Subgroups — C₄×C₃₁×#335 #21465031

Order 7936.

470 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#335	C₂	1	2	yes	maximal
4	C₃₁×#1277	C₂	1	2	yes	maximal
5	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₃₁	—	2	2	no	—
7	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₄³×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#336	—	2	2	no	—
13	C₃₁×#335	C₄	1	8	yes	direct product factor
14	C₂²×C₈×C₃₁	—	2	4	no	—
15	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
22	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
23	C₃₁×C₄⋊C₈	—	2	4	no	—
24	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
25	C₄×C₈×C₃₁	—	2	4	no	—
26	C₄×C₈×C₃₁	—	2	2	no	—
27	C₄×C₈×C₃₁	—	4	4	no	—
28	C₄×C₈×C₃₁	C₈	1	8	yes	complemented
29	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₃₁	—	2	2	no	—
36	C₂×C₈×C₃₁	—	2	2	no	—
37	C₂×C₈×C₃₁	—	2	2	no	—
38	C₂×C₈×C₃₁	—	2	8	no	—
39	C₂×C₈×C₃₁	—	4	4	no	—
40	C₂×C₈×C₃₁	—	4	4	no	—
41	C₂×C₈×C₃₁	C₄²	1	2	yes	—
42	C₂×C₈×C₃₁	C₄²	1	2	yes	—
43	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
44	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
45	C₄²×C₃₁	—	2	4	no	—
46	C₄²×C₃₁	—	2	2	no	—
47	C₄²×C₃₁	C₂×C₈	1	4	yes	—
48	C₄²×C₃₁	M₁₆	1	4	yes	—
49	C₄²×C₃₁	D₈	1	2	yes	—
50	C₄²×C₃₁	Q₁₆	1	2	yes	—
51	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
52	C₄²×C₃₁	#41	1	2	yes	—
53	C₄×#335	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₃₁	—	2	2	no	—
56	C₂×C₄×C₃₁	—	2	2	no	—
57	C₂×C₄×C₃₁	—	2	2	no	—
58	C₂×C₄×C₃₁	C₄⋊C₈	1	4	yes	—
59	C₂×C₄×C₃₁	C₄⋊C₈	1	4	yes	—
60	C₂×C₄×C₃₁	#107	1	4	yes	—
61	C₂×C₄×C₃₁	#108	1	4	yes	—
62	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
64	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
65	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
66	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
67	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
68	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₃₁	#135	1	2	yes	—
71	C₈×C₃₁	—	2	2	no	—
72	C₈×C₃₁	—	2	4	no	—
73	C₈×C₃₁	—	4	8	no	—
74	C₈×C₃₁	C₄×C₈	1	4	yes	complemented
75	C₄×C₄⋊C₈	C₂×C₃₁	1	2	yes	—
76	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
77	C₂×#335	C₂×C₃₁	1	2	yes	—
78	#1277	C₂×C₃₁	1	2	yes	—
79	C₂²×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
81	C₂²×C₃₁	C₂×#107	1	1	yes	characteristic
82	C₂²×C₃₁	#427	1	1	yes	characteristic
83	C₂²×C₃₁	C₂×#108	1	2	yes	—
84	C₄×C₃₁	—	2	4	no	—
85	C₄×C₃₁	—	2	2	no	—
86	C₄×C₃₁	#335	1	8	yes	direct product factor
87	C₄×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
88	C₄×C₃₁	C₄×M₁₆	1	1	yes	characteristic
89	C₄×C₃₁	#431	1	2	yes	—
90	C₄×C₃₁	C₄×D₈	1	2	yes	—
91	C₄×C₃₁	C₄×Q₁₆	1	2	yes	—
92	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
93	C₂×C₄×C₈	—	2	2	no	—
94	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
95	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
96	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
98	C₄³	C₂²×C₃₁	1	1	yes	characteristic
99	#336	—	2	2	no	—
100	#335	C₄×C₃₁	1	8	yes	direct product factor
101	C₂×C₃₁	C₂×#335	1	2	yes	—
102	C₂×C₃₁	#1564	1	2	yes	—
103	C₂×C₃₁	C₄×C₄⋊C₈	1	1	yes	characteristic
104	C₂×C₃₁	C₄×#107	1	1	yes	characteristic
105	C₂×C₃₁	C₄×#108	1	1	yes	characteristic
106	C₂²×C₈	—	2	4	no	—
107	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
108	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
109	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
