Subgroups — C₃₁×#33895 #21490654

Order 7936.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3342	C₂	1	1	yes	maximal, characteristic
2	D₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2241	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2237	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2225	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2219	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2209	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2188	C₂	1	2	yes	maximal, complemented
9	C₄×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2169	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#2141	C₂	1	2	yes	maximal, complemented
12	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#556	—	2	1	no	—
14	D₄×Q₈×C₃₁	—	2	1	no	—
15	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×Q₃₂	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×#485	—	2	2	no	—
21	C₃₁×#475	—	2	1	no	—
22	C₃₁×#467	C₂²	1	2	yes	—
23	C₃₁×#467	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×Q₁₆×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	2	yes	—
31	C₃₁×#366	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
38	C₃₁×#358	C₂²	1	2	yes	—
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	1	yes	characteristic
47	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
48	C₂×Q₁₆×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×#126	—	4	2	no	—
54	C₃₁×#124	—	4	1	no	—
55	C₃₁×C₂²⋊Q₈	—	2	2	no	—
56	C₃₁×C₂²⋊Q₈	—	2	2	no	—
57	C₃₁×C₂²⋊Q₈	—	4	2	no	—
58	C₄×Q₈×C₃₁	—	2	1	no	—
59	C₄×Q₈×C₃₁	—	2	1	no	—
60	C₄×Q₈×C₃₁	—	4	1	no	—
61	C₄×D₄×C₃₁	—	4	1	no	—
62	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
63	C₃₁×#117	—	2	2	no	—
64	C₃₁×Q₃₂	—	2	2	no	—
65	C₃₁×Q₃₂	—	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	1	no	—
78	C₃₁×#103	—	2	2	no	—
79	C₃₁×#103	—	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	1	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂×Q₈×C₃₁	—	4	4	no	—
86	C₂×Q₈×C₃₁	—	4	1	no	—
87	C₂×Q₈×C₃₁	—	4	2	no	—
88	C₂×Q₈×C₃₁	—	8	2	no	—
89	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	—	4	2	no	—
91	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
92	Q₁₆×C₃₁	—	2	1	no	—
93	Q₁₆×C₃₁	—	2	2	no	—
94	Q₁₆×C₃₁	—	2	1	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	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₃₁	—	2	1	no	—
106	C₄⋊C₄×C₃₁	—	2	1	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	4	1	no	—
109	C₄⋊C₄×C₃₁	—	4	1	no	—
110	C₄⋊C₄×C₃₁	—	4	1	no	—
111	C₄⋊C₄×C₃₁	—	4	2	no	—
112	C₄⋊C₄×C₃₁	—	4	2	no	—
113	C₄⋊C₄×C₃₁	—	4	1	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	D₈	1	2	yes	—
120	C₄²×C₃₁	—	4	2	no	—
121	C₄²×C₃₁	—	4	1	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	#33895	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₃₁	C₂×D₈	1	2	yes	—
129	Q₈×C₃₁	—	2	2	no	—
130	Q₈×C₃₁	—	4	1	no	—
131	Q₈×C₃₁	—	4	2	no	—
132	Q₈×C₃₁	—	4	1	no	—
133	Q₈×C₃₁	—	4	2	no	—
134	Q₈×C₃₁	—	4	1	no	—
135	Q₈×C₃₁	—	8	4	no	—
136	Q₈×C₃₁	—	8	2	no	—
137	D₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	4	1	no	—
141	C₂×C₄×C₃₁	—	4	1	no	—
142	C₂×C₄×C₃₁	—	4	1	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	1	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	8	4	no	—
147	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₃₁	—	2	2	no	—
152	C₈×C₃₁	—	2	1	no	—
153	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
154	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
155	#3342	C₂×C₃₁	1	1	yes	characteristic
156	D₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
157	#2241	C₂×C₃₁	1	1	yes	characteristic, complemented
158	#2237	C₂×C₃₁	1	2	yes	complemented
159	#2225	C₂×C₃₁	1	1	yes	characteristic, complemented
160	#2219	C₂×C₃₁	1	2	yes	complemented
161	#2209	C₂×C₃₁	1	1	yes	characteristic, complemented
162	#2188	C₂×C₃₁	1	2	yes	complemented
163	C₄×Q₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
164	#2169	C₂×C₃₁	1	1	yes	characteristic
165	#2141	C₂×C₃₁	1	2	yes	complemented
166	C₂²×C₃₁	—	2	2	no	—
167	C₂²×C₃₁	—	2	2	no	—
168	C₂²×C₃₁	—	2	2	no	—
169	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
170	C₄×C₃₁	—	2	1	no	—
171	C₄×C₃₁	—	2	2	no	—
172	C₄×C₃₁	—	4	2	no	—
173	C₄×C₃₁	—	8	2	no	—
174	C₄×C₃₁	—	8	1	no	—
175	C₄×C₃₁	—	8	1	no	—
176	C₄×C₃₁	—	8	2	no	—
177	C₄×C₃₁	—	8	1	no	—
178	C₄×C₃₁	#510	1	2	yes	—
179	C₄×C₃₁	#546	1	1	yes	characteristic
180	C₄×C₃₁	#577	1	1	yes	characteristic
181	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
182	#556	—	2	1	no	—
183	D₄×Q₈	—	2	1	no	—
184	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
185	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
186	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
187	#500	C₂²×C₃₁	1	1	yes	characteristic
188	#494	C₂²×C₃₁	1	1	yes	characteristic
189	#485	—	2	2	no	—
190	#475	—	2	1	no	—
