Subgroups — C₃₁×#33607 #21490366

Order 7936.

714 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3058	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2218	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2215	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2140	C₂	1	2	yes	maximal, complemented
8	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
9	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
10	C₃₁×#535	—	2	2	no	—
11	C₂×C₃₁×D₁₆	—	2	4	no	—
12	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
13	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
14	C₃₁×#504	—	2	4	no	—
15	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×M₃₂	C₂²	1	2	yes	complemented
17	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	complemented
19	C₃₁×#469	—	2	4	no	—
20	C₃₁×#466	C₂²	1	8	yes	complemented
21	C₃₁×#427	C₂²	1	2	yes	—
22	C₃₁×#418	—	2	2	no	—
23	C₃₁×#367	C₂²	1	2	yes	complemented
24	C₃₁×#366	C₂²	1	2	yes	complemented
25	C₃₁×#357	C₂²	1	8	yes	complemented
26	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₃₁	—	2	2	no	—
28	C₂×D₈×C₃₁	—	2	4	no	—
29	C₂×D₈×C₃₁	—	2	2	no	—
30	C₂×D₈×C₃₁	—	2	4	no	—
31	C₂×D₈×C₃₁	C₂³	1	4	yes	—
32	C₃₁×#131	—	2	2	no	—
33	C₃₁×#131	—	2	2	no	—
34	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
38	C₃₁×#127	—	4	2	no	—
39	C₃₁×#124	—	4	2	no	—
40	C₃₁×#121	—	2	8	no	—
41	C₃₁×#120	—	4	4	no	—
42	C₃₁×#117	—	2	2	no	—
43	C₃₁×D₁₆	—	2	4	no	—
44	C₃₁×D₁₆	—	2	4	no	—
45	C₃₁×M₃₂	—	2	2	no	—
46	C₃₁×M₃₂	—	2	4	no	—
47	C₂×C₁₆×C₃₁	—	2	4	no	—
48	C₂×C₁₆×C₃₁	D₄	1	4	yes	complemented
49	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
50	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#107	C₂³	1	4	yes	—
53	C₃₁×#106	—	2	2	no	—
54	C₃₁×#102	—	2	8	no	—
55	C₂⁴×C₃₁	—	4	2	no	—
56	#41×C₃₁	—	2	2	no	—
57	#41×C₃₁	—	2	2	no	—
58	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
59	C₂×D₄×C₃₁	—	2	4	no	—
60	C₂×D₄×C₃₁	—	4	8	no	—
61	C₂×D₄×C₃₁	—	4	4	no	—
62	C₂×D₄×C₃₁	—	4	2	no	—
63	C₂×D₄×C₃₁	—	4	4	no	—
64	C₂×D₄×C₃₁	D₈	1	2	yes	—
65	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	D₈×C₃₁	—	2	4	no	—
69	D₈×C₃₁	—	2	4	no	—
70	D₈×C₃₁	—	4	4	no	—
71	M₁₆×C₃₁	—	2	2	no	—
72	M₁₆×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₃₁	#41	1	2	yes	—
79	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₂²⋊C₄×C₃₁	—	4	2	no	—
82	C₂²⋊C₄×C₃₁	—	4	8	no	—
83	C₄²×C₃₁	—	4	2	no	—
84	C₁₆×C₃₁	—	2	2	no	—
85	C₁₆×C₃₁	C₂×D₄	1	2	yes	—
86	C₁₆×C₃₁	C₂×D₄	1	2	yes	—
87	#33607	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	—	4	4	no	—
89	C₂³×C₃₁	—	8	4	no	—
90	C₂³×C₃₁	—	16	4	no	—
91	C₂³×C₃₁	C₂×D₈	1	2	yes	—
92	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
93	Q₈×C₃₁	—	2	2	no	—
94	D₄×C₃₁	—	2	4	no	—
95	D₄×C₃₁	—	2	2	no	—
96	D₄×C₃₁	—	4	4	no	—
97	D₄×C₃₁	—	4	4	no	—
98	D₄×C₃₁	—	8	4	no	—
99	D₄×C₃₁	—	16	8	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	#121	1	2	yes	—
104	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₃₁	#135	1	2	yes	—
107	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₃₁	—	2	2	no	—
109	C₈×C₃₁	#121	1	2	yes	—
110	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
111	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
112	C₂²×D₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
113	C₂×#504	C₂×C₃₁	1	1	yes	characteristic, complemented
114	#3058	C₂×C₃₁	1	2	yes	complemented
115	#2218	C₂×C₃₁	1	4	yes	complemented
116	#2215	C₂×C₃₁	1	4	yes	complemented
117	#2158	C₂×C₃₁	1	1	yes	characteristic, complemented
118	#2140	C₂×C₃₁	1	2	yes	complemented
119	C₂²×C₃₁	—	2	2	no	—
120	C₂²×C₃₁	—	2	4	no	—
121	C₂²×C₃₁	—	2	2	no	—
122	C₂²×C₃₁	—	8	8	no	—
123	C₂²×C₃₁	—	8	4	no	—
124	C₂²×C₃₁	—	16	8	no	—
125	C₂²×C₃₁	#466	1	2	yes	—
126	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
127	C₄×C₃₁	—	2	2	no	—
128	C₄×C₃₁	—	8	4	no	—
129	C₄×C₃₁	#466	1	2	yes	—
130	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
131	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
132	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
133	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
134	#535	—	2	2	no	—
135	C₂×D₁₆	—	2	4	no	—
136	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
137	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
138	#504	—	2	4	no	—
139	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
140	C₂×M₃₂	C₂²×C₃₁	1	2	yes	complemented
141	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
