Subgroups — C₃₁×#9702 #21466461

Order 7936.

538 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2868	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#333	C₂	1	4	yes	maximal
5	C₂×C₃₁×C₄⋊Q₈	C₄	1	4	yes	—
6	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₃₁×#97	—	2	1	no	—
10	C₃₁×#384	—	2	1	no	—
11	C₃₁×#384	—	2	2	no	—
12	C₃₁×#384	—	2	1	no	—
13	C₄³×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#336	—	2	2	no	—
15	C₃₁×#336	—	2	1	no	—
16	C₃₁×#333	—	2	8	no	—
17	C₂²×Q₈×C₃₁	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₃₁	—	2	2	no	—
19	C₂²×C₈×C₃₁	—	2	2	no	—
20	C₃₁×C₄⋊Q₈	—	2	4	no	—
21	C₃₁×C₄⋊Q₈	—	2	4	no	—
22	C₃₁×C₄⋊Q₈	—	4	8	no	—
23	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
24	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₃₁	D₄	1	2	yes	—
30	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
32	C₃₁×C₄⋊C₈	—	2	4	no	—
33	C₃₁×C₄⋊C₈	—	2	2	no	—
34	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
35	C₃₁×#97	—	4	2	no	—
36	C₂×Q₈×C₃₁	—	2	8	no	—
37	C₂×Q₈×C₃₁	—	4	8	no	—
38	C₂²×C₄×C₃₁	—	2	4	no	—
39	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₃₁	—	2	4	no	—
45	C₂×C₈×C₃₁	—	4	2	no	—
46	C₂×C₈×C₃₁	—	4	2	no	—
47	C₂×C₈×C₃₁	—	4	2	no	—
48	C₂×C₈×C₃₁	—	4	4	no	—
49	C₄⋊C₄×C₃₁	—	4	8	no	—
50	C₄⋊C₄×C₃₁	—	4	8	no	—
51	C₄⋊C₄×C₃₁	—	4	4	no	—
52	C₄⋊C₄×C₃₁	—	4	2	no	—
53	C₄⋊C₄×C₃₁	—	4	2	no	—
54	C₄²×C₃₁	—	2	4	no	—
55	C₄²×C₃₁	—	2	2	no	—
56	C₄²×C₃₁	—	2	4	no	—
57	C₄²×C₃₁	SD₁₆	1	2	yes	—
58	C₄²×C₃₁	Q₁₆	1	2	yes	—
59	C₄²×C₃₁	C₂×D₄	1	2	yes	—
60	C₄²×C₃₁	#41	1	2	yes	—
61	#9702	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	Q₈×C₃₁	—	8	8	no	—
64	C₂×C₄×C₃₁	—	2	2	no	—
65	C₂×C₄×C₃₁	—	2	2	no	—
66	C₂×C₄×C₃₁	—	2	2	no	—
67	C₂×C₄×C₃₁	—	2	2	no	—
68	C₂×C₄×C₃₁	—	2	2	no	—
69	C₂×C₄×C₃₁	—	4	4	no	—
70	C₂×C₄×C₃₁	—	4	8	no	—
71	C₂×C₄×C₃₁	#100	1	4	yes	—
72	C₂×C₄×C₃₁	#103	1	4	yes	—
73	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₃₁	#121	1	2	yes	—
75	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
76	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
77	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₃₁	#137	1	2	yes	—
80	C₂×C₄×C₃₁	#137	1	4	yes	—
81	C₈×C₃₁	—	4	4	no	—
82	C₈×C₃₁	—	8	2	no	—
83	#2868	C₂×C₃₁	1	1	yes	characteristic
84	#1831	C₂×C₃₁	1	1	yes	characteristic
85	#1767	C₂×C₃₁	1	1	yes	characteristic
86	C₂×#333	C₂×C₃₁	1	4	yes	—
87	C₂²×C₃₁	#390	1	1	yes	characteristic
88	C₂²×C₃₁	C₂×#100	1	2	yes	—
89	C₂²×C₃₁	C₂×#103	1	1	yes	characteristic
90	C₂²×C₃₁	#419	1	2	yes	—
91	C₂²×C₃₁	#419	1	1	yes	characteristic
92	C₄×C₃₁	—	2	4	no	—
93	C₄×C₃₁	—	2	4	no	—
94	C₄×C₃₁	—	2	2	no	—
95	C₄×C₃₁	—	8	4	no	—
96	C₄×C₃₁	#461	1	2	yes	—
97	C₄×C₃₁	#462	1	2	yes	—
98	C₄×C₃₁	#491	1	2	yes	—
99	C₄×C₃₁	#497	1	2	yes	—
100	C₂×C₄⋊Q₈	C₄×C₃₁	1	4	yes	—
101	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
102	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
103	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
104	C₂×#97	—	2	1	no	—
105	#384	—	2	1	no	—
