Subgroups — C₃₁×#2430 #16842363

Order 3968.

914 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#123	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#121	C₂	1	2	yes	maximal, complemented
4	C₃₁×#395	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#394	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#393	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#392	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#386	C₂	1	2	yes	maximal, complemented
9	C₃₁×#386	C₂	1	2	yes	maximal, complemented
10	C₃₁×#385	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#95	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	—	2	8	no	—
13	C₂×#41×C₃₁	—	2	4	no	—
14	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
18	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
19	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
20	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
21	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#123	—	2	4	no	—
23	C₃₁×#121	—	2	8	no	—
24	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₃₁×#95	C₂²	1	1	yes	characteristic
35	C₃₁×#95	C₂²	1	2	yes	—
36	C₃₁×#95	C₂²	1	2	yes	—
37	C₃₁×#95	C₂²	1	1	yes	characteristic
38	C₃₁×#95	C₂²	1	2	yes	—
39	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
40	C₂⁴×C₃₁	C₂³	1	2	yes	—
41	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
42	#41×C₃₁	—	2	32	no	—
43	C₂×Q₈×C₃₁	—	2	4	no	—
44	C₂×Q₈×C₃₁	—	2	2	no	—
45	C₂×D₄×C₃₁	—	2	4	no	—
46	C₂×D₄×C₃₁	—	2	4	no	—
47	C₂×D₄×C₃₁	—	2	4	no	—
48	C₂×D₄×C₃₁	—	2	4	no	—
49	C₂×D₄×C₃₁	—	2	4	no	—
50	C₂×D₄×C₃₁	—	2	2	no	—
51	C₂×D₄×C₃₁	—	2	2	no	—
52	C₂×D₄×C₃₁	—	2	4	no	—
53	C₂×D₄×C₃₁	—	4	2	no	—
54	C₂×D₄×C₃₁	D₄	1	8	yes	complemented
55	C₂²×C₄×C₃₁	—	2	4	no	—
56	C₂²×C₄×C₃₁	—	2	4	no	—
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	—	4	2	no	—
62	C₂²×C₄×C₃₁	D₄	1	4	yes	complemented
63	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂³	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	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
71	C₄⋊C₄×C₃₁	—	2	4	no	—
72	C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₂²⋊C₄×C₃₁	—	2	4	no	—
75	C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂²⋊C₄×C₃₁	—	2	4	no	—
77	C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂²⋊C₄×C₃₁	—	2	4	no	—
80	C₂²⋊C₄×C₃₁	—	4	2	no	—
81	C₂³×C₃₁	—	2	2	no	—
82	C₂³×C₃₁	—	2	1	no	—
83	C₂³×C₃₁	—	2	1	no	—
84	C₂³×C₃₁	—	2	2	no	—
85	C₂³×C₃₁	—	2	2	no	—
86	C₂³×C₃₁	—	2	2	no	—
87	C₂³×C₃₁	—	2	2	no	—
88	C₂³×C₃₁	—	2	2	no	—
89	C₂³×C₃₁	—	2	2	no	—
90	C₂³×C₃₁	—	2	1	no	—
91	C₂³×C₃₁	—	2	2	no	—
92	C₂³×C₃₁	—	2	2	no	—
93	C₂³×C₃₁	—	2	1	no	—
94	C₂³×C₃₁	—	2	1	no	—
95	C₂³×C₃₁	C₂×D₄	1	4	yes	—
96	C₂³×C₃₁	C₂×D₄	1	2	yes	—
97	C₂³×C₃₁	C₂×D₄	1	4	yes	—
98	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
99	Q₈×C₃₁	—	2	8	no	—
100	D₄×C₃₁	—	2	8	no	—
101	D₄×C₃₁	—	2	16	no	—
102	D₄×C₃₁	—	4	8	no	—
103	D₄×C₃₁	—	4	4	no	—
104	C₂×C₄×C₃₁	—	2	16	no	—
105	C₂×C₄×C₃₁	—	2	8	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	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	1	no	—
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	2	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	—	2	1	no	—
