Subgroups — C₂²×C₄×C₂²⋊D₃₁ #16858880

Order 3968.

2528 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊D₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₂²⋊D₃₁	C₂	1	24	yes	maximal, direct product factor
4	C₂²×Dic₃₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₂²⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×(C₂×D₃₁)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴×C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×C₃₁	C₂²	1	1	yes	characteristic
10	C₂⁴×D₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₃₁	—	2	8	no	—
12	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₃₁	C₂²	1	12	yes	complemented
15	C₂²×C₂²⋊D₃₁	C₄	1	16	yes	direct product factor
16	C₂²×C₂²⋊D₃₁	C₂²	1	12	yes	—
17	C₂³×Dic₃₁	C₂²	1	1	yes	characteristic
18	C₂³×Dic₃₁	C₂²	1	1	yes	characteristic
19	C₂³×Dic₃₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄×D₃₁	—	2	8	no	—
21	C₂²×C₄×D₃₁	C₂²	1	12	yes	complemented
22	C₂×C₂²⋊Dic₃₁	C₂²	1	12	yes	complemented
23	C₄×C₂²⋊D₃₁	C₂²	1	64	yes	direct product factor
24	C₂×(C₂×D₃₁)⋊C₄	C₂²	1	12	yes	complemented
25	C₂×Dic₃₁⋊C₄	C₂²	1	12	yes	complemented
26	C₂×C₄×Dic₃₁	C₂²	1	12	yes	complemented
27	C₂⁴×C₃₁	—	2	4	no	—
28	C₂⁴×C₃₁	—	2	4	no	—
29	C₂⁴×C₃₁	C₂×C₄	1	8	yes	complemented
30	C₂⁴×C₃₁	C₂³	1	6	yes	—
31	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
32	C₂³×D₃₁	—	2	4	no	—
33	C₂³×D₃₁	—	2	4	no	—
34	C₂³×D₃₁	C₂×C₄	1	8	yes	complemented
35	C₂³×D₃₁	C₂³	1	6	yes	—
36	C₂²×C₄×C₃₁	—	2	4	no	—
37	C₂²×C₄×C₃₁	—	2	48	no	—
38	C₂²×C₄×C₃₁	D₄	1	8	yes	complemented
39	C₂²×C₄×C₃₁	C₂³	1	6	yes	—
40	C₂²×C₄×C₃₁	C₂³	1	6	yes	—
41	C₂²×C₄×C₃₁	C₂³	1	16	yes	complemented
42	C₂×C₂²⋊D₃₁	C₂×C₄	1	96	yes	direct product factor
43	C₂×C₂²⋊D₃₁	C₂³	1	16	yes	—
44	C₂²×Dic₃₁	—	2	4	no	—
45	C₂²×Dic₃₁	—	2	4	no	—
46	C₂²×Dic₃₁	C₂×C₄	1	8	yes	complemented
47	C₂²×Dic₃₁	C₂³	1	6	yes	—
48	C₂²×Dic₃₁	C₂³	1	6	yes	—
49	C₂²×Dic₃₁	C₂³	1	6	yes	—
50	C₂×C₄×D₃₁	—	2	48	no	—
51	C₂×C₄×D₃₁	C₂³	1	16	yes	complemented
52	C₂²⋊Dic₃₁	C₂³	1	16	yes	complemented
53	(C₂×D₃₁)⋊C₄	C₂³	1	16	yes	complemented
54	Dic₃₁⋊C₄	C₂³	1	16	yes	complemented
55	C₄×Dic₃₁	C₂³	1	16	yes	complemented
56	C₂³×C₃₁	—	2	32	no	—
57	C₂³×C₃₁	—	2	12	no	—
58	C₂³×C₃₁	—	2	12	no	—
59	C₂³×C₃₁	C₂²×C₄	1	4	yes	—
