Subgroups — C₃₁×#10839 #21467598

Order 7936.

1110 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#339	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
6	C₂×C₃₁×#117	C₂²	1	2	yes	—
7	C₂²×C₄²×C₃₁	—	2	2	no	—
8	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
9	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#423	—	2	2	no	—
11	C₃₁×#423	C₄	1	4	yes	complemented
12	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
13	C₃₁×#384	C₄	1	2	yes	complemented
14	C₃₁×#382	C₄	1	2	yes	complemented
15	C₃₁×#380	—	2	2	no	—
16	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
17	C₄³×C₃₁	—	2	2	no	—
18	C₄³×C₃₁	C₄	1	4	yes	complemented
19	C₃₁×#339	—	2	8	no	—
20	C₂³×C₄×C₃₁	—	2	2	no	—
21	C₂³×C₄×C₃₁	D₄	1	2	yes	—
22	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	—	2	4	no	—
25	C₂×M₁₆×C₃₁	—	2	4	no	—
26	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₃₁	—	2	2	no	—
28	C₃₁×#117	—	2	4	no	—
29	C₃₁×#117	—	2	4	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
33	C₂×C₄²×C₃₁	—	2	16	no	—
34	C₂×C₄²×C₃₁	—	2	8	no	—
35	C₂×C₄²×C₃₁	—	2	2	no	—
36	C₂×C₄²×C₃₁	—	2	2	no	—
37	C₂×C₄²×C₃₁	—	2	2	no	—
38	C₂×C₄²×C₃₁	—	2	2	no	—
39	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₃₁	D₄	1	4	yes	—
45	C₂×C₄²×C₃₁	D₄	1	2	yes	—
46	C₂×C₄²×C₃₁	D₄	1	2	yes	—
47	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
48	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
49	C₃₁×C₄⋊C₈	—	2	8	no	—
50	C₃₁×#95	—	2	4	no	—
51	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	—	2	16	no	—
53	C₂²×C₄×C₃₁	—	2	4	no	—
54	C₂²×C₄×C₃₁	—	2	4	no	—
55	C₂²×C₄×C₃₁	—	2	4	no	—
56	C₂²×C₄×C₃₁	—	2	8	no	—
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₃₁	#41	1	2	yes	—
71	C₂²×C₄×C₃₁	#41	1	2	yes	—
72	M₁₆×C₃₁	—	2	8	no	—
73	M₁₆×C₃₁	—	4	4	no	—
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	—	4	2	no	—
76	C₂×C₈×C₃₁	—	4	2	no	—
77	C₄⋊C₄×C₃₁	—	2	8	no	—
78	C₄⋊C₄×C₃₁	—	4	8	no	—
79	C₂²⋊C₄×C₃₁	—	4	4	no	—
80	C₄²×C₃₁	—	2	32	no	—
81	C₄²×C₃₁	—	2	4	no	—
82	C₄²×C₃₁	—	2	4	no	—
83	C₄²×C₃₁	—	2	4	no	—
84	C₄²×C₃₁	—	4	2	no	—
85	C₄²×C₃₁	—	4	2	no	—
86	C₄²×C₃₁	C₄²	1	2	yes	—
87	C₄²×C₃₁	C₄²	1	2	yes	—
88	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
89	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
90	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
91	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
92	C₄²×C₃₁	C₄⋊C₄	1	8	yes	complemented
93	#10839	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	—	2	4	no	—
95	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₃₁	#121	1	2	yes	—
99	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₃₁	—	2	16	no	—
101	C₂×C₄×C₃₁	—	2	16	no	—
102	C₂×C₄×C₃₁	—	2	16	no	—
103	C₂×C₄×C₃₁	—	2	8	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	8	no	—
106	C₂×C₄×C₃₁	—	2	8	no	—
107	C₂×C₄×C₃₁	—	4	4	no	—
108	C₂×C₄×C₃₁	—	4	8	no	—
109	C₂×C₄×C₃₁	#95	1	4	yes	—
110	C₂×C₄×C₃₁	#95	1	4	yes	—
111	C₂×C₄×C₃₁	#104	1	8	yes	—
112	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	#117	1	2	yes	—
118	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₃₁	#121	1	2	yes	—
123	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
125	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#125	1	2	yes	—
127	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
130	C₈×C₃₁	—	4	4	no	—
131	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
132	#2285	C₂×C₃₁	1	1	yes	characteristic
133	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic
134	C₂×#339	C₂×C₃₁	1	4	yes	—
135	C₂²×C₃₁	—	2	4	no	—
136	C₂²×C₃₁	—	2	2	no	—
137	C₂²×C₃₁	—	2	2	no	—
138	C₂²×C₃₁	—	2	4	no	—
139	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
140	C₂²×C₃₁	#380	1	2	yes	—
141	C₂²×C₃₁	#389	1	2	yes	—
142	C₂²×C₃₁	#390	1	1	yes	characteristic
143	C₂²×C₃₁	#391	1	1	yes	characteristic
144	C₂²×C₃₁	C₂×#104	1	4	yes	—
145	C₄×C₃₁	—	2	16	no	—
146	C₄×C₃₁	—	2	4	no	—
147	C₄×C₃₁	—	8	4	no	—
148	C₄×C₃₁	#339	1	8	yes	complemented
149	C₄×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
150	C₄×C₃₁	#382	1	1	yes	characteristic
151	C₄×C₃₁	#384	1	1	yes	characteristic
152	C₄×C₃₁	#389	1	2	yes	—
153	C₄×C₃₁	#390	1	1	yes	characteristic
154	C₄×C₃₁	#391	1	1	yes	characteristic
155	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
156	C₂×#117	C₂²×C₃₁	1	2	yes	—
157	C₂²×C₄²	—	2	2	no	—
158	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
159	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
160	#423	—	2	2	no	—
161	#423	C₄×C₃₁	1	4	yes	complemented
162	C₂×C₄⋊C₈	—	2	2	no	—
163	#384	C₄×C₃₁	1	2	yes	complemented
164	#382	C₄×C₃₁	1	2	yes	complemented
165	#380	—	2	2	no	—
166	C₄×C₄⋊C₄	—	2	2	no	—
167	C₄³	—	2	2	no	—
168	C₄³	C₄×C₃₁	1	4	yes	complemented
