Subgroups — C₃₁×#33278 #21490037

Order 7936.

734 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2862	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1990	C₂	1	2	yes	maximal
4	C₃₁×#1956	C₂	1	2	yes	maximal
5	C₃₁×#1933	C₂	1	8	yes	maximal
6	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#125	—	2	1	no	—
10	C₂×C₃₁×#103	C₂²	1	8	yes	—
11	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#391	—	2	2	no	—
14	C₃₁×#391	C₂²	1	4	yes	—
15	C₃₁×#389	—	2	2	no	—
16	C₃₁×#389	C₂²	1	4	yes	—
17	C₃₁×#384	C₂²	1	2	yes	—
18	C₃₁×#382	—	2	1	no	—
19	C₃₁×#382	—	2	1	no	—
20	C₃₁×#382	—	2	1	no	—
21	C₄³×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#340	C₂²	1	8	yes	—
23	C₂²×Q₈×C₃₁	—	2	1	no	—
24	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
25	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
26	C₃₁×C₄⋊Q₈	—	2	4	no	—
27	C₃₁×C₄⋊Q₈	—	2	4	no	—
28	C₃₁×#125	—	4	4	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
36	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
37	C₂×C₄²×C₃₁	D₄	1	2	yes	—
38	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
41	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×#103	—	2	16	no	—
43	C₄×C₈×C₃₁	—	2	8	no	—
44	C₄×C₈×C₃₁	—	2	4	no	—
45	C₃₁×#95	—	2	4	no	—
46	C₃₁×#95	—	2	4	no	—
47	C₃₁×#95	—	2	4	no	—
48	C₂×Q₈×C₃₁	—	2	8	no	—
49	C₂×Q₈×C₃₁	—	2	4	no	—
50	C₂×Q₈×C₃₁	—	4	2	no	—
51	C₂×Q₈×C₃₁	—	4	4	no	—
52	C₂²×C₄×C₃₁	—	2	2	no	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	2	4	no	—
55	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₃₁	—	2	4	no	—
61	C₂×C₈×C₃₁	—	2	4	no	—
62	C₂×C₈×C₃₁	#41	1	8	yes	—
63	C₄⋊C₄×C₃₁	—	2	8	no	—
64	C₄⋊C₄×C₃₁	—	2	4	no	—
65	C₄⋊C₄×C₃₁	—	4	2	no	—
66	C₄⋊C₄×C₃₁	—	4	2	no	—
67	C₄⋊C₄×C₃₁	—	4	2	no	—
68	C₄⋊C₄×C₃₁	—	4	8	no	—
69	C₄⋊C₄×C₃₁	—	4	4	no	—
70	C₄⋊C₄×C₃₁	—	4	4	no	—
71	C₄²×C₃₁	—	2	4	no	—
72	C₄²×C₃₁	—	2	4	no	—
73	C₄²×C₃₁	SD₁₆	1	4	yes	—
74	C₄²×C₃₁	Q₁₆	1	4	yes	—
75	C₄²×C₃₁	#41	1	4	yes	—
76	#33278	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
78	Q₈×C₃₁	—	4	8	no	—
79	Q₈×C₃₁	—	8	4	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	4	2	no	—
85	C₂×C₄×C₃₁	—	4	2	no	—
86	C₂×C₄×C₃₁	—	4	4	no	—
87	C₂×C₄×C₃₁	—	4	4	no	—
88	C₂×C₄×C₃₁	—	4	4	no	—
89	C₂×C₄×C₃₁	—	4	4	no	—
90	C₂×C₄×C₃₁	—	4	4	no	—
91	C₂×C₄×C₃₁	#124	1	2	yes	—
92	C₂×C₄×C₃₁	#124	1	2	yes	—
93	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
94	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
96	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
97	C₂×C₄×C₃₁	#135	1	4	yes	—
98	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
99	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
100	C₈×C₃₁	—	2	8	no	—
101	#2862	C₂×C₃₁	1	1	yes	characteristic
102	#2859	C₂×C₃₁	1	1	yes	characteristic
103	#1990	C₂×C₃₁	1	2	yes	—
104	#1956	C₂×C₃₁	1	2	yes	—
105	#1933	C₂×C₃₁	1	8	yes	—
106	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
107	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
108	C₂²×C₃₁	#532	1	2	yes	—
109	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
110	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
111	C₂²×C₃₁	C₂×#135	1	2	yes	—
112	C₄×C₃₁	—	2	4	no	—
113	C₄×C₃₁	—	2	4	no	—
114	C₄×C₃₁	—	8	2	no	—
115	C₄×C₃₁	—	8	2	no	—
116	C₄×C₃₁	—	8	4	no	—
117	C₄×C₃₁	#487	1	4	yes	—
118	C₄×C₃₁	#488	1	4	yes	—
119	C₄×C₃₁	#529	1	4	yes	—
120	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
121	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
122	C₂×#125	—	2	1	no	—
123	C₂×#103	C₂²×C₃₁	1	8	yes	—