191	#467	C₂²×C₃₁	1	2	yes	—
192	#467	C₂²×C₃₁	1	2	yes	—
193	#462	—	2	1	no	—
194	#451	—	2	2	no	—
195	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
196	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
197	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
198	#427	C₂²×C₃₁	1	2	yes	—
199	#367	C₂²×C₃₁	1	2	yes	—
200	#366	C₂²×C₃₁	1	1	yes	characteristic
201	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
202	#358	C₂²×C₃₁	1	1	yes	characteristic
203	#358	C₂²×C₃₁	1	2	yes	—
204	#358	C₂²×C₃₁	1	2	yes	—
205	#358	C₂²×C₃₁	1	2	yes	—
206	#358	C₂²×C₃₁	1	1	yes	characteristic
207	#358	C₂²×C₃₁	1	2	yes	—
208	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
209	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
210	C₂×C₃₁	—	4	2	no	—
211	C₂×C₃₁	#3294	1	1	yes	characteristic
212	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
213	C₂×C₃₁	#3417	1	1	yes	characteristic
214	C₂²×Q₈	—	2	2	no	—
215	C₂×Q₁₆	—	2	4	no	—
216	C₂×Q₁₆	—	2	2	no	—
217	C₂×Q₁₆	—	2	1	no	—
218	C₂×Q₁₆	—	4	2	no	—
219	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
220	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
221	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
222	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
223	C₄⋊Q₈	—	2	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	4	1	no	—
226	#126	—	4	2	no	—
227	#124	—	4	1	no	—
228	C₂²⋊Q₈	—	2	2	no	—
229	C₂²⋊Q₈	—	2	2	no	—
230	C₂²⋊Q₈	—	4	2	no	—
231	C₄×Q₈	—	2	1	no	—
232	C₄×Q₈	—	2	1	no	—
233	C₄×Q₈	—	4	1	no	—
234	C₄×D₄	—	4	1	no	—
235	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
236	#117	—	2	2	no	—
237	Q₃₂	—	2	2	no	—
238	Q₃₂	—	4	2	no	—
239	M₃₂	—	4	2	no	—
240	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
241	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
242	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
243	#107	—	2	1	no	—
244	#107	C₂³×C₃₁	1	1	yes	characteristic
245	#107	C₂³×C₃₁	1	2	yes	—
246	#107	C₂³×C₃₁	1	1	yes	characteristic
247	#106	—	2	2	no	—
248	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
249	#103	—	2	1	no	—
250	#103	—	2	1	no	—
251	#103	—	2	2	no	—
252	#103	—	2	2	no	—
253	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
254	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
255	C₃₁	#33895	1	1	yes	characteristic, direct product factor, Sylow
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	4	4	no	—
260	C₂×Q₈	—	4	1	no	—
261	C₂×Q₈	—	4	2	no	—
262	C₂×Q₈	—	8	2	no	—
263	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	2	1	no	—
269	Q₁₆	—	2	1	no	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	4	4	no	—
272	Q₁₆	—	8	1	no	—
273	Q₁₆	#41×C₃₁	1	2	yes	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
278	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
294	C₄²	—	4	2	no	—
295	C₄²	—	4	1	no	—
296	C₄²	—	4	1	no	—
297	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
298	C₁₆	—	2	1	no	—
299	C₁₆	—	2	1	no	—
300	C₁₆	—	2	2	no	—
301	C₂³	C₂×D₈×C₃₁	1	2	yes	—
302	Q₈	—	2	2	no	—
303	Q₈	—	4	1	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	4	2	no	—
307	Q₈	—	4	1	no	—
308	Q₈	—	8	4	no	—
309	Q₈	—	8	2	no	—
310	D₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	8	4	no	—
320	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
321	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
322	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	1	no	—
326	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
327	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	2	no	—
334	C₄	—	4	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	1	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	2	no	—
339	C₄	—	8	1	no	—
340	C₄	C₃₁×#510	1	2	yes	—
341	C₄	C₃₁×#546	1	1	yes	characteristic
342	C₄	C₃₁×#577	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	C₃₁×#3294	1	1	yes	characteristic
345	C₂	C₂×C₃₁×#510	1	1	yes	characteristic
346	C₂	C₃₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×D₄×C₃₁	D₈
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
3. C₃₁×#2241	C₂	semidirect
4. C₃₁×#2237	C₂	semidirect
5. C₃₁×#2225	C₂	semidirect
6. C₃₁×#2219	C₂	semidirect
7. C₃₁×#2209	C₂	semidirect
8. C₃₁×#2188	C₂	semidirect
9. C₄×C₃₁×Q₃₂	C₂	semidirect
11. C₃₁×#2141	C₂	semidirect
127. #33895	C₃₁	direct (N × H)
157. #2241	C₂×C₃₁	semidirect
158. #2237	C₂×C₃₁	semidirect
159. #2225	C₂×C₃₁	semidirect
160. #2219	C₂×C₃₁	semidirect
161. #2209	C₂×C₃₁	semidirect
162. #2188	C₂×C₃₁	semidirect
163. C₄×Q₃₂	C₂×C₃₁	semidirect
165. #2141	C₂×C₃₁	semidirect
255. C₃₁	#33895	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×#3342	2
D₄×Q₁₆×C₃₁	2
C₃₁×#2241	2
C₃₁×#2237	2
C₃₁×#2225	2
C₃₁×#2219	2
C₃₁×#2209	2
C₃₁×#2188	2
C₄×C₃₁×Q₃₂	2
C₃₁×#2169	2
C₃₁×#2141	2
#33895	31

Subgroups — C31×#33895 #21490654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33895 #21490654