Subgroups — C₃₁×#33195 #21489954

Order 7936.

1018 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#474	C₂	1	2	yes	maximal
2	C₃₁×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1988	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1947	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1802	C₂	1	4	yes	maximal
8	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
13	C₂²×C₄²×C₃₁	—	2	1	no	—
14	C₃₁×#474	—	2	8	no	—
15	C₂×C₃₁×#107	C₂²	1	4	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×#102	C₂²	1	4	yes	—
19	C₃₁×#384	—	2	2	no	—
20	C₃₁×#384	C₂²	1	4	yes	—
21	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
24	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
25	C₄³×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#340	C₂²	1	8	yes	—
27	C₂²×Q₈×C₃₁	—	2	2	no	—
28	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₃₁	—	2	1	no	—
30	C₂³×C₄×C₃₁	—	2	2	no	—
31	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
32	C₃₁×C₄⋊Q₈	—	2	4	no	—
33	C₃₁×C₄⋊Q₈	—	2	4	no	—
34	C₃₁×C₄⋊Q₈	—	4	4	no	—
35	C₃₁×C₄⋊Q₈	—	4	4	no	—
36	C₄×D₄×C₃₁	—	2	2	no	—
37	C₄×D₄×C₃₁	—	2	4	no	—
38	C₄×D₄×C₃₁	D₄	1	4	yes	—
39	C₄×D₄×C₃₁	Q₈	1	8	yes	complemented
40	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
46	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
47	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄²×C₃₁	—	2	1	no	—
49	C₂×C₄²×C₃₁	—	2	2	no	—
50	C₂×C₄²×C₃₁	—	4	4	no	—
51	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₃₁	D₄	1	2	yes	—
53	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
56	C₃₁×#107	—	2	8	no	—
57	C₃₁×C₄⋊C₈	—	2	4	no	—
58	C₃₁×C₄⋊C₈	—	2	8	no	—
59	C₃₁×#102	—	2	8	no	—
60	C₃₁×#95	—	2	1	no	—
61	C₃₁×#95	—	2	2	no	—
62	C₂⁴×C₃₁	—	2	1	no	—
63	C₂×Q₈×C₃₁	—	4	4	no	—
64	C₂×Q₈×C₃₁	—	4	4	no	—
65	C₂×Q₈×C₃₁	—	4	4	no	—
66	C₂×D₄×C₃₁	—	2	2	no	—
67	C₂×D₄×C₃₁	—	2	1	no	—
68	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
69	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₃₁	—	2	1	no	—
71	C₂²×C₄×C₃₁	—	2	2	no	—
72	C₂²×C₄×C₃₁	—	2	4	no	—
73	C₂²×C₄×C₃₁	—	2	8	no	—
74	C₂²×C₄×C₃₁	—	2	4	no	—
75	C₂²×C₄×C₃₁	—	4	2	no	—
76	C₂²×C₄×C₃₁	—	4	4	no	—
77	C₂²×C₄×C₃₁	—	4	2	no	—
78	C₂²×C₄×C₃₁	—	4	4	no	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₃₁	—	2	4	no	—
85	C₂×C₈×C₃₁	—	2	8	no	—
86	C₄⋊C₄×C₃₁	—	2	1	no	—
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	2	4	no	—
89	C₄⋊C₄×C₃₁	—	2	4	no	—
90	C₄⋊C₄×C₃₁	—	4	8	no	—
91	C₄⋊C₄×C₃₁	—	4	4	no	—
92	C₄⋊C₄×C₃₁	—	4	8	no	—
93	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₃₁	C₂×Q₈	1	4	yes	—
95	C₂²⋊C₄×C₃₁	—	2	4	no	—
96	C₂²⋊C₄×C₃₁	—	2	8	no	—
97	C₄²×C₃₁	—	2	2	no	—
98	C₄²×C₃₁	—	2	4	no	—
99	C₄²×C₃₁	—	4	2	no	—
100	C₄²×C₃₁	—	4	4	no	—
101	C₄²×C₃₁	D₈	1	4	yes	—
102	C₄²×C₃₁	C₂×D₄	1	2	yes	—
103	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
104	C₄²×C₃₁	#41	1	2	yes	—
105	C₄²×C₃₁	#41	1	4	yes	—
106	#33195	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	—	2	2	no	—
108	C₂³×C₃₁	—	2	4	no	—
109	C₂³×C₃₁	—	4	1	no	—
110	C₂³×C₃₁	—	4	1	no	—
111	C₂³×C₃₁	—	4	2	no	—
112	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₃₁	—	8	8	no	—
114	D₄×C₃₁	—	2	2	no	—
115	D₄×C₃₁	—	2	4	no	—
116	D₄×C₃₁	C₄⋊Q₈	1	4	yes	complemented
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	4	no	—
122	C₂×C₄×C₃₁	—	4	1	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	1	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	4	8	no	—
127	C₂×C₄×C₃₁	—	4	16	no	—
128	C₂×C₄×C₃₁	—	4	4	no	—
129	C₂×C₄×C₃₁	—	4	4	no	—
130	C₂×C₄×C₃₁	—	4	4	no	—
131	C₂×C₄×C₃₁	#121	1	2	yes	—
