Subgroups — C₃₁×#29326 #21486085

Order 7936.

606 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2647	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2330	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2083	C₂	1	2	yes	maximal
4	C₃₁×#2059	C₂	1	2	yes	maximal
5	C₃₁×#1949	C₂	1	2	yes	maximal
6	C₃₁×#1947	C₂	1	2	yes	maximal
7	C₃₁×#1944	C₂	1	2	yes	maximal
8	C₃₁×#1942	C₂	1	2	yes	maximal
9	C₃₁×#1768	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#107	C₂²	1	2	yes	—
12	C₂×C₃₁×#107	C₂²	1	2	yes	—
13	C₂×C₃₁×#107	C₂²	1	2	yes	—
14	C₂×C₃₁×#106	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#400	—	2	2	no	—
19	C₃₁×#398	—	2	2	no	—
20	C₃₁×#389	—	2	2	no	—
21	C₃₁×#389	C₂²	1	2	yes	—
22	C₃₁×#389	C₂²	1	2	yes	—
23	C₃₁×#389	C₂²	1	2	yes	—
24	C₃₁×#387	—	2	2	no	—
25	C₃₁×#384	C₂²	1	1	yes	characteristic
26	C₃₁×#384	C₂²	1	1	yes	characteristic
27	C₃₁×#382	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#376	C₂²	1	1	yes	characteristic
30	C₃₁×#340	C₂²	1	2	yes	—
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₃₁×#125	—	2	4	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄²×C₃₁	—	2	2	no	—
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#107	—	2	4	no	—
53	C₃₁×#107	—	4	2	no	—
54	C₃₁×#107	—	4	2	no	—
55	C₃₁×#106	—	2	4	no	—
56	C₃₁×C₄⋊C₈	Q₈	1	8	yes	—
57	C₃₁×#97	—	4	1	no	—
58	C₄×C₈×C₃₁	—	4	1	no	—
59	C₃₁×#95	—	2	2	no	—
60	C₃₁×#95	—	2	2	no	—
61	C₃₁×#95	—	2	2	no	—
62	C₃₁×#95	—	2	2	no	—
63	C₃₁×#95	—	2	2	no	—
64	C₃₁×#95	D₄	1	2	yes	—
65	C₃₁×#95	D₄	1	2	yes	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	—	2	2	no	—
68	C₂²×C₄×C₃₁	—	2	2	no	—
69	C₂²×C₄×C₃₁	—	2	2	no	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
80	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
81	C₄⋊C₄×C₃₁	—	2	8	no	—
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	—	2	2	no	—
84	C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₄⋊C₄×C₃₁	—	2	2	no	—
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₄⋊C₄×C₃₁	—	8	2	no	—
90	C₄⋊C₄×C₃₁	—	8	2	no	—
91	C₄⋊C₄×C₃₁	—	8	2	no	—
92	C₄⋊C₄×C₃₁	—	8	2	no	—
93	C₄⋊C₄×C₃₁	—	8	2	no	—
94	C₄⋊C₄×C₃₁	#41	1	4	yes	—
95	C₄²×C₃₁	—	2	2	no	—
96	C₄²×C₃₁	—	2	1	no	—
97	C₄²×C₃₁	—	2	1	no	—
98	C₄²×C₃₁	—	8	2	no	—
99	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
100	#29326	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	1	no	—
107	C₂×C₄×C₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	4	2	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	4	2	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
124	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
125	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
126	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
128	C₈×C₃₁	—	2	4	no	—
129	C₈×C₃₁	—	2	4	no	—
130	#2647	C₂×C₃₁	1	1	yes	characteristic
131	#2330	C₂×C₃₁	1	1	yes	characteristic
132	#2083	C₂×C₃₁	1	2	yes	—
133	#2059	C₂×C₃₁	1	2	yes	—
134	#1949	C₂×C₃₁	1	2	yes	—
135	#1947	C₂×C₃₁	1	2	yes	—
136	#1944	C₂×C₃₁	1	2	yes	—
137	#1942	C₂×C₃₁	1	2	yes	—
138	#1768	C₂×C₃₁	1	1	yes	characteristic
139	C₂²×C₃₁	#525	1	1	yes	characteristic
140	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
141	C₂²×C₃₁	#537	1	1	yes	characteristic
142	C₂²×C₃₁	#576	1	2	yes	—
143	C₂²×C₃₁	#578	1	2	yes	—
144	C₄×C₃₁	—	2	4	no	—
145	C₄×C₃₁	—	4	2	no	—
146	C₄×C₃₁	—	4	4	no	—
147	C₄×C₃₁	—	4	1	no	—
148	C₄×C₃₁	—	4	1	no	—
149	C₄×C₃₁	—	8	2	no	—
