Subgroups — C₃₁×#33896 #21490655

Order 7936.

546 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3341	C₂	1	1	yes	maximal, characteristic
2	D₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2241	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2237	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2227	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2220	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2208	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2185	C₂	1	2	yes	maximal, complemented
9	C₄×C₃₁×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2169	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2141	C₂	1	2	yes	maximal, complemented
12	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#555	—	2	1	no	—
14	D₄×Q₈×C₃₁	—	2	1	no	—
15	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
17	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
18	C₃₁×#500	C₂²	1	1	yes	characteristic
19	C₃₁×#494	C₂²	1	1	yes	characteristic
20	C₃₁×#482	—	2	2	no	—
21	C₃₁×#476	—	2	1	no	—
22	C₃₁×#467	C₂²	1	2	yes	—
23	C₃₁×#466	C₂²	1	2	yes	—
24	C₃₁×#462	—	2	1	no	—
25	C₃₁×#451	—	2	2	no	—
26	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#427	C₂²	1	2	yes	—
30	C₃₁×#367	C₂²	1	1	yes	characteristic
31	C₃₁×#366	C₂²	1	2	yes	—
32	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₃₁×#358	C₂²	1	1	yes	characteristic
34	C₃₁×#358	C₂²	1	2	yes	—
35	C₃₁×#358	C₂²	1	2	yes	—
36	C₃₁×#358	C₂²	1	2	yes	—
37	C₃₁×#358	C₂²	1	2	yes	—
38	C₃₁×#357	C₂²	1	1	yes	characteristic
39	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₃₁	—	2	2	no	—
42	C₂×Q₁₆×C₃₁	—	2	4	no	—
43	C₂×Q₁₆×C₃₁	—	2	2	no	—
44	C₂×Q₁₆×C₃₁	—	2	1	no	—
45	C₂×Q₁₆×C₃₁	—	4	2	no	—
46	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
47	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₃₁×C₄⋊Q₈	—	2	1	no	—
51	C₃₁×C₄⋊Q₈	—	2	1	no	—
52	C₃₁×C₄⋊Q₈	—	4	1	no	—
53	C₃₁×#124	—	4	2	no	—
54	C₃₁×C₂²⋊Q₈	—	2	2	no	—
55	C₃₁×C₂²⋊Q₈	—	4	2	no	—
56	C₃₁×#121	—	2	2	no	—
57	C₃₁×#121	—	4	1	no	—
58	C₄×Q₈×C₃₁	—	2	1	no	—
59	C₄×D₄×C₃₁	—	2	1	no	—
60	C₄×D₄×C₃₁	—	4	1	no	—
61	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
62	C₃₁×#117	—	2	2	no	—
63	C₃₁×#117	—	4	1	no	—
64	C₃₁×SD₃₂	—	2	2	no	—
65	C₃₁×SD₃₂	—	4	2	no	—
66	C₃₁×M₃₂	—	4	2	no	—
67	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
69	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×#107	—	2	1	no	—
71	C₃₁×#107	C₂³	1	1	yes	characteristic
72	C₃₁×#107	C₂³	1	2	yes	—
73	C₃₁×#107	C₂³	1	1	yes	characteristic
74	C₃₁×#106	—	2	2	no	—
75	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
76	C₃₁×#103	—	2	1	no	—
77	C₃₁×#103	—	2	2	no	—
78	C₃₁×#102	—	2	1	no	—
79	C₃₁×#102	—	2	2	no	—
80	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
81	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₃₁	—	2	1	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	—	4	4	no	—
85	C₂×Q₈×C₃₁	—	4	2	no	—
86	C₂×Q₈×C₃₁	—	4	1	no	—
87	C₂×Q₈×C₃₁	—	8	2	no	—
88	C₂×D₄×C₃₁	—	2	1	no	—
89	C₂×D₄×C₃₁	—	4	2	no	—
90	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	—	4	2	no	—
92	C₂²×C₄×C₃₁	—	4	1	no	—
93	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
94	Q₁₆×C₃₁	—	2	2	no	—
95	Q₁₆×C₃₁	—	2	1	no	—
96	Q₁₆×C₃₁	—	2	2	no	—
97	Q₁₆×C₃₁	—	4	4	no	—
98	Q₁₆×C₃₁	—	8	1	no	—
99	Q₁₆×C₃₁	#41	1	2	yes	—
100	D₈×C₃₁	—	2	1	no	—
101	D₈×C₃₁	—	2	1	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₃₁	—	2	1	no	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	2	1	no	—
110	C₄⋊C₄×C₃₁	—	4	1	no	—
111	C₄⋊C₄×C₃₁	—	4	1	no	—
112	C₄⋊C₄×C₃₁	—	4	1	no	—
113	C₄⋊C₄×C₃₁	—	4	2	no	—
114	C₄⋊C₄×C₃₁	—	4	1	no	—
115	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
116	C₂²⋊C₄×C₃₁	—	4	2	no	—
117	C₂²⋊C₄×C₃₁	—	4	2	no	—
118	C₂²⋊C₄×C₃₁	—	4	2	no	—
119	C₂²⋊C₄×C₃₁	—	4	1	no	—
120	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
