Subgroups — C₃₁×#62637 #21519396

Order 7936.

2026 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3584	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#578	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3417	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3416	C₂	1	8	yes	maximal, complemented
5	C₃₁×#3414	C₂	1	6	yes	maximal, complemented
6	C₂×C₃₁×#510	C₂	1	6	yes	maximal, complemented
7	C₃₁×#3407	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#585	—	2	1	no	—
9	C₂×C₃₁×2₋¹⁺⁴	—	2	1	no	—
10	C₃₁×#578	—	2	4	no	—
11	C₃₁×#578	C₂²	1	8	yes	complemented
12	C₃₁×#578	C₂²	1	4	yes	complemented
13	C₃₁×#577	—	2	4	no	—
14	C₃₁×#576	C₂²	1	4	yes	complemented
15	C₃₁×#575	—	2	3	no	—
16	C₂×C₃₁×#137	—	2	3	no	—
17	C₂×C₃₁×#135	C₂²	1	6	yes	—
18	C₂×C₃₁×#135	C₂²	1	3	yes	complemented
19	C₂²×Q₁₆×C₃₁	C₂²	1	3	yes	complemented
20	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
21	C₃₁×#510	C₂²	1	12	yes	complemented
22	C₃₁×#510	C₂²	1	24	yes	complemented
23	C₃₁×#510	C₂²	1	12	yes	complemented
24	C₃₁×#509	C₂²	1	4	yes	complemented
25	C₃₁×#509	C₂²	1	12	yes	complemented
26	C₃₁×#508	C₂²	1	24	yes	complemented
27	C₂×C₃₁×Q₃₂	C₂²	1	12	yes	complemented
28	C₂×C₃₁×SD₃₂	C₂²	1	12	yes	complemented
29	C₃₁×#504	C₂²	1	8	yes	complemented
30	C₂×C₃₁×M₃₂	C₂²	1	6	yes	complemented
31	C₃₁×2₋¹⁺⁴	—	2	4	no	—
32	C₃₁×2₋¹⁺⁴	—	2	8	no	—
33	C₃₁×2₋¹⁺⁴	—	4	4	no	—
34	C₃₁×2₋¹⁺⁴	—	4	1	no	—
35	C₃₁×H(5,𝔽₂)	—	2	4	no	—
36	C₃₁×H(5,𝔽₂)	—	4	3	no	—
37	C₂×#41×C₃₁	—	2	6	no	—
38	C₂×#41×C₃₁	—	2	3	no	—
39	C₂×#41×C₃₁	—	4	4	no	—
40	C₂×#41×C₃₁	—	4	3	no	—
41	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₃₁	—	2	3	no	—
43	C₂²×Q₈×C₃₁	—	4	1	no	—
44	C₃₁×#137	—	2	24	no	—
45	C₃₁×#137	—	2	12	no	—
46	C₃₁×AGL(1,ℤ/8)	—	2	12	no	—
47	C₃₁×#135	—	2	6	no	—
48	C₃₁×#135	—	2	6	no	—
49	C₃₁×#135	—	2	6	no	—
50	C₃₁×#135	C₂³	1	12	yes	—
51	C₃₁×#135	C₂³	1	12	yes	—
52	C₃₁×#135	C₂³	1	4	yes	—
53	C₂×Q₁₆×C₃₁	—	2	3	no	—
54	C₂×Q₁₆×C₃₁	—	2	6	no	—
55	C₂×Q₁₆×C₃₁	C₂³	1	12	yes	—
56	C₂×Q₁₆×C₃₁	C₂³	1	6	yes	—
57	C₂×Q₁₆×C₃₁	C₂³	1	3	yes	—
58	C₂×SD₁₆×C₃₁	—	2	6	no	—
59	C₂×SD₁₆×C₃₁	—	2	3	no	—
60	C₂×D₈×C₃₁	C₂³	1	6	yes	—
61	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₃₁×#131	D₄	1	4	yes	—
63	C₃₁×#131	C₂³	1	4	yes	—
64	C₂×M₁₆×C₃₁	D₄	1	3	yes	—
65	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
66	C₃₁×Q₃₂	C₂³	1	24	yes	—
67	C₃₁×SD₃₂	C₂³	1	24	yes	—
