Subgroups — C₂²×C₄×Q₁₆×C₃₁ #21516909

Order 7936.

2830 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₃₁	C₂	1	2	yes	maximal
3	C₂×C₄×Q₁₆×C₃₁	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃₁×#107	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₃₁×#103	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
8	C₂²×Q₁₆×C₃₁	C₄	1	16	yes	direct product factor
9	C₂²×Q₁₆×C₃₁	C₂²	1	12	yes	—
10	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₁	—	2	12	no	—
13	C₂×C₄×Q₈×C₃₁	C₂²	1	24	yes	—
14	C₂²×C₄⋊C₄×C₃₁	—	2	2	no	—
15	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂²×C₄²×C₃₁	—	2	2	no	—
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₄×Q₁₆×C₃₁	C₂²	1	64	yes	direct product factor
19	C₂×C₃₁×#107	C₂²	1	12	yes	—
20	C₂×C₃₁×#103	C₂²	1	24	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	12	yes	—
22	C₂²×Q₈×C₃₁	—	2	6	no	—
23	C₂²×Q₈×C₃₁	—	2	8	no	—
24	C₂²×Q₈×C₃₁	C₂×C₄	1	16	yes	—
25	C₂²×Q₈×C₃₁	C₂³	1	12	yes	—
26	C₂³×C₄×C₃₁	—	2	2	no	—
27	C₂³×C₄×C₃₁	—	2	2	no	—
28	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂×Q₁₆×C₃₁	C₂×C₄	1	96	yes	direct product factor
32	C₂×Q₁₆×C₃₁	C₂³	1	16	yes	—
33	C₂²×C₈×C₃₁	—	2	4	no	—
34	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
35	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
37	C₄×Q₈×C₃₁	—	2	48	no	—
38	C₄×Q₈×C₃₁	C₂³	1	32	yes	—
39	C₂×C₄⋊C₄×C₃₁	—	2	24	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
41	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
42	C₂×C₄²×C₃₁	—	2	24	no	—
43	C₂×C₄²×C₃₁	D₄	1	6	yes	—
44	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
45	C₃₁×#107	C₂³	1	16	yes	—
46	C₃₁×#103	C₂³	1	32	yes	—
47	C₄×C₈×C₃₁	C₂³	1	16	yes	—
48	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×Q₈×C₃₁	—	2	48	no	—
50	C₂×Q₈×C₃₁	—	2	12	no	—
51	C₂×Q₈×C₃₁	—	2	24	no	—
52	C₂×Q₈×C₃₁	C₂²×C₄	1	48	yes	—
53	C₂×Q₈×C₃₁	C₂⁴	1	8	yes	—
54	C₂²×C₄×C₃₁	—	2	16	no	—
55	C₂²×C₄×C₃₁	—	2	4	no	—
56	C₂²×C₄×C₃₁	—	2	12	no	—
57	C₂²×C₄×C₃₁	—	2	12	no	—
58	C₂²×C₄×C₃₁	—	4	8	no	—
59	C₂²×C₄×C₃₁	Q₁₆	1	8	yes	direct product factor
60	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
63	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
64	C₂²×C₄×C₃₁	#41	1	4	yes	—
65	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
66	Q₁₆×C₃₁	C₂²×C₄	1	64	yes	direct product factor
67	C₂×C₈×C₃₁	—	2	12	no	—
68	C₂×C₈×C₃₁	C₂²×C₄	1	24	yes	—
69	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
70	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
71	C₄⋊C₄×C₃₁	—	2	32	no	—
72	C₄⋊C₄×C₃₁	—	2	12	no	—
73	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
74	C₄²×C₃₁	—	2	32	no	—
75	C₄²×C₃₁	C₂×D₄	1	12	yes	—
76	C₄²×C₃₁	C₂⁴	1	4	yes	—
77	C₂²×C₄×Q₁₆	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₃₁	C₄×D₄	1	4	yes	—
79	C₂³×C₃₁	C₂×Q₁₆	1	4	yes	—
80	C₂³×C₃₁	#135	1	4	yes	—
81	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
82	Q₈×C₃₁	—	2	48	no	—
83	Q₈×C₃₁	—	2	8	no	—
84	Q₈×C₃₁	C₂³×C₄	1	16	yes	—
85	C₂×C₄×C₃₁	—	2	48	no	—
86	C₂×C₄×C₃₁	—	2	12	no	—
87	C₂×C₄×C₃₁	—	2	8	no	—
88	C₂×C₄×C₃₁	—	2	8	no	—
89	C₂×C₄×C₃₁	—	4	24	no	—
90	C₂×C₄×C₃₁	C₄×D₄	1	12	yes	—
91	C₂×C₄×C₃₁	C₂×Q₁₆	1	24	yes	direct product factor
92	C₂×C₄×C₃₁	C₂³×C₄	1	6	yes	—
93	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
94	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
