Subgroups — C₃₁×#29318 #21486077

Order 7936.

622 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2864	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#125	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2085	C₂	1	2	yes	maximal
4	C₃₁×#2059	C₂	1	2	yes	maximal
5	C₃₁×#1949	C₂	1	2	yes	maximal
6	C₃₁×#1949	C₂	1	2	yes	maximal
7	C₃₁×#1944	C₂	1	2	yes	maximal
8	C₃₁×#1943	C₂	1	2	yes	maximal
9	C₃₁×#1767	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×#106	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×C₃₁×#97	C₂²	1	1	yes	characteristic
18	C₃₁×#389	C₂²	1	2	yes	—
19	C₃₁×#389	C₂²	1	2	yes	—
20	C₃₁×#389	C₂²	1	2	yes	—
21	C₃₁×#387	—	2	2	no	—
22	C₃₁×#387	—	2	2	no	—
23	C₃₁×#387	—	2	2	no	—
24	C₃₁×#382	C₂²	1	1	yes	characteristic
25	C₃₁×#382	C₂²	1	1	yes	characteristic
26	C₃₁×#382	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₄³×C₃₁	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	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
46	C₂×C₄²×C₃₁	—	2	2	no	—
47	C₂×C₄²×C₃₁	D₄	1	2	yes	—
48	C₂×C₄²×C₃₁	D₄	1	2	yes	—
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×#106	—	2	4	no	—
55	C₃₁×#106	—	4	2	no	—
56	C₃₁×C₄⋊C₈	Q₈	1	8	yes	—
57	C₃₁×#97	—	4	1	no	—
58	C₃₁×#97	—	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	1	no	—
63	C₃₁×#95	—	2	1	no	—
64	C₃₁×#95	—	2	1	no	—
65	C₃₁×#95	—	2	1	no	—
66	C₃₁×#95	—	2	2	no	—
67	C₃₁×#95	—	2	2	no	—
68	C₂²×C₄×C₃₁	—	2	2	no	—
69	C₂²×C₄×C₃₁	—	2	2	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	2	2	no	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
82	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
83	C₄⋊C₄×C₃₁	—	2	8	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	—	2	2	no	—
90	C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₄⋊C₄×C₃₁	—	4	4	no	—
92	C₄⋊C₄×C₃₁	—	4	4	no	—
93	C₄⋊C₄×C₃₁	—	4	4	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	—	4	2	no	—
99	C₄²×C₃₁	—	4	2	no	—
100	C₄²×C₃₁	—	4	4	no	—
101	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
102	#29318	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	4	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	—	2	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	—	4	2	no	—
124	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
127	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
128	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
130	C₈×C₃₁	—	2	4	no	—
131	C₈×C₃₁	—	2	4	no	—
132	#2864	C₂×C₃₁	1	1	yes	characteristic
133	C₄×#125	C₂×C₃₁	1	1	yes	characteristic
134	#2085	C₂×C₃₁	1	2	yes	—
135	#2059	C₂×C₃₁	1	2	yes	—
136	#1949	C₂×C₃₁	1	2	yes	—
137	#1949	C₂×C₃₁	1	2	yes	—
138	#1944	C₂×C₃₁	1	2	yes	—
139	#1943	C₂×C₃₁	1	2	yes	—
140	#1767	C₂×C₃₁	1	1	yes	characteristic
141	C₂²×C₃₁	#525	1	1	yes	characteristic
142	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
143	C₂²×C₃₁	#537	1	1	yes	characteristic
144	C₂²×C₃₁	#575	1	2	yes	—
145	C₂²×C₃₁	#575	1	2	yes	—
146	C₄×C₃₁	—	2	2	no	—
147	C₄×C₃₁	—	2	2	no	—
148	C₄×C₃₁	—	4	2	no	—
149	C₄×C₃₁	—	4	4	no	—
150	C₄×C₃₁	—	4	2	no	—
151	C₄×C₃₁	—	8	2	no	—
152	C₄×C₃₁	—	8	2	no	—
153	C₄×C₃₁	—	8	2	no	—
154	C₄×C₃₁	#554	1	2	yes	—
155	C₄×C₃₁	#557	1	2	yes	—
156	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×#107	C₂²×C₃₁	1	2	yes	—
158	C₂×#107	C₂²×C₃₁	1	2	yes	—
159	C₂×#106	C₂²×C₃₁	1	2	yes	—
160	C₂×#106	C₂²×C₃₁	1	2	yes	—