60	C₂³×C₃₁	C₂²×C₄	1	24	yes	complemented
61	C₂³×C₃₁	C₂×D₄	1	4	yes	—
62	C₂³×C₃₁	#41	1	4	yes	—
63	C₂³×C₃₁	C₂⁴	1	4	yes	—
64	C₂³×C₃₁	C₂⁴	1	3	yes	—
65	C₂²×D₃₁	—	2	32	no	—
66	C₂²×D₃₁	—	2	12	no	—
67	C₂²×D₃₁	—	2	12	no	—
68	C₂²×D₃₁	C₂²×C₄	1	24	yes	complemented
69	C₂²×D₃₁	C₂⁴	1	4	yes	—
70	C₂×C₄×C₃₁	—	2	12	no	—
71	C₂×C₄×C₃₁	—	2	32	no	—
72	C₂×C₄×C₃₁	C₂×D₄	1	24	yes	complemented
73	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
74	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
75	C₂²⋊D₃₁	C₂²×C₄	1	64	yes	direct product factor
76	C₂×Dic₃₁	—	2	12	no	—
77	C₂×Dic₃₁	—	2	12	no	—
78	C₂×Dic₃₁	C₂²×C₄	1	24	yes	complemented
79	C₂×Dic₃₁	C₂⁴	1	4	yes	—
80	C₂×Dic₃₁	C₂⁴	1	4	yes	—
81	C₂×Dic₃₁	C₂⁴	1	4	yes	—
82	C₄×D₃₁	—	2	32	no	—
83	C₂²×C₄×D₄	—	31	1	no	maximal, Sylow
84	C₂²×C₃₁	—	2	48	no	—
85	C₂²×C₃₁	—	2	4	no	—
86	C₂²×C₃₁	—	2	4	no	—
87	C₂²×C₃₁	C₄×D₄	1	16	yes	complemented
88	C₂²×C₃₁	C₂³×C₄	1	6	yes	—
89	C₂²×C₃₁	C₂³×C₄	1	8	yes	complemented
90	C₂²×C₃₁	C₂²×D₄	1	6	yes	—
91	C₂²×C₃₁	C₂×#41	1	6	yes	—
92	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
93	C₂×D₃₁	—	2	48	no	—
94	C₂×D₃₁	—	2	4	no	—
95	C₂×D₃₁	—	2	4	no	—
96	C₂×D₃₁	C₂³×C₄	1	8	yes	complemented
97	C₄×C₃₁	—	2	4	no	—
98	C₄×C₃₁	C₂²×D₄	1	8	yes	complemented
99	Dic₃₁	—	2	4	no	—
100	Dic₃₁	—	2	4	no	—
101	Dic₃₁	C₂³×C₄	1	8	yes	complemented
102	C₂³×D₄	—	31	1	no	—
103	C₂⁴×C₄	—	31	1	no	—
104	C₂⁴×C₄	D₃₁	1	1	yes	characteristic, complemented
105	C₂×C₄×D₄	—	31	24	no	—
106	C₂²×C₄⋊C₄	—	31	1	no	—
107	C₂²×C₂²⋊C₄	—	31	1	no	—
108	C₂²×C₂²⋊C₄	—	31	1	no	—
109	C₂²×C₄²	—	31	1	no	—
110	C₂×C₃₁	—	2	8	no	—
111	C₂×C₃₁	C₂×C₄×D₄	1	12	yes	complemented
112	C₂×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
113	C₂×C₃₁	C₂³×D₄	1	1	yes	characteristic
114	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
115	D₃₁	—	2	8	no	—
116	C₂⁵	—	31	1	no	—
117	C₂⁵	C₂×D₃₁	1	1	yes	characteristic
118	C₂²×D₄	—	31	16	no	—
119	C₂²×D₄	—	31	12	no	—
120	C₂³×C₄	—	2	8	no	—
121	C₂³×C₄	—	31	1	no	—
122	C₂³×C₄	—	31	1	no	—
123	C₂³×C₄	—	31	1	no	—
124	C₂³×C₄	—	31	12	no	—
125	C₂³×C₄	—	62	8	no	—
126	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
127	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
128	C₂³×C₄	C₂×D₃₁	1	12	yes	complemented
129	C₄×D₄	—	31	64	no	—
130	C₂×C₄⋊C₄	—	31	12	no	—
