Subgroups — C₃₁×#13470 #21470229

Order 7936.

326 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#368	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#361	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#357	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1978	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1952	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1387	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
13	C₃₁×#428	—	2	1	no	—
14	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
15	C₃₁×#418	—	2	1	no	—
16	C₂×C₃₁×#100	—	2	1	no	—
17	C₃₁×#405	C₂²	1	1	yes	characteristic
18	C₃₁×#368	—	2	2	no	—
19	C₃₁×C₄⋊C₁₆	—	2	2	no	—
20	C₃₁×#361	—	2	2	no	—
21	C₃₁×#357	—	2	2	no	—
22	C₃₁×#343	—	2	1	no	—
23	C₂²×D₄×C₃₁	—	2	1	no	—
24	C₂×D₈×C₃₁	—	2	1	no	—
25	C₂×D₈×C₃₁	—	2	1	no	—
26	C₂×D₈×C₃₁	—	2	1	no	—
27	C₂×D₈×C₃₁	—	2	1	no	—
28	C₂×D₈×C₃₁	—	2	1	no	—
29	C₂×D₈×C₃₁	Q₈	1	2	yes	—
30	C₂×M₁₆×C₃₁	—	2	1	no	—
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×#117	D₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
36	C₃₁×M₃₂	—	2	1	no	—
37	C₃₁×M₃₂	—	2	1	no	—
38	C₂×C₁₆×C₃₁	—	2	1	no	—
39	C₂×C₁₆×C₃₁	—	2	1	no	—
40	C₂×C₁₆×C₃₁	—	2	1	no	—
41	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
42	C₃₁×#108	—	2	1	no	—
43	C₃₁×#108	—	2	1	no	—
44	C₃₁×#108	—	2	1	no	—
45	C₃₁×#108	D₄	1	2	yes	—
46	C₃₁×#107	—	2	1	no	—
47	C₃₁×#107	—	2	1	no	—
48	C₃₁×#107	—	4	1	no	—
49	C₃₁×#106	—	4	1	no	—
50	C₃₁×#102	—	4	1	no	—
51	C₃₁×#102	—	4	1	no	—
52	C₃₁×#100	—	4	2	no	—
53	C₃₁×#97	—	2	1	no	—
54	C₄×C₈×C₃₁	—	2	1	no	—
55	C₂⁴×C₃₁	—	4	1	no	—
56	C₂×D₄×C₃₁	—	2	2	no	—
57	C₂×D₄×C₃₁	—	4	1	no	—
58	C₂×D₄×C₃₁	—	4	1	no	—
59	C₂×D₄×C₃₁	—	4	1	no	—
60	C₂×D₄×C₃₁	—	4	1	no	—
61	C₂×D₄×C₃₁	—	4	1	no	—
62	C₂²×C₄×C₃₁	—	4	1	no	—
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	D₈×C₃₁	—	2	2	no	—
65	D₈×C₃₁	—	2	2	no	—
66	D₈×C₃₁	—	4	1	no	—
67	D₈×C₃₁	—	4	1	no	—
68	M₁₆×C₃₁	—	2	2	no	—
69	M₁₆×C₃₁	—	2	1	no	—
70	M₁₆×C₃₁	—	2	1	no	—
71	M₁₆×C₃₁	—	4	1	no	—
72	M₁₆×C₃₁	—	4	1	no	—
73	M₁₆×C₃₁	—	4	1	no	—
74	C₂×C₈×C₃₁	—	2	1	no	—
75	C₂×C₈×C₃₁	—	4	1	no	—
76	C₂×C₈×C₃₁	—	4	1	no	—
77	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
81	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
82	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
83	C₄⋊C₄×C₃₁	—	4	1	no	—
84	C₄⋊C₄×C₃₁	—	4	1	no	—
85	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
86	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
87	C₂²⋊C₄×C₃₁	—	2	1	no	—
88	C₄²×C₃₁	—	2	1	no	—
89	C₁₆×C₃₁	—	2	2	no	—
90	C₁₆×C₃₁	—	4	1	no	—
91	#13470	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	4	2	no	—
93	C₂³×C₃₁	—	8	1	no	—
94	C₂³×C₃₁	—	8	1	no	—
95	C₂³×C₃₁	—	16	1	no	—
96	C₂³×C₃₁	—	16	1	no	—
97	C₂³×C₃₁	#135	1	1	yes	characteristic
98	D₄×C₃₁	—	4	2	no	—
99	D₄×C₃₁	—	4	2	no	—
100	D₄×C₃₁	—	8	1	no	—
101	D₄×C₃₁	—	8	1	no	—
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	—	8	1	no	—
105	C₂×C₄×C₃₁	—	16	1	no	—
106	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
107	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
112	C₈×C₃₁	—	4	2	no	—
113	C₈×C₃₁	—	4	1	no	—
114	C₈×C₃₁	—	8	1	no	—
115	C₈×C₃₁	#121	1	1	yes	characteristic
116	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
117	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
118	C₈×C₃₁	#126	1	1	yes	characteristic
119	C₂×#368	C₂×C₃₁	1	1	yes	characteristic, complemented
120	#2151	C₂×C₃₁	1	1	yes	characteristic, complemented
121	C₂×#361	C₂×C₃₁	1	1	yes	characteristic
122	C₂×#357	C₂×C₃₁	1	1	yes	characteristic
123	#1978	C₂×C₃₁	1	1	yes	characteristic
124	#1952	C₂×C₃₁	1	1	yes	characteristic, complemented
125	#1387	C₂×C₃₁	1	1	yes	characteristic, complemented
126	C₂²×C₃₁	—	2	1	no	—
127	C₂²×C₃₁	—	2	1	no	—
128	C₂²×C₃₁	—	8	2	no	—
129	C₂²×C₃₁	—	16	2	no	—
130	C₂²×C₃₁	—	16	2	no	—
131	C₂²×C₃₁	—	16	1	no	—
132	C₂²×C₃₁	—	16	1	no	—
133	C₂²×C₃₁	#464	1	1	yes	characteristic
134	C₂²×C₃₁	#466	1	1	yes	characteristic
135	C₂²×C₃₁	#478	1	1	yes	characteristic
136	C₄×C₃₁	—	4	1	no	—
137	C₄×C₃₁	—	4	1	no	—
138	C₄×C₃₁	—	16	1	no	—
139	C₄×C₃₁	#396	1	1	yes	characteristic
140	C₄×C₃₁	#459	1	1	yes	characteristic
141	C₄×C₃₁	#468	1	1	yes	characteristic
142	C₄×C₃₁	#474	1	1	yes	characteristic
143	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
144	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
