Subgroups — C₄×C₄⋊C₄×D₃₁ #16848044

Order 3968.

928 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×D₃₁	C₂	1	2	yes	maximal
2	C₂×C₄²×D₃₁	C₂	1	2	yes	maximal
3	C₂×C₄²×D₃₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₃₁	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₃₁⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#14520900	C₂	1	2	yes	maximal, complemented
8	#95×D₃₁	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₃₁	C₂²	1	4	yes	—
12	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
18	C₂×Dic₃₁⋊C₄	C₂²	1	4	yes	—
19	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
20	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
21	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
24	C₄⋊C₄×D₃₁	C₄	1	16	yes	direct product factor
25	C₄²×D₃₁	—	2	8	no	—
26	C₄²×D₃₁	C₄	1	8	yes	complemented
27	C₃₁×#95	C₂²	1	2	yes	—
28	C₃₁⋊#95	C₂²	1	2	yes	—
29	C₃₁⋊#95	C₂²	1	4	yes	—
30	C₂³×D₃₁	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
33	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
36	C₂²×Dic₃₁	C₂³	1	4	yes	—
37	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
38	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×D₃₁	—	2	8	no	—
40	C₂×C₄×D₃₁	—	2	4	no	—
41	C₂×C₄×D₃₁	—	2	8	no	—
42	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
43	C₂×C₄×D₃₁	C₂×C₄	1	16	yes	—
44	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
45	C₂×C₄×D₃₁	C₂×C₄	1	8	yes	—
46	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
47	C₂×C₄×D₃₁	D₄	1	4	yes	—
48	C₂×C₄×D₃₁	Q₈	1	4	yes	—
49	C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
50	C₄²×C₃₁	—	2	4	no	—
51	C₄²×C₃₁	C₂×C₄	1	4	yes	complemented
52	C₄⋊Dic₃₁	C₂×C₄	1	8	yes	complemented
53	Dic₃₁⋊C₄	C₂×C₄	1	16	yes	complemented
54	C₄×Dic₃₁	—	2	4	no	—
55	C₄×Dic₃₁	—	2	8	no	—
56	C₄×Dic₃₁	C₂×C₄	1	4	yes	complemented
57	C₄×Dic₃₁	C₂×C₄	1	4	yes	complemented
58	C₄×Dic₃₁	C₂×C₄	1	4	yes	complemented
59	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
60	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
61	C₂²×D₃₁	C₂²×C₄	1	4	yes	—
62	C₂²×D₃₁	C₂×D₄	1	2	yes	—
63	C₂²×D₃₁	C₂×Q₈	1	2	yes	—
64	C₂²×D₃₁	#41	1	4	yes	—
65	C₂×C₄×C₃₁	—	2	4	no	—
66	C₂×C₄×C₃₁	—	2	2	no	—
67	C₂×C₄×C₃₁	—	2	4	no	—
68	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
70	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
72	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
75	C₂×Dic₃₁	—	2	4	no	—
76	C₂×Dic₃₁	—	2	2	no	—
77	C₂×Dic₃₁	—	2	4	no	—
78	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
79	C₂×Dic₃₁	C₂²×C₄	1	8	yes	—
80	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
81	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
82	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
