Subgroups — C₃₁×#33420 #21490179

Order 7936.

598 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2164	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2162	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2151	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2113	C₂	1	2	yes	maximal
6	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#131	C₄	1	1	yes	characteristic
9	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
10	C₄×#41×C₃₁	C₄	1	1	yes	characteristic
11	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#405	C₄	1	2	yes	—
14	C₃₁×#405	C₄	1	1	yes	characteristic
15	C₃₁×#405	C₂²	1	1	yes	characteristic
16	C₃₁×#405	C₂²	1	2	yes	—
17	C₄×M₁₆×C₃₁	C₄	1	1	yes	characteristic
18	C₄×M₁₆×C₃₁	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	C₃₁×C₄⋊C₁₆	C₂²	1	8	yes	—
22	C₃₁×C₂²⋊C₁₆	C₂²	1	8	yes	—
23	C₃₁×#346	C₂²	1	2	yes	—
24	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
25	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
26	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
27	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₃₁×#131	—	2	4	no	—
29	C₃₁×#131	—	2	4	no	—
30	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
38	C₄×Q₈×C₃₁	C₈	1	2	yes	—
39	C₄×D₄×C₃₁	C₈	1	4	yes	—
40	C₄×D₄×C₃₁	C₈	1	2	yes	—
41	C₃₁×#117	C₂×C₄	1	2	yes	—
42	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₃₁×M₃₂	—	2	4	no	—
46	C₂×C₁₆×C₃₁	—	2	4	no	—
47	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
48	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
49	C₃₁×#97	C₂×C₄	1	2	yes	—
50	C₃₁×#97	C₂×C₄	1	2	yes	—
51	C₃₁×#97	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₃₁	C₂³	1	2	yes	—
58	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	#41×C₃₁	—	2	2	no	—
60	#41×C₃₁	—	2	2	no	—
61	C₂×Q₈×C₃₁	C₂×C₈	1	1	yes	characteristic
62	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
63	C₂×D₄×C₃₁	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
65	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	M₁₆×C₃₁	—	2	2	no	—
69	M₁₆×C₃₁	—	2	2	no	—
70	M₁₆×C₃₁	—	2	4	no	—
71	M₁₆×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	M₁₆	1	4	yes	—
75	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₃₁	#41	1	4	yes	—
81	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
83	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
84	C₂²⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
85	C₂²⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
86	C₄²×C₃₁	C₂×C₈	1	4	yes	—
87	C₄²×C₃₁	C₂×C₈	1	2	yes	—
88	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
91	C₁₆×C₃₁	—	2	4	no	—
92	C₁₆×C₃₁	—	2	4	no	—
93	#33420	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	C₂²×C₈	1	2	yes	—
95	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
96	Q₈×C₃₁	—	2	2	no	—
97	D₄×C₃₁	—	2	4	no	—
98	D₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	2	8	no	—
103	C₂×C₄×C₃₁	#117	1	2	yes	—
104	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
111	C₈×C₃₁	—	2	2	no	—
112	C₈×C₃₁	—	2	2	no	—
113	C₈×C₃₁	#117	1	2	yes	—
114	C₈×C₃₁	C₂×M₁₆	1	2	yes	—
115	C₈×C₃₁	C₂×M₁₆	1	2	yes	—
116	C₈×C₃₁	C₂×#41	1	2	yes	—
117	C₄×#131	C₂×C₃₁	1	1	yes	characteristic
118	#2164	C₂×C₃₁	1	4	yes	complemented
119	#2162	C₂×C₃₁	1	4	yes	complemented
120	#2151	C₂×C₃₁	1	2	yes	complemented
121	#2113	C₂×C₃₁	1	2	yes	—
122	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
123	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
124	C₂²×C₃₁	—	2	2	no	—
125	C₂²×C₃₁	—	2	4	no	—
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	#431	1	2	yes	—
128	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
129	C₄×C₃₁	—	2	2	no	—
130	C₄×C₃₁	—	2	4	no	—
131	C₄×C₃₁	—	2	2	no	—
132	C₄×C₃₁	#431	1	2	yes	—
133	C₄×C₃₁	#504	1	4	yes	—
134	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
135	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
136	C₂×#131	C₄×C₃₁	1	1	yes	characteristic
137	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
138	C₄×#41	C₄×C₃₁	1	1	yes	characteristic
139	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
140	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
141	#405	C₄×C₃₁	1	2	yes	—
142	#405	C₄×C₃₁	1	1	yes	characteristic
143	#405	C₂²×C₃₁	1	1	yes	characteristic
144	#405	C₂²×C₃₁	1	2	yes	—
145	C₄×M₁₆	C₄×C₃₁	1	1	yes	characteristic
146	C₄×M₁₆	C₄×C₃₁	1	2	yes	—
147	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
149	C₄⋊C₁₆	C₂²×C₃₁	1	8	yes	—
150	C₂²⋊C₁₆	C₂²×C₃₁	1	8	yes	—
151	#346	C₂²×C₃₁	1	2	yes	—
152	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
153	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
154	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
155	C₂×C₃₁	—	2	2	no	—
