Subgroups — C₃₁×#33609 #21490368

Order 7936.

682 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3059	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3058	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2220	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2217	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2141	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2140	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
13	C₃₁×#536	—	2	1	no	—
14	C₃₁×#535	—	2	1	no	—
15	C₂×C₃₁×SD₃₂	—	2	4	no	—
16	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
17	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
18	C₃₁×#504	—	2	4	no	—
19	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×M₃₂	C₂²	1	2	yes	complemented
21	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	complemented
23	C₃₁×#470	—	2	2	no	—
24	C₃₁×#469	—	2	2	no	—
25	C₃₁×#467	C₂²	1	4	yes	complemented
26	C₃₁×#466	C₂²	1	4	yes	—
27	C₃₁×#427	C₂²	1	1	yes	characteristic
28	C₃₁×#427	C₂²	1	1	yes	characteristic
29	C₃₁×#419	—	2	1	no	—
30	C₃₁×#418	—	2	1	no	—
31	C₃₁×#367	C₂²	1	2	yes	complemented
32	C₃₁×#366	C₂²	1	2	yes	complemented
33	C₃₁×#358	C₂²	1	4	yes	complemented
34	C₃₁×#357	C₂²	1	4	yes	—
35	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₃₁	—	2	1	no	—
37	C₂²×D₄×C₃₁	—	2	1	no	—
38	C₂×Q₁₆×C₃₁	—	2	2	no	—
39	C₂×Q₁₆×C₃₁	—	2	1	no	—
40	C₂×Q₁₆×C₃₁	—	2	2	no	—
41	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
42	C₂×D₈×C₃₁	—	2	2	no	—
43	C₂×D₈×C₃₁	—	2	1	no	—
44	C₂×D₈×C₃₁	—	2	2	no	—
45	C₂×D₈×C₃₁	C₂³	1	2	yes	—
46	C₃₁×#131	—	2	2	no	—
47	C₃₁×#131	—	2	2	no	—
48	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
49	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×C₄⋊Q₈	—	4	1	no	—
53	C₃₁×#127	—	4	1	no	—
54	C₃₁×#124	—	4	1	no	—
55	C₃₁×#124	—	4	1	no	—
56	C₃₁×#123	—	4	2	no	—
57	C₃₁×C₂²⋊Q₈	—	2	4	no	—
58	C₃₁×#121	—	2	4	no	—
59	C₃₁×#120	—	4	2	no	—
60	C₃₁×#117	—	2	1	no	—
61	C₃₁×#117	—	2	1	no	—
62	C₃₁×SD₃₂	—	2	4	no	—
63	C₃₁×SD₃₂	—	2	4	no	—
64	C₃₁×M₃₂	—	2	2	no	—
65	C₃₁×M₃₂	—	2	4	no	—
66	C₂×C₁₆×C₃₁	—	2	4	no	—
67	C₂×C₁₆×C₃₁	D₄	1	4	yes	complemented
68	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
69	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#107	C₂³	1	2	yes	—
72	C₃₁×#107	C₂³	1	2	yes	—
73	C₃₁×#106	—	2	1	no	—
74	C₃₁×#106	—	2	1	no	—
75	C₃₁×#103	—	2	4	no	—
76	C₃₁×#102	—	2	4	no	—
77	C₂⁴×C₃₁	—	4	1	no	—
78	#41×C₃₁	—	2	2	no	—
79	#41×C₃₁	—	2	2	no	—
80	C₂×Q₈×C₃₁	—	2	2	no	—
81	C₂×Q₈×C₃₁	—	4	1	no	—
82	C₂×Q₈×C₃₁	—	4	2	no	—
83	C₂×Q₈×C₃₁	—	4	2	no	—
84	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	4	4	no	—
87	C₂×D₄×C₃₁	—	4	2	no	—
88	C₂×D₄×C₃₁	—	4	1	no	—
89	C₂×D₄×C₃₁	—	4	2	no	—
90	C₂×D₄×C₃₁	D₈	1	2	yes	—
91	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	—	4	1	no	—
93	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
95	Q₁₆×C₃₁	—	2	2	no	—
96	Q₁₆×C₃₁	—	2	2	no	—
97	Q₁₆×C₃₁	—	4	2	no	—
98	D₈×C₃₁	—	2	2	no	—
99	D₈×C₃₁	—	2	2	no	—
100	D₈×C₃₁	—	4	2	no	—
101	M₁₆×C₃₁	—	2	2	no	—
102	M₁₆×C₃₁	—	2	4	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₃₁	#41	1	2	yes	—
109	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₄⋊C₄×C₃₁	—	4	4	no	—
113	C₂²⋊C₄×C₃₁	—	4	1	no	—
114	C₂²⋊C₄×C₃₁	—	4	1	no	—
115	C₂²⋊C₄×C₃₁	—	4	4	no	—
116	C₂²⋊C₄×C₃₁	—	4	4	no	—
117	C₄²×C₃₁	—	4	1	no	—
118	C₄²×C₃₁	—	4	1	no	—
119	C₁₆×C₃₁	—	2	2	no	—
120	C₁₆×C₃₁	C₂×D₄	1	2	yes	—
121	C₁₆×C₃₁	C₂×D₄	1	2	yes	—
122	#33609	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂³×C₃₁	—	4	2	no	—
124	C₂³×C₃₁	—	8	2	no	—
125	C₂³×C₃₁	—	16	2	no	—
126	C₂³×C₃₁	C₂×D₈	1	2	yes	—
127	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
128	Q₈×C₃₁	—	2	2	no	—
129	Q₈×C₃₁	—	4	2	no	—
130	Q₈×C₃₁	—	4	2	no	—
131	Q₈×C₃₁	—	8	2	no	—
132	D₄×C₃₁	—	2	4	no	—
133	D₄×C₃₁	—	2	2	no	—
