Subgroups — C₃₁×#22690 #21479449

Order 7936.

846 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#443	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2951	C₂	1	2	yes	maximal
3	C₃₁×#2912	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1952	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1860	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1815	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#131	C₂²	1	2	yes	—
11	C₃₁×#520	C₂²	1	4	yes	—
12	C₃₁×#443	C₄	1	8	yes	complemented
13	C₃₁×#440	—	2	4	no	—
14	C₄×Q₁₆×C₃₁	—	2	2	no	—
15	C₄×D₈×C₃₁	—	2	2	no	—
16	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
18	C₃₁×#428	—	2	4	no	—
19	C₃₁×#428	C₂²	1	4	yes	—
20	C₃₁×#427	—	2	2	no	—
21	C₃₁×#427	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#107	C₂²	1	2	yes	—
23	C₂×C₃₁×#106	—	2	2	no	—
24	C₂×C₃₁×#106	C₂²	1	2	yes	—
25	C₂×C₃₁×#104	C₂²	1	2	yes	—
26	C₃₁×#417	C₂²	1	4	yes	—
27	C₃₁×#405	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#382	C₂²	1	2	yes	—
30	C₃₁×#343	C₂²	1	4	yes	—
31	C₃₁×#339	C₂²	1	2	yes	—
32	C₂×#41×C₃₁	C₂³	1	2	yes	—
33	C₃₁×#135	C₂×C₄	1	4	yes	—
34	C₃₁×#135	Q₈	1	4	yes	—
35	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
36	C₂×SD₁₆×C₃₁	D₄	1	2	yes	—
37	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
38	C₃₁×#131	—	2	4	no	—
39	C₃₁×#131	C₂×C₄	1	8	yes	complemented
40	C₂×M₁₆×C₃₁	—	2	2	no	—
41	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
42	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₃₁	—	2	2	no	—
44	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₃₁	—	2	4	no	—
47	C₄×D₄×C₃₁	—	2	4	no	—
48	C₄×D₄×C₃₁	—	4	4	no	—
49	C₃₁×#117	—	2	4	no	—
50	C₃₁×#117	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
52	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#108	—	2	2	no	—
55	C₃₁×#108	C₂×C₄	1	4	yes	complemented
56	C₃₁×#107	—	2	2	no	—
57	C₃₁×#107	—	2	2	no	—
58	C₃₁×#107	—	2	4	no	—
59	C₃₁×#107	D₄	1	2	yes	—
60	C₃₁×#106	—	2	2	no	—
61	C₃₁×#106	—	2	2	no	—
62	C₃₁×#106	—	2	4	no	—
63	C₃₁×#106	D₄	1	1	yes	characteristic
64	C₃₁×#106	D₄	1	1	yes	characteristic
65	C₃₁×#104	C₂×C₄	1	8	yes	complemented
66	C₃₁×#103	—	2	4	no	—
67	C₃₁×#102	—	2	4	no	—
68	C₃₁×#97	—	2	2	no	—
69	C₄×C₈×C₃₁	—	2	2	no	—
70	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
71	C₃₁×#95	—	4	2	no	—
72	#41×C₃₁	—	2	2	no	—
73	#41×C₃₁	—	2	2	no	—
74	#41×C₃₁	C₂²×C₄	1	4	yes	—
75	#41×C₃₁	C₂×Q₈	1	4	yes	—
76	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₁	—	2	2	no	—
80	C₂²×C₄×C₃₁	—	2	1	no	—
81	C₂²×C₄×C₃₁	—	2	4	no	—
82	C₂²×C₄×C₃₁	—	4	4	no	—
83	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
84	Q₁₆×C₃₁	C₄⋊C₄	1	4	yes	—
85	SD₁₆×C₃₁	C₄⋊C₄	1	8	yes	—
86	D₈×C₃₁	C₄⋊C₄	1	4	yes	—
87	M₁₆×C₃₁	—	2	2	no	—
88	M₁₆×C₃₁	—	2	8	no	—
89	M₁₆×C₃₁	—	2	2	no	—
90	M₁₆×C₃₁	—	2	2	no	—
91	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	—	2	8	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
102	C₂×C₈×C₃₁	#41	1	2	yes	—
103	C₄⋊C₄×C₃₁	—	2	4	no	—
104	C₄⋊C₄×C₃₁	—	2	4	no	—
105	C₄⋊C₄×C₃₁	—	4	4	no	—
106	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
107	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
108	C₂²⋊C₄×C₃₁	—	2	2	no	—
109	C₂²⋊C₄×C₃₁	—	4	4	no	—
110	C₄²×C₃₁	—	2	2	no	—
111	C₄²×C₃₁	—	2	1	no	—
112	C₄²×C₃₁	—	4	4	no	—
113	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
114	#22690	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₃₁	—	2	2	no	—
116	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
117	Q₈×C₃₁	—	2	2	no	—