150	C₄×C₃₁	—	8	2	no	—
151	C₄×C₃₁	—	8	2	no	—
152	C₄×C₃₁	#554	1	2	yes	—
153	C₄×C₃₁	#557	1	2	yes	—
154	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
155	C₂×#107	C₂²×C₃₁	1	2	yes	—
156	C₂×#107	C₂²×C₃₁	1	2	yes	—
157	C₂×#107	C₂²×C₃₁	1	2	yes	—
158	C₂×#106	C₂²×C₃₁	1	2	yes	—
159	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
160	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
162	#400	—	2	2	no	—
163	#398	—	2	2	no	—
164	#389	—	2	2	no	—
165	#389	C₂²×C₃₁	1	2	yes	—
166	#389	C₂²×C₃₁	1	2	yes	—
167	#389	C₂²×C₃₁	1	2	yes	—
168	#387	—	2	2	no	—
169	#384	C₂²×C₃₁	1	1	yes	characteristic
170	#384	C₂²×C₃₁	1	1	yes	characteristic
171	#382	C₂²×C₃₁	1	1	yes	characteristic
172	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
173	#376	C₂²×C₃₁	1	1	yes	characteristic
174	#340	C₂²×C₃₁	1	2	yes	—
175	#340	C₂²×C₃₁	1	2	yes	—
176	#340	C₂²×C₃₁	1	2	yes	—
177	#340	C₂²×C₃₁	1	2	yes	—
178	#336	C₂²×C₃₁	1	2	yes	—
179	C₂×C₃₁	#2584	1	1	yes	characteristic
180	C₂×C₃₁	#3124	1	2	yes	—
181	C₂×C₃₁	#3164	1	2	yes	—
182	C₂×C₃₁	#3259	1	1	yes	characteristic
183	C₂×C₃₁	#3261	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
185	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
186	#125	—	2	4	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
194	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
196	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
201	#107	—	2	4	no	—
202	#107	—	4	2	no	—
203	#107	—	4	2	no	—
204	#106	—	2	4	no	—
205	C₄⋊C₈	Q₈×C₃₁	1	8	yes	—
206	#97	—	4	1	no	—
207	C₄×C₈	—	4	1	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	#95	—	2	2	no	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	D₄×C₃₁	1	2	yes	—
214	#95	D₄×C₃₁	1	2	yes	—
215	C₃₁	#29326	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
224	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
230	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	8	2	no	—
240	C₄⋊C₄	—	8	2	no	—
241	C₄⋊C₄	—	8	2	no	—
242	C₄⋊C₄	—	8	2	no	—
243	C₄⋊C₄	—	8	2	no	—
244	C₄⋊C₄	#41×C₃₁	1	4	yes	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	8	2	no	—
249	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
250	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
273	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
274	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
275	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
276	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
277	C₈	—	2	4	no	—
278	C₈	—	2	4	no	—
279	C₂²	C₃₁×#525	1	1	yes	characteristic
280	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
281	C₂²	C₃₁×#537	1	1	yes	characteristic
282	C₂²	C₃₁×#576	1	2	yes	—
283	C₂²	C₃₁×#578	1	2	yes	—
284	C₄	—	2	4	no	—
285	C₄	—	4	2	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	8	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	C₃₁×#554	1	2	yes	—
293	C₄	C₃₁×#557	1	2	yes	—
294	C₂	C₃₁×#2584	1	1	yes	characteristic
295	C₂	C₃₁×#3124	1	2	yes	—
296	C₂	C₃₁×#3164	1	2	yes	—
297	C₂	C₃₁×#3259	1	1	yes	characteristic
298	C₂	C₃₁×#3261	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
100. #29326	C₃₁	direct (N × H)
215. C₃₁	#29326	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×#2647	2
C₃₁×#2330	2
C₃₁×#2083	2
C₃₁×#2059	2
C₃₁×#1949	2
C₃₁×#1947	2
C₃₁×#1944	2
C₃₁×#1942	2
C₃₁×#1768	2
#29326	31

Subgroups — C31×#29326 #21486085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29326 #21486085