Subgroups — C₄×D₄×Dic₃₁ #16848058

Order 3968.

924 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Dic₃₁	C₂	1	2	yes	maximal
2	C₂²×C₄×Dic₃₁	C₂	1	2	yes	maximal, complemented
3	C₄⋊C₄×Dic₃₁	C₂	1	2	yes	maximal, complemented
4	C₄×C₄⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊C₄×Dic₃₁	C₂	1	4	yes	maximal, complemented
6	C₄×C₂²⋊Dic₃₁	C₂	1	2	yes	maximal, complemented
7	C₄²×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
11	C₂³×Dic₃₁	C₂²	1	4	yes	—
12	C₄×D₄×C₃₁	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Dic₃₁	C₂²	1	4	yes	—
17	D₄×Dic₃₁	C₄	1	16	yes	direct product factor
18	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
19	C₂×C₄×Dic₃₁	—	2	8	no	—
20	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
22	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
23	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
24	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
25	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
26	C₃₁⋊#95	C₂²	1	4	yes	—
27	C₃₁⋊#95	C₂²	1	2	yes	—
28	C₂⁴×C₃₁	C₂³	1	2	yes	—
29	C₂×D₄×C₃₁	C₂×C₄	1	8	yes	—
30	C₂×D₄×C₃₁	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₃₁	—	2	4	no	—
32	C₂²×C₄×C₃₁	—	2	4	no	—
33	C₂²×C₄×C₃₁	C₂×C₄	1	8	yes	complemented
34	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²×Dic₃₁	—	2	8	no	—
39	C₂²×Dic₃₁	—	2	8	no	—
40	C₂²×Dic₃₁	C₂×C₄	1	16	yes	complemented
41	C₂²×Dic₃₁	C₂³	1	2	yes	—
42	C₂²×Dic₃₁	C₂³	1	4	yes	—
43	C₂²×Dic₃₁	C₂³	1	2	yes	—
44	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	complemented
45	C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
46	C₄²×C₃₁	C₂×C₄	1	4	yes	complemented
47	C₂²⋊Dic₃₁	C₂×C₄	1	16	yes	complemented
48	C₄⋊Dic₃₁	C₂×C₄	1	8	yes	complemented
49	C₄×Dic₃₁	—	2	4	no	—
50	C₄×Dic₃₁	—	2	2	no	—
51	C₄×Dic₃₁	—	2	8	no	—
52	C₄×Dic₃₁	—	2	4	no	—
53	C₄×Dic₃₁	C₂×C₄	1	8	yes	complemented
54	C₄×Dic₃₁	D₄	1	4	yes	direct product factor
55	C₂³×C₃₁	—	2	2	no	—
56	C₂³×C₃₁	—	2	4	no	—
57	C₂³×C₃₁	—	2	2	no	—
58	C₂³×C₃₁	C₂²×C₄	1	8	yes	—
59	C₂³×C₃₁	C₂²×C₄	1	4	yes	—
60	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
61	D₄×C₃₁	C₄²	1	16	yes	complemented
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	16	no	—
66	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
67	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
69	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₃₁	#41	1	2	yes	—
73	C₂×Dic₃₁	—	2	4	no	—
74	C₂×Dic₃₁	—	2	2	no	—
75	C₂×Dic₃₁	—	2	2	no	—
76	C₂×Dic₃₁	—	2	4	no	—
77	C₂×Dic₃₁	—	2	32	no	—
78	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
79	C₂×Dic₃₁	C₂²×C₄	1	8	yes	—
80	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
81	C₂×Dic₃₁	C₂×D₄	1	4	yes	—
82	C₂×Dic₃₁	#41	1	4	yes	—
83	C₄²×D₄	—	31	1	no	maximal, Sylow
84	C₂²×C₃₁	—	2	8	no	—
