Subgroups — C₃₁×#3367 #16843300

Order 3968.

372 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#489	C₂	1	2	yes	maximal, complemented
4	C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#477	C₂	1	2	yes	maximal, complemented
6	C₃₁×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#463	C₂	1	2	yes	maximal
8	C₃₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#440	C₂	1	2	yes	maximal, complemented
11	C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₃₁	—	2	1	no	—
13	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₃₁×#126	—	2	2	no	—
19	C₃₁×#124	C₂²	1	2	yes	—
20	C₃₁×C₂²⋊Q₈	—	2	1	no	—
21	C₃₁×C₂²⋊Q₈	—	2	1	no	—
22	C₃₁×C₂²⋊Q₈	—	2	2	no	—
23	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
24	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#117	—	2	1	no	—
29	C₃₁×#107	C₂²	1	1	yes	characteristic
30	C₃₁×#107	C₂²	1	2	yes	—
31	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
33	C₃₁×#103	C₂²	1	2	yes	—
34	C₃₁×#103	C₂²	1	2	yes	—
35	C₃₁×#103	C₂²	1	2	yes	—
36	C₃₁×#102	C₂²	1	1	yes	characteristic
37	C₃₁×#102	C₂²	1	1	yes	characteristic
38	C₃₁×#102	C₂²	1	2	yes	—
39	C₃₁×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₃₁	—	2	2	no	—
42	C₂×Q₈×C₃₁	—	2	1	no	—
43	C₂×Q₈×C₃₁	—	4	4	no	—
44	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
45	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
46	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	—	2	1	no	—
49	C₂²×C₄×C₃₁	—	2	2	no	—
50	C₂²×C₄×C₃₁	—	2	1	no	—
51	SD₁₆×C₃₁	—	2	4	no	—
52	D₈×C₃₁	—	2	2	no	—
53	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₃₁	C₂³	1	2	yes	—
56	C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₃₁	D₄	1	2	yes	—
63	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂²⋊C₄×C₃₁	—	2	1	no	—
69	C₂²⋊C₄×C₃₁	—	2	1	no	—
70	C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₄²×C₃₁	—	2	2	no	—
72	C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₄²×C₃₁	C₂³	1	2	yes	—
74	C₂³×C₃₁	—	2	1	no	—
75	C₂³×C₃₁	—	2	1	no	—
76	Q₈×C₃₁	—	2	2	no	—
77	Q₈×C₃₁	—	2	2	no	—
78	Q₈×C₃₁	—	2	2	no	—
79	Q₈×C₃₁	—	4	4	no	—
80	Q₈×C₃₁	—	4	2	no	—
81	D₄×C₃₁	—	2	1	no	—
82	D₄×C₃₁	—	2	1	no	—
83	D₄×C₃₁	—	2	1	no	—
84	D₄×C₃₁	#41	1	2	yes	—
85	C₂×C₄×C₃₁	—	2	1	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	1	no	—
89	C₂×C₄×C₃₁	—	4	4	no	—
90	C₂×C₄×C₃₁	—	4	2	no	—
91	C₂×C₄×C₃₁	—	4	2	no	—
92	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
97	C₈×C₃₁	—	2	1	no	—
98	C₈×C₃₁	—	2	2	no	—
99	C₈×C₃₁	—	2	1	no	—
100	#3367	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₃₁	—	2	2	no	—
102	C₂²×C₃₁	—	4	1	no	—
103	C₂²×C₃₁	—	4	1	no	—
104	C₂²×C₃₁	—	4	1	no	—
105	C₂²×C₃₁	—	4	1	no	—
106	C₂²×C₃₁	—	4	1	no	—
107	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₃₁	—	2	2	no	—
109	C₄×C₃₁	—	2	2	no	—
110	C₄×C₃₁	—	2	1	no	—
111	C₄×C₃₁	—	4	2	no	—
112	C₄×C₃₁	—	4	1	no	—
113	C₄×C₃₁	—	4	2	no	—
114	C₄×C₃₁	—	4	1	no	—
115	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
116	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
117	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
118	#556	C₂×C₃₁	1	1	yes	characteristic, complemented
119	D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
120	#489	C₂×C₃₁	1	2	yes	complemented
121	#487	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#477	C₂×C₃₁	1	2	yes	complemented
123	#474	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#463	C₂×C₃₁	1	2	yes	—
125	#461	C₂×C₃₁	1	1	yes	characteristic, complemented
126	#446	C₂×C₃₁	1	1	yes	characteristic, complemented
127	#440	C₂×C₃₁	1	2	yes	complemented
128	C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
129	C₂×C₃₁	—	4	2	no	—
130	C₂×C₃₁	—	8	1	no	—
131	C₂×C₃₁	#548	1	1	yes	characteristic
132	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₃₁	#578	1	1	yes	characteristic