121	C₄²×C₃₁	—	4	2	no	—
122	C₄²×C₃₁	—	4	1	no	—
123	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₃₁	—	2	1	no	—
125	C₁₆×C₃₁	—	2	1	no	—
126	C₁₆×C₃₁	—	2	2	no	—
127	#33896	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₃₁	—	4	1	no	—
129	C₂³×C₃₁	C₂×D₈	1	2	yes	—
130	Q₈×C₃₁	—	2	2	no	—
131	Q₈×C₃₁	—	4	1	no	—
132	Q₈×C₃₁	—	4	2	no	—
133	Q₈×C₃₁	—	4	2	no	—
134	Q₈×C₃₁	—	8	4	no	—
135	Q₈×C₃₁	—	16	1	no	—
136	D₄×C₃₁	—	2	2	no	—
137	D₄×C₃₁	—	4	1	no	—
138	D₄×C₃₁	—	4	1	no	—
139	D₄×C₃₁	—	16	2	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	4	1	no	—
143	C₂×C₄×C₃₁	—	4	1	no	—
144	C₂×C₄×C₃₁	—	4	1	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	—	8	4	no	—
148	C₂×C₄×C₃₁	—	8	2	no	—
149	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
150	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₃₁	—	2	2	no	—
154	C₈×C₃₁	—	2	1	no	—
155	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
156	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
157	#3341	C₂×C₃₁	1	1	yes	characteristic
158	D₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
159	#2241	C₂×C₃₁	1	1	yes	characteristic, complemented
160	#2237	C₂×C₃₁	1	2	yes	complemented
161	#2227	C₂×C₃₁	1	1	yes	characteristic, complemented
162	#2220	C₂×C₃₁	1	2	yes	complemented
163	#2208	C₂×C₃₁	1	1	yes	characteristic, complemented
164	#2185	C₂×C₃₁	1	2	yes	complemented
165	C₄×SD₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
166	#2169	C₂×C₃₁	1	1	yes	characteristic, complemented
167	#2141	C₂×C₃₁	1	2	yes	complemented
168	C₂²×C₃₁	—	2	2	no	—
169	C₂²×C₃₁	—	2	2	no	—
170	C₂²×C₃₁	—	2	2	no	—
171	C₂²×C₃₁	—	8	1	no	—
172	C₂²×C₃₁	—	8	1	no	—
173	C₂²×C₃₁	—	8	1	no	—
174	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
175	C₄×C₃₁	—	2	1	no	—
176	C₄×C₃₁	—	2	2	no	—
177	C₄×C₃₁	—	4	2	no	—
178	C₄×C₃₁	—	8	1	no	—
179	C₄×C₃₁	—	8	1	no	—
180	C₄×C₃₁	—	8	2	no	—
181	C₄×C₃₁	—	8	2	no	—
182	C₄×C₃₁	#510	1	2	yes	—
183	C₄×C₃₁	#546	1	1	yes	characteristic
184	C₄×C₃₁	#577	1	1	yes	characteristic
185	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
186	#555	—	2	1	no	—
187	D₄×Q₈	—	2	1	no	—
188	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
189	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
190	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
191	#500	C₂²×C₃₁	1	1	yes	characteristic
192	#494	C₂²×C₃₁	1	1	yes	characteristic
193	#482	—	2	2	no	—
194	#476	—	2	1	no	—
195	#467	C₂²×C₃₁	1	2	yes	—
196	#466	C₂²×C₃₁	1	2	yes	—
197	#462	—	2	1	no	—
198	#451	—	2	2	no	—
199	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
200	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
201	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
202	#427	C₂²×C₃₁	1	2	yes	—
203	#367	C₂²×C₃₁	1	1	yes	characteristic
204	#366	C₂²×C₃₁	1	2	yes	—
205	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
206	#358	C₂²×C₃₁	1	1	yes	characteristic
207	#358	C₂²×C₃₁	1	2	yes	—
208	#358	C₂²×C₃₁	1	2	yes	—
209	#358	C₂²×C₃₁	1	2	yes	—
210	#358	C₂²×C₃₁	1	2	yes	—
211	#357	C₂²×C₃₁	1	1	yes	characteristic
212	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
213	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
214	C₂×C₃₁	—	4	2	no	—
215	C₂×C₃₁	—	16	1	no	—
216	C₂×C₃₁	#3294	1	1	yes	characteristic
217	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
218	C₂×C₃₁	#3416	1	1	yes	characteristic
219	C₂²×Q₈	—	2	2	no	—
220	C₂×Q₁₆	—	2	4	no	—
221	C₂×Q₁₆	—	2	2	no	—
222	C₂×Q₁₆	—	2	1	no	—
223	C₂×Q₁₆	—	4	2	no	—
224	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
225	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
226	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
227	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	4	1	no	—
231	#124	—	4	2	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	C₂²⋊Q₈	—	4	2	no	—