132	C₂×C₄×C₃₁	#121	1	2	yes	—
133	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
134	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
135	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
137	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₃₁	#137	1	4	yes	—
139	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
141	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
142	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
143	C₈×C₃₁	—	4	4	no	—
144	C₂×#474	C₂×C₃₁	1	2	yes	—
145	#2868	C₂×C₃₁	1	1	yes	characteristic, complemented
146	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic
147	#1988	C₂×C₃₁	1	2	yes	complemented
148	#1947	C₂×C₃₁	1	4	yes	complemented
149	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#1802	C₂×C₃₁	1	4	yes	—
151	C₂²×C₃₁	—	2	4	no	—
152	C₂²×C₃₁	—	4	4	no	—
153	C₂²×C₃₁	—	4	2	no	—
154	C₂²×C₃₁	—	4	2	no	—
155	C₂²×C₃₁	—	4	4	no	—
156	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
157	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
158	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
159	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₂²×C₃₁	C₂×#137	1	2	yes	—
161	C₄×C₃₁	—	2	4	no	—
162	C₄×C₃₁	—	2	2	no	—
163	C₄×C₃₁	—	4	2	no	—
164	C₄×C₃₁	—	4	4	no	—
165	C₄×C₃₁	—	8	4	no	—
166	C₄×C₃₁	#459	1	4	yes	—
167	C₄×C₃₁	#474	1	8	yes	complemented
168	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
169	C₄×C₃₁	#532	1	1	yes	characteristic
170	C₄×C₃₁	#533	1	2	yes	—
171	C₂×C₄⋊Q₈	—	2	1	no	—
172	C₂×C₄⋊Q₈	—	2	1	no	—
173	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
174	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
176	C₂²×C₄²	—	2	1	no	—
177	#474	—	2	8	no	—
178	C₂×#107	C₂²×C₃₁	1	4	yes	—
179	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
180	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
181	C₂×#102	C₂²×C₃₁	1	4	yes	—
182	#384	—	2	2	no	—
183	#384	C₂²×C₃₁	1	4	yes	—
184	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
185	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
186	C₄×C₂²⋊C₄	—	2	1	no	—
187	C₄×C₂²⋊C₄	—	2	2	no	—
188	C₄³	C₂²×C₃₁	1	1	yes	characteristic
189	#340	C₂²×C₃₁	1	8	yes	—
190	C₂×C₃₁	—	4	4	no	—
191	C₂×C₃₁	#2857	1	1	yes	characteristic
192	C₂×C₃₁	C₂×#459	1	1	yes	characteristic
193	C₂×C₃₁	#3039	1	1	yes	characteristic
194	C₂×C₃₁	C₂×#474	1	2	yes	—
195	C₂×C₃₁	#3079	1	2	yes	—
196	C₂²×Q₈	—	2	2	no	—
197	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
198	C₂³×C₄	—	2	1	no	—
199	C₂³×C₄	—	2	2	no	—
200	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
201	C₄⋊Q₈	—	2	4	no	—
202	C₄⋊Q₈	—	2	4	no	—
203	C₄⋊Q₈	—	4	4	no	—
204	C₄⋊Q₈	—	4	4	no	—
205	C₄×D₄	—	2	2	no	—
206	C₄×D₄	—	2	4	no	—
207	C₄×D₄	D₄×C₃₁	1	4	yes	—
208	C₄×D₄	Q₈×C₃₁	1	8	yes	complemented
209	C₂×C₄⋊C₄	—	2	4	no	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
213	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
215	C₂×C₂²⋊C₄	—	2	1	no	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₄²	—	2	1	no	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	—	4	4	no	—
220	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
221	C₂×C₄²	D₄×C₃₁	1	2	yes	—
222	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
223	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
224	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
225	#107	—	2	8	no	—
226	C₄⋊C₈	—	2	4	no	—
227	C₄⋊C₈	—	2	8	no	—
228	#102	—	2	8	no	—
229	#95	—	2	1	no	—
230	#95	—	2	2	no	—
231	C₃₁	#33195	1	1	yes	characteristic, direct product factor, Sylow
232	C₂⁴	—	2	1	no	—
233	C₂×Q₈	—	4	4	no	—
