Subgroups — C₃₁×#3056 #16842989

Order 3968.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#525	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#470	C₂	1	2	yes	maximal, complemented
6	C₃₁×#468	C₂	1	2	yes	maximal, complemented
7	C₃₁×#467	C₂	1	2	yes	maximal, complemented
8	C₃₁×#465	C₂	1	2	yes	maximal, complemented
9	C₃₁×#428	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#419	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#417	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#137	—	2	4	no	—
16	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
17	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
18	C₂×M₁₆×C₃₁	—	2	4	no	—
19	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
21	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#123	—	2	1	no	—
24	C₃₁×C₂²⋊Q₈	—	2	1	no	—
25	C₃₁×C₂²⋊Q₈	—	2	1	no	—
26	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
28	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
29	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
30	C₃₁×#121	C₂²	1	1	yes	characteristic
31	C₃₁×#121	C₂²	1	1	yes	characteristic
32	C₃₁×#120	—	2	1	no	—
33	C₄×D₄×C₃₁	—	2	1	no	—
34	C₄×D₄×C₃₁	—	2	1	no	—
35	C₃₁×#117	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
37	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
39	C₃₁×#107	C₂²	1	2	yes	complemented
40	C₃₁×#106	C₂²	1	2	yes	complemented
41	C₃₁×#103	C₂²	1	2	yes	complemented
42	C₃₁×#103	C₂²	1	2	yes	complemented
43	C₃₁×#103	C₂²	1	2	yes	complemented
44	C₃₁×#102	C₂²	1	2	yes	complemented
45	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
46	#41×C₃₁	—	2	2	no	—
47	#41×C₃₁	—	2	2	no	—
48	C₂×Q₈×C₃₁	—	2	1	no	—
49	C₂×Q₈×C₃₁	—	2	2	no	—
50	C₂×Q₈×C₃₁	—	2	2	no	—
51	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
52	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₃₁	—	2	1	no	—
54	C₂×D₄×C₃₁	—	2	1	no	—
55	C₂×D₄×C₃₁	—	2	1	no	—
56	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	—	2	2	no	—
58	C₂²×C₄×C₃₁	—	2	2	no	—
59	C₂²×C₄×C₃₁	—	2	1	no	—
60	C₂²×C₄×C₃₁	—	2	1	no	—
61	C₂²×C₄×C₃₁	—	2	1	no	—
62	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	Q₁₆×C₃₁	—	2	4	no	—
69	SD₁₆×C₃₁	—	2	4	no	—
70	M₁₆×C₃₁	—	2	2	no	—
71	M₁₆×C₃₁	—	2	4	no	—
72	M₁₆×C₃₁	D₄	1	4	yes	complemented
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	C₂³	1	2	yes	—
75	C₂×C₈×C₃₁	C₂³	1	2	yes	—
76	C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₄⋊C₄×C₃₁	—	2	1	no	—
81	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
82	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂²⋊C₄×C₃₁	—	2	2	no	—
85	C₂²⋊C₄×C₃₁	—	2	2	no	—
86	C₂²⋊C₄×C₃₁	—	2	1	no	—
87	C₂²⋊C₄×C₃₁	—	2	1	no	—
88	C₂²⋊C₄×C₃₁	—	2	1	no	—
89	C₂²⋊C₄×C₃₁	—	2	1	no	—
90	C₂²⋊C₄×C₃₁	—	2	1	no	—
91	C₄²×C₃₁	—	2	1	no	—
92	C₂³×C₃₁	—	2	2	no	—
93	C₂³×C₃₁	—	2	2	no	—
94	C₂³×C₃₁	—	2	1	no	—
95	C₂³×C₃₁	—	4	1	no	—
96	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
99	Q₈×C₃₁	—	2	2	no	—
100	Q₈×C₃₁	—	2	1	no	—
101	Q₈×C₃₁	—	2	2	no	—
102	Q₈×C₃₁	—	2	1	no	—
103	Q₈×C₃₁	—	4	2	no	—
104	D₄×C₃₁	—	2	1	no	—
105	D₄×C₃₁	—	2	1	no	—
106	D₄×C₃₁	—	4	2	no	—
107	D₄×C₃₁	—	4	2	no	—
108	D₄×C₃₁	—	4	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	1	no	—