131	C₂×C₂²⋊C₄	—	31	12	no	—
132	C₂×C₂²⋊C₄	—	31	12	no	—
133	C₂×C₄²	—	31	12	no	—
134	C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
135	C₂⁴	—	2	4	no	—
136	C₂⁴	—	2	4	no	—
137	C₂⁴	—	31	8	no	—
138	C₂⁴	—	31	6	no	—
139	C₂⁴	—	62	4	no	—
140	C₂⁴	—	62	4	no	—
141	C₂⁴	C₄×D₃₁	1	8	yes	complemented
142	C₂⁴	C₂²×D₃₁	1	6	yes	—
143	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
144	C₂×D₄	—	31	96	no	—
145	C₂×D₄	—	31	16	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	2	48	no	—
148	C₂²×C₄	—	31	8	no	—
149	C₂²×C₄	—	31	6	no	—
150	C₂²×C₄	—	31	6	no	—
151	C₂²×C₄	—	31	6	no	—
152	C₂²×C₄	—	31	16	no	—
153	C₂²×C₄	—	62	4	no	—
154	C₂²×C₄	—	62	4	no	—
155	C₂²×C₄	—	62	48	no	—
156	C₂²×C₄	C₂²⋊D₃₁	1	8	yes	direct product factor
157	C₂²×C₄	C₂²×D₃₁	1	6	yes	—
158	C₂²×C₄	C₂²×D₃₁	1	6	yes	—
159	C₂²×C₄	C₂²×D₃₁	1	16	yes	complemented
160	C₄⋊C₄	—	31	16	no	—
161	C₂²⋊C₄	—	31	16	no	—
162	C₂²⋊C₄	—	31	16	no	—
163	C₄²	—	31	16	no	—
164	C₂³	—	2	32	no	—
165	C₂³	—	2	12	no	—
166	C₂³	—	2	12	no	—
167	C₂³	—	31	24	no	—
168	C₂³	—	31	4	no	—
169	C₂³	—	62	32	no	—
170	C₂³	—	62	12	no	—
171	C₂³	—	62	12	no	—
172	C₂³	C₂×C₄×D₃₁	1	4	yes	—
173	C₂³	C₂×C₄×D₃₁	1	24	yes	complemented
174	C₂³	#92495	1	4	yes	—
175	C₂³	C₂×C₂²⋊D₃₁	1	4	yes	—
176	C₂³	C₂³×D₃₁	1	4	yes	—
177	C₂³	C₂³×D₃₁	1	3	yes	—
178	D₄	—	31	64	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	2	32	no	—
181	C₂×C₄	—	31	24	no	—
182	C₂×C₄	—	31	4	no	—
183	C₂×C₄	—	31	4	no	—
184	C₂×C₄	—	31	4	no	—
185	C₂×C₄	—	62	12	no	—
186	C₂×C₄	—	62	12	no	—
187	C₂×C₄	—	62	32	no	—
188	C₂×C₄	C₂×C₂²⋊D₃₁	1	24	yes	direct product factor
189	C₂×C₄	C₂³×D₃₁	1	4	yes	—
190	C₂×C₄	C₂³×D₃₁	1	4	yes	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	48	no	—
194	C₂²	—	31	8	no	—
195	C₂²	—	62	4	no	—
196	C₂²	—	62	4	no	—
197	C₂²	—	62	48	no	—
198	C₂²	C₄×C₂²⋊D₃₁	1	16	yes	direct product factor
199	C₂²	C₂²×C₄×D₃₁	1	6	yes	—
200	C₂²	C₂²×C₄×D₃₁	1	8	yes	complemented
201	C₂²	C₂×#92495	1	6	yes	—
202	C₂²	C₂²×C₂²⋊D₃₁	1	6	yes	—
203	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
204	C₄	—	2	4	no	—
205	C₄	—	31	8	no	—
206	C₄	—	62	4	no	—
207	C₄	—	62	4	no	—
208	C₄	C₂²×C₂²⋊D₃₁	1	8	yes	direct product factor
209	C₂	—	2	8	no	—
210	C₂	—	62	8	no	—
211	C₂	C₂×C₄×C₂²⋊D₃₁	1	12	yes	direct product factor