234	C₂×Q₈	—	4	4	no	—
235	C₂×Q₈	—	4	4	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
239	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	8	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	4	4	no	—
249	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
253	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	8	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	4	8	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	4	8	no	—
263	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
264	C₄⋊C₄	C₂×Q₈×C₃₁	1	4	yes	—
265	C₂²⋊C₄	—	2	4	no	—
266	C₂²⋊C₄	—	2	8	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	4	no	—
269	C₄²	—	4	2	no	—
270	C₄²	—	4	4	no	—
271	C₄²	D₈×C₃₁	1	4	yes	—
272	C₄²	C₂×D₄×C₃₁	1	2	yes	—
273	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
274	C₄²	#41×C₃₁	1	2	yes	—
275	C₄²	#41×C₃₁	1	4	yes	—
276	C₂³	—	2	2	no	—
277	C₂³	—	2	4	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	1	no	—
280	C₂³	—	4	2	no	—
281	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
282	Q₈	—	8	8	no	—
283	D₄	—	2	2	no	—
284	D₄	—	2	4	no	—
285	D₄	C₃₁×C₄⋊Q₈	1	4	yes	complemented
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	8	no	—
296	C₂×C₄	—	4	16	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	C₃₁×#121	1	2	yes	—
301	C₂×C₄	C₃₁×#121	1	2	yes	—
302	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
303	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
304	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
305	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
306	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
307	C₂×C₄	C₃₁×#137	1	4	yes	—
308	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
309	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
310	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
311	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
312	C₈	—	4	4	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	4	no	—
318	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
319	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
320	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
321	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
322	C₂²	C₂×C₃₁×#137	1	2	yes	—
323	C₄	—	2	4	no	—
324	C₄	—	2	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	4	no	—
327	C₄	—	8	4	no	—
328	C₄	C₃₁×#459	1	4	yes	—
329	C₄	C₃₁×#474	1	8	yes	complemented
330	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
331	C₄	C₃₁×#532	1	1	yes	characteristic
332	C₄	C₃₁×#533	1	2	yes	—
333	C₂	—	4	4	no	—
334	C₂	C₃₁×#2857	1	1	yes	characteristic
335	C₂	C₂×C₃₁×#459	1	1	yes	characteristic
336	C₂	C₃₁×#3039	1	1	yes	characteristic
337	C₂	C₂×C₃₁×#474	1	2	yes	—
338	C₂	C₃₁×#3079	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
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
2. C₃₁×#2868	C₂	semidirect
4. C₃₁×#1988	C₂	semidirect
5. C₃₁×#1947	C₂	semidirect
6. C₃₁×#1842	C₂	semidirect
39. C₄×D₄×C₃₁	Q₈	semidirect
106. #33195	C₃₁	direct (N × H)
116. D₄×C₃₁	C₄⋊Q₈	semidirect
145. #2868	C₂×C₃₁	semidirect
147. #1988	C₂×C₃₁	semidirect
148. #1947	C₂×C₃₁	semidirect
149. #1842	C₂×C₃₁	semidirect
167. C₄×C₃₁	#474	semidirect
208. C₄×D₄	Q₈×C₃₁	semidirect
231. C₃₁	#33195	direct (N × H)
285. D₄	C₃₁×C₄⋊Q₈	semidirect
329. C₄	C₃₁×#474	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#474	2
C₃₁×#2868	2
C₄²×D₄×C₃₁	2
C₃₁×#1988	2
C₃₁×#1947	2
C₃₁×#1842	2
C₃₁×#1802	2
#33195	31

Subgroups — C31×#33195 #21489954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33195 #21489954