Subgroups — C₃₁×#33426 #21490185

Order 7936.

520 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2875	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2166	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2165	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2153	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2115	C₂	1	2	yes	maximal
6	C₂×C₃₁×#347	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#131	C₄	1	1	yes	characteristic
8	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
9	C₄×#41×C₃₁	C₄	1	1	yes	characteristic
10	C₂×C₃₁×M₃₂	C₂²	1	4	yes	—
11	C₃₁×#405	C₄	1	2	yes	—
12	C₃₁×#405	C₄	1	1	yes	characteristic
13	C₃₁×#405	C₂²	1	2	yes	—
14	C₃₁×#405	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₃₁	C₄	1	1	yes	characteristic
16	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
17	C₂×C₃₁×#97	C₄	1	2	yes	—
18	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
19	C₃₁×#364	C₂²	1	8	yes	—
20	C₃₁×#349	C₂²	1	8	yes	—
21	C₃₁×#347	C₂²	1	4	yes	—
22	C₃₁×#347	C₂²	1	4	yes	—
23	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₃₁×#131	—	2	4	no	—
25	C₃₁×#131	—	2	4	no	—
26	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
30	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
34	C₄×Q₈×C₃₁	—	2	1	no	—
35	C₄×D₄×C₃₁	—	2	2	no	—
36	C₄×D₄×C₃₁	—	2	1	no	—
37	C₃₁×#117	C₂×C₄	1	2	yes	—
38	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₃₁×M₃₂	—	2	4	no	—
42	C₃₁×M₃₂	C₂³	1	8	yes	—
43	C₂×C₁₆×C₃₁	—	2	4	no	—
44	C₃₁×#97	C₂×C₄	1	1	yes	characteristic
45	C₃₁×#97	C₂×C₄	1	2	yes	—
46	C₃₁×#97	C₂×C₄	1	2	yes	—
47	C₃₁×#97	C₂×C₄	1	2	yes	—
48	C₃₁×#97	C₂×C₄	1	1	yes	characteristic
49	C₃₁×#97	C₂³	1	1	yes	characteristic
50	C₃₁×#97	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₃₁	C₂³	1	2	yes	—
54	#41×C₃₁	—	2	2	no	—
55	#41×C₃₁	—	2	2	no	—
56	C₂×Q₈×C₃₁	M₁₆	1	1	yes	characteristic
57	C₂×D₄×C₃₁	M₁₆	1	2	yes	—
58	C₂×D₄×C₃₁	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	—	2	1	no	—
61	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	M₁₆×C₃₁	—	2	2	no	—
64	M₁₆×C₃₁	—	2	4	no	—
65	M₁₆×C₃₁	—	2	4	no	—
66	M₁₆×C₃₁	M₁₆	1	4	yes	—
67	C₂×C₈×C₃₁	—	2	2	no	—
68	C₂×C₈×C₃₁	—	2	4	no	—
69	C₂×C₈×C₃₁	—	2	4	no	—
70	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₃₁	#41	1	4	yes	—
76	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
77	C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂²⋊C₄×C₃₁	—	2	1	no	—
81	C₄²×C₃₁	—	2	2	no	—
82	C₄²×C₃₁	—	2	1	no	—
83	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
85	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₁₆×C₃₁	—	2	8	no	—
87	#33426	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
89	C₂³×C₃₁	C₂×M₁₆	1	1	yes	characteristic
90	Q₈×C₃₁	—	2	2	no	—
91	D₄×C₃₁	—	2	4	no	—
92	D₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	8	1	no	—
99	C₂×C₄×C₃₁	—	8	2	no	—
100	C₂×C₄×C₃₁	#117	1	2	yes	—
101	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
102	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
105	C₈×C₃₁	—	2	2	no	—
106	C₈×C₃₁	—	2	1	no	—
107	C₈×C₃₁	—	2	1	no	—
108	C₈×C₃₁	#117	1	2	yes	—
109	C₈×C₃₁	C₂×M₁₆	1	4	yes	—
110	C₈×C₃₁	C₂×#41	1	2	yes	—
111	#2875	C₂×C₃₁	1	1	yes	characteristic
112	#2166	C₂×C₃₁	1	4	yes	complemented
113	#2165	C₂×C₃₁	1	4	yes	complemented
114	#2153	C₂×C₃₁	1	2	yes	complemented
115	#2115	C₂×C₃₁	1	2	yes	—
116	C₂×#347	C₂×C₃₁	1	2	yes	complemented
117	C₂²×C₃₁	—	2	2	no	—
118	C₂²×C₃₁	—	4	1	no	—
119	C₂²×C₃₁	—	4	2	no	—
120	C₂²×C₃₁	#432	1	2	yes	—
121	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
122	C₄×C₃₁	—	2	2	no	—
123	C₄×C₃₁	—	4	1	no	—
124	C₄×C₃₁	—	4	2	no	—
125	C₄×C₃₁	—	4	1	no	—
126	C₄×C₃₁	#432	1	2	yes	—
127	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
128	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
129	C₂×#131	C₄×C₃₁	1	1	yes	characteristic
130	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
131	C₄×#41	C₄×C₃₁	1	1	yes	characteristic
132	C₂×M₃₂	C₂²×C₃₁	1	4	yes	—
133	#405	C₄×C₃₁	1	2	yes	—
134	#405	C₄×C₃₁	1	1	yes	characteristic
135	#405	C₂²×C₃₁	1	2	yes	—
136	#405	C₂²×C₃₁	1	1	yes	characteristic
137	C₄×M₁₆	C₄×C₃₁	1	1	yes	characteristic
138	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
139	C₂×#97	C₄×C₃₁	1	2	yes	—