68	C₃₁×SD₃₂	C₂³	1	8	yes	—
69	C₃₁×D₁₆	C₂³	1	8	yes	—
70	C₃₁×M₃₂	C₂³	1	4	yes	—
71	C₃₁×M₃₂	C₂³	1	12	yes	—
72	C₂×C₁₆×C₃₁	C₂³	1	12	yes	—
73	#41×C₃₁	—	2	4	no	—
74	#41×C₃₁	—	2	12	no	—
75	#41×C₃₁	—	2	12	no	—
76	#41×C₃₁	—	4	4	no	—
77	#41×C₃₁	—	4	12	no	—
78	#41×C₃₁	—	4	24	no	—
79	#41×C₃₁	—	4	6	no	—
80	#41×C₃₁	—	4	6	no	—
81	#41×C₃₁	—	4	6	no	—
82	#41×C₃₁	D₈	1	4	yes	—
83	#41×C₃₁	C₂×D₄	1	4	yes	—
84	C₂×Q₈×C₃₁	—	2	3	no	—
85	C₂×Q₈×C₃₁	—	2	6	no	—
86	C₂×Q₈×C₃₁	—	2	12	no	—
87	C₂×Q₈×C₃₁	—	4	8	no	—
88	C₂×Q₈×C₃₁	—	4	4	no	—
89	C₂×Q₈×C₃₁	—	4	3	no	—
90	C₂×Q₈×C₃₁	—	4	6	no	—
91	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
92	C₂×D₄×C₃₁	—	2	1	no	—
93	C₂×D₄×C₃₁	—	2	6	no	—
94	C₂×D₄×C₃₁	—	4	3	no	—
95	C₂×D₄×C₃₁	—	4	6	no	—
96	C₂×D₄×C₃₁	—	4	12	no	—
97	C₂×D₄×C₃₁	D₈	1	3	yes	—
98	C₂²×C₄×C₃₁	—	4	3	no	—
99	C₂²×C₄×C₃₁	—	4	3	no	—
100	C₂²×C₄×C₃₁	—	4	6	no	—
101	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
102	Q₁₆×C₃₁	—	2	3	no	—
103	Q₁₆×C₃₁	—	2	12	no	—
104	Q₁₆×C₃₁	—	2	6	no	—
105	Q₁₆×C₃₁	C₂⁴	1	6	yes	—
106	Q₁₆×C₃₁	C₂⁴	1	4	yes	—
107	SD₁₆×C₃₁	—	2	4	no	—
108	SD₁₆×C₃₁	—	2	12	no	—
109	SD₁₆×C₃₁	—	2	12	no	—
110	D₈×C₃₁	—	2	1	no	—
111	D₈×C₃₁	—	2	6	no	—
112	D₈×C₃₁	C₂⁴	1	2	yes	—
113	D₈×C₃₁	C₂⁴	1	4	yes	—
114	M₁₆×C₃₁	C₂×D₄	1	12	yes	—
115	C₂×C₈×C₃₁	C₂×D₄	1	3	yes	—
116	C₂×C₈×C₃₁	C₂×D₄	1	6	yes	—
117	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
118	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
119	C₁₆×C₃₁	C₂⁴	1	8	yes	—
120	#62637	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₃₁	—	4	1	no	—
122	C₂³×C₃₁	—	4	6	no	—
123	C₂³×C₃₁	C₂×D₈	1	3	yes	—
124	Q₈×C₃₁	—	2	4	no	—
125	Q₈×C₃₁	—	2	6	no	—
126	Q₈×C₃₁	—	4	3	no	—
127	Q₈×C₃₁	—	4	12	no	—
128	Q₈×C₃₁	—	4	6	no	—
129	Q₈×C₃₁	C₂×D₈	1	4	yes	—
130	D₄×C₃₁	—	2	4	no	—
131	D₄×C₃₁	—	2	2	no	—
132	D₄×C₃₁	—	4	4	no	—
133	D₄×C₃₁	—	4	12	no	—
134	D₄×C₃₁	—	4	12	no	—
135	D₄×C₃₁	—	4	6	no	—
136	D₄×C₃₁	—	4	1	no	—
137	D₄×C₃₁	C₂×D₈	1	12	yes	—
138	C₂×C₄×C₃₁	—	4	3	no	—
139	C₂×C₄×C₃₁	—	4	6	no	—
140	C₂×C₄×C₃₁	—	4	12	no	—
141	C₂×C₄×C₃₁	—	4	4	no	—
142	C₂×C₄×C₃₁	—	4	12	no	—
143	C₂×C₄×C₃₁	—	4	12	no	—
144	C₂×C₄×C₃₁	C₂×D₈	1	3	yes	—
145	C₂×C₄×C₃₁	C₂×D₈	1	6	yes	—
146	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
148	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₃₁	C₂²×D₄	1	6	yes	—
150	C₈×C₃₁	C₂⁵	1	1	yes	characteristic
151	#3584	C₂×C₃₁	1	1	yes	characteristic, complemented
