Subgroups — C₃₁×#13194 #21469953

Order 7936.

466 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#349	C₂	1	4	yes	maximal, complemented
4	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
6	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
8	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
9	C₃₁×#436	—	2	1	no	—
10	C₃₁×#436	—	2	1	no	—
11	C₃₁×#436	—	2	1	no	—
12	C₃₁×#436	—	2	1	no	—
13	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊C₈	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×C₄⋊C₁₆	—	2	1	no	—
17	C₃₁×C₄⋊C₁₆	—	2	1	no	—
18	C₃₁×#349	—	2	8	no	—
19	C₃₁×#346	—	2	1	no	—
20	C₄×C₁₆×C₃₁	—	2	1	no	—
21	C₂²×D₄×C₃₁	C₈	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	4	no	—
26	C₂×M₁₆×C₃₁	—	2	4	no	—
27	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
30	C₄×D₄×C₃₁	—	2	1	no	—
31	C₄×D₄×C₃₁	—	2	1	no	—
32	C₄×D₄×C₃₁	—	2	1	no	—
33	C₄×D₄×C₃₁	—	2	1	no	—
34	C₂×C₄⋊C₄×C₃₁	C₈	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₃₁	C₈	1	2	yes	—
36	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₃₁×M₃₂	—	2	2	no	—
38	C₃₁×M₃₂	—	2	4	no	—
39	C₃₁×M₃₂	D₄	1	4	yes	complemented
40	C₂×C₁₆×C₃₁	—	2	2	no	—
41	C₂×C₁₆×C₃₁	—	2	2	no	—
42	C₃₁×C₄⋊C₈	—	2	1	no	—
43	C₃₁×C₄⋊C₈	—	2	1	no	—
44	C₃₁×C₂²⋊C₈	—	2	2	no	—
45	C₃₁×C₂²⋊C₈	—	2	2	no	—
46	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
48	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
49	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
50	C₂⁴×C₃₁	C₂×C₈	1	2	yes	—
51	C₂×D₄×C₃₁	—	2	1	no	—
52	C₂×D₄×C₃₁	—	2	1	no	—
53	C₂×D₄×C₃₁	—	4	2	no	—
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	—	2	4	no	—
56	C₂²×C₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	M₁₆×C₃₁	—	4	4	no	—
63	M₁₆×C₃₁	—	4	2	no	—
64	M₁₆×C₃₁	—	4	2	no	—
65	C₂×C₈×C₃₁	—	2	2	no	—
66	C₂×C₈×C₃₁	—	2	2	no	—
67	C₂×C₈×C₃₁	—	2	4	no	—
68	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₃₁	#41	1	2	yes	—
74	C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₃₁	M₁₆	1	1	yes	characteristic
81	C₄²×C₃₁	M₁₆	1	1	yes	characteristic
82	C₁₆×C₃₁	—	2	4	no	—
83	C₁₆×C₃₁	—	4	2	no	—
84	#13194	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₃₁	—	2	2	no	—
86	C₂³×C₃₁	—	2	4	no	—
87	C₂³×C₃₁	—	4	2	no	—
88	C₂³×C₃₁	—	4	2	no	—
89	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
90	D₄×C₃₁	—	4	2	no	—
91	D₄×C₃₁	—	4	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	4	no	—
95	C₂×C₄×C₃₁	—	4	1	no	—
96	C₂×C₄×C₃₁	—	4	2	no	—
97	C₂×C₄×C₃₁	—	4	2	no	—
98	C₂×C₄×C₃₁	—	4	4	no	—
99	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₃₁	#131	1	2	yes	—
107	C₈×C₃₁	—	2	1	no	—
108	C₈×C₃₁	—	2	1	no	—
109	C₈×C₃₁	—	4	2	no	—
110	C₈×C₃₁	#121	1	2	yes	—
111	C₈×C₃₁	#124	1	1	yes	characteristic
112	C₈×C₃₁	#127	1	1	yes	characteristic
113	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
114	#2150	C₂×C₃₁	1	1	yes	characteristic, complemented
115	C₂×#349	C₂×C₃₁	1	4	yes	complemented
116	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
117	C₂²×C₃₁	—	2	1	no	—
118	C₂²×C₃₁	—	2	1	no	—
119	C₂²×C₃₁	—	4	2	no	—
120	C₂²×C₃₁	—	4	2	no	—
121	C₂²×C₃₁	—	4	2	no	—
122	C₂²×C₃₁	—	8	4	no	—
123	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₂²×C₃₁	C₈×D₄	1	2	yes	—
125	C₄×C₃₁	—	2	1	no	—
126	C₄×C₃₁	—	2	1	no	—
127	C₄×C₃₁	—	2	1	no	—
128	C₄×C₃₁	—	4	2	no	—
129	C₄×C₃₁	#349	1	2	yes	complemented
130	C₄×C₃₁	#390	1	1	yes	characteristic
131	C₄×C₃₁	#408	1	1	yes	characteristic
132	C₄×C₃₁	#436	1	2	yes	—
133	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
134	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
135	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
136	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
137	#436	—	2	1	no	—