106	#384	—	2	2	no	—
107	#384	—	2	1	no	—
108	C₄³	C₂²×C₃₁	1	1	yes	characteristic
109	#336	—	2	2	no	—
110	#336	—	2	1	no	—
111	#333	—	2	8	no	—
112	C₂×C₃₁	#1548	1	2	yes	—
113	C₂×C₃₁	#1558	1	2	yes	—
114	C₂×C₃₁	#1960	1	1	yes	characteristic
115	C₂×C₃₁	#1965	1	1	yes	characteristic
116	C₂×C₃₁	#1981	1	1	yes	characteristic
117	C₂²×Q₈	C₂×C₄×C₃₁	1	4	yes	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₄⋊Q₈	—	2	4	no	—
121	C₄⋊Q₈	—	2	4	no	—
122	C₄⋊Q₈	—	4	8	no	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₄⋊C₄	—	2	4	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
128	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
129	C₂×C₄²	D₄×C₃₁	1	2	yes	—
130	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
131	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
132	C₄⋊C₈	—	2	4	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
135	#97	—	4	2	no	—
136	C₃₁	#9702	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×Q₈	—	2	8	no	—
138	C₂×Q₈	—	4	8	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
141	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
142	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
143	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	4	4	no	—
150	C₄⋊C₄	—	4	8	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₄⋊C₄	—	4	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	4	no	—
158	C₄²	SD₁₆×C₃₁	1	2	yes	—
159	C₄²	Q₁₆×C₃₁	1	2	yes	—
160	C₄²	C₂×D₄×C₃₁	1	2	yes	—
161	C₄²	#41×C₃₁	1	2	yes	—
162	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
163	Q₈	—	8	8	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	8	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	C₃₁×#100	1	4	yes	—
172	C₂×C₄	C₃₁×#103	1	4	yes	—
173	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
174	C₂×C₄	C₃₁×#121	1	2	yes	—
175	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
176	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
177	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
178	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
179	C₂×C₄	C₃₁×#137	1	2	yes	—
180	C₂×C₄	C₃₁×#137	1	4	yes	—
181	C₈	—	4	4	no	—
182	C₈	—	8	2	no	—
183	C₂²	C₃₁×#390	1	1	yes	characteristic
184	C₂²	C₂×C₃₁×#100	1	2	yes	—
185	C₂²	C₂×C₃₁×#103	1	1	yes	characteristic
186	C₂²	C₃₁×#419	1	2	yes	—
187	C₂²	C₃₁×#419	1	1	yes	characteristic
188	C₄	—	2	4	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	8	4	no	—
192	C₄	C₃₁×#461	1	2	yes	—
193	C₄	C₃₁×#462	1	2	yes	—
194	C₄	C₃₁×#491	1	2	yes	—
195	C₄	C₃₁×#497	1	2	yes	—
196	C₂	C₃₁×#1548	1	2	yes	—
197	C₂	C₃₁×#1558	1	2	yes	—
198	C₂	C₃₁×#1960	1	1	yes	characteristic
199	C₂	C₃₁×#1965	1	1	yes	characteristic
200	C₂	C₃₁×#1981	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
61. #9702	C₃₁	direct (N × H)
136. C₃₁	#9702	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×#2868	2
C₃₁×#1831	2
C₃₁×#1767	2
C₂×C₃₁×#333	2
#9702	31

Subgroups — C31×#9702 #21466461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9702 #21466461