152	C₂×#578	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#3417	C₂×C₃₁	1	8	yes	complemented
154	#3416	C₂×C₃₁	1	8	yes	complemented
155	#3414	C₂×C₃₁	1	6	yes	complemented
156	C₂×#510	C₂×C₃₁	1	6	yes	complemented
157	#3407	C₂×C₃₁	1	1	yes	characteristic, complemented
158	C₂²×C₃₁	—	4	3	no	—
159	C₂²×C₃₁	—	4	2	no	—
160	C₂²×C₃₁	—	4	4	no	—
161	C₂²×C₃₁	—	8	2	no	—
162	C₂²×C₃₁	—	16	6	no	—
163	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
164	C₂²×C₃₁	C₂²×D₈	1	6	yes	—
165	C₄×C₃₁	—	4	4	no	—
166	C₄×C₃₁	—	4	6	no	—
167	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
168	C₄×C₃₁	C₂²×D₈	1	6	yes	—
169	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
170	#585	—	2	1	no	—
171	C₂×2₋¹⁺⁴	—	2	1	no	—
172	#578	—	2	4	no	—
173	#578	C₂²×C₃₁	1	4	yes	complemented
174	#578	C₂²×C₃₁	1	8	yes	complemented
175	#577	—	2	4	no	—
176	#576	C₂²×C₃₁	1	4	yes	complemented
177	#575	—	2	3	no	—
178	C₂×#137	—	2	3	no	—
179	C₂×#135	C₂²×C₃₁	1	6	yes	—
180	C₂×#135	C₂²×C₃₁	1	3	yes	complemented
181	C₂²×Q₁₆	C₂²×C₃₁	1	3	yes	complemented
182	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
183	#510	C₂²×C₃₁	1	12	yes	complemented
184	#510	C₂²×C₃₁	1	24	yes	complemented
185	#510	C₂²×C₃₁	1	12	yes	complemented
186	#509	C₂²×C₃₁	1	4	yes	complemented
187	#509	C₂²×C₃₁	1	12	yes	complemented
188	#508	C₂²×C₃₁	1	24	yes	complemented
189	C₂×Q₃₂	C₂²×C₃₁	1	12	yes	complemented
190	C₂×SD₃₂	C₂²×C₃₁	1	12	yes	complemented
191	#504	C₂²×C₃₁	1	8	yes	complemented
192	C₂×M₃₂	C₂²×C₃₁	1	6	yes	complemented
193	C₂×C₃₁	—	2	1	no	—
194	C₂×C₃₁	—	2	6	no	—
195	C₂×C₃₁	—	8	2	no	—
196	C₂×C₃₁	—	8	4	no	—
197	C₂×C₃₁	C₂³×D₈	1	1	yes	characteristic
198	2₋¹⁺⁴	—	2	4	no	—
199	2₋¹⁺⁴	—	2	8	no	—
200	2₋¹⁺⁴	—	4	4	no	—
201	2₋¹⁺⁴	—	4	1	no	—
202	H(5,𝔽₂)	—	2	4	no	—
203	H(5,𝔽₂)	—	4	3	no	—
204	C₂×#41	—	2	6	no	—
205	C₂×#41	—	2	3	no	—
206	C₂×#41	—	4	4	no	—
207	C₂×#41	—	4	3	no	—
208	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
209	C₂²×Q₈	—	2	3	no	—
210	C₂²×Q₈	—	4	1	no	—
211	#137	—	2	24	no	—
212	#137	—	2	12	no	—
213	AGL(1,ℤ/8)	—	2	12	no	—
214	#135	—	2	6	no	—
215	#135	—	2	6	no	—
216	#135	—	2	6	no	—
217	#135	C₂³×C₃₁	1	4	yes	—
218	#135	C₂³×C₃₁	1	12	yes	—
219	#135	C₂³×C₃₁	1	12	yes	—
220	C₂×Q₁₆	—	2	3	no	—
221	C₂×Q₁₆	—	2	6	no	—
222	C₂×Q₁₆	C₂³×C₃₁	1	3	yes	—
223	C₂×Q₁₆	C₂³×C₃₁	1	12	yes	—
224	C₂×Q₁₆	C₂³×C₃₁	1	6	yes	—
225	C₂×SD₁₆	—	2	6	no	—
226	C₂×SD₁₆	—	2	3	no	—
227	C₂×D₈	C₂³×C₃₁	1	6	yes	—