234	#121	—	2	2	no	—
235	#121	—	4	1	no	—
236	C₄×Q₈	—	2	1	no	—
237	C₄×D₄	—	2	1	no	—
238	C₄×D₄	—	4	1	no	—
239	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
240	#117	—	2	2	no	—
241	#117	—	4	1	no	—
242	SD₃₂	—	2	2	no	—
243	SD₃₂	—	4	2	no	—
244	M₃₂	—	4	2	no	—
245	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
246	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
247	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
248	#107	—	2	1	no	—
249	#107	C₂³×C₃₁	1	1	yes	characteristic
250	#107	C₂³×C₃₁	1	2	yes	—
251	#107	C₂³×C₃₁	1	1	yes	characteristic
252	#106	—	2	2	no	—
253	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
254	#103	—	2	1	no	—
255	#103	—	2	2	no	—
256	#102	—	2	1	no	—
257	#102	—	2	2	no	—
258	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
259	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
260	C₃₁	#33896	1	1	yes	characteristic, direct product factor, Sylow
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	4	4	no	—
264	C₂×Q₈	—	4	2	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	8	2	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	4	2	no	—
269	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
273	Q₁₆	—	2	2	no	—
274	Q₁₆	—	2	1	no	—
275	Q₁₆	—	2	2	no	—
276	Q₁₆	—	4	4	no	—
277	Q₁₆	—	8	1	no	—
278	Q₁₆	#41×C₃₁	1	2	yes	—
279	D₈	—	2	1	no	—
280	D₈	—	2	1	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
284	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
285	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
300	C₄²	—	4	2	no	—
301	C₄²	—	4	1	no	—
302	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
303	C₁₆	—	2	1	no	—
304	C₁₆	—	2	1	no	—
305	C₁₆	—	2	2	no	—
306	C₂³	—	4	1	no	—
307	C₂³	C₂×D₈×C₃₁	1	2	yes	—
308	Q₈	—	2	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	8	4	no	—
313	Q₈	—	16	1	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	1	no	—
316	D₄	—	4	1	no	—
317	D₄	—	16	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	4	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
328	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
329	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	2	1	no	—
333	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
334	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	8	1	no	—
339	C₂²	—	8	1	no	—
340	C₂²	—	8	1	no	—
341	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	2	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	2	no	—
349	C₄	C₃₁×#510	1	2	yes	—
350	C₄	C₃₁×#546	1	1	yes	characteristic
351	C₄	C₃₁×#577	1	1	yes	characteristic
352	C₂	—	4	2	no	—
353	C₂	—	16	1	no	—
354	C₂	C₃₁×#3294	1	1	yes	characteristic
355	C₂	C₂×C₃₁×#510	1	1	yes	characteristic
356	C₂	C₃₁×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#3341	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ C₈×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. D₄×Q₁₆×C₃₁	C₂	semidirect
3. C₃₁×#2241	C₂	semidirect
4. C₃₁×#2237	C₂	semidirect
5. C₃₁×#2227	C₂	semidirect
6. C₃₁×#2220	C₂	semidirect
7. C₃₁×#2208	C₂	semidirect
8. C₃₁×#2185	C₂	semidirect
9. C₄×C₃₁×SD₃₂	C₂	semidirect
10. C₃₁×#2169	C₂	semidirect
11. C₃₁×#2141	C₂	semidirect
127. #33896	C₃₁	direct (N × H)
158. D₄×Q₁₆	C₂×C₃₁	semidirect
159. #2241	C₂×C₃₁	semidirect
160. #2237	C₂×C₃₁	semidirect
161. #2227	C₂×C₃₁	semidirect
162. #2220	C₂×C₃₁	semidirect
163. #2208	C₂×C₃₁	semidirect
164. #2185	C₂×C₃₁	semidirect
165. C₄×SD₃₂	C₂×C₃₁	semidirect
166. #2169	C₂×C₃₁	semidirect
167. #2141	C₂×C₃₁	semidirect
260. C₃₁	#33896	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×#3341	2
D₄×Q₁₆×C₃₁	2
C₃₁×#2241	2
C₃₁×#2237	2
C₃₁×#2227	2
C₃₁×#2220	2
C₃₁×#2208	2
C₃₁×#2185	2
C₄×C₃₁×SD₃₂	2
C₃₁×#2169	2
C₃₁×#2141	2
#33896	31

Subgroups — C31×#33896 #21490655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33896 #21490655