110	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
111	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
112	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
113	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
114	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
117	C₄×C₈	—	2	4	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	4	4	no	—
120	C₄×C₈	C₈×C₃₁	1	8	yes	complemented
121	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
122	C₃₁	C₄×#335	1	1	yes	characteristic, direct product factor, Sylow
123	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
124	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
125	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
126	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
127	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	8	no	—
132	C₂×C₈	—	4	4	no	—
133	C₂×C₈	—	4	4	no	—
134	C₂×C₈	C₄²×C₃₁	1	2	yes	—
135	C₂×C₈	C₄²×C₃₁	1	2	yes	—
136	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
137	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
138	C₄²	—	2	4	no	—
139	C₄²	—	2	2	no	—
140	C₄²	C₂×C₈×C₃₁	1	4	yes	—
141	C₄²	M₁₆×C₃₁	1	4	yes	—
142	C₄²	D₈×C₃₁	1	2	yes	—
143	C₄²	Q₁₆×C₃₁	1	2	yes	—
144	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
145	C₄²	#41×C₃₁	1	2	yes	—
146	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	C₃₁×C₄⋊C₈	1	4	yes	—
151	C₂×C₄	C₃₁×C₄⋊C₈	1	4	yes	—
152	C₂×C₄	C₃₁×#107	1	4	yes	—
153	C₂×C₄	C₃₁×#108	1	4	yes	—
154	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
155	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
156	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
157	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
158	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
159	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
160	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
161	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
162	C₂×C₄	C₃₁×#135	1	2	yes	—
163	C₈	—	2	2	no	—
164	C₈	—	2	4	no	—
165	C₈	—	4	8	no	—
166	C₈	C₄×C₈×C₃₁	1	4	yes	complemented
167	C₂²	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
168	C₂²	C₂×C₃₁×C₄⋊C₈	1	2	yes	—
169	C₂²	C₂×C₃₁×#107	1	1	yes	characteristic
170	C₂²	C₃₁×#427	1	1	yes	characteristic
171	C₂²	C₂×C₃₁×#108	1	2	yes	—
172	C₄	—	2	4	no	—
173	C₄	—	2	2	no	—
174	C₄	C₃₁×#335	1	8	yes	direct product factor
175	C₄	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
176	C₄	C₄×M₁₆×C₃₁	1	1	yes	characteristic
177	C₄	C₃₁×#431	1	2	yes	—
178	C₄	C₄×D₈×C₃₁	1	2	yes	—
179	C₄	C₄×Q₁₆×C₃₁	1	2	yes	—
180	C₂	C₂×C₃₁×#335	1	2	yes	—
181	C₂	C₃₁×#1564	1	2	yes	—
182	C₂	C₄×C₃₁×C₄⋊C₈	1	1	yes	characteristic
183	C₂	C₄×C₃₁×#107	1	1	yes	characteristic
184	C₂	C₄×C₃₁×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₃₁	D₄
commutator	C₄	C₂×C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₂×C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₃₁×#335	C₄	direct (N × H)
28. C₄×C₈×C₃₁	C₈	semidirect
53. C₄×#335	C₃₁	direct (N × H)
74. C₈×C₃₁	C₄×C₈	semidirect
86. C₄×C₃₁	#335	direct (N × H)
100. #335	C₄×C₃₁	direct (N × H)
120. C₄×C₈	C₈×C₃₁	semidirect
122. C₃₁	C₄×#335	direct (N × H)
166. C₈	C₄×C₈×C₃₁	semidirect
174. C₄	C₃₁×#335	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×C₄⋊C₈	2
C₄²×C₈×C₃₁	2
C₂×C₃₁×#335	2
C₃₁×#1277	2
C₄×#335	31

Subgroups — C4×C31×#335 #21465031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₃₁×#335 #21465031