118	Q₈×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
119	D₄×C₃₁	—	2	4	no	—
120	D₄×C₃₁	—	2	4	no	—
121	D₄×C₃₁	—	2	2	no	—
122	D₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	2	4	no	—
125	C₂×C₄×C₃₁	—	2	4	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	4	4	no	—
131	C₂×C₄×C₃₁	—	8	4	no	—
132	C₂×C₄×C₃₁	—	16	4	no	—
133	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
139	C₈×C₃₁	—	2	4	no	—
140	C₈×C₃₁	—	4	2	no	—
141	C₈×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
142	C₈×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
143	C₈×C₃₁	C₄×D₄	1	2	yes	—
144	C₈×C₃₁	C₄×D₄	1	2	yes	—
145	C₂×#443	C₂×C₃₁	1	1	yes	characteristic
146	#2951	C₂×C₃₁	1	2	yes	—
147	#2912	C₂×C₃₁	1	2	yes	complemented
148	#1952	C₂×C₃₁	1	2	yes	complemented
149	#1860	C₂×C₃₁	1	4	yes	complemented
150	#1846	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#1815	C₂×C₃₁	1	2	yes	complemented
152	#1772	C₂×C₃₁	1	1	yes	characteristic, complemented
153	C₂²×C₃₁	—	2	2	no	—
154	C₂²×C₃₁	—	2	4	no	—
155	C₂²×C₃₁	—	4	4	no	—
156	C₂²×C₃₁	#520	1	1	yes	characteristic
157	C₂²×C₃₁	D₄²	1	1	yes	characteristic
158	C₂²×C₃₁	#554	1	1	yes	characteristic
159	C₄×C₃₁	—	2	4	no	—
160	C₄×C₃₁	—	4	2	no	—
161	C₄×C₃₁	—	4	2	no	—
162	C₄×C₃₁	—	4	2	no	—
163	C₄×C₃₁	—	8	4	no	—
164	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
166	C₄×C₃₁	#547	1	1	yes	characteristic
167	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
168	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
169	C₂×#131	C₂²×C₃₁	1	2	yes	—
170	#520	C₂²×C₃₁	1	4	yes	—
171	#443	C₄×C₃₁	1	8	yes	complemented
172	#440	—	2	4	no	—
173	C₄×Q₁₆	—	2	2	no	—
174	C₄×D₈	—	2	2	no	—
175	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
176	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
177	#428	—	2	4	no	—
178	#428	C₂²×C₃₁	1	4	yes	—
179	#427	—	2	2	no	—
180	#427	C₂²×C₃₁	1	1	yes	characteristic
181	C₂×#107	C₂²×C₃₁	1	2	yes	—
182	C₂×#106	—	2	2	no	—
183	C₂×#106	C₂²×C₃₁	1	2	yes	—
184	C₂×#104	C₂²×C₃₁	1	2	yes	—
185	#417	C₂²×C₃₁	1	4	yes	—
186	#405	C₂²×C₃₁	1	2	yes	—
187	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
188	#382	C₂²×C₃₁	1	2	yes	—
189	#343	C₂²×C₃₁	1	4	yes	—
190	#339	C₂²×C₃₁	1	2	yes	—
191	C₂×C₃₁	—	2	1	no	—
192	C₂×C₃₁	—	2	1	no	—
193	C₂×C₃₁	—	4	4	no	—
194	C₂×C₃₁	D₄×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₃₁	#3290	1	2	yes	—
196	C₂×#41	C₂³×C₃₁	1	2	yes	—
197	#135	C₂×C₄×C₃₁	1	4	yes	—
198	#135	Q₈×C₃₁	1	4	yes	—
199	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
200	C₂×SD₁₆	D₄×C₃₁	1	2	yes	—
201	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
202	#131	—	2	4	no	—
203	#131	C₂×C₄×C₃₁	1	8	yes	complemented
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
206	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
210	C₄×Q₈	—	2	4	no	—
211	C₄×D₄	—	2	4	no	—
212	C₄×D₄	—	4	4	no	—
213	#117	—	2	4	no	—
214	#117	C₂³×C₃₁	1	2	yes	—
215	C₂×C₄⋊C₄	—	2	4	no	—
216	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
217	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
218	#108	—	2	2	no	—
219	#108	C₂×C₄×C₃₁	1	4	yes	complemented
220	#107	—	2	2	no	—
221	#107	—	2	2	no	—
222	#107	—	2	4	no	—
223	#107	D₄×C₃₁	1	2	yes	—
224	#106	—	2	2	no	—
225	#106	—	2	2	no	—
226	#106	—	2	4	no	—
227	#106	D₄×C₃₁	1	1	yes	characteristic
228	#106	D₄×C₃₁	1	1	yes	characteristic
229	#104	C₂×C₄×C₃₁	1	8	yes	complemented
230	#103	—	2	4	no	—
231	#102	—	2	4	no	—
232	#97	—	2	2	no	—
233	C₄×C₈	—	2	2	no	—
234	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