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	2	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	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
128	C₂×C₄×C₃₁	#41	1	2	yes	—
129	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
130	C₈×C₃₁	—	2	2	no	—
131	C₈×C₃₁	C₂×D₄	1	4	yes	—
132	#3056	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₃₁	—	2	1	no	—
134	C₂²×C₃₁	—	2	1	no	—
135	C₂²×C₃₁	—	2	1	no	—
136	C₂²×C₃₁	—	2	1	no	—
137	C₂²×C₃₁	—	2	1	no	—
138	C₂²×C₃₁	—	2	1	no	—
139	C₂²×C₃₁	—	2	1	no	—
140	C₂²×C₃₁	—	4	4	no	—
141	C₂²×C₃₁	—	4	1	no	—
142	C₂²×C₃₁	—	4	1	no	—
143	C₂²×C₃₁	—	4	1	no	—
144	C₂²×C₃₁	#121	1	2	yes	—
145	C₂²×C₃₁	#137	1	2	yes	complemented
146	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
147	C₄×C₃₁	—	2	1	no	—
148	C₄×C₃₁	—	2	1	no	—
149	C₄×C₃₁	—	4	1	no	—
150	C₄×C₃₁	—	4	2	no	—
151	C₄×C₃₁	—	4	1	no	—
152	C₄×C₃₁	—	4	2	no	—
153	C₄×C₃₁	—	4	1	no	—
154	C₄×C₃₁	#121	1	2	yes	—
155	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
156	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
157	C₂×#137	C₂×C₃₁	1	1	yes	characteristic, complemented
158	C₂²×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
159	#525	C₂×C₃₁	1	1	yes	characteristic
160	C₂×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
161	#470	C₂×C₃₁	1	2	yes	complemented
162	#468	C₂×C₃₁	1	2	yes	complemented
163	#467	C₂×C₃₁	1	2	yes	complemented
164	#465	C₂×C₃₁	1	2	yes	complemented
165	#428	C₂×C₃₁	1	1	yes	characteristic, complemented
166	#419	C₂×C₃₁	1	1	yes	characteristic, complemented
167	#417	C₂×C₃₁	1	1	yes	characteristic, complemented
168	C₂×C₃₁	—	2	2	no	—
169	C₂×C₃₁	—	2	2	no	—
170	C₂×C₃₁	—	4	1	no	—
171	C₂×C₃₁	—	8	1	no	—
172	C₂×C₃₁	C₂×#121	1	1	yes	characteristic
173	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
174	C₂×C₃₁	#575	1	1	yes	characteristic
175	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
176	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
177	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
178	#137	—	2	4	no	—
179	C₂×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
180	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
183	C₂×M₁₆	C₂²×C₃₁	1	2	yes	—
184	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
185	C₂²×C₈	C₂²×C₃₁	1	2	yes	—
186	#123	—	2	1	no	—
187	C₂²⋊Q₈	—	2	1	no	—
188	C₂²⋊Q₈	—	2	1	no	—
189	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
190	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
191	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
192	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
193	#121	C₂²×C₃₁	1	1	yes	characteristic
194	#121	C₂²×C₃₁	1	1	yes	characteristic
195	#120	—	2	1	no	—
196	C₄×D₄	—	2	1	no	—
197	C₄×D₄	—	2	1	no	—
198	#117	C₂²×C₃₁	1	1	yes	characteristic
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	—	2	1	no	—
202	#107	C₂²×C₃₁	1	2	yes	complemented
203	#106	C₂²×C₃₁	1	2	yes	complemented
204	#103	C₂²×C₃₁	1	2	yes	complemented
205	#103	C₂²×C₃₁	1	2	yes	complemented
206	#103	C₂²×C₃₁	1	2	yes	complemented
207	#102	C₂²×C₃₁	1	2	yes	complemented
208	C₃₁	#3056	1	1	yes	characteristic, direct product factor, Sylow
209	C₂⁴	D₄×C₃₁	1	1	yes	characteristic
210	#41	—	2	2	no	—
211	#41	—	2	2	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
216	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