156	C₂×C₃₁	—	4	2	no	—
157	C₂×C₃₁	C₂×#431	1	1	yes	characteristic
158	C₂×C₃₁	C₂×#504	1	2	yes	—
159	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
160	#131	—	2	4	no	—
161	#131	—	2	4	no	—
162	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
163	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
164	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
165	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
166	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
167	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
169	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
170	C₄×Q₈	C₈×C₃₁	1	2	yes	—
171	C₄×D₄	C₈×C₃₁	1	4	yes	—
172	C₄×D₄	C₈×C₃₁	1	2	yes	—
173	#117	C₂×C₄×C₃₁	1	2	yes	—
174	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
176	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
177	M₃₂	—	2	4	no	—
178	C₂×C₁₆	—	2	4	no	—
179	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
180	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
181	#97	C₂×C₄×C₃₁	1	2	yes	—
182	#97	C₂×C₄×C₃₁	1	2	yes	—
183	#97	C₂×C₄×C₃₁	1	2	yes	—
184	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
185	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
186	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
187	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
189	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
190	C₄×C₈	C₂³×C₃₁	1	2	yes	—
191	C₃₁	#33420	1	1	yes	characteristic, direct product factor, Sylow
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	C₂×Q₈	C₂×C₈×C₃₁	1	1	yes	characteristic
195	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
196	C₂×D₄	C₂×C₈×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
198	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
199	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
200	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	M₁₆×C₃₁	1	4	yes	—
208	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
209	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
210	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
211	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
212	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
213	C₂×C₈	#41×C₃₁	1	4	yes	—
214	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
215	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
216	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
217	C₂²⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
218	C₂²⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
219	C₄²	C₂×C₈×C₃₁	1	4	yes	—
220	C₄²	C₂×C₈×C₃₁	1	2	yes	—
221	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
222	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
223	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
224	C₁₆	—	2	4	no	—
225	C₁₆	—	2	4	no	—
226	C₂³	C₂²×C₈×C₃₁	1	2	yes	—
227	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
228	Q₈	—	2	2	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	C₃₁×#117	1	2	yes	—
236	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
237	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
238	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
239	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
241	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
242	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	C₃₁×#117	1	2	yes	—
246	C₈	C₂×M₁₆×C₃₁	1	2	yes	—
247	C₈	C₂×M₁₆×C₃₁	1	2	yes	—
248	C₈	C₂×#41×C₃₁	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	C₃₁×#431	1	2	yes	—
253	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	4	no	—
256	C₄	—	2	2	no	—
257	C₄	C₃₁×#431	1	2	yes	—
258	C₄	C₃₁×#504	1	4	yes	—
259	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
260	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	4	2	no	—
263	C₂	C₂×C₃₁×#431	1	1	yes	characteristic
264	C₂	C₂×C₃₁×#504	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₃₁	C₂³
commutator	C₂²	C₂³×C₈×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂³×C₈
prime radical	C₂×#41×C₃₁	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2164	C₂	semidirect
3. C₃₁×#2162	C₂	semidirect
4. C₃₁×#2151	C₂	semidirect
6. C₄×C₃₁×M₃₂	C₂	semidirect
7. C₂×C₄×C₁₆×C₃₁	C₂	semidirect
93. #33420	C₃₁	direct (N × H)
118. #2164	C₂×C₃₁	semidirect
119. #2162	C₂×C₃₁	semidirect
120. #2151	C₂×C₃₁	semidirect
122. C₄×M₃₂	C₂×C₃₁	semidirect
123. C₂×C₄×C₁₆	C₂×C₃₁	semidirect
191. C₃₁	#33420	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#131	2
C₃₁×#2164	2
C₃₁×#2162	2
C₃₁×#2151	2
C₃₁×#2113	2
C₄×C₃₁×M₃₂	2
C₂×C₄×C₁₆×C₃₁	2
#33420	31

Subgroups — C31×#33420 #21490179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33420 #21490179