145	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
146	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
147	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
148	#428	—	2	1	no	—
149	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
150	#418	—	2	1	no	—
151	C₂×#100	—	2	1	no	—
152	#405	C₂²×C₃₁	1	1	yes	characteristic
153	#368	—	2	2	no	—
154	C₄⋊C₁₆	—	2	2	no	—
155	#361	—	2	2	no	—
156	#357	—	2	2	no	—
157	#343	—	2	1	no	—
158	C₂×C₃₁	—	2	1	no	—
159	C₂×C₃₁	—	2	1	no	—
160	C₂×C₃₁	—	16	2	no	—
161	C₂×C₃₁	#2067	1	1	yes	characteristic
162	C₂×C₃₁	#2221	1	1	yes	characteristic
163	C₂×C₃₁	#2223	1	1	yes	characteristic
164	C₂²×D₄	—	2	1	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	—	2	1	no	—
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	—	2	1	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	Q₈×C₃₁	1	2	yes	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
175	#117	D₄×C₃₁	1	1	yes	characteristic
176	C₂×C₄⋊C₄	—	2	1	no	—
177	M₃₂	—	2	1	no	—
178	M₃₂	—	2	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
183	#108	—	2	1	no	—
184	#108	—	2	1	no	—
185	#108	—	2	1	no	—
186	#108	D₄×C₃₁	1	2	yes	—
187	#107	—	2	1	no	—
188	#107	—	2	1	no	—
189	#107	—	4	1	no	—
190	#106	—	4	1	no	—
191	#102	—	4	1	no	—
192	#102	—	4	1	no	—
193	#100	—	4	2	no	—
194	#97	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₃₁	#13470	1	1	yes	characteristic, direct product factor, Sylow
197	C₂⁴	—	4	1	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂×D₄	—	4	1	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂×D₄	—	4	1	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
206	D₈	—	2	2	no	—
207	D₈	—	2	2	no	—
208	D₈	—	4	1	no	—
209	D₈	—	4	1	no	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	1	no	—
212	M₁₆	—	2	1	no	—
213	M₁₆	—	4	1	no	—
214	M₁₆	—	4	1	no	—
215	M₁₆	—	4	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
222	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
223	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
224	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
228	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
229	C₂²⋊C₄	—	2	1	no	—
230	C₄²	—	2	1	no	—
231	C₁₆	—	2	2	no	—
232	C₁₆	—	4	1	no	—
233	C₂³	—	4	2	no	—
234	C₂³	—	8	1	no	—
235	C₂³	—	8	1	no	—
236	C₂³	—	16	1	no	—
237	C₂³	—	16	1	no	—
238	C₂³	C₃₁×#135	1	1	yes	characteristic
239	D₄	—	4	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	8	1	no	—
242	D₄	—	8	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	8	1	no	—
246	C₂×C₄	—	16	1	no	—
247	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
248	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
249	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
250	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
251	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
252	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
253	C₈	—	4	2	no	—
254	C₈	—	4	1	no	—
255	C₈	—	8	1	no	—
256	C₈	C₃₁×#121	1	1	yes	characteristic
257	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
258	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
259	C₈	C₃₁×#126	1	1	yes	characteristic
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	—	16	2	no	—
264	C₂²	—	16	2	no	—
265	C₂²	—	16	1	no	—
266	C₂²	—	16	1	no	—
267	C₂²	C₃₁×#464	1	1	yes	characteristic
268	C₂²	C₃₁×#466	1	1	yes	characteristic
269	C₂²	C₃₁×#478	1	1	yes	characteristic
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	16	1	no	—
273	C₄	C₃₁×#396	1	1	yes	characteristic
274	C₄	C₃₁×#459	1	1	yes	characteristic
275	C₄	C₃₁×#468	1	1	yes	characteristic
276	C₄	C₃₁×#474	1	1	yes	characteristic
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	16	2	no	—
280	C₂	C₃₁×#2067	1	1	yes	characteristic
281	C₂	C₃₁×#2221	1	1	yes	characteristic
282	C₂	C₃₁×#2223	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#368	C₂	semidirect
2. C₃₁×#2151	C₂	semidirect
6. C₃₁×#1952	C₂	semidirect
7. C₃₁×#1387	C₂	semidirect
91. #13470	C₃₁	direct (N × H)
119. C₂×#368	C₂×C₃₁	semidirect
120. #2151	C₂×C₃₁	semidirect
124. #1952	C₂×C₃₁	semidirect
125. #1387	C₂×C₃₁	semidirect
196. C₃₁	#13470	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×C₃₁×#368	2
C₃₁×#2151	2
C₂×C₃₁×#361	2
C₂×C₃₁×#357	2
C₃₁×#1978	2
C₃₁×#1952	2
C₃₁×#1387	2
#13470	31

Subgroups — C31×#13470 #21470229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13470 #21470229