	C₂²×C₃₁	1	2	yes	—
127	#488	—	2	16	no	—
128	C₂×#103	C₂²×C₃₁	1	8	yes	—
129	C₂×#102	C₂²×C₃₁	1	8	yes	—
130	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
131	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
132	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
133	#390	—	2	4	no	—
134	#389	—	2	4	no	—
135	#384	C₂²×C₃₁	1	2	yes	—
136	#382	C₂²×C₃₁	1	2	yes	—
137	C₄³	C₂²×C₃₁	1	1	yes	characteristic
138	C₂×C₃₁	—	16	2	no	—
139	C₂×C₃₁	#2858	1	1	yes	characteristic
140	C₂×C₃₁	C₂×#488	1	2	yes	—
141	C₂×C₃₁	#3131	1	2	yes	—
142	C₂×C₃₁	#3147	1	2	yes	—
143	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
144	C₂²×D₄	—	2	2	no	—
145	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
146	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
147	C₄⋊Q₈	—	2	4	no	—
148	#127	—	2	4	no	—
149	#125	—	2	8	no	—
150	#124	—	2	8	no	—
151	#124	—	2	8	no	—
152	C₂×C₄⋊C₄	—	2	4	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
156	C₂×C₂²⋊C₄	—	2	4	no	—
157	C₂×C₂²⋊C₄	—	2	4	no	—
158	C₂×C₄²	D₄×C₃₁	1	4	yes	—
159	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
160	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
162	#103	—	2	16	no	—
163	#102	—	2	16	no	—
164	C₄×C₈	—	2	4	no	—
165	C₄×C₈	—	2	4	no	—
166	C₄×C₈	D₄×C₃₁	1	8	yes	complemented
167	#95	—	2	4	no	—
168	C₃₁	#33226	1	1	yes	characteristic, direct product factor, Sylow
169	C₂⁴	—	2	2	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	4	4	no	—
177	C₂×D₄	—	4	8	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
183	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
187	C₄⋊C₄	—	2	8	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	8	no	—
190	C₄⋊C₄	—	4	8	no	—
191	C₂²⋊C₄	—	4	8	no	—
192	C₂²⋊C₄	—	4	8	no	—
193	C₄²	—	2	8	no	—
194	C₄²	C₂×D₄×C₃₁	1	4	yes	—
195	C₄²	#41×C₃₁	1	4	yes	—
196	C₄²	#41×C₃₁	1	4	yes	—
197	C₂³	—	4	2	no	—
198	C₂³	—	4	2	no	—
199	C₂³	—	4	2	no	—
200	C₂³	—	4	8	no	—
201	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
202	Q₈	—	4	8	no	—
203	D₄	—	4	8	no	—
204	D₄	—	8	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	8	no	—
212	C₂×C₄	C₃₁×#124	1	4	yes	—
213	C₂×C₄	C₃₁×#124	1	4	yes	—
214	C₂×C₄	C₃₁×#127	1	4	yes	—
215	C₂×C₄	C₃₁×#135	1	8	yes	—
216	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
217	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
218	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
219	C₈	—	2	8	no	—
220	C₂²	—	8	2	no	—
221	C₂²	—	8	4	no	—
222	C₂²	—	8	4	no	—
223	C₂²	—	8	4	no	—
224	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
225	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
226	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
227	C₂²	C₂×C₃₁×#135	1	4	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	4	no	—
230	C₄	—	8	2	no	—
231	C₄	—	8	4	no	—
232	C₄	C₃₁×#488	1	8	yes	complemented
233	C₄	C₃₁×#532	1	4	yes	—
234	C₂	—	16	2	no	—
235	C₂	C₃₁×#2858	1	1	yes	characteristic
236	C₂	C₂×C₃₁×#488	1	2	yes	—
237	C₂	C₃₁×#3131	1	2	yes	—
238	C₂	C₃₁×#3147	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#488	C₂	semidirect
2. C₃₁×#2859	C₂	semidirect
4. C₃₁×#1958	C₂	semidirect
6. C₃₁×#1956	C₂	semidirect
7. C₄²×C₈×C₃₁	C₂	semidirect
47. C₄×C₈×C₃₁	D₄	semidirect
77. #33226	C₃₁	direct (N × H)
101. C₂×#488	C₂×C₃₁	semidirect
102. #2859	C₂×C₃₁	semidirect
104. #1958	C₂×C₃₁	semidirect
106. #1956	C₂×C₃₁	semidirect
107. C₄²×C₈	C₂×C₃₁	semidirect
120. C₄×C₃₁	#488	semidirect
166. C₄×C₈	D₄×C₃₁	semidirect
168. C₃₁	#33226	direct (N × H)
232. C₄	C₃₁×#488	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#488	2
C₃₁×#2859	2
C₃₁×#2858	2
C₃₁×#1958	2
C₃₁×#1957	2
C₃₁×#1956	2
C₄²×C₈×C₃₁	2
#33226	31

Subgroups — C31×#33226 #21489985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33226 #21489985