Subgroups — C₃₁×#11345 #21468104

Order 7936.

698 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2905	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#424	C₂	1	2	yes	maximal
3	C₂×C₃₁×#344	C₂	1	2	yes	maximal
4	C₃₁×#1740	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	2	yes	—
10	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
11	C₃₁×#424	—	2	4	no	—
12	C₃₁×#424	—	2	4	no	—
13	C₃₁×#424	—	2	4	no	—
14	C₃₁×#423	—	2	2	no	—
15	C₂×C₃₁×C₄⋊C₈	—	2	4	no	—
16	C₃₁×#344	—	2	4	no	—
17	C₃₁×#342	C₄	1	4	yes	—
18	C₃₁×#336	—	2	2	no	—
19	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₃₁	—	2	2	no	—
21	C₂×M₁₆×C₃₁	—	2	2	no	—
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	2	no	—
26	C₂×M₁₆×C₃₁	—	2	1	no	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	Q₈	1	2	yes	—
30	C₂²×C₈×C₃₁	—	2	2	no	—
31	C₂²×C₈×C₃₁	—	2	2	no	—
32	C₂²×C₈×C₃₁	—	2	1	no	—
33	C₂²×C₈×C₃₁	—	2	1	no	—
34	C₂²×C₈×C₃₁	—	2	1	no	—
35	C₂²×C₈×C₃₁	D₄	1	4	yes	—
36	C₂²×C₈×C₃₁	D₄	1	2	yes	—
37	C₃₁×#117	—	2	2	no	—
38	C₃₁×#117	—	2	2	no	—
39	C₃₁×#117	—	2	2	no	—
40	C₃₁×#117	—	2	2	no	—
41	C₃₁×#117	D₄	1	4	yes	—
42	C₃₁×#117	Q₈	1	4	yes	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	Q₈	1	2	yes	—
45	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₃₁	—	2	2	no	—
48	C₂×C₄²×C₃₁	D₄	1	2	yes	—
49	C₃₁×C₄⋊C₈	—	2	16	no	—
50	C₃₁×C₄⋊C₈	—	2	8	no	—
51	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	—	2	4	no	—
53	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	#41	1	2	yes	—
63	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
64	M₁₆×C₃₁	—	2	4	no	—
65	M₁₆×C₃₁	—	4	2	no	—
66	M₁₆×C₃₁	—	4	2	no	—
67	M₁₆×C₃₁	—	4	2	no	—
68	M₁₆×C₃₁	—	4	2	no	—
69	M₁₆×C₃₁	—	4	2	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	—	4	1	no	—
77	C₂×C₈×C₃₁	—	4	2	no	—
78	C₂×C₈×C₃₁	—	4	1	no	—
79	C₂×C₈×C₃₁	—	4	1	no	—
80	C₂×C₈×C₃₁	—	4	1	no	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	2	4	no	—
83	C₄⋊C₄×C₃₁	—	2	4	no	—
84	C₂²⋊C₄×C₃₁	—	2	4	no	—
85	C₂²⋊C₄×C₃₁	—	2	4	no	—
86	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
87	C₄²×C₃₁	—	2	4	no	—
88	C₄²×C₃₁	—	2	2	no	—
89	C₄²×C₃₁	—	2	4	no	—
90	C₄²×C₃₁	—	2	2	no	—
91	#11345	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	4	2	no	—
93	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₃₁	C₄×D₄	1	2	yes	—
96	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
97	C₂³×C₃₁	C₄×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₃₁	—	2	4	no	—
99	C₂×C₄×C₃₁	—	2	8	no	—
100	C₂×C₄×C₃₁	—	4	8	no	—
101	C₂×C₄×C₃₁	—	4	4	no	—
102	C₂×C₄×C₃₁	#120	1	2	yes	—
103	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
104	C₂×C₄×C₃₁	#121	1	2	yes	—
105	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
112	C₂×C₄×C₃₁	#123	1	2	yes	—
113	C₂×C₄×C₃₁	#125	1	1	yes	characteristic
114	C₂×C₄×C₃₁	#125	1	2	yes	—
115	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
118	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
120	C₈×C₃₁	—	4	2	no	—
121	C₈×C₃₁	—	4	4	no	—
122	C₈×C₃₁	—	4	2	no	—
123	#2905	C₂×C₃₁	1	1	yes	characteristic
124	C₂×#424	C₂×C₃₁	1	2	yes	—
125	C₂×#344	C₂×C₃₁	1	2	yes	—
126	#1740	C₂×C₃₁	1	1	yes	characteristic
127	#1729	C₂×C₃₁	1	1	yes	characteristic
128	C₂²×C₃₁	—	2	2	no	—
129	C₂²×C₃₁	—	2	4	no	—
130	C₂²×C₃₁	—	4	2	no	—
131	C₂²×C₃₁	—	4	1	no	—
132	C₂²×C₃₁	—	4	1	no	—
133	C₂²×C₃₁	#384	1	1	yes	characteristic
134	C₂²×C₃₁	#385	1	2	yes	—
135	C₂²×C₃₁	#385	1	1	yes	characteristic
136	C₂²×C₃₁	#389	1	2	yes	—
137	C₂²×C₃₁	#389	1	1	yes	characteristic
138	C₄×C₃₁	—	4	8	no	—
139	C₄×C₃₁	—	4	4	no	—
140	C₄×C₃₁	#392	1	1	yes	characteristic
141	C₄×C₃₁	#395	1	2	yes	—
142	C₄×C₃₁	#395	1	1	yes	characteristic
143	C₄×C₃₁	#398	1	2	yes	—
144	C₄×C₃₁	#398	1	1	yes	characteristic
145	C₄×C₃₁	#399	1	1	yes	characteristic
146	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