83	C₂×Dic₃₁	C₂×D₄	1	2	yes	—
84	C₂×Dic₃₁	C₂×Q₈	1	2	yes	—
85	C₄×D₃₁	—	2	8	no	—
86	C₄×D₃₁	—	2	32	no	—
87	C₄×D₃₁	C₄²	1	16	yes	complemented
88	C₄×D₃₁	C₄⋊C₄	1	16	yes	direct product factor
89	C₂×C₄×C₄⋊C₄	—	31	1	no	maximal, Sylow
90	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
91	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
92	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
94	C₂²×C₃₁	C₂×#41	1	2	yes	—
95	C₂×D₃₁	C₂×C₄²	1	4	yes	—
96	C₂×D₃₁	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×D₃₁	#117	1	4	yes	—
98	C₂×D₃₁	C₄×D₄	1	8	yes	—
99	C₂×D₃₁	C₄×Q₈	1	8	yes	—
100	C₄×C₃₁	—	2	2	no	—
101	C₄×C₃₁	—	2	8	no	—
102	C₄×C₃₁	C₂×C₄²	1	4	yes	complemented
103	C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	complemented
104	Dic₃₁	—	2	2	no	—
105	Dic₃₁	—	2	8	no	—
106	Dic₃₁	C₂×C₄²	1	4	yes	complemented
107	Dic₃₁	C₂×C₄⋊C₄	1	4	yes	complemented
108	C₂²×C₄⋊C₄	—	31	2	no	—
109	C₂²×C₄²	—	31	1	no	—
110	C₂²×C₄²	—	31	2	no	—
111	C₄×C₄⋊C₄	—	31	2	no	—
112	C₄×C₄⋊C₄	—	31	1	no	—
113	C₄×C₄⋊C₄	—	31	2	no	—
114	C₄×C₄⋊C₄	—	31	2	no	—
115	C₄×C₄⋊C₄	D₃₁	1	1	yes	characteristic, direct product factor
116	C₂×#95	—	31	2	no	—
117	C₂×C₃₁	C₂²×C₄²	1	1	yes	characteristic
118	C₂×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
120	C₂×C₃₁	C₂×C₄×D₄	1	2	yes	—
121	C₂×C₃₁	C₂×C₄×Q₈	1	2	yes	—
122	D₃₁	C₄×C₄⋊C₄	1	8	yes	direct product factor
123	C₂³×C₄	—	31	1	no	—
124	C₂³×C₄	—	31	1	no	—
125	C₂³×C₄	—	31	4	no	—
126	C₂³×C₄	—	31	1	no	—
127	C₂×C₄⋊C₄	—	31	4	no	—
128	C₂×C₄⋊C₄	—	31	2	no	—
129	C₂×C₄⋊C₄	—	31	16	no	—
130	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
131	C₂×C₄²	—	31	1	no	—
132	C₂×C₄²	—	31	1	no	—
133	C₂×C₄²	—	31	1	no	—
134	C₂×C₄²	—	31	8	no	—
135	C₂×C₄²	—	31	4	no	—
136	C₂×C₄²	—	31	2	no	—
137	C₂×C₄²	—	62	8	no	—
138	C₂×C₄²	C₂×D₃₁	1	2	yes	—
139	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
140	#95	—	31	4	no	—
141	#95	—	31	2	no	—
142	#95	C₂×D₃₁	1	2	yes	—
143	C₃₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
144	C₂⁴	—	31	1	no	—
145	C₂²×C₄	—	31	1	no	—
146	C₂²×C₄	—	31	1	no	—
147	C₂²×C₄	—	31	4	no	—
148	C₂²×C₄	—	31	4	no	—
149	C₂²×C₄	—	31	8	no	—
150	C₂²×C₄	—	31	4	no	—
151	C₂²×C₄	—	31	4	no	—
152	C₂²×C₄	—	31	1	no	—
153	C₂²×C₄	—	31	4	no	—
154	C₂²×C₄	—	31	4	no	—
155	C₂²×C₄	—	31	16	no	—
156	C₂²×C₄	—	62	8	no	—
157	C₂²×C₄	—	62	8	no	—
158	C₂²×C₄	—	62	4	no	—
159	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
160	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
161	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
163	C₄⋊C₄	—	31	16	no	—
164	C₄⋊C₄	—	31	8	no	—
165	C₄⋊C₄	C₄×D₃₁	1	8	yes	direct product factor
166	C₄²	—	2	4	no	—
167	C₄²	—	31	4	no	—
168	C₄²	—	31	4	no	—
169	C₄²	—	31	4	no	—