217	C₂×D₄	—	2	1	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
228	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
229	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
232	Q₁₆	—	2	4	no	—
233	SD₁₆	—	2	4	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	D₄×C₃₁	1	4	yes	complemented
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	C₂³×C₃₁	1	2	yes	—
239	C₂×C₈	C₂³×C₃₁	1	2	yes	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
246	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
247	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	4	1	no	—
260	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
262	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
263	Q₈	—	2	2	no	—
264	Q₈	—	2	1	no	—
265	Q₈	—	2	2	no	—
266	Q₈	—	2	1	no	—
267	Q₈	—	4	2	no	—
268	D₄	—	2	1	no	—
269	D₄	—	2	1	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	4	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
289	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
290	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
292	C₂×C₄	#41×C₃₁	1	2	yes	—
293	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	C₂×D₄×C₃₁	1	4	yes	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	C₃₁×#121	1	2	yes	—
308	C₂²	C₃₁×#137	1	2	yes	complemented
309	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	1	no	—
317	C₄	C₃₁×#121	1	2	yes	—
318	C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
319	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	2	2	no	—
322	C₂	—	4	1	no	—
323	C₂	—	8	1	no	—
324	C₂	C₂×C₃₁×#121	1	1	yes	characteristic
325	C₂	C₂×C₃₁×#137	1	1	yes	characteristic
326	C₂	C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#137	C₂	semidirect
2. C₂²×M₁₆×C₃₁	C₂	semidirect
4. C₂×C₃₁×C₂²⋊Q₈	C₂	semidirect
5. C₃₁×#470	C₂	semidirect
6. C₃₁×#468	C₂	semidirect
7. C₃₁×#467	C₂	semidirect
8. C₃₁×#465	C₂	semidirect
9. C₃₁×#428	C₂	semidirect
10. C₃₁×#419	C₂	semidirect
11. C₃₁×#417	C₂	semidirect
16. C₂×Q₁₆×C₃₁	C₂²	semidirect
17. C₂×SD₁₆×C₃₁	C₂²	semidirect
39. C₃₁×#107	C₂²	semidirect
40. C₃₁×#106	C₂²	semidirect
41. C₃₁×#103	C₂²	semidirect
42. C₃₁×#103	C₂²	semidirect
43. C₃₁×#103	C₂²	semidirect
44. C₃₁×#102	C₂²	semidirect
72. M₁₆×C₃₁	D₄	semidirect
132. #3056	C₃₁	direct (N × H)
145. C₂²×C₃₁	#137	semidirect
157. C₂×#137	C₂×C₃₁	semidirect
158. C₂²×M₁₆	C₂×C₃₁	semidirect
160. C₂×C₂²⋊Q₈	C₂×C₃₁	semidirect
161. #470	C₂×C₃₁	semidirect
162. #468	C₂×C₃₁	semidirect
163. #467	C₂×C₃₁	semidirect
164. #465	C₂×C₃₁	semidirect
165. #428	C₂×C₃₁	semidirect
166. #419	C₂×C₃₁	semidirect
167. #417	C₂×C₃₁	semidirect
179. C₂×Q₁₆	C₂²×C₃₁	semidirect
180. C₂×SD₁₆	C₂²×C₃₁	semidirect
202. #107	C₂²×C₃₁	semidirect
203. #106	C₂²×C₃₁	semidirect
204. #103	C₂²×C₃₁	semidirect
205. #103	C₂²×C₃₁	semidirect
206. #103	C₂²×C₃₁	semidirect
207. #102	C₂²×C₃₁	semidirect
208. C₃₁	#3056	direct (N × H)
236. M₁₆	D₄×C₃₁	semidirect
308. C₂²	C₃₁×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#137	2
C₂²×M₁₆×C₃₁	2
C₃₁×#525	2
C₂×C₃₁×C₂²⋊Q₈	2
C₃₁×#470	2
C₃₁×#468	2
C₃₁×#467	2
C₃₁×#465	2
C₃₁×#428	2
C₃₁×#419	2
C₃₁×#417	2
#3056	31

Subgroups — C31×#3056 #16842989

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3056 #16842989