Subgroups — C₃₁×#2855 #16842788

Order 3968.

564 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#391	C₂	1	2	yes	maximal, complemented
4	C₃₁×#389	C₂	1	2	yes	maximal, complemented
5	C₃₁×#387	C₂	1	4	yes	maximal, complemented
6	C₃₁×#385	C₂	1	2	yes	maximal
7	C₃₁×#381	C₂	1	2	yes	maximal
8	C₃₁×#380	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×C₂²⋊Q₈	—	2	4	no	—
14	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
18	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₃₁	—	2	4	no	—
21	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#95	C₂²	1	2	yes	—
24	C₃₁×#95	C₂²	1	4	yes	—
25	C₃₁×#95	C₂²	1	4	yes	—
26	C₃₁×#95	C₂²	1	4	yes	—
27	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×Q₈×C₃₁	—	2	2	no	—
29	C₂×Q₈×C₃₁	—	2	4	no	—
30	C₂²×C₄×C₃₁	—	2	2	no	—
31	C₂²×C₄×C₃₁	—	2	2	no	—
32	C₂²×C₄×C₃₁	—	2	2	no	—
33	C₂²×C₄×C₃₁	—	2	4	no	—
34	C₂²×C₄×C₃₁	—	2	4	no	—
35	C₂²×C₄×C₃₁	D₄	1	4	yes	—
36	C₂²×C₄×C₃₁	Q₈	1	4	yes	complemented
37	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
38	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₄⋊C₄×C₃₁	—	2	4	no	—
46	C₄⋊C₄×C₃₁	—	2	8	no	—
47	C₄⋊C₄×C₃₁	—	2	4	no	—
48	C₂²⋊C₄×C₃₁	—	2	4	no	—
49	C₂²⋊C₄×C₃₁	—	2	4	no	—
50	C₄²×C₃₁	—	2	4	no	—
51	C₄²×C₃₁	—	2	4	no	—
52	C₂³×C₃₁	—	2	2	no	—
53	C₂³×C₃₁	—	2	2	no	—
54	C₂³×C₃₁	C₂×D₄	1	2	yes	—
55	C₂³×C₃₁	C₂×Q₈	1	2	yes	—
56	C₂³×C₃₁	#41	1	2	yes	—
57	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
58	Q₈×C₃₁	—	4	4	no	—
59	C₂×C₄×C₃₁	—	2	8	no	—
60	C₂×C₄×C₃₁	—	2	8	no	—
61	C₂×C₄×C₃₁	—	2	2	no	—
62	C₂×C₄×C₃₁	—	2	2	no	—
63	C₂×C₄×C₃₁	—	2	2	no	—
64	C₂×C₄×C₃₁	—	2	2	no	—
65	C₂×C₄×C₃₁	—	2	2	no	—
66	C₂×C₄×C₃₁	—	2	8	no	—
67	C₂×C₄×C₃₁	—	2	2	no	—
68	C₂×C₄×C₃₁	—	2	2	no	—
69	C₂×C₄×C₃₁	—	2	4	no	—
70	C₂×C₄×C₃₁	—	2	4	no	—
71	C₂×C₄×C₃₁	—	2	4	no	—
72	C₂×C₄×C₃₁	—	2	4	no	—
73	C₂×C₄×C₃₁	—	2	4	no	—
74	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
77	C₂×C₄×C₃₁	#41	1	4	yes	—
78	C₂×C₄×C₃₁	#41	1	4	yes	—
79	#2855	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂²×C₃₁	—	2	2	no	—
81	C₂²×C₃₁	—	2	2	no	—
82	C₂²×C₃₁	—	2	4	no	—
83	C₂²×C₃₁	—	2	4	no	—
84	C₂²×C₃₁	C₂²⋊Q₈	1	4	yes	—
85	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
86	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
87	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
88	C₂²×C₃₁	C₂×#41	1	2	yes	—
89	C₂²×C₃₁	C₂×#41	1	2	yes	—
90	C₄×C₃₁	—	2	4	no	—
91	C₄×C₃₁	—	2	2	no	—
92	C₄×C₃₁	—	2	4	no	—
93	C₄×C₃₁	—	4	2	no	—
94	C₄×C₃₁	—	4	2	no	—
95	C₄×C₃₁	—	4	4	no	—
96	C₄×C₃₁	#123	1	4	yes	—
97	C₂×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
98	C₂²×C₄²	C₂×C₃₁	1	1	yes	characteristic
99	#391	C₂×C₃₁	1	2	yes	complemented
100	#389	C₂×C₃₁	1	2	yes	complemented
101	#387	C₂×C₃₁	1	4	yes	complemented
102	#385	C₂×C₃₁	1	2	yes	—
103	#381	C₂×C₃₁	1	2	yes	—
104	#380	C₂×C₃₁	1	1	yes	characteristic
105	C₂×C₃₁	—	2	4	no	—
106	C₂×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₃₁	C₂×#123	1	1	yes	characteristic
108	C₂×C₃₁	#525	1	1	yes	characteristic
109	C₂×C₃₁	#529	1	2	yes	—
110	C₂×C₃₁	#533	1	2	yes	—
111	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
112	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
113	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