138	#436	—	2	1	no	—
139	#436	—	2	1	no	—
140	#436	—	2	1	no	—
141	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
142	C₂×C₂²⋊C₈	C₄×C₃₁	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
144	C₄⋊C₁₆	—	2	1	no	—
145	C₄⋊C₁₆	—	2	1	no	—
146	#349	—	2	8	no	—
147	#346	—	2	1	no	—
148	C₄×C₁₆	—	2	1	no	—
149	C₂×C₃₁	—	2	2	no	—
150	C₂×C₃₁	—	8	2	no	—
151	C₂×C₃₁	#1954	1	1	yes	characteristic
152	C₂×C₃₁	C₂×#349	1	1	yes	characteristic
153	C₂×C₃₁	#2115	1	1	yes	characteristic
154	C₂²×D₄	C₈×C₃₁	1	1	yes	characteristic
155	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	—	2	2	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	4	no	—
160	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
161	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
163	C₄×D₄	—	2	1	no	—
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	C₂×C₄⋊C₄	C₈×C₃₁	1	1	yes	characteristic
168	C₂×C₂²⋊C₄	C₈×C₃₁	1	2	yes	—
169	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	—	2	4	no	—
172	M₃₂	D₄×C₃₁	1	4	yes	complemented
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₂²⋊C₈	—	2	2	no	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
180	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
181	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
182	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
183	C₃₁	#13194	1	1	yes	characteristic, direct product factor, Sylow
184	C₂⁴	C₂×C₈×C₃₁	1	2	yes	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
193	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
196	M₁₆	—	4	4	no	—
197	M₁₆	—	4	2	no	—
198	M₁₆	—	4	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
206	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
207	C₂×C₈	#41×C₃₁	1	2	yes	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
213	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
214	C₄²	M₁₆×C₃₁	1	1	yes	characteristic
215	C₄²	M₁₆×C₃₁	1	1	yes	characteristic
216	C₁₆	—	2	4	no	—
217	C₁₆	—	4	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	2	no	—
222	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
223	D₄	—	4	2	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
233	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
234	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
235	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
238	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
239	C₂×C₄	C₃₁×#131	1	2	yes	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	4	2	no	—
243	C₈	C₃₁×#121	1	2	yes	—
244	C₈	C₃₁×#124	1	1	yes	characteristic
245	C₈	C₃₁×#127	1	1	yes	characteristic
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
253	C₂²	C₈×D₄×C₃₁	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	2	no	—
258	C₄	C₃₁×#349	1	2	yes	complemented
259	C₄	C₃₁×#390	1	1	yes	characteristic
260	C₄	C₃₁×#408	1	1	yes	characteristic
261	C₄	C₃₁×#436	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	8	2	no	—
264	C₂	C₃₁×#1954	1	1	yes	characteristic
265	C₂	C₂×C₃₁×#349	1	1	yes	characteristic
266	C₂	C₃₁×#2115	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₃₁	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2150	C₂	semidirect
3. C₂×C₃₁×#349	C₂	semidirect
4. C₄×C₃₁×M₃₂	C₂	semidirect
39. C₃₁×M₃₂	D₄	semidirect
84. #13194	C₃₁	direct (N × H)
114. #2150	C₂×C₃₁	semidirect
115. C₂×#349	C₂×C₃₁	semidirect
116. C₄×M₃₂	C₂×C₃₁	semidirect
129. C₄×C₃₁	#349	semidirect
172. M₃₂	D₄×C₃₁	semidirect
183. C₃₁	#13194	direct (N × H)
258. C₄	C₃₁×#349	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#436	2
C₃₁×#2150	2
C₂×C₃₁×#349	2
C₄×C₃₁×M₃₂	2
#13194	31

Subgroups — C31×#13194 #21469953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13194 #21469953