85	C₂²×C₃₁	—	2	4	no	—
86	C₂²×C₃₁	—	2	8	no	—
87	C₂²×C₃₁	—	2	4	no	—
88	C₂²×C₃₁	C₂×C₄²	1	8	yes	complemented
89	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
90	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
91	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
92	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
93	C₂²×C₃₁	C₂×#41	1	2	yes	—
94	C₄×C₃₁	—	2	4	no	—
95	C₄×C₃₁	—	2	2	no	—
96	C₄×C₃₁	C₂×C₄²	1	4	yes	complemented
97	C₄×C₃₁	C₄×D₄	1	4	yes	complemented
98	Dic₃₁	—	2	8	no	—
99	Dic₃₁	—	2	4	no	—
100	Dic₃₁	C₄×D₄	1	8	yes	direct product factor
101	C₂×C₄×D₄	—	31	2	no	—
102	C₂×C₄×D₄	D₃₁	1	1	yes	characteristic
103	C₂²×C₄²	—	31	2	no	—
104	C₄×C₄⋊C₄	—	31	2	no	—
105	C₄×C₄⋊C₄	—	31	1	no	—
106	C₄×C₂²⋊C₄	—	31	2	no	—
107	C₄×C₂²⋊C₄	—	31	4	no	—
108	C₄³	—	31	1	no	—
109	C₂×C₃₁	—	2	8	no	—
110	C₂×C₃₁	C₂²×C₄²	1	1	yes	characteristic
111	C₂×C₃₁	C₂×C₄×D₄	1	2	yes	—
112	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂×C₃₁	C₄×#41	1	2	yes	—
114	C₂×C₃₁	C₄×#41	1	1	yes	characteristic
115	C₂²×D₄	C₂×D₃₁	1	1	yes	characteristic
116	C₂³×C₄	—	31	4	no	—
117	C₂³×C₄	C₂×D₃₁	1	2	yes	—
118	C₄×D₄	—	31	16	no	—
119	C₄×D₄	Dic₃₁	1	8	yes	direct product factor
120	C₂×C₄⋊C₄	—	31	2	no	—
121	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
122	C₂×C₂²⋊C₄	—	31	4	no	—
123	C₂×C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
124	C₂×C₄²	—	31	2	no	—
125	C₂×C₄²	—	31	1	no	—
126	C₂×C₄²	—	31	2	no	—
127	C₂×C₄²	—	31	4	no	—
128	C₂×C₄²	—	31	1	no	—
129	C₂×C₄²	—	31	2	no	—
130	C₂×C₄²	—	62	8	no	—
131	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
132	#95	—	31	4	no	—
133	#95	—	31	2	no	—
134	C₃₁	C₄²×D₄	1	1	yes	characteristic, complemented, Sylow
135	C₂⁴	C₂²×D₃₁	1	2	yes	—
136	C₂×D₄	C₄×D₃₁	1	8	yes	—
137	C₂×D₄	C₂×Dic₃₁	1	4	yes	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	31	2	no	—
141	C₂²×C₄	—	31	16	no	—
142	C₂²×C₄	—	31	2	no	—
143	C₂²×C₄	—	31	4	no	—
144	C₂²×C₄	—	62	8	no	—
145	C₂²×C₄	—	62	8	no	—
146	C₂²×C₄	C₂×Dic₃₁	1	8	yes	complemented
147	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
149	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
151	C₄⋊C₄	—	31	8	no	—
152	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	complemented
153	C₂²⋊C₄	—	31	16	no	—
154	C₂²⋊C₄	C₂×Dic₃₁	1	8	yes	complemented
155	C₄²	—	31	8	no	—
156	C₄²	—	31	4	no	—
157	C₄²	—	62	4	no	—
158	C₄²	—	62	8	no	—
159	C₄²	—	62	4	no	—
160	C₄²	—	62	2	no	—
161	C₄²	C₂×Dic₃₁	1	4	yes	complemented
162	C₂³	—	2	2	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	4	no	—
165	C₂³	C₂×C₄×D₃₁	1	8	yes	—
166	C₂³	C₂²×Dic₃₁	1	4	yes	—
167	C₂³	C₂³×D₃₁	1	1	yes	characteristic
168	D₄	C₄×Dic₃₁	1	16	yes	direct product factor
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	16	no	—
173	C₂×C₄	—	31	4	no	—
174	C₂×C₄	—	31	4	no	—
