Subgroups — C₃₁×#33169 #21489928

Order 7936.

1046 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#464	C₂	1	3	yes	maximal
2	C₂×C₃₁×#462	C₂	1	3	yes	maximal, complemented
3	C₃₁×#2658	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2276	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1983	C₂	1	3	yes	maximal
6	C₃₁×#1981	C₂	1	3	yes	maximal, complemented
7	C₃₁×#1843	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₃₁	C₂²	1	3	yes	—
9	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
11	C₂×C₃₁×#124	C₂²	1	3	yes	—
12	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
13	C₃₁×#464	—	2	12	no	—
14	C₃₁×#462	—	2	12	no	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	—
17	C₂×C₃₁×#103	C₂²	1	3	yes	—
18	C₂×C₃₁×#103	C₂²	1	6	yes	—
19	C₂×C₃₁×#102	C₂²	1	3	yes	—
20	C₃₁×#399	—	2	1	no	—
21	C₃₁×#399	—	2	3	no	—
22	C₄×C₄⋊C₄×C₃₁	—	2	3	no	—
23	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
24	C₃₁×#376	—	2	1	no	—
25	C₃₁×#376	C₂²	1	1	yes	characteristic
26	C₂²×Q₈×C₃₁	C₂³	1	3	yes	—
27	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂×Q₁₆×C₃₁	—	2	12	no	—
30	C₂×Q₁₆×C₃₁	—	2	6	no	—
31	C₂×SD₁₆×C₃₁	—	2	6	no	—
32	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×C₄⋊Q₈	—	2	6	no	—
35	C₃₁×C₄⋊Q₈	—	2	6	no	—
36	C₃₁×#124	—	2	6	no	—
37	C₃₁×#124	—	2	6	no	—
38	C₄×Q₈×C₃₁	—	2	6	no	—
39	C₄×Q₈×C₃₁	D₄	1	12	yes	—
40	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
43	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
45	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
46	C₂×C₄²×C₃₁	—	2	3	no	—
47	C₂×C₄²×C₃₁	—	2	3	no	—
48	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
49	C₃₁×C₄⋊C₈	—	2	6	no	—
50	C₃₁×C₄⋊C₈	—	2	6	no	—
51	C₃₁×#103	—	2	12	no	—
52	C₃₁×#103	—	2	6	no	—
53	C₃₁×#102	—	2	6	no	—
54	C₃₁×#95	—	2	3	no	—
55	C₃₁×#95	D₄	1	1	yes	characteristic
56	C₃₁×#95	D₄	1	3	yes	—
57	C₂⁴×C₃₁	—	2	1	no	—
58	C₂×Q₈×C₃₁	—	2	3	no	—
59	C₂×Q₈×C₃₁	—	2	3	no	—
60	C₂×Q₈×C₃₁	—	2	6	no	—
61	C₂×Q₈×C₃₁	—	2	3	no	—
62	C₂×Q₈×C₃₁	—	2	6	no	—
63	C₂×Q₈×C₃₁	C₂×D₄	1	6	yes	—
64	C₂×D₄×C₃₁	—	2	3	no	—
65	C₂×D₄×C₃₁	—	2	3	no	—
66	C₂²×C₄×C₃₁	—	2	1	no	—
67	C₂²×C₄×C₃₁	—	2	3	no	—
68	C₂²×C₄×C₃₁	—	2	3	no	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
71	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
72	Q₁₆×C₃₁	—	4	12	no	—
73	SD₁₆×C₃₁	—	4	4	no	—
74	C₂×C₈×C₃₁	—	2	3	no	—
75	C₂×C₈×C₃₁	—	2	3	no	—
76	C₂×C₈×C₃₁	—	2	6	no	—
77	C₄⋊C₄×C₃₁	—	2	12	no	—
78	C₄⋊C₄×C₃₁	—	2	3	no	—
79	C₄⋊C₄×C₃₁	—	4	6	no	—
80	C₄⋊C₄×C₃₁	—	4	6	no	—
81	C₄⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
82	C₂²⋊C₄×C₃₁	—	4	6	no	—
83	C₂²⋊C₄×C₃₁	—	4	6	no	—
84	C₄²×C₃₁	—	2	12	no	—
85	C₄²×C₃₁	—	8	3	no	—
86	C₄²×C₃₁	C₂×D₄	1	6	yes	—
87	C₄²×C₃₁	#41	1	6	yes	—
88	#33169	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₃₁	—	4	3	no	—
90	C₂³×C₃₁	—	4	1	no	—