170	C₄²	—	62	8	no	—
171	C₄²	—	62	4	no	—
172	C₄²	C₄×D₃₁	1	4	yes	complemented
173	C₂³	—	31	2	no	—
174	C₂³	—	31	4	no	—
175	C₂³	—	31	2	no	—
176	C₂³	—	31	2	no	—
177	C₂³	—	31	4	no	—
178	C₂³	C₂³×D₃₁	1	1	yes	characteristic
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	31	2	no	—
183	C₂×C₄	—	31	2	no	—
184	C₂×C₄	—	31	4	no	—
185	C₂×C₄	—	31	2	no	—
186	C₂×C₄	—	31	2	no	—
187	C₂×C₄	—	31	16	no	—
188	C₂×C₄	—	31	16	no	—
189	C₂×C₄	—	31	2	no	—
190	C₂×C₄	—	31	8	no	—
191	C₂×C₄	—	62	8	no	—
192	C₂×C₄	—	62	32	no	—
193	C₂×C₄	—	62	4	no	—
194	C₂×C₄	—	62	2	no	—
195	C₂×C₄	—	62	4	no	—
196	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
197	C₂×C₄	C₂×C₄×D₃₁	1	8	yes	—
198	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
200	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
201	C₂×C₄	D₄×D₃₁	1	2	yes	—
202	C₂×C₄	Q₈×D₃₁	1	2	yes	—
203	C₂²	—	31	4	no	—
204	C₂²	—	31	4	no	—
205	C₂²	—	31	4	no	—
206	C₂²	—	31	8	no	—
207	C₂²	—	31	8	no	—
208	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
209	C₂²	C₂²×C₄×D₃₁	1	2	yes	—
210	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
211	C₂²	C₂×Q₈×D₃₁	1	1	yes	characteristic
212	C₂²	#41×D₃₁	1	2	yes	—
213	C₄	—	2	2	no	—
214	C₄	—	2	8	no	—
215	C₄	—	31	4	no	—
216	C₄	—	31	4	no	—
217	C₄	—	62	2	no	—
218	C₄	—	62	8	no	—
219	C₄	C₄²×D₃₁	1	4	yes	complemented
220	C₄	C₄⋊C₄×D₃₁	1	4	yes	direct product factor
221	C₂	—	31	8	no	—
222	C₂	C₂×C₄²×D₃₁	1	1	yes	characteristic
223	C₂	C₂×C₄⋊C₄×D₃₁	1	1	yes	characteristic
224	C₂	#117×D₃₁	1	1	yes	characteristic
225	C₂	C₄×D₄×D₃₁	1	2	yes	—
226	C₂	C₄×Q₈×D₃₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄⋊C₄×Dic₃₁	C₂	semidirect
5. C₄×Dic₃₁⋊C₄	C₂	semidirect
6. C₄×C₄⋊Dic₃₁	C₂	semidirect
7. #14520900	C₂	semidirect
9. C₄×C₄⋊C₄×C₃₁	C₂	semidirect
24. C₄⋊C₄×D₃₁	C₄	direct (N × H)
26. C₄²×D₃₁	C₄	semidirect
49. C₄⋊C₄×C₃₁	C₂×C₄	semidirect
51. C₄²×C₃₁	C₂×C₄	semidirect
52. C₄⋊Dic₃₁	C₂×C₄	semidirect
53. Dic₃₁⋊C₄	C₂×C₄	semidirect
56. C₄×Dic₃₁	C₂×C₄	semidirect
57. C₄×Dic₃₁	C₂×C₄	semidirect
58. C₄×Dic₃₁	C₂×C₄	semidirect
87. C₄×D₃₁	C₄²	semidirect
88. C₄×D₃₁	C₄⋊C₄	direct (N × H)
102. C₄×C₃₁	C₂×C₄²	semidirect
103. C₄×C₃₁	C₂×C₄⋊C₄	semidirect
106. Dic₃₁	C₂×C₄²	semidirect
107. Dic₃₁	C₂×C₄⋊C₄	semidirect
115. C₄×C₄⋊C₄	D₃₁	direct (N × H)
122. D₃₁	C₄×C₄⋊C₄	direct (N × H)
143. C₃₁	C₂×C₄×C₄⋊C₄	semidirect
165. C₄⋊C₄	C₄×D₃₁	direct (N × H)
172. C₄²	C₄×D₃₁	semidirect
219. C₄	C₄²×D₃₁	semidirect
220. C₄	C₄⋊C₄×D₃₁	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×C₄⋊C₄×D₃₁	2
C₂×C₄²×D₃₁	2
C₄⋊C₄×Dic₃₁	2
C₄×Dic₃₁⋊C₄	2
C₄×C₄⋊Dic₃₁	2
#14520900	2
#95×D₃₁	2
C₄×C₄⋊C₄×C₃₁	2
C₂×C₄×C₄⋊C₄	31

Subgroups — C4×C4⋊C4×D31 #16848044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄⋊C₄×D₃₁ #16848044