161	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
162	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
164	#389	C₂²×C₃₁	1	2	yes	—
165	#389	C₂²×C₃₁	1	2	yes	—
166	#389	C₂²×C₃₁	1	2	yes	—
167	#387	—	2	2	no	—
168	#387	—	2	2	no	—
169	#387	—	2	2	no	—
170	#382	C₂²×C₃₁	1	1	yes	characteristic
171	#382	C₂²×C₃₁	1	1	yes	characteristic
172	#382	C₂²×C₃₁	1	1	yes	characteristic
173	C₄×C₄⋊C₄	—	2	2	no	—
174	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
175	C₄³	C₂²×C₃₁	1	1	yes	characteristic
176	#340	C₂²×C₃₁	1	2	yes	—
177	#340	C₂²×C₃₁	1	2	yes	—
178	#340	C₂²×C₃₁	1	2	yes	—
179	#340	C₂²×C₃₁	1	2	yes	—
180	#336	C₂²×C₃₁	1	2	yes	—
181	C₂×C₃₁	#2584	1	1	yes	characteristic
182	C₂×C₃₁	#3111	1	2	yes	—
183	C₂×C₃₁	#3162	1	1	yes	characteristic
184	C₂×C₃₁	#3166	1	1	yes	characteristic
185	C₂×C₃₁	#3256	1	2	yes	—
186	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
187	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
188	#125	—	2	4	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	D₄×C₃₁	1	2	yes	—
199	C₂×C₄²	D₄×C₃₁	1	2	yes	—
200	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
203	#107	—	2	4	no	—
204	#107	—	4	2	no	—
205	#106	—	2	4	no	—
206	#106	—	4	2	no	—
207	C₄⋊C₈	Q₈×C₃₁	1	8	yes	—
208	#97	—	4	1	no	—
209	#97	—	4	1	no	—
210	#95	—	2	2	no	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	—	2	1	no	—
214	#95	—	2	1	no	—
215	#95	—	2	1	no	—
216	#95	—	2	1	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	C₃₁	#29318	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
228	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
234	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
235	C₄⋊C₄	—	2	8	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	—	4	4	no	—
246	C₄⋊C₄	#41×C₃₁	1	4	yes	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	4	2	no	—
251	C₄²	—	4	2	no	—
252	C₄²	—	4	4	no	—
253	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
254	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	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	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
276	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
277	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
278	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
279	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
280	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₂²	C₃₁×#525	1	1	yes	characteristic
284	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
285	C₂²	C₃₁×#537	1	1	yes	characteristic
286	C₂²	C₃₁×#575	1	2	yes	—
287	C₂²	C₃₁×#575	1	2	yes	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	4	2	no	—
291	C₄	—	4	4	no	—
292	C₄	—	4	2	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	C₃₁×#554	1	2	yes	—
297	C₄	C₃₁×#557	1	2	yes	—
298	C₂	C₃₁×#2584	1	1	yes	characteristic
299	C₂	C₃₁×#3111	1	2	yes	—
300	C₂	C₃₁×#3162	1	1	yes	characteristic
301	C₂	C₃₁×#3166	1	1	yes	characteristic
302	C₂	C₃₁×#3256	1	2	yes	—

	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 ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
102. #29318	C₃₁	direct (N × H)
219. C₃₁	#29318	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×#2864	2
C₄×C₃₁×#125	2
C₃₁×#2085	2
C₃₁×#2059	2
C₃₁×#1949	2
C₃₁×#1944	2
C₃₁×#1943	2
C₃₁×#1767	2
#29318	31

Subgroups — C31×#29318 #21486077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29318 #21486077