134	D₄×C₃₁	—	4	2	no	—
135	D₄×C₃₁	—	4	2	no	—
136	D₄×C₃₁	—	8	2	no	—
137	D₄×C₃₁	—	16	4	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	4	2	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	—	8	2	no	—
144	C₂×C₄×C₃₁	#121	1	2	yes	—
145	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₃₁	#135	1	2	yes	—
148	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₃₁	—	2	2	no	—
150	C₈×C₃₁	#121	1	2	yes	—
151	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
153	C₂²×SD₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
154	C₂×#504	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#3059	C₂×C₃₁	1	1	yes	characteristic, complemented
156	#3058	C₂×C₃₁	1	1	yes	characteristic
157	#2220	C₂×C₃₁	1	4	yes	complemented
158	#2217	C₂×C₃₁	1	4	yes	complemented
159	#2158	C₂×C₃₁	1	1	yes	characteristic, complemented
160	#2141	C₂×C₃₁	1	1	yes	characteristic, complemented
161	#2140	C₂×C₃₁	1	1	yes	characteristic, complemented
162	C₂²×C₃₁	—	2	2	no	—
163	C₂²×C₃₁	—	2	4	no	—
164	C₂²×C₃₁	—	2	2	no	—
165	C₂²×C₃₁	—	8	4	no	—
166	C₂²×C₃₁	—	8	2	no	—
167	C₂²×C₃₁	—	16	4	no	—
168	C₂²×C₃₁	#466	1	2	yes	—
169	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
170	C₄×C₃₁	—	2	2	no	—
171	C₄×C₃₁	—	8	2	no	—
172	C₄×C₃₁	—	8	2	no	—
173	C₄×C₃₁	—	8	2	no	—
174	C₄×C₃₁	#466	1	2	yes	—
175	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
176	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
177	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
178	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
179	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
180	#536	—	2	1	no	—
181	#535	—	2	1	no	—
182	C₂×SD₃₂	—	2	4	no	—
183	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
184	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
185	#504	—	2	4	no	—
186	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
187	C₂×M₃₂	C₂²×C₃₁	1	2	yes	complemented
188	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
189	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	complemented
190	#470	—	2	2	no	—
191	#469	—	2	2	no	—
192	#467	C₂²×C₃₁	1	4	yes	complemented
193	#466	C₂²×C₃₁	1	4	yes	—
194	#427	C₂²×C₃₁	1	1	yes	characteristic
195	#427	C₂²×C₃₁	1	1	yes	characteristic
196	#419	—	2	1	no	—
197	#418	—	2	1	no	—
198	#367	C₂²×C₃₁	1	2	yes	complemented
199	#366	C₂²×C₃₁	1	2	yes	complemented
200	#358	C₂²×C₃₁	1	4	yes	complemented
201	#357	C₂²×C₃₁	1	4	yes	—
202	C₂×C₃₁	—	2	2	no	—
203	C₂×C₃₁	—	4	2	no	—
204	C₂×C₃₁	—	16	2	no	—
205	C₂×C₃₁	C₂×#466	1	1	yes	characteristic
206	C₂×C₃₁	#3416	1	2	yes	—
207	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
208	C₂²×Q₈	—	2	1	no	—
209	C₂²×D₄	—	2	1	no	—
210	C₂×Q₁₆	—	2	2	no	—
211	C₂×Q₁₆	—	2	1	no	—
212	C₂×Q₁₆	—	2	2	no	—
213	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
214	C₂×D₈	—	2	2	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	2	2	no	—
217	C₂×D₈	C₂³×C₃₁	1	2	yes	—
218	#131	—	2	2	no	—
219	#131	—	2	2	no	—
220	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
221	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
222	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
223	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
224	C₄⋊Q₈	—	4	1	no	—
225	#127	—	4	1	no	—
226	#124	—	4	1	no	—
227	#124	—	4	1	no	—
228	#123	—	4	2	no	—
229	C₂²⋊Q₈	—	2	4	no	—
230	#121	—	2	4	no	—
231	#120	—	4	2	no	—
232	#117	—	2	1	no	—
233	#117	—	2	1	no	—
234	SD₃₂	—	2	4	no	—
235	SD₃₂	—	2	4	no	—
236	M₃₂	—	2	2	no	—
237	M₃₂	—	2	4	no	—
238	C₂×C₁₆	—	2	4	no	—
239	C₂×C₁₆	D₄×C₃₁	1	4	yes	complemented
240	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
241	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
242	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
243	#107	C₂³×C₃₁	1	2	yes	—
244	#107	C₂³×C₃₁	1	2	yes	—