124	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
125	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
126	#391	—	2	2	no	—
127	#391	C₂²×C₃₁	1	4	yes	—
128	#389	—	2	2	no	—
129	#389	C₂²×C₃₁	1	4	yes	—
130	#384	C₂²×C₃₁	1	2	yes	—
131	#382	—	2	1	no	—
132	#382	—	2	1	no	—
133	#382	—	2	1	no	—
134	C₄³	C₂²×C₃₁	1	1	yes	characteristic
135	#340	C₂²×C₃₁	1	8	yes	—
136	C₂×C₃₁	#2861	1	1	yes	characteristic
137	C₂×C₃₁	C₂×#487	1	1	yes	characteristic
138	C₂×C₃₁	C₂×#488	1	1	yes	characteristic
139	C₂×C₃₁	#3167	1	2	yes	—
140	C₂×C₃₁	#3170	1	2	yes	—
141	C₂²×Q₈	—	2	1	no	—
142	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
143	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
144	C₄⋊Q₈	—	2	4	no	—
145	C₄⋊Q₈	—	2	4	no	—
146	#125	—	4	4	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	4	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
154	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
155	C₂×C₄²	D₄×C₃₁	1	2	yes	—
156	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
157	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
158	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
159	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
160	#103	—	2	16	no	—
161	C₄×C₈	—	2	8	no	—
162	C₄×C₈	—	2	4	no	—
163	#95	—	2	4	no	—
164	#95	—	2	4	no	—
165	#95	—	2	4	no	—
166	C₃₁	#33278	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×Q₈	—	2	8	no	—
168	C₂×Q₈	—	2	4	no	—
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
175	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
176	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
178	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	#41×C₃₁	1	8	yes	—
182	C₄⋊C₄	—	2	8	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	8	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄²	—	2	4	no	—
191	C₄²	—	2	4	no	—
192	C₄²	SD₁₆×C₃₁	1	4	yes	—
193	C₄²	Q₁₆×C₃₁	1	4	yes	—
194	C₄²	#41×C₃₁	1	4	yes	—
195	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
196	Q₈	—	4	8	no	—
197	Q₈	—	8	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	C₃₁×#124	1	2	yes	—
210	C₂×C₄	C₃₁×#124	1	2	yes	—
211	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
212	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
213	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
214	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
215	C₂×C₄	C₃₁×#135	1	4	yes	—
216	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
217	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
218	C₈	—	2	8	no	—
219	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
220	C₂²	C₃₁×#532	1	2	yes	—
221	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
222	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
223	C₂²	C₂×C₃₁×#135	1	2	yes	—
224	C₄	—	2	4	no	—
225	C₄	—	2	4	no	—
226	C₄	—	8	2	no	—
227	C₄	—	8	2	no	—
228	C₄	—	8	4	no	—
229	C₄	C₃₁×#487	1	4	yes	—
230	C₄	C₃₁×#488	1	4	yes	—
231	C₄	C₃₁×#529	1	4	yes	—
232	C₂	C₃₁×#2861	1	1	yes	characteristic
233	C₂	C₂×C₃₁×#487	1	1	yes	characteristic
234	C₂	C₂×C₃₁×#488	1	1	yes	characteristic
235	C₂	C₃₁×#3167	1	2	yes	—
236	C₂	C₃₁×#3170	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
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
76. #33278	C₃₁	direct (N × H)
166. C₃₁	#33278	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×#2862	2
C₃₁×#2859	2
C₃₁×#1990	2
C₃₁×#1956	2
C₃₁×#1933	2
C₄²×C₈×C₃₁	2
#33278	31

Subgroups — C31×#33278 #21490037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33278 #21490037