134	C₂²×Q₈	—	2	1	no	—
135	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
136	C₂×D₈	C₂²×C₃₁	1	1	yes	characteristic
137	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
138	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
139	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
140	#126	—	2	2	no	—
141	#124	C₂²×C₃₁	1	2	yes	—
142	C₂²⋊Q₈	—	2	1	no	—
143	C₂²⋊Q₈	—	2	1	no	—
144	C₂²⋊Q₈	—	2	2	no	—
145	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
146	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
147	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
148	C₄×D₄	C₂²×C₃₁	1	2	yes	—
149	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
150	#117	—	2	1	no	—
151	#107	C₂²×C₃₁	1	1	yes	characteristic
152	#107	C₂²×C₃₁	1	2	yes	—
153	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
154	C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
155	#103	C₂²×C₃₁	1	2	yes	—
156	#103	C₂²×C₃₁	1	2	yes	—
157	#103	C₂²×C₃₁	1	2	yes	—
158	#102	C₂²×C₃₁	1	1	yes	characteristic
159	#102	C₂²×C₃₁	1	1	yes	characteristic
160	#102	C₂²×C₃₁	1	2	yes	—
161	#97	C₂²×C₃₁	1	2	yes	—
162	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
163	C₃₁	#3367	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	4	4	no	—
167	C₂×Q₈	D₄×C₃₁	1	1	yes	characteristic
168	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
169	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
170	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	1	no	—
174	SD₁₆	—	2	4	no	—
175	D₈	—	2	2	no	—
176	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₃₁	1	2	yes	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
185	C₄⋊C₄	D₄×C₃₁	1	2	yes	—
186	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
187	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
188	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	1	no	—
192	C₂²⋊C₄	—	2	1	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	C₂³×C₃₁	1	1	yes	characteristic
196	C₄²	C₂³×C₃₁	1	2	yes	—
197	C₂³	—	2	1	no	—
198	C₂³	—	2	1	no	—
199	Q₈	—	2	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	—	2	2	no	—
202	Q₈	—	4	4	no	—
203	Q₈	—	4	2	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	D₄	—	2	1	no	—
207	D₄	#41×C₃₁	1	2	yes	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
216	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
217	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
220	C₈	—	2	1	no	—
221	C₈	—	2	2	no	—
222	C₈	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	1	no	—
237	C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
238	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
239	C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	—	8	1	no	—
242	C₂	C₃₁×#548	1	1	yes	characteristic
243	C₂	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₂	C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₂×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#556	C₂	semidirect
2. D₄×Q₈×C₃₁	C₂	semidirect
3. C₃₁×#489	C₂	semidirect
4. C₃₁×#487	C₂	semidirect
5. C₃₁×#477	C₂	semidirect
6. C₃₁×#474	C₂	semidirect
8. C₃₁×#461	C₂	semidirect
9. C₃₁×#446	C₂	semidirect
10. C₃₁×#440	C₂	semidirect
100. #3367	C₃₁	direct (N × H)
118. #556	C₂×C₃₁	semidirect
119. D₄×Q₈	C₂×C₃₁	semidirect
120. #489	C₂×C₃₁	semidirect
121. #487	C₂×C₃₁	semidirect
122. #477	C₂×C₃₁	semidirect
123. #474	C₂×C₃₁	semidirect
125. #461	C₂×C₃₁	semidirect
126. #446	C₂×C₃₁	semidirect
127. #440	C₂×C₃₁	semidirect
163. C₃₁	#3367	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#556	2
D₄×Q₈×C₃₁	2
C₃₁×#489	2
C₃₁×#487	2
C₃₁×#477	2
C₃₁×#474	2
C₃₁×#463	2
C₃₁×#461	2
C₃₁×#446	2
C₃₁×#440	2
C₄×D₈×C₃₁	2
#3367	31

Subgroups — C31×#3367 #16843300

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3367 #16843300