245	#106	—	2	1	no	—
246	#106	—	2	1	no	—
247	#103	—	2	4	no	—
248	#102	—	2	4	no	—
249	C₃₁	#33609	1	1	yes	characteristic, direct product factor, Sylow
250	C₂⁴	—	4	1	no	—
251	#41	—	2	2	no	—
252	#41	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×Q₈	—	4	2	no	—
257	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	4	4	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	D₈×C₃₁	1	2	yes	—
264	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
265	C₂²×C₄	—	4	1	no	—
266	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
267	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
268	Q₁₆	—	2	2	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	4	2	no	—
271	D₈	—	2	2	no	—
272	D₈	—	2	2	no	—
273	D₈	—	4	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	4	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
279	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
280	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
281	C₂×C₈	#41×C₃₁	1	2	yes	—
282	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	4	4	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	4	1	no	—
288	C₂²⋊C₄	—	4	4	no	—
289	C₂²⋊C₄	—	4	4	no	—
290	C₄²	—	4	1	no	—
291	C₄²	—	4	1	no	—
292	C₁₆	—	2	2	no	—
293	C₁₆	C₂×D₄×C₃₁	1	2	yes	—
294	C₁₆	C₂×D₄×C₃₁	1	2	yes	—
295	C₂³	—	4	2	no	—
296	C₂³	—	8	2	no	—
297	C₂³	—	16	2	no	—
298	C₂³	C₂×D₈×C₃₁	1	2	yes	—
299	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	2	no	—
304	D₄	—	2	4	no	—
305	D₄	—	2	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	2	no	—
309	D₄	—	16	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	C₃₁×#121	1	2	yes	—
317	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
318	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
319	C₂×C₄	C₃₁×#135	1	2	yes	—
320	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	C₃₁×#121	1	2	yes	—
323	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
324	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	2	4	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	8	4	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	16	4	no	—
331	C₂²	C₃₁×#466	1	2	yes	—
332	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	C₃₁×#466	1	2	yes	—
338	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
339	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	4	2	no	—
342	C₂	—	16	2	no	—
343	C₂	C₂×C₃₁×#466	1	1	yes	characteristic
344	C₂	C₃₁×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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. C₂²×C₃₁×SD₃₂	C₂	semidirect
2. C₂×C₃₁×#504	C₂	semidirect
3. C₃₁×#3059	C₂	semidirect
5. C₃₁×#2220	C₂	semidirect
6. C₃₁×#2217	C₂	semidirect
7. C₃₁×#2158	C₂	semidirect
8. C₃₁×#2141	C₂	semidirect
9. C₃₁×#2140	C₂	semidirect
20. C₂×C₃₁×M₃₂	C₂²	semidirect
22. C₂²×C₁₆×C₃₁	C₂²	semidirect
25. C₃₁×#467	C₂²	semidirect
31. C₃₁×#367	C₂²	semidirect
32. C₃₁×#366	C₂²	semidirect
33. C₃₁×#358	C₂²	semidirect
67. C₂×C₁₆×C₃₁	D₄	semidirect
122. #33609	C₃₁	direct (N × H)
153. C₂²×SD₃₂	C₂×C₃₁	semidirect
154. C₂×#504	C₂×C₃₁	semidirect
155. #3059	C₂×C₃₁	semidirect
157. #2220	C₂×C₃₁	semidirect
158. #2217	C₂×C₃₁	semidirect
159. #2158	C₂×C₃₁	semidirect
160. #2141	C₂×C₃₁	semidirect
161. #2140	C₂×C₃₁	semidirect
187. C₂×M₃₂	C₂²×C₃₁	semidirect
189. C₂²×C₁₆	C₂²×C₃₁	semidirect
192. #467	C₂²×C₃₁	semidirect
198. #367	C₂²×C₃₁	semidirect
199. #366	C₂²×C₃₁	semidirect
200. #358	C₂²×C₃₁	semidirect
239. C₂×C₁₆	D₄×C₃₁	semidirect
249. C₃₁	#33609	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×SD₃₂	2
C₂×C₃₁×#504	2
C₃₁×#3059	2
C₃₁×#3058	2
C₃₁×#2220	2
C₃₁×#2217	2
C₃₁×#2158	2
C₃₁×#2141	2
C₃₁×#2140	2
#33609	31

Subgroups — C31×#33609 #21490368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33609 #21490368