169	#339	—	2	8	no	—
170	C₂×C₃₁	—	2	2	no	—
171	C₂×C₃₁	—	2	2	no	—
172	C₂×C₃₁	C₄⋊#95	1	1	yes	characteristic
173	C₂×C₃₁	C₂×#339	1	2	yes	—
174	C₂×C₃₁	#1962	1	2	yes	—
175	C₂×C₃₁	#1994	1	2	yes	—
176	C₂³×C₄	—	2	2	no	—
177	C₂³×C₄	D₄×C₃₁	1	2	yes	—
178	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
183	C₂²×C₈	—	2	2	no	—
184	#117	—	2	4	no	—
185	#117	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₄²	—	2	16	no	—
190	C₂×C₄²	—	2	8	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
196	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
197	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
198	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
199	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
200	C₂×C₄²	D₄×C₃₁	1	4	yes	—
201	C₂×C₄²	D₄×C₃₁	1	2	yes	—
202	C₂×C₄²	D₄×C₃₁	1	2	yes	—
203	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
204	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
205	C₄⋊C₈	—	2	8	no	—
206	#95	—	2	4	no	—
207	C₃₁	#10839	1	1	yes	characteristic, direct product factor, Sylow
208	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
209	C₂²×C₄	—	2	16	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
218	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
219	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
225	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
226	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
227	C₂²×C₄	#41×C₃₁	1	2	yes	—
228	C₂²×C₄	#41×C₃₁	1	2	yes	—
229	M₁₆	—	2	8	no	—
230	M₁₆	—	4	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	4	8	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	—	2	32	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	4	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	C₄²×C₃₁	1	2	yes	—
244	C₄²	C₄²×C₃₁	1	2	yes	—
245	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
246	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
247	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
248	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
249	C₄²	C₄⋊C₄×C₃₁	1	8	yes	complemented
250	C₂³	—	2	4	no	—
251	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
252	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
253	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
254	C₂³	C₃₁×#121	1	2	yes	—
255	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	8	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	C₃₁×#95	1	4	yes	—
266	C₂×C₄	C₃₁×#95	1	4	yes	—
267	C₂×C₄	C₃₁×#104	1	8	yes	—
268	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
271	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
273	C₂×C₄	C₃₁×#117	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
275	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
276	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
277	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
278	C₂×C₄	C₃₁×#121	1	2	yes	—
279	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
280	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
281	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
282	C₂×C₄	C₃₁×#125	1	2	yes	—
283	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
284	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
285	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
286	C₈	—	4	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
292	C₂²	C₃₁×#380	1	2	yes	—
293	C₂²	C₃₁×#389	1	2	yes	—
294	C₂²	C₃₁×#390	1	1	yes	characteristic
295	C₂²	C₃₁×#391	1	1	yes	characteristic
296	C₂²	C₂×C₃₁×#104	1	4	yes	—
297	C₄	—	2	16	no	—
298	C₄	—	2	4	no	—
299	C₄	—	8	4	no	—
300	C₄	C₃₁×#339	1	8	yes	complemented
301	C₄	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
302	C₄	C₃₁×#382	1	1	yes	characteristic
303	C₄	C₃₁×#384	1	1	yes	characteristic
304	C₄	C₃₁×#389	1	2	yes	—
305	C₄	C₃₁×#390	1	1	yes	characteristic
306	C₄	C₃₁×#391	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	C₃₁×C₄⋊#95	1	1	yes	characteristic
310	C₂	C₂×C₃₁×#339	1	2	yes	—
311	C₂	C₃₁×#1962	1	2	yes	—
312	C₂	C₃₁×#1994	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₃₁×#423	C₄	semidirect
13. C₃₁×#384	C₄	semidirect
14. C₃₁×#382	C₄	semidirect
18. C₄³×C₃₁	C₄	semidirect
92. C₄²×C₃₁	C₄⋊C₄	semidirect
93. #10839	C₃₁	direct (N × H)
148. C₄×C₃₁	#339	semidirect
161. #423	C₄×C₃₁	semidirect
163. #384	C₄×C₃₁	semidirect
164. #382	C₄×C₃₁	semidirect
168. C₄³	C₄×C₃₁	semidirect
207. C₃₁	#10839	direct (N × H)
249. C₄²	C₄⋊C₄×C₃₁	semidirect
300. C₄	C₃₁×#339	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#423	2
C₃₁×#2285	2
C₂×C₄³×C₃₁	2
C₂×C₃₁×#339	2
#10839	31

Subgroups — C31×#10839 #21467598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10839 #21467598