212	C₂	C₂³×C₄×D₃₁	1	1	yes	characteristic
213	C₂	C₂³×C₂²⋊D₃₁	1	1	yes	characteristic
214	C₂	C₂²×#92495	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×D₃₁
commutator	C₂×C₃₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₂⁴×C₄×C₃₁	C₂
socle	C₂⁴×C₃₁	C₂³
prime radical	C₂³×C₂²⋊D₃₁	C₂
cosocle	C₂²×C₃₁	C₂⁵
hypercenter	C₂⁴×C₄	D₃₁
hypocenter	C₃₁	C₂²×C₄×D₄
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄×D₃₁	C₂	semidirect
3. C₂×C₄×C₂²⋊D₃₁	C₂	direct (N × H)
4. C₂²×Dic₃₁⋊C₄	C₂	semidirect
5. C₂²×C₂²⋊Dic₃₁	C₂	semidirect
6. C₂²×(C₂×D₃₁)⋊C₄	C₂	semidirect
7. C₂²×C₄×Dic₃₁	C₂	semidirect
8. C₂⁴×C₄×C₃₁	C₂	semidirect
14. C₂³×C₄×C₃₁	C₂²	semidirect
15. C₂²×C₂²⋊D₃₁	C₄	direct (N × H)
21. C₂²×C₄×D₃₁	C₂²	semidirect
22. C₂×C₂²⋊Dic₃₁	C₂²	semidirect
23. C₄×C₂²⋊D₃₁	C₂²	direct (N × H)
24. C₂×(C₂×D₃₁)⋊C₄	C₂²	semidirect
25. C₂×Dic₃₁⋊C₄	C₂²	semidirect
26. C₂×C₄×Dic₃₁	C₂²	semidirect
29. C₂⁴×C₃₁	C₂×C₄	semidirect
34. C₂³×D₃₁	C₂×C₄	semidirect
38. C₂²×C₄×C₃₁	D₄	semidirect
41. C₂²×C₄×C₃₁	C₂³	semidirect
42. C₂×C₂²⋊D₃₁	C₂×C₄	direct (N × H)
46. C₂²×Dic₃₁	C₂×C₄	semidirect
51. C₂×C₄×D₃₁	C₂³	semidirect
52. C₂²⋊Dic₃₁	C₂³	semidirect
53. (C₂×D₃₁)⋊C₄	C₂³	semidirect
54. Dic₃₁⋊C₄	C₂³	semidirect
55. C₄×Dic₃₁	C₂³	semidirect
60. C₂³×C₃₁	C₂²×C₄	semidirect
68. C₂²×D₃₁	C₂²×C₄	semidirect
72. C₂×C₄×C₃₁	C₂×D₄	semidirect
75. C₂²⋊D₃₁	C₂²×C₄	direct (N × H)
78. C₂×Dic₃₁	C₂²×C₄	semidirect
87. C₂²×C₃₁	C₄×D₄	semidirect
89. C₂²×C₃₁	C₂³×C₄	semidirect
96. C₂×D₃₁	C₂³×C₄	semidirect
98. C₄×C₃₁	C₂²×D₄	semidirect
101. Dic₃₁	C₂³×C₄	semidirect
104. C₂⁴×C₄	D₃₁	semidirect
111. C₂×C₃₁	C₂×C₄×D₄	semidirect
128. C₂³×C₄	C₂×D₃₁	semidirect
134. C₃₁	C₂²×C₄×D₄	semidirect
141. C₂⁴	C₄×D₃₁	semidirect
156. C₂²×C₄	C₂²⋊D₃₁	direct (N × H)
159. C₂²×C₄	C₂²×D₃₁	semidirect
173. C₂³	C₂×C₄×D₃₁	semidirect
188. C₂×C₄	C₂×C₂²⋊D₃₁	direct (N × H)
198. C₂²	C₄×C₂²⋊D₃₁	direct (N × H)
200. C₂²	C₂²×C₄×D₃₁	semidirect
208. C₄	C₂²×C₂²⋊D₃₁	direct (N × H)
211. C₂	C₂×C₄×C₂²⋊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
C₂³×C₂²⋊D₃₁	2
C₂³×C₄×D₃₁	2
C₂×C₄×C₂²⋊D₃₁	2
C₂²×Dic₃₁⋊C₄	2
C₂²×C₂²⋊Dic₃₁	2
C₂²×(C₂×D₃₁)⋊C₄	2
C₂²×C₄×Dic₃₁	2
C₂⁴×C₄×C₃₁	2
C₂²×C₄×D₄	31

Subgroups — C22×C4×C22⋊D31 #16858880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₂²⋊D₃₁ #16858880