122	C₂×C₄×C₃₁	—	4	8	no	—
123	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
124	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
125	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
126	C₂×C₄×C₃₁	#41	1	4	yes	complemented
127	#2430	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₃₁	—	2	2	no	—
129	C₂²×C₃₁	—	2	2	no	—
130	C₂²×C₃₁	—	2	4	no	—
131	C₂²×C₃₁	—	2	4	no	—
132	C₂²×C₃₁	—	2	4	no	—
133	C₂²×C₃₁	—	2	4	no	—
134	C₂²×C₃₁	—	2	4	no	—
135	C₂²×C₃₁	—	4	2	no	—
136	C₂²×C₃₁	—	4	2	no	—
137	C₂²×C₃₁	—	4	2	no	—
138	C₂²×C₃₁	—	4	2	no	—
139	C₂²×C₃₁	—	4	1	no	—
140	C₂²×C₃₁	—	4	1	no	—
141	C₂²×C₃₁	—	4	1	no	—
142	C₂²×C₃₁	—	4	2	no	—
143	C₂²×C₃₁	—	4	2	no	—
144	C₂²×C₃₁	#120	1	4	yes	—
145	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
146	C₂²×C₃₁	C₂²×D₄	1	2	yes	—
147	C₂²×C₃₁	C₂×#41	1	2	yes	—
148	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
150	C₄×C₃₁	—	2	2	no	—
151	C₄×C₃₁	—	2	4	no	—
152	C₄×C₃₁	—	2	2	no	—
153	C₄×C₃₁	—	2	4	no	—
154	C₄×C₃₁	—	4	1	no	—
155	C₄×C₃₁	—	4	1	no	—
156	C₄×C₃₁	—	4	1	no	—
157	C₄×C₃₁	—	4	2	no	—
158	C₂²×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
159	C₂×#123	C₂×C₃₁	1	1	yes	characteristic, complemented
160	C₂×#121	C₂×C₃₁	1	2	yes	complemented
161	#395	C₂×C₃₁	1	1	yes	characteristic, complemented
162	#394	C₂×C₃₁	1	1	yes	characteristic, complemented
163	#393	C₂×C₃₁	1	1	yes	characteristic, complemented
164	#392	C₂×C₃₁	1	1	yes	characteristic, complemented
165	#386	C₂×C₃₁	1	2	yes	complemented
166	#386	C₂×C₃₁	1	2	yes	complemented
167	#385	C₂×C₃₁	1	2	yes	complemented
168	C₂×#95	C₂×C₃₁	1	1	yes	characteristic, complemented
169	C₂×C₃₁	—	2	4	no	—
170	C₂×C₃₁	—	4	4	no	—
171	C₂×C₃₁	—	8	1	no	—
172	C₂×C₃₁	C₂×#120	1	1	yes	characteristic
173	C₂×C₃₁	#532	1	1	yes	characteristic
174	C₂×C₃₁	#537	1	1	yes	characteristic
175	C₂×C₃₁	#546	1	2	yes	—
176	C₂×C₃₁	#547	1	2	yes	—
177	C₂×#41	—	2	8	no	—
178	C₂×#41	—	2	4	no	—
179	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
180	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
181	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
182	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
183	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
184	C₂³×C₄	C₂²×C₃₁	1	2	yes	—
185	C₂³×C₄	C₂²×C₃₁	1	2	yes	—
186	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
187	#123	—	2	4	no	—
188	#121	—	2	8	no	—
189	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
190	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
193	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
194	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
195	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
196	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
197	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
198	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
199	#95	C₂²×C₃₁	1	2	yes	—
200	#95	C₂²×C₃₁	1	1	yes	characteristic
201	#95	C₂²×C₃₁	1	1	yes	characteristic
202	#95	C₂²×C₃₁	1	2	yes	—
203	#95	C₂²×C₃₁	1	2	yes	—
204	C₃₁	#2430	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
206	C₂⁴	C₂³×C₃₁	1	2	yes	—
207	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