91	C₂³×C₃₁	—	4	3	no	—
92	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
93	Q₈×C₃₁	—	2	6	no	—
94	Q₈×C₃₁	—	4	3	no	—
95	Q₈×C₃₁	—	4	3	no	—
96	Q₈×C₃₁	—	4	6	no	—
97	Q₈×C₃₁	#127	1	4	yes	—
98	D₄×C₃₁	—	4	1	no	—
99	D₄×C₃₁	—	4	3	no	—
100	C₂×C₄×C₃₁	—	2	6	no	—
101	C₂×C₄×C₃₁	—	2	3	no	—
102	C₂×C₄×C₃₁	—	2	6	no	—
103	C₂×C₄×C₃₁	—	2	3	no	—
104	C₂×C₄×C₃₁	—	2	6	no	—
105	C₂×C₄×C₃₁	—	4	3	no	—
106	C₂×C₄×C₃₁	—	4	3	no	—
107	C₂×C₄×C₃₁	—	4	3	no	—
108	C₂×C₄×C₃₁	—	4	6	no	—
109	C₂×C₄×C₃₁	#121	1	6	yes	—
110	C₂×C₄×C₃₁	#121	1	6	yes	—
111	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
112	C₂×C₄×C₃₁	C₂×#41	1	3	yes	—
113	C₈×C₃₁	—	4	1	no	—
114	C₈×C₃₁	—	4	3	no	—
115	C₂×#464	C₂×C₃₁	1	3	yes	—
116	C₂×#462	C₂×C₃₁	1	3	yes	complemented
117	#2658	C₂×C₃₁	1	1	yes	characteristic
118	#2276	C₂×C₃₁	1	1	yes	characteristic, complemented
119	#1983	C₂×C₃₁	1	3	yes	—
120	#1981	C₂×C₃₁	1	3	yes	complemented
121	#1843	C₂×C₃₁	1	1	yes	characteristic, complemented
122	C₂²×C₃₁	—	8	1	no	—
123	C₂²×C₃₁	—	8	3	no	—
124	C₂²×C₃₁	—	8	3	no	—
125	C₂²×C₃₁	C₂×#121	1	3	yes	—
126	C₂²×C₃₁	#577	1	1	yes	characteristic
127	C₂²×C₃₁	#578	1	3	yes	—
128	C₄×C₃₁	—	2	4	no	—
129	C₄×C₃₁	—	2	6	no	—
130	C₄×C₃₁	—	2	6	no	—
131	C₄×C₃₁	—	4	6	no	—
132	C₄×C₃₁	—	8	3	no	—
133	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
134	C₄×C₃₁	#532	1	3	yes	—
135	C₂²×Q₁₆	C₂²×C₃₁	1	3	yes	—
136	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
137	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
138	C₂×#124	C₂²×C₃₁	1	3	yes	—
139	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
140	#464	—	2	12	no	—
141	#462	—	2	12	no	—
142	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	—
143	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	—
144	C₂×#103	C₂²×C₃₁	1	3	yes	—
145	C₂×#103	C₂²×C₃₁	1	6	yes	—
146	C₂×#102	C₂²×C₃₁	1	3	yes	—
147	#399	—	2	1	no	—
148	#399	—	2	3	no	—
149	C₄×C₄⋊C₄	—	2	3	no	—
150	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
151	#376	—	2	1	no	—
152	#376	C₂²×C₃₁	1	1	yes	characteristic
153	C₂×C₃₁	—	16	1	no	—
154	C₂×C₃₁	#2856	1	1	yes	characteristic
155	C₂×C₃₁	#3044	1	3	yes	—
156	C₂×C₃₁	#3046	1	3	yes	—
157	C₂²×Q₈	C₂³×C₃₁	1	3	yes	—
158	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
159	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
160	C₂×Q₁₆	—	2	12	no	—
161	C₂×Q₁₆	—	2	6	no	—
162	C₂×SD₁₆	—	2	6	no	—
163	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
165	C₄⋊Q₈	—	2	6	no	—
166	C₄⋊Q₈	—	2	6	no	—
167	#124	—	2	6	no	—
168	#124	—	2	6	no	—
169	C₄×Q₈	—	2	6	no	—
170	C₄×Q₈	D₄×C₃₁	1	12	yes	—
171	C₂×C₄⋊C₄	—	2	3	no	—
172	C₂×C₄⋊C₄	—	2	3	no	—
173	C₂×C₄⋊C₄	—	2	3	no	—
174	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
175	C₂×C₂²⋊C₄	—	2	3	no	—
176	C₂×C₂²⋊C₄	—	2	3	no	—
177	C₂×C₄²	—	2	3	no	—
178	C₂×C₄²	—	2	3	no	—
179	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