235	#95	—	4	2	no	—
236	C₃₁	#22690	1	1	yes	characteristic, direct product factor, Sylow
237	#41	—	2	2	no	—
238	#41	—	2	2	no	—
239	#41	C₂²×C₄×C₃₁	1	4	yes	—
240	#41	C₂×Q₈×C₃₁	1	4	yes	—
241	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	4	4	no	—
248	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
249	Q₁₆	C₄⋊C₄×C₃₁	1	4	yes	—
250	SD₁₆	C₄⋊C₄×C₃₁	1	8	yes	—
251	D₈	C₄⋊C₄×C₃₁	1	4	yes	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	8	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	8	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
261	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
262	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
264	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
266	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
267	C₂×C₈	#41×C₃₁	1	2	yes	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	4	4	no	—
271	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
272	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	4	4	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	4	4	no	—
278	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
279	C₂³	—	2	2	no	—
280	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
281	Q₈	—	2	2	no	—
282	Q₈	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
283	D₄	—	2	4	no	—
284	D₄	—	2	4	no	—
285	D₄	—	2	2	no	—
286	D₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	8	4	no	—
296	C₂×C₄	—	16	4	no	—
297	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
298	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
300	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
302	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
303	C₈	—	2	4	no	—
304	C₈	—	4	2	no	—
305	C₈	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
306	C₈	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
307	C₈	C₄×D₄×C₃₁	1	2	yes	—
308	C₈	C₄×D₄×C₃₁	1	2	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	C₃₁×#520	1	1	yes	characteristic
313	C₂²	D₄²×C₃₁	1	1	yes	characteristic
314	C₂²	C₃₁×#554	1	1	yes	characteristic
315	C₄	—	2	4	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	2	no	—
319	C₄	—	8	4	no	—
320	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
321	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
322	C₄	C₃₁×#547	1	1	yes	characteristic
323	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	4	4	no	—
327	C₂	D₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
328	C₂	C₃₁×#3290	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2912	C₂	semidirect
4. C₃₁×#1952	C₂	semidirect
5. C₃₁×#1860	C₂	semidirect
6. C₃₁×#1846	C₂	semidirect
7. C₃₁×#1815	C₂	semidirect
8. C₃₁×#1772	C₂	semidirect
12. C₃₁×#443	C₄	semidirect
39. C₃₁×#131	C₂×C₄	semidirect
55. C₃₁×#108	C₂×C₄	semidirect
65. C₃₁×#104	C₂×C₄	semidirect
70. C₄×C₈×C₃₁	C₂×C₄	semidirect
114. #22690	C₃₁	direct (N × H)
147. #2912	C₂×C₃₁	semidirect
148. #1952	C₂×C₃₁	semidirect
149. #1860	C₂×C₃₁	semidirect
150. #1846	C₂×C₃₁	semidirect
151. #1815	C₂×C₃₁	semidirect
152. #1772	C₂×C₃₁	semidirect
171. #443	C₄×C₃₁	semidirect
203. #131	C₂×C₄×C₃₁	semidirect
219. #108	C₂×C₄×C₃₁	semidirect
229. #104	C₂×C₄×C₃₁	semidirect
234. C₄×C₈	C₂×C₄×C₃₁	semidirect
236. C₃₁	#22690	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#443	2
C₃₁×#2951	2
C₃₁×#2912	2
C₃₁×#1952	2
C₃₁×#1860	2
C₃₁×#1846	2
C₃₁×#1815	2
C₃₁×#1772	2
#22690	31

Subgroups — C31×#22690 #21479449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22690 #21479449