175	C₂×C₄	—	31	4	no	—
176	C₂×C₄	—	31	4	no	—
177	C₂×C₄	—	31	8	no	—
178	C₂×C₄	—	62	32	no	—
179	C₂×C₄	—	62	4	no	—
180	C₂×C₄	—	62	2	no	—
181	C₂×C₄	—	62	2	no	—
182	C₂×C₄	—	62	4	no	—
183	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
184	C₂×C₄	D₄×D₃₁	1	2	yes	—
185	C₂×C₄	#92497	1	2	yes	—
186	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
187	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
188	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
189	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
190	C₂²	—	2	8	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	8	no	—
193	C₂²	—	2	4	no	—
194	C₂²	C₂×C₄×Dic₃₁	1	8	yes	complemented
195	C₂²	C₂²×C₄×D₃₁	1	2	yes	—
196	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
197	C₂²	C₂×#92497	1	1	yes	characteristic
198	C₂²	#41×D₃₁	1	2	yes	—
199	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	—	2	2	no	—
202	C₄	—	31	8	no	—
203	C₄	—	62	8	no	—
204	C₄	—	62	4	no	—
205	C₄	C₂×C₄×Dic₃₁	1	4	yes	complemented
206	C₄	D₄×Dic₃₁	1	4	yes	direct product factor
207	C₂	—	2	8	no	—
208	C₂	C₂²×C₄×Dic₃₁	1	1	yes	characteristic
209	C₂	C₄×D₄×D₃₁	1	2	yes	—
210	C₂	C₂×D₄×Dic₃₁	1	1	yes	characteristic
211	C₂	C₄×#92497	1	2	yes	—
212	C₂	#41×Dic₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂×C₃₁	C₂²×C₄²
Frattini	C₂³	C₂³×D₃₁
Fitting	C₂×C₄×D₄×C₃₁	C₂
socle, cosocle	C₂³×C₃₁	C₂⁴
prime radical	C₂²×D₄×C₃₁	C₂²
hypercenter	C₂×C₄×D₄	D₃₁
hypocenter	C₃₁	C₄²×D₄
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Dic₃₁	C₂	semidirect
3. C₄⋊C₄×Dic₃₁	C₂	semidirect
4. C₄×C₄⋊Dic₃₁	C₂	semidirect
5. C₂²⋊C₄×Dic₃₁	C₂	semidirect
6. C₄×C₂²⋊Dic₃₁	C₂	semidirect
7. C₄²×Dic₃₁	C₂	semidirect
12. C₄×D₄×C₃₁	C₄	semidirect
17. D₄×Dic₃₁	C₄	direct (N × H)
33. C₂²×C₄×C₃₁	C₂×C₄	semidirect
40. C₂²×Dic₃₁	C₂×C₄	semidirect
44. C₄⋊C₄×C₃₁	C₂×C₄	semidirect
45. C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
46. C₄²×C₃₁	C₂×C₄	semidirect
47. C₂²⋊Dic₃₁	C₂×C₄	semidirect
48. C₄⋊Dic₃₁	C₂×C₄	semidirect
53. C₄×Dic₃₁	C₂×C₄	semidirect
54. C₄×Dic₃₁	D₄	direct (N × H)
61. D₄×C₃₁	C₄²	semidirect
88. C₂²×C₃₁	C₂×C₄²	semidirect
96. C₄×C₃₁	C₂×C₄²	semidirect
97. C₄×C₃₁	C₄×D₄	semidirect
100. Dic₃₁	C₄×D₄	direct (N × H)
119. C₄×D₄	Dic₃₁	direct (N × H)
134. C₃₁	C₄²×D₄	semidirect
146. C₂²×C₄	C₂×Dic₃₁	semidirect
152. C₄⋊C₄	C₂×Dic₃₁	semidirect
154. C₂²⋊C₄	C₂×Dic₃₁	semidirect
161. C₄²	C₂×Dic₃₁	semidirect
168. D₄	C₄×Dic₃₁	direct (N × H)
194. C₂²	C₂×C₄×Dic₃₁	semidirect
205. C₄	C₂×C₄×Dic₃₁	semidirect
206. C₄	D₄×Dic₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₄×Dic₃₁	2
C₂²×C₄×Dic₃₁	2
C₄⋊C₄×Dic₃₁	2
C₄×C₄⋊Dic₃₁	2
C₂²⋊C₄×Dic₃₁	2
C₄×C₂²⋊Dic₃₁	2
C₄²×Dic₃₁	2
C₂×C₄×D₄×C₃₁	2
C₄²×D₄	31

Subgroups — C4×D4×Dic31 #16848058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₄×Dic₃₁ #16848058