95	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
96	C₂×C₄×C₃₁	C₂×#41	1	6	yes	—
97	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
98	C₈×C₃₁	—	2	4	no	—
99	C₈×C₃₁	C₂³×C₄	1	8	yes	—
100	C₂³×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
101	C₂²×C₄×Q₈	C₂×C₃₁	1	2	yes	—
102	C₂×C₄×Q₁₆	C₂×C₃₁	1	24	yes	direct product factor
103	C₂²×#107	C₂×C₃₁	1	1	yes	characteristic
104	C₂²×#103	C₂×C₃₁	1	2	yes	—
105	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
106	C₂²×C₃₁	C₄×Q₁₆	1	16	yes	direct product factor
107	C₂²×C₃₁	C₂×C₄×D₄	1	6	yes	—
108	C₂²×C₃₁	C₂²×Q₁₆	1	6	yes	—
109	C₂²×C₃₁	C₂×#135	1	6	yes	—
110	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
111	C₄×C₃₁	—	2	16	no	—
112	C₄×C₃₁	—	2	4	no	—
113	C₄×C₃₁	—	4	8	no	—
114	C₄×C₃₁	C₂×C₄×D₄	1	6	yes	—
115	C₄×C₃₁	C₂²×Q₁₆	1	8	yes	direct product factor
116	C₄×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
117	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
118	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
119	C₂²×Q₁₆	C₄×C₃₁	1	16	yes	direct product factor
120	C₂²×Q₁₆	C₂²×C₃₁	1	12	yes	—
121	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
122	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
123	C₂×C₄×Q₈	—	2	12	no	—
124	C₂×C₄×Q₈	C₂²×C₃₁	1	24	yes	—
125	C₂²×C₄⋊C₄	—	2	2	no	—
126	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
127	C₂²×C₄²	—	2	2	no	—
128	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
129	C₄×Q₁₆	C₂²×C₃₁	1	64	yes	direct product factor
130	C₂×#107	C₂²×C₃₁	1	12	yes	—
131	C₂×#103	C₂²×C₃₁	1	24	yes	—
132	C₂×C₄×C₈	C₂²×C₃₁	1	12	yes	—
133	C₂×C₃₁	C₂×C₄×Q₁₆	1	12	yes	direct product factor
134	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
135	C₂×C₃₁	C₂³×Q₁₆	1	1	yes	characteristic
136	C₂×C₃₁	C₂²×#135	1	1	yes	characteristic
137	C₂²×Q₈	—	2	6	no	—
138	C₂²×Q₈	—	2	8	no	—
139	C₂²×Q₈	C₂×C₄×C₃₁	1	16	yes	—
140	C₂²×Q₈	C₂³×C₃₁	1	12	yes	—
141	C₂³×C₄	—	2	2	no	—
142	C₂³×C₄	—	2	2	no	—
143	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
144	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
145	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
146	C₂×Q₁₆	C₂×C₄×C₃₁	1	96	yes	direct product factor
147	C₂×Q₁₆	C₂³×C₃₁	1	16	yes	—
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
150	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
151	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
152	C₄×Q₈	—	2	48	no	—
153	C₄×Q₈	C₂³×C₃₁	1	32	yes	—
154	C₂×C₄⋊C₄	—	2	24	no	—
155	C₂×C₄⋊C₄	—	2	6	no	—
156	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
157	C₂×C₄²	—	2	24	no	—
158	C₂×C₄²	D₄×C₃₁	1	6	yes	—
159	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
160	#107	C₂³×C₃₁	1	16	yes	—
161	#103	C₂³×C₃₁	1	32	yes	—
162	C₄×C₈	C₂³×C₃₁	1	16	yes	—
163	C₃₁	C₂²×C₄×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
164	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
165	C₂×Q₈	—	2	48	no	—
166	C₂×Q₈	—	2	12	no	—
167	C₂×Q₈	—	2	24	no	—
168	C₂×Q₈	C₂²×C₄×C₃₁	1	48	yes	—
169	C₂×Q₈	C₂⁴×C₃₁	1	8	yes	—
170	C₂²×C₄	—	2	16	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	12	no	—
173	C₂²×C₄	—	2	12	no	—
174	C₂²×C₄	—	4	8	no	—
175	C₂²×C₄	Q₁₆×C₃₁	1	8	yes	direct product factor
176	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