228	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
229	#131	D₄×C₃₁	1	4	yes	—
230	#131	C₂³×C₃₁	1	4	yes	—
231	C₂×M₁₆	D₄×C₃₁	1	3	yes	—
232	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
233	Q₃₂	C₂³×C₃₁	1	24	yes	—
234	SD₃₂	C₂³×C₃₁	1	24	yes	—
235	SD₃₂	C₂³×C₃₁	1	8	yes	—
236	D₁₆	C₂³×C₃₁	1	8	yes	—
237	M₃₂	C₂³×C₃₁	1	4	yes	—
238	M₃₂	C₂³×C₃₁	1	12	yes	—
239	C₂×C₁₆	C₂³×C₃₁	1	12	yes	—
240	C₃₁	#62637	1	1	yes	characteristic, direct product factor, Sylow
241	#41	—	2	4	no	—
242	#41	—	2	12	no	—
243	#41	—	2	12	no	—
244	#41	—	4	4	no	—
245	#41	—	4	12	no	—
246	#41	—	4	24	no	—
247	#41	—	4	6	no	—
248	#41	—	4	6	no	—
249	#41	—	4	6	no	—
250	#41	D₈×C₃₁	1	4	yes	—
251	#41	C₂×D₄×C₃₁	1	4	yes	—
252	C₂×Q₈	—	2	3	no	—
253	C₂×Q₈	—	2	6	no	—
254	C₂×Q₈	—	2	12	no	—
255	C₂×Q₈	—	4	8	no	—
256	C₂×Q₈	—	4	4	no	—
257	C₂×Q₈	—	4	3	no	—
258	C₂×Q₈	—	4	6	no	—
259	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	6	no	—
262	C₂×D₄	—	4	3	no	—
263	C₂×D₄	—	4	6	no	—
264	C₂×D₄	—	4	12	no	—
265	C₂×D₄	D₈×C₃₁	1	3	yes	—
266	C₂²×C₄	—	4	3	no	—
267	C₂²×C₄	—	4	3	no	—
268	C₂²×C₄	—	4	6	no	—
269	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
270	Q₁₆	—	2	3	no	—
271	Q₁₆	—	2	12	no	—
272	Q₁₆	—	2	6	no	—
273	Q₁₆	C₂⁴×C₃₁	1	6	yes	—
274	Q₁₆	C₂⁴×C₃₁	1	4	yes	—
275	SD₁₆	—	2	4	no	—
276	SD₁₆	—	2	12	no	—
277	SD₁₆	—	2	12	no	—
278	D₈	—	2	1	no	—
279	D₈	—	2	6	no	—
280	D₈	C₂⁴×C₃₁	1	2	yes	—
281	D₈	C₂⁴×C₃₁	1	4	yes	—
282	M₁₆	C₂×D₄×C₃₁	1	12	yes	—
283	C₂×C₈	C₂×D₄×C₃₁	1	3	yes	—
284	C₂×C₈	C₂×D₄×C₃₁	1	6	yes	—
285	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
286	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
287	C₁₆	C₂⁴×C₃₁	1	8	yes	—
288	C₂³	—	4	1	no	—
289	C₂³	—	4	6	no	—
290	C₂³	C₂×D₈×C₃₁	1	3	yes	—
291	Q₈	—	2	4	no	—
292	Q₈	—	2	6	no	—
293	Q₈	—	4	3	no	—
294	Q₈	—	4	12	no	—
295	Q₈	—	4	6	no	—
296	Q₈	C₂×D₈×C₃₁	1	4	yes	—
297	D₄	—	2	4	no	—
298	D₄	—	2	2	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	12	no	—
301	D₄	—	4	12	no	—
302	D₄	—	4	6	no	—
303	D₄	—	4	1	no	—
304	D₄	C₂×D₈×C₃₁	1	12	yes	—
305	C₂×C₄	—	4	3	no	—
306	C₂×C₄	—	4	6	no	—
307	C₂×C₄	—	4	12	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	12	no	—
310	C₂×C₄	—	4	12	no	—
311	C₂×C₄	C₂×D₈×C₃₁	1	3	yes	—
312	C₂×C₄	C₂×D₈×C₃₁	1	6	yes	—
313	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
315	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