180	C₄⋊C₈	—	2	6	no	—
181	C₄⋊C₈	—	2	6	no	—
182	#103	—	2	12	no	—
183	#103	—	2	6	no	—
184	#102	—	2	6	no	—
185	#95	—	2	3	no	—
186	#95	D₄×C₃₁	1	1	yes	characteristic
187	#95	D₄×C₃₁	1	3	yes	—
188	C₃₁	#33169	1	1	yes	characteristic, direct product factor, Sylow
189	C₂⁴	—	2	1	no	—
190	C₂×Q₈	—	2	3	no	—
191	C₂×Q₈	—	2	3	no	—
192	C₂×Q₈	—	2	6	no	—
193	C₂×Q₈	—	2	3	no	—
194	C₂×Q₈	—	2	6	no	—
195	C₂×Q₈	C₂×D₄×C₃₁	1	6	yes	—
196	C₂×D₄	—	2	3	no	—
197	C₂×D₄	—	2	3	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	3	no	—
200	C₂²×C₄	—	2	3	no	—
201	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
202	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
203	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
204	Q₁₆	—	4	12	no	—
205	SD₁₆	—	4	4	no	—
206	C₂×C₈	—	2	3	no	—
207	C₂×C₈	—	2	3	no	—
208	C₂×C₈	—	2	6	no	—
209	C₄⋊C₄	—	2	12	no	—
210	C₄⋊C₄	—	2	3	no	—
211	C₄⋊C₄	—	4	6	no	—
212	C₄⋊C₄	—	4	6	no	—
213	C₄⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
214	C₂²⋊C₄	—	4	6	no	—
215	C₂²⋊C₄	—	4	6	no	—
216	C₄²	—	2	12	no	—
217	C₄²	—	8	3	no	—
218	C₄²	C₂×D₄×C₃₁	1	6	yes	—
219	C₄²	#41×C₃₁	1	6	yes	—
220	C₂³	—	4	3	no	—
221	C₂³	—	4	1	no	—
222	C₂³	—	4	3	no	—
223	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
224	Q₈	—	2	6	no	—
225	Q₈	—	4	3	no	—
226	Q₈	—	4	3	no	—
227	Q₈	—	4	6	no	—
228	Q₈	C₃₁×#127	1	4	yes	—
229	D₄	—	4	1	no	—
230	D₄	—	4	3	no	—
231	C₂×C₄	—	2	6	no	—
232	C₂×C₄	—	2	3	no	—
233	C₂×C₄	—	2	6	no	—
234	C₂×C₄	—	2	3	no	—
235	C₂×C₄	—	2	6	no	—
236	C₂×C₄	—	4	3	no	—
237	C₂×C₄	—	4	3	no	—
238	C₂×C₄	—	4	3	no	—
239	C₂×C₄	—	4	6	no	—
240	C₂×C₄	C₃₁×#121	1	6	yes	—
241	C₂×C₄	C₃₁×#121	1	6	yes	—
242	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
243	C₂×C₄	C₂×#41×C₃₁	1	3	yes	—
244	C₈	—	4	1	no	—
245	C₈	—	4	3	no	—
246	C₂²	—	8	1	no	—
247	C₂²	—	8	3	no	—
248	C₂²	—	8	3	no	—
249	C₂²	C₂×C₃₁×#121	1	3	yes	—
250	C₂²	C₃₁×#577	1	1	yes	characteristic
251	C₂²	C₃₁×#578	1	3	yes	—
252	C₄	—	2	4	no	—
253	C₄	—	2	6	no	—
254	C₄	—	2	6	no	—
255	C₄	—	4	6	no	—
256	C₄	—	8	3	no	—
257	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
258	C₄	C₃₁×#532	1	3	yes	—
259	C₂	—	16	1	no	—
260	C₂	C₃₁×#2856	1	1	yes	characteristic
261	C₂	C₃₁×#3044	1	3	yes	—
262	C₂	C₃₁×#3046	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#462	C₂	semidirect
4. C₃₁×#2276	C₂	semidirect
6. C₃₁×#1981	C₂	semidirect
7. C₃₁×#1843	C₂	semidirect
88. #33169	C₃₁	direct (N × H)
116. C₂×#462	C₂×C₃₁	semidirect
118. #2276	C₂×C₃₁	semidirect
120. #1981	C₂×C₃₁	semidirect
121. #1843	C₂×C₃₁	semidirect
188. C₃₁	#33169	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#464	2
C₂×C₃₁×#462	2
C₃₁×#2658	2
C₃₁×#2276	2
C₃₁×#1983	2
C₃₁×#1981	2
C₃₁×#1843	2
#33169	31

Subgroups — C31×#33169 #21489928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33169 #21489928