140	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
141	#364	C₂²×C₃₁	1	8	yes	—
142	#349	C₂²×C₃₁	1	8	yes	—
143	#347	C₂²×C₃₁	1	4	yes	—
144	#347	C₂²×C₃₁	1	4	yes	—
145	C₂×C₃₁	—	2	1	no	—
146	C₂×C₃₁	—	2	2	no	—
147	C₂×C₃₁	—	4	2	no	—
148	C₂×C₃₁	C₂×#432	1	1	yes	characteristic
149	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
150	#131	—	2	4	no	—
151	#131	—	2	4	no	—
152	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
153	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
154	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
155	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
156	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
157	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
158	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
159	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
160	C₄×Q₈	—	2	1	no	—
161	C₄×D₄	—	2	2	no	—
162	C₄×D₄	—	2	1	no	—
163	#117	C₂×C₄×C₃₁	1	2	yes	—
164	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
165	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
166	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
167	M₃₂	—	2	4	no	—
168	M₃₂	C₂³×C₃₁	1	8	yes	—
169	C₂×C₁₆	—	2	4	no	—
170	#97	C₂×C₄×C₃₁	1	1	yes	characteristic
171	#97	C₂×C₄×C₃₁	1	2	yes	—
172	#97	C₂×C₄×C₃₁	1	2	yes	—
173	#97	C₂×C₄×C₃₁	1	2	yes	—
174	#97	C₂×C₄×C₃₁	1	1	yes	characteristic
175	#97	C₂³×C₃₁	1	1	yes	characteristic
176	#97	C₂³×C₃₁	1	1	yes	characteristic
177	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
178	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
179	C₄×C₈	C₂³×C₃₁	1	2	yes	—
180	C₃₁	#33426	1	1	yes	characteristic, direct product factor, Sylow
181	#41	—	2	2	no	—
182	#41	—	2	2	no	—
183	C₂×Q₈	M₁₆×C₃₁	1	1	yes	characteristic
184	C₂×D₄	M₁₆×C₃₁	1	2	yes	—
185	C₂×D₄	M₁₆×C₃₁	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
189	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
190	M₁₆	—	2	2	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	2	4	no	—
193	M₁₆	M₁₆×C₃₁	1	4	yes	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
198	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
199	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
200	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
201	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
202	C₂×C₈	#41×C₃₁	1	4	yes	—
203	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
211	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
212	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
213	C₁₆	—	2	8	no	—
214	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
215	C₂³	C₂×M₁₆×C₃₁	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	D₄	—	2	4	no	—
218	D₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	C₃₁×#117	1	2	yes	—
227	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
228	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
229	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
230	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	C₃₁×#117	1	2	yes	—
235	C₈	C₂×M₁₆×C₃₁	1	4	yes	—
236	C₈	C₂×#41×C₃₁	1	2	yes	—
237	C₂²	—	2	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	2	no	—
240	C₂²	C₃₁×#432	1	2	yes	—
241	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	2	no	—
246	C₄	C₃₁×#432	1	2	yes	—
247	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
248	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	2	2	no	—
251	C₂	—	4	2	no	—
252	C₂	C₂×C₃₁×#432	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	C₂×#117
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂×C₃₁	C₂×#432
prime radical	C₂×#41×C₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂×C₃₁×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2166	C₂	semidirect
3. C₃₁×#2165	C₂	semidirect
4. C₃₁×#2153	C₂	semidirect
6. C₂×C₃₁×#347	C₂	semidirect
87. #33426	C₃₁	direct (N × H)
112. #2166	C₂×C₃₁	semidirect
113. #2165	C₂×C₃₁	semidirect
114. #2153	C₂×C₃₁	semidirect
116. C₂×#347	C₂×C₃₁	semidirect
180. C₃₁	#33426	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×#2875	2
C₃₁×#2166	2
C₃₁×#2165	2
C₃₁×#2153	2
C₃₁×#2115	2
C₂×C₃₁×#347	2
#33426	31

Subgroups — C31×#33426 #21490185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33426 #21490185