316	C₈	C₂²×D₄×C₃₁	1	6	yes	—
317	C₈	C₂⁵×C₃₁	1	1	yes	characteristic
318	C₂²	—	4	3	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	4	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	16	6	no	—
323	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
324	C₂²	C₂²×D₈×C₃₁	1	6	yes	—
325	C₄	—	4	4	no	—
326	C₄	—	4	6	no	—
327	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
328	C₄	C₂²×D₈×C₃₁	1	6	yes	—
329	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	6	no	—
332	C₂	—	8	2	no	—
333	C₂	—	8	4	no	—
334	C₂	C₂³×D₈×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	C₂³×D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₃₁
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂×#41×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3584	C₂	semidirect
2. C₂×C₃₁×#578	C₂	semidirect
3. C₃₁×#3417	C₂	semidirect
4. C₃₁×#3416	C₂	semidirect
5. C₃₁×#3414	C₂	semidirect
6. C₂×C₃₁×#510	C₂	semidirect
7. C₃₁×#3407	C₂	semidirect
11. C₃₁×#578	C₂²	semidirect
12. C₃₁×#578	C₂²	semidirect
14. C₃₁×#576	C₂²	semidirect
18. C₂×C₃₁×#135	C₂²	semidirect
19. C₂²×Q₁₆×C₃₁	C₂²	semidirect
21. C₃₁×#510	C₂²	semidirect
22. C₃₁×#510	C₂²	semidirect
23. C₃₁×#510	C₂²	semidirect
24. C₃₁×#509	C₂²	semidirect
25. C₃₁×#509	C₂²	semidirect
26. C₃₁×#508	C₂²	semidirect
27. C₂×C₃₁×Q₃₂	C₂²	semidirect
28. C₂×C₃₁×SD₃₂	C₂²	semidirect
29. C₃₁×#504	C₂²	semidirect
30. C₂×C₃₁×M₃₂	C₂²	semidirect
120. #62637	C₃₁	direct (N × H)
151. #3584	C₂×C₃₁	semidirect
152. C₂×#578	C₂×C₃₁	semidirect
153. #3417	C₂×C₃₁	semidirect
154. #3416	C₂×C₃₁	semidirect
155. #3414	C₂×C₃₁	semidirect
156. C₂×#510	C₂×C₃₁	semidirect
157. #3407	C₂×C₃₁	semidirect
173. #578	C₂²×C₃₁	semidirect
174. #578	C₂²×C₃₁	semidirect
176. #576	C₂²×C₃₁	semidirect
180. C₂×#135	C₂²×C₃₁	semidirect
181. C₂²×Q₁₆	C₂²×C₃₁	semidirect
183. #510	C₂²×C₃₁	semidirect
184. #510	C₂²×C₃₁	semidirect
185. #510	C₂²×C₃₁	semidirect
186. #509	C₂²×C₃₁	semidirect
187. #509	C₂²×C₃₁	semidirect
188. #508	C₂²×C₃₁	semidirect
189. C₂×Q₃₂	C₂²×C₃₁	semidirect
190. C₂×SD₃₂	C₂²×C₃₁	semidirect
191. #504	C₂²×C₃₁	semidirect
192. C₂×M₃₂	C₂²×C₃₁	semidirect
240. C₃₁	#62637	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3584	2
C₂×C₃₁×#578	2
C₃₁×#3417	2
C₃₁×#3416	2
C₃₁×#3414	2
C₂×C₃₁×#510	2
C₃₁×#3407	2
#62637	31

Subgroups — C31×#62637 #21519396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62637 #21519396