147	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
148	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×#117	C₂²×C₃₁	1	2	yes	—
150	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
151	#424	—	2	4	no	—
152	#424	—	2	4	no	—
153	#424	—	2	4	no	—
154	#423	—	2	2	no	—
155	C₂×C₄⋊C₈	—	2	4	no	—
156	#344	—	2	4	no	—
157	#342	C₄×C₃₁	1	4	yes	—
158	#336	—	2	2	no	—
159	C₂×C₃₁	—	2	2	no	—
160	C₂×C₃₁	—	2	1	no	—
161	C₂×C₃₁	—	2	1	no	—
162	C₂×C₃₁	—	2	2	no	—
163	C₂×C₃₁	#1444	1	1	yes	characteristic
164	C₂×C₃₁	#1868	1	2	yes	—
165	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	Q₈×C₃₁	1	2	yes	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	D₄×C₃₁	1	2	yes	—
182	C₂²×C₈	D₄×C₃₁	1	4	yes	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	D₄×C₃₁	1	4	yes	—
188	#117	Q₈×C₃₁	1	4	yes	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	Q₈×C₃₁	1	2	yes	—
191	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
192	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	D₄×C₃₁	1	2	yes	—
195	C₄⋊C₈	—	2	16	no	—
196	C₄⋊C₈	—	2	8	no	—
197	C₃₁	#11345	1	1	yes	characteristic, direct product factor, Sylow
198	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
201	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
205	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
207	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
208	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
209	C₂²×C₄	#41×C₃₁	1	2	yes	—
210	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	2	no	—
238	C₂³	—	4	2	no	—
239	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
240	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
241	C₂³	C₄×D₄×C₃₁	1	2	yes	—
242	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
243	C₂³	C₄×Q₈×C₃₁	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	C₃₁×#120	1	2	yes	—
249	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
250	C₂×C₄	C₃₁×#121	1	2	yes	—
251	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
252	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
253	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
254	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
255	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
256	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
257	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
258	C₂×C₄	C₃₁×#123	1	2	yes	—
259	C₂×C₄	C₃₁×#125	1	1	yes	characteristic
260	C₂×C₄	C₃₁×#125	1	2	yes	—
261	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
262	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
263	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
264	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
265	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
266	C₈	—	4	2	no	—
267	C₈	—	4	2	no	—
268	C₈	—	4	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	C₃₁×#384	1	1	yes	characteristic
275	C₂²	C₃₁×#385	1	2	yes	—
276	C₂²	C₃₁×#385	1	1	yes	characteristic
277	C₂²	C₃₁×#389	1	2	yes	—
278	C₂²	C₃₁×#389	1	1	yes	characteristic
279	C₄	—	4	8	no	—
280	C₄	—	4	4	no	—
281	C₄	C₃₁×#392	1	1	yes	characteristic
282	C₄	C₃₁×#395	1	2	yes	—
283	C₄	C₃₁×#395	1	1	yes	characteristic
284	C₄	C₃₁×#398	1	2	yes	—
285	C₄	C₃₁×#398	1	1	yes	characteristic
286	C₄	C₃₁×#399	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	C₃₁×#1444	1	1	yes	characteristic
292	C₂	C₃₁×#1868	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#120
commutator	C₂⁴	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	#385
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
91. #11345	C₃₁	direct (N × H)
197. C₃₁	#11345	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×#2905	2
C₂×C₃₁×#424	2
C₂×C₃₁×#344	2
C₃₁×#1740	2
C₃₁×#1729	2
#11345	31

Subgroups — C31×#11345 #21468104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11345 #21468104