Subgroups — C₄×C₃₁×#334 #21465030

Order 7936.

470 conjugacy classes of subgroups.

The subgroup lattice has 180 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×#334	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×#334	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	SD₁₆	1	4	yes	—
50	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
51	C₄²×C₃₁	#41	1	2	yes	—
52	C₄×#334	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₃₁	—	2	2	no	—
55	C₂×C₄×C₃₁	—	2	2	no	—
56	C₂×C₄×C₃₁	—	2	2	no	—
57	C₂×C₄×C₃₁	C₄⋊C₈	1	4	yes	—
58	C₂×C₄×C₃₁	C₄⋊C₈	1	4	yes	—
59	C₂×C₄×C₃₁	#106	1	4	yes	—
60	C₂×C₄×C₃₁	#108	1	4	yes	—
61	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
63	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
64	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
65	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
68	C₂×C₄×C₃₁	#135	1	2	yes	—
69	C₈×C₃₁	—	2	2	no	—
70	C₈×C₃₁	—	2	4	no	—
71	C₈×C₃₁	—	4	8	no	—
72	C₈×C₃₁	C₄×C₈	1	4	yes	complemented
73	C₄×C₄⋊C₈	C₂×C₃₁	1	2	yes	—
74	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
75	C₂×#334	C₂×C₃₁	1	2	yes	—
76	#1277	C₂×C₃₁	1	2	yes	—
77	C₂²×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
79	C₂²×C₃₁	C₂×#106	1	1	yes	characteristic
80	C₂²×C₃₁	#427	1	1	yes	characteristic
81	C₂²×C₃₁	C₂×#108	1	2	yes	—
82	C₄×C₃₁	—	2	4	no	—
83	C₄×C₃₁	—	2	2	no	—
84	C₄×C₃₁	#334	1	8	yes	direct product factor
85	C₄×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
86	C₄×C₃₁	C₄×M₁₆	1	1	yes	characteristic
87	C₄×C₃₁	#431	1	2	yes	—
88	C₄×C₃₁	C₄×SD₁₆	1	4	yes	—
89	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
90	C₂×C₄×C₈	—	2	2	no	—
91	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
92	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
93	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
95	C₄³	C₂²×C₃₁	1	1	yes	characteristic
96	#336	—	2	2	no	—
97	#334	C₄×C₃₁	1	8	yes	direct product factor
98	C₂×C₃₁	C₂×#334	1	2	yes	—
99	C₂×C₃₁	#1564	1	2	yes	—
100	C₂×C₃₁	C₄×C₄⋊C₈	1	1	yes	characteristic
101	C₂×C₃₁	C₄×#106	1	1	yes	characteristic
102	C₂×C₃₁	C₄×#108	1	1	yes	characteristic
103	C₂²×C₈	—	2	4	no	—
104	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
105	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
106	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
107	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
108	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
109	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
110	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
111	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
112	C₄⋊C₈	—	2	4	no	—
113	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
114	C₄×C₈	—	2	4	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	4	4	no	—
117	C₄×C₈	C₈×C₃₁	1	8	yes	complemented
118	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
119	C₃₁	C₄×#334	1	1	yes	characteristic, direct product factor, Sylow
120	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
121	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
122	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
123	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
124	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	8	no	—
129	C₂×C₈	—	4	4	no	—
130	C₂×C₈	—	4	4	no	—
131	C₂×C₈	C₄²×C₃₁	1	2	yes	—
132	C₂×C₈	C₄²×C₃₁	1	2	yes	—
133	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
134	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
135	C₄²	—	2	4	no	—
136	C₄²	—	2	2	no	—
137	C₄²	C₂×C₈×C₃₁	1	4	yes	—
138	C₄²	M₁₆×C₃₁	1	4	yes	—
139	C₄²	SD₁₆×C₃₁	1	4	yes	—
140	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
141	C₄²	#41×C₃₁	1	2	yes	—
142	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	C₃₁×C₄⋊C₈	1	4	yes	—
147	C₂×C₄	C₃₁×C₄⋊C₈	1	4	yes	—
148	C₂×C₄	C₃₁×#106	1	4	yes	—
149	C₂×C₄	C₃₁×#108	1	4	yes	—
150	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
151	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
152	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
153	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
154	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
155	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
156	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
157	C₂×C₄	C₃₁×#135	1	2	yes	—
158	C₈	—	2	2	no	—
159	C₈	—	2	4	no	—
160	C₈	—	4	8	no	—
161	C₈	C₄×C₈×C₃₁	1	4	yes	complemented
162	C₂²	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
163	C₂²	C₂×C₃₁×C₄⋊C₈	1	2	yes	—
164	C₂²	C₂×C₃₁×#106	1	1	yes	characteristic
165	C₂²	C₃₁×#427	1	1	yes	characteristic
166	C₂²	C₂×C₃₁×#108	1	2	yes	—
167	C₄	—	2	4	no	—
168	C₄	—	2	2	no	—
169	C₄	C₃₁×#334	1	8	yes	direct product factor
170	C₄	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
171	C₄	C₄×M₁₆×C₃₁	1	1	yes	characteristic
172	C₄	C₃₁×#431	1	2	yes	—
173	C₄	C₄×SD₁₆×C₃₁	1	4	yes	—
174	C₂	C₂×C₃₁×#334	1	2	yes	—
175	C₂	C₃₁×#1564	1	2	yes	—
176	C₂	C₄×C₃₁×C₄⋊C₈	1	1	yes	characteristic
177	C₂	C₄×C₃₁×#106	1	1	yes	characteristic
178	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×#334	C₄	direct (N × H)
28. C₄×C₈×C₃₁	C₈	semidirect
52. C₄×#334	C₃₁	direct (N × H)
72. C₈×C₃₁	C₄×C₈	semidirect
84. C₄×C₃₁	#334	direct (N × H)
97. #334	C₄×C₃₁	direct (N × H)
117. C₄×C₈	C₈×C₃₁	semidirect
119. C₃₁	C₄×#334	direct (N × H)
161. C₈	C₄×C₈×C₃₁	semidirect
169. C₄	C₃₁×#334	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×#334	2
C₃₁×#1277	2
C₄×#334	31

Subgroups — C4×C31×#334 #21465030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₃₁×#334 #21465030