114	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
115	C₂²⋊Q₈	—	2	4	no	—
116	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
117	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
119	C₂×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
120	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
121	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
122	C₂×C₄²	—	2	4	no	—
123	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
124	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
125	#95	C₂²×C₃₁	1	2	yes	—
126	#95	C₂²×C₃₁	1	4	yes	—
127	#95	C₂²×C₃₁	1	4	yes	—
128	#95	C₂²×C₃₁	1	4	yes	—
129	C₃₁	#2855	1	1	yes	characteristic, direct product factor, Sylow
130	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	—	2	4	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	D₄×C₃₁	1	4	yes	—
139	C₂²×C₄	Q₈×C₃₁	1	4	yes	complemented
140	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
141	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
142	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
143	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
144	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
145	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
146	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	8	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₂²⋊C₄	—	2	4	no	—
152	C₂²⋊C₄	—	2	4	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	2	4	no	—
155	C₂³	—	2	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	C₂×D₄×C₃₁	1	2	yes	—
158	C₂³	C₂×Q₈×C₃₁	1	2	yes	—
159	C₂³	#41×C₃₁	1	2	yes	—
160	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
161	Q₈	—	4	4	no	—
162	C₂×C₄	—	2	8	no	—
163	C₂×C₄	—	2	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	—	2	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
178	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
179	C₂×C₄	C₂×Q₈×C₃₁	1	4	yes	—
180	C₂×C₄	#41×C₃₁	1	4	yes	—
181	C₂×C₄	#41×C₃₁	1	4	yes	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	4	no	—
185	C₂²	—	2	4	no	—
186	C₂²	C₃₁×C₂²⋊Q₈	1	4	yes	—
187	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
188	C₂²	C₂²×Q₈×C₃₁	1	1	yes	characteristic
189	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
190	C₂²	C₂×#41×C₃₁	1	2	yes	—
191	C₂²	C₂×#41×C₃₁	1	2	yes	—
192	C₄	—	2	4	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	4	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	4	no	—
198	C₄	C₃₁×#123	1	4	yes	—
199	C₂	—	2	4	no	—
200	C₂	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
201	C₂	C₂×C₃₁×#123	1	1	yes	characteristic
202	C₂	C₃₁×#525	1	1	yes	characteristic
203	C₂	C₃₁×#529	1	2	yes	—
204	C₂	C₃₁×#533	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#391	C₂	semidirect
4. C₃₁×#389	C₂	semidirect
5. C₃₁×#387	C₂	semidirect
36. C₂²×C₄×C₃₁	Q₈	semidirect
79. #2855	C₃₁	direct (N × H)
99. #391	C₂×C₃₁	semidirect
100. #389	C₂×C₃₁	semidirect
101. #387	C₂×C₃₁	semidirect
129. C₃₁	#2855	direct (N × H)
139. C₂²×C₄	Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×C₂²⋊Q₈	2
C₂²×C₄²×C₃₁	2
C₃₁×#391	2
C₃₁×#389	2
C₃₁×#387	2
C₃₁×#385	2
C₃₁×#381	2
C₃₁×#380	2
#2855	31

Subgroups — C31×#2855 #16842788

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2855 #16842788