208	#41	—	2	32	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	2	4	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	2	4	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	D₄×C₃₁	1	8	yes	complemented
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	D₄×C₃₁	1	4	yes	complemented
229	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
230	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
233	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
234	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
235	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
236	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	4	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	1	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	1	no	—
260	C₂³	—	2	1	no	—
261	C₂³	C₂×D₄×C₃₁	1	4	yes	—
262	C₂³	C₂×D₄×C₃₁	1	2	yes	—
263	C₂³	C₂×D₄×C₃₁	1	4	yes	—
264	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
265	Q₈	—	2	8	no	—
266	D₄	—	2	8	no	—
267	D₄	—	2	16	no	—
268	D₄	—	4	8	no	—
269	D₄	—	4	4	no	—
270	C₂×C₄	—	2	16	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	8	no	—
289	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
290	C₂×C₄	C₂×D₄×C₃₁	1	4	yes	—
291	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
292	C₂×C₄	#41×C₃₁	1	4	yes	complemented
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₃₁×#120	1	4	yes	—
310	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
311	C₂²	C₂²×D₄×C₃₁	1	2	yes	—
312	C₂²	C₂×#41×C₃₁	1	2	yes	—
313	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
314	C₂²	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	2	4	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	4	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	2	no	—
323	C₂	—	2	4	no	—
324	C₂	—	4	4	no	—
325	C₂	—	8	1	no	—
326	C₂	C₂×C₃₁×#120	1	1	yes	characteristic
327	C₂	C₃₁×#532	1	1	yes	characteristic
328	C₂	C₃₁×#537	1	1	yes	characteristic
329	C₂	C₃₁×#546	1	2	yes	—
330	C₂	C₃₁×#547	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₃₁	C₂	semidirect
2. C₂×C₃₁×#123	C₂	semidirect
3. C₂×C₃₁×#121	C₂	semidirect
4. C₃₁×#395	C₂	semidirect
5. C₃₁×#394	C₂	semidirect
6. C₃₁×#393	C₂	semidirect
7. C₃₁×#392	C₂	semidirect
8. C₃₁×#386	C₂	semidirect
9. C₃₁×#386	C₂	semidirect
10. C₃₁×#385	C₂	semidirect
11. C₂×C₃₁×#95	C₂	semidirect
54. C₂×D₄×C₃₁	D₄	semidirect
62. C₂²×C₄×C₃₁	D₄	semidirect
126. C₂×C₄×C₃₁	#41	semidirect
127. #2430	C₃₁	direct (N × H)
158. C₂²×#41	C₂×C₃₁	semidirect
159. C₂×#123	C₂×C₃₁	semidirect
160. C₂×#121	C₂×C₃₁	semidirect
161. #395	C₂×C₃₁	semidirect
162. #394	C₂×C₃₁	semidirect
163. #393	C₂×C₃₁	semidirect
164. #392	C₂×C₃₁	semidirect
165. #386	C₂×C₃₁	semidirect
166. #386	C₂×C₃₁	semidirect
167. #385	C₂×C₃₁	semidirect
168. C₂×#95	C₂×C₃₁	semidirect
204. C₃₁	#2430	direct (N × H)
220. C₂×D₄	D₄×C₃₁	semidirect
228. C₂²×C₄	D₄×C₃₁	semidirect
292. C₂×C₄	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×C₃₁	2
C₂×C₃₁×#123	2
C₂×C₃₁×#121	2
C₃₁×#395	2
C₃₁×#394	2
C₃₁×#393	2
C₃₁×#392	2
C₃₁×#386	2
C₃₁×#385	2
C₂×C₃₁×#95	2
#2430	31

Subgroups — C31×#2430 #16842363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2430 #16842363