177	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
178	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
179	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
180	C₂²×C₄	#41×C₃₁	1	4	yes	—
181	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
182	Q₁₆	C₂²×C₄×C₃₁	1	64	yes	direct product factor
183	C₂×C₈	—	2	12	no	—
184	C₂×C₈	C₂²×C₄×C₃₁	1	24	yes	—
185	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
186	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
187	C₄⋊C₄	—	2	32	no	—
188	C₄⋊C₄	—	2	12	no	—
189	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
190	C₄²	—	2	32	no	—
191	C₄²	C₂×D₄×C₃₁	1	12	yes	—
192	C₄²	C₂⁴×C₃₁	1	4	yes	—
193	C₂³	C₄×D₄×C₃₁	1	4	yes	—
194	C₂³	C₂×Q₁₆×C₃₁	1	4	yes	—
195	C₂³	C₃₁×#135	1	4	yes	—
196	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
197	Q₈	—	2	48	no	—
198	Q₈	—	2	8	no	—
199	Q₈	C₂³×C₄×C₃₁	1	16	yes	—
200	C₂×C₄	—	2	48	no	—
201	C₂×C₄	—	2	12	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	4	24	no	—
205	C₂×C₄	C₄×D₄×C₃₁	1	12	yes	—
206	C₂×C₄	C₂×Q₁₆×C₃₁	1	24	yes	direct product factor
207	C₂×C₄	C₂³×C₄×C₃₁	1	6	yes	—
208	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
209	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
210	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
211	C₂×C₄	C₂×#41×C₃₁	1	6	yes	—
212	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
213	C₈	—	2	4	no	—
214	C₈	C₂³×C₄×C₃₁	1	8	yes	—
215	C₂²	C₄×Q₁₆×C₃₁	1	16	yes	direct product factor
216	C₂²	C₂×C₄×D₄×C₃₁	1	6	yes	—
217	C₂²	C₂²×Q₁₆×C₃₁	1	6	yes	—
218	C₂²	C₂×C₃₁×#135	1	6	yes	—
219	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
220	C₄	—	2	16	no	—
221	C₄	—	2	4	no	—
222	C₄	—	4	8	no	—
223	C₄	C₂×C₄×D₄×C₃₁	1	6	yes	—
224	C₄	C₂²×Q₁₆×C₃₁	1	8	yes	direct product factor
225	C₄	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
226	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
227	C₂	C₂×C₄×Q₁₆×C₃₁	1	12	yes	direct product factor
228	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
229	C₂	C₂³×Q₁₆×C₃₁	1	1	yes	characteristic
230	C₂	C₂²×C₃₁×#135	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₃₁	D₄
commutator	C₄	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle, prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Q₁₆×C₃₁	C₂	direct (N × H)
8. C₂²×Q₁₆×C₃₁	C₄	direct (N × H)
18. C₄×Q₁₆×C₃₁	C₂²	direct (N × H)
31. C₂×Q₁₆×C₃₁	C₂×C₄	direct (N × H)
59. C₂²×C₄×C₃₁	Q₁₆	direct (N × H)
66. Q₁₆×C₃₁	C₂²×C₄	direct (N × H)
77. C₂²×C₄×Q₁₆	C₃₁	direct (N × H)
91. C₂×C₄×C₃₁	C₂×Q₁₆	direct (N × H)
102. C₂×C₄×Q₁₆	C₂×C₃₁	direct (N × H)
106. C₂²×C₃₁	C₄×Q₁₆	direct (N × H)
115. C₄×C₃₁	C₂²×Q₁₆	direct (N × H)
119. C₂²×Q₁₆	C₄×C₃₁	direct (N × H)
129. C₄×Q₁₆	C₂²×C₃₁	direct (N × H)
133. C₂×C₃₁	C₂×C₄×Q₁₆	direct (N × H)
146. C₂×Q₁₆	C₂×C₄×C₃₁	direct (N × H)
163. C₃₁	C₂²×C₄×Q₁₆	direct (N × H)
175. C₂²×C₄	Q₁₆×C₃₁	direct (N × H)
182. Q₁₆	C₂²×C₄×C₃₁	direct (N × H)
206. C₂×C₄	C₂×Q₁₆×C₃₁	direct (N × H)
215. C₂²	C₄×Q₁₆×C₃₁	direct (N × H)
224. C₄	C₂²×Q₁₆×C₃₁	direct (N × H)
227. C₂	C₂×C₄×Q₁₆×C₃₁	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×Q₁₆×C₃₁	2
C₂²×C₄×Q₈×C₃₁	2
C₂×C₄×Q₁₆×C₃₁	2
C₂²×C₃₁×#107	2
C₂²×C₃₁×#103	2
C₂²×C₄×C₈×C₃₁	2
C₂²×C₄×Q₁₆	31

Subgroups — C22×C4×Q16×C31 #21516909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×Q₁₆×C₃₁ #21516909