142	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	complemented
143	#469	—	2	4	no	—
144	#466	C₂²×C₃₁	1	8	yes	complemented
145	#427	C₂²×C₃₁	1	2	yes	—
146	#418	—	2	2	no	—
147	#367	C₂²×C₃₁	1	2	yes	complemented
148	#366	C₂²×C₃₁	1	2	yes	complemented
149	#357	C₂²×C₃₁	1	8	yes	complemented
150	C₂×C₃₁	—	2	2	no	—
151	C₂×C₃₁	—	4	2	no	—
152	C₂×C₃₁	—	16	4	no	—
153	C₂×C₃₁	C₂×#466	1	1	yes	characteristic
154	C₂×C₃₁	#3415	1	2	yes	—
155	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
156	C₂²×D₄	—	2	2	no	—
157	C₂×D₈	—	2	4	no	—
158	C₂×D₈	—	2	2	no	—
159	C₂×D₈	—	2	4	no	—
160	C₂×D₈	C₂³×C₃₁	1	4	yes	—
161	#131	—	2	2	no	—
162	#131	—	2	2	no	—
163	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
164	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
165	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
166	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
167	#127	—	4	2	no	—
168	#124	—	4	2	no	—
169	#121	—	2	8	no	—
170	#120	—	4	4	no	—
171	#117	—	2	2	no	—
172	D₁₆	—	2	4	no	—
173	D₁₆	—	2	4	no	—
174	M₃₂	—	2	2	no	—
175	M₃₂	—	2	4	no	—
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	D₄×C₃₁	1	4	yes	complemented
178	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
179	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
180	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
181	#107	C₂³×C₃₁	1	4	yes	—
182	#106	—	2	2	no	—
183	#102	—	2	8	no	—
184	C₃₁	#33607	1	1	yes	characteristic, direct product factor, Sylow
185	C₂⁴	—	4	2	no	—
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	4	8	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	4	no	—
194	C₂×D₄	D₈×C₃₁	1	2	yes	—
195	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
198	D₈	—	2	4	no	—
199	D₈	—	2	4	no	—
200	D₈	—	4	4	no	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	4	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
206	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
208	C₂×C₈	#41×C₃₁	1	2	yes	—
209	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
210	C₄⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₂²⋊C₄	—	4	8	no	—
213	C₄²	—	4	2	no	—
214	C₁₆	—	2	2	no	—
215	C₁₆	C₂×D₄×C₃₁	1	2	yes	—
216	C₁₆	C₂×D₄×C₃₁	1	2	yes	—
217	C₂³	—	4	4	no	—
218	C₂³	—	8	4	no	—
219	C₂³	—	16	4	no	—
220	C₂³	C₂×D₈×C₃₁	1	2	yes	—
221	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	2	2	no	—
225	D₄	—	4	4	no	—
226	D₄	—	4	4	no	—
227	D₄	—	8	4	no	—
228	D₄	—	16	8	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	C₃₁×#121	1	2	yes	—
233	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
234	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
235	C₂×C₄	C₃₁×#135	1	2	yes	—
236	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	C₃₁×#121	1	2	yes	—
239	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
240	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	8	8	no	—
245	C₂²	—	8	4	no	—
246	C₂²	—	16	8	no	—
247	C₂²	C₃₁×#466	1	2	yes	—
248	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	8	4	no	—
251	C₄	C₃₁×#466	1	2	yes	—
252	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
253	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	4	2	no	—
256	C₂	—	16	4	no	—
257	C₂	C₂×C₃₁×#466	1	1	yes	characteristic
258	C₂	C₃₁×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×#41×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×D₁₆	C₂	semidirect
2. C₂×C₃₁×#504	C₂	semidirect
3. C₃₁×#3058	C₂	semidirect
4. C₃₁×#2218	C₂	semidirect
5. C₃₁×#2215	C₂	semidirect
6. C₃₁×#2158	C₂	semidirect
7. C₃₁×#2140	C₂	semidirect
16. C₂×C₃₁×M₃₂	C₂²	semidirect
18. C₂²×C₁₆×C₃₁	C₂²	semidirect
20. C₃₁×#466	C₂²	semidirect
23. C₃₁×#367	C₂²	semidirect
24. C₃₁×#366	C₂²	semidirect
25. C₃₁×#357	C₂²	semidirect
48. C₂×C₁₆×C₃₁	D₄	semidirect
87. #33607	C₃₁	direct (N × H)
112. C₂²×D₁₆	C₂×C₃₁	semidirect
113. C₂×#504	C₂×C₃₁	semidirect
114. #3058	C₂×C₃₁	semidirect
115. #2218	C₂×C₃₁	semidirect
116. #2215	C₂×C₃₁	semidirect
117. #2158	C₂×C₃₁	semidirect
118. #2140	C₂×C₃₁	semidirect
140. C₂×M₃₂	C₂²×C₃₁	semidirect
142. C₂²×C₁₆	C₂²×C₃₁	semidirect
144. #466	C₂²×C₃₁	semidirect
147. #367	C₂²×C₃₁	semidirect
148. #366	C₂²×C₃₁	semidirect
149. #357	C₂²×C₃₁	semidirect
177. C₂×C₁₆	D₄×C₃₁	semidirect
184. C₃₁	#33607	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×D₁₆	2
C₂×C₃₁×#504	2
C₃₁×#3058	2
C₃₁×#2218	2
C₃₁×#2215	2
C₃₁×#2158	2
C₃₁×#2140	2
#33607	31

Subgroups — C31×#33607 #21490366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33607 #21490366