Subgroups — C₂×C₃₁×#2175 #21490207

Order 7936.

734 conjugacy classes of subgroups.

The subgroup lattice has 316 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
2	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2175	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#358	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#357	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
12	C₂×C₄×D₄×C₃₁	—	2	1	no	—
13	C₂×C₃₁×SD₃₂	C₄	1	8	yes	complemented
14	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	—
15	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₄×Q₁₆×C₃₁	—	2	2	no	—
18	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	complemented
19	C₄×D₈×C₃₁	—	2	2	no	—
20	C₄×D₈×C₃₁	C₂²	1	4	yes	—
21	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#103	—	2	1	no	—
24	C₂×C₃₁×#102	—	2	1	no	—
25	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#366	C₂²	1	4	yes	complemented
27	C₃₁×#358	C₂²	1	4	yes	complemented
28	C₃₁×#357	C₂²	1	4	yes	—
29	C₃₁×#346	C₂²	1	4	yes	complemented
30	C₂²×Q₈×C₃₁	—	2	1	no	—
31	C₂²×D₄×C₃₁	—	2	1	no	—
32	C₂³×C₄×C₃₁	—	4	1	no	—
33	C₂×Q₁₆×C₃₁	—	2	1	no	—
34	C₂×Q₁₆×C₃₁	—	2	2	no	—
35	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
36	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
37	C₂×D₈×C₃₁	—	2	1	no	—
38	C₂×D₈×C₃₁	—	2	2	no	—
39	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
40	C₂×D₈×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₃₁	—	2	4	no	—
44	C₄×Q₈×C₃₁	—	4	2	no	—
45	C₄×D₄×C₃₁	—	2	4	no	—
46	C₄×D₄×C₃₁	—	4	2	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
49	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
50	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
51	C₂×C₄²×C₃₁	—	4	1	no	—
52	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
53	C₃₁×SD₃₂	C₂×C₄	1	16	yes	complemented
54	C₂×C₁₆×C₃₁	—	2	2	no	—
55	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
56	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
57	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
58	C₃₁×#107	—	2	1	no	—
59	C₃₁×#107	—	2	1	no	—
60	C₃₁×#107	C₂³	1	2	yes	—
61	C₃₁×#107	C₂³	1	2	yes	—
62	C₃₁×#103	—	2	4	no	—
63	C₃₁×#102	—	2	4	no	—
64	C₄×C₈×C₃₁	D₄	1	2	yes	—
65	C₄×C₈×C₃₁	C₂³	1	2	yes	—
66	C₂⁴×C₃₁	—	4	1	no	—
67	C₂×Q₈×C₃₁	—	2	4	no	—
68	C₂×Q₈×C₃₁	—	2	2	no	—
69	C₂×Q₈×C₃₁	—	4	1	no	—
70	C₂×Q₈×C₃₁	—	4	2	no	—
71	C₂×D₄×C₃₁	—	2	4	no	—
72	C₂×D₄×C₃₁	—	2	2	no	—
73	C₂×D₄×C₃₁	—	4	1	no	—
74	C₂×D₄×C₃₁	—	4	2	no	—
75	C₂²×C₄×C₃₁	—	4	1	no	—
76	C₂²×C₄×C₃₁	—	4	1	no	—
77	C₂²×C₄×C₃₁	—	4	1	no	—
78	C₂²×C₄×C₃₁	—	4	4	no	—
79	C₂²×C₄×C₃₁	—	8	4	no	—
80	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₃₁	—	2	4	no	—
84	Q₁₆×C₃₁	—	2	2	no	—
85	Q₁₆×C₃₁	C₂²×C₄	1	4	yes	complemented
86	D₈×C₃₁	—	2	4	no	—
87	D₈×C₃₁	—	2	2	no	—
88	D₈×C₃₁	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₃₁	#41	1	2	yes	—
94	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	4	1	no	—
98	C₄⋊C₄×C₃₁	—	4	1	no	—
99	C₄⋊C₄×C₃₁	—	4	4	no	—
100	C₂²⋊C₄×C₃₁	—	4	4	no	—
101	C₄²×C₃₁	—	4	4	no	—
102	C₄²×C₃₁	D₈	1	2	yes	—
103	C₄²×C₃₁	C₂×D₄	1	2	yes	—
104	C₁₆×C₃₁	—	2	2	no	—
105	C₁₆×C₃₁	C₂²×C₄	1	4	yes	complemented
106	C₂×#2175	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	—	4	2	no	—
108	C₂³×C₃₁	—	4	4	no	—
109	C₂³×C₃₁	—	8	2	no	—
110	C₂³×C₃₁	—	8	2	no	—
111	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
112	Q₈×C₃₁	—	2	4	no	—
113	Q₈×C₃₁	—	4	2	no	—
114	Q₈×C₃₁	—	4	4	no	—
115	D₄×C₃₁	—	2	4	no	—
116	D₄×C₃₁	—	4	2	no	—
117	D₄×C₃₁	—	4	4	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	4	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	8	8	no	—
125	C₂×C₄×C₃₁	—	8	2	no	—
126	C₂×C₄×C₃₁	—	8	2	no	—
127	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₃₁	#135	1	2	yes	—
132	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
133	C₈×C₃₁	—	2	2	no	—
134	C₈×C₃₁	C₄×D₄	1	2	yes	—
135	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
136	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
137	C₂²×SD₃₂	C₂×C₃₁	1	1	yes	characteristic
138	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
139	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
140	#2175	C₂×C₃₁	1	8	yes	direct product factor
141	C₂×#366	C₂×C₃₁	1	1	yes	characteristic, complemented
142	C₂×#358	C₂×C₃₁	1	1	yes	characteristic, complemented
143	C₂×#357	C₂×C₃₁	1	1	yes	characteristic
144	C₂×#346	C₂×C₃₁	1	1	yes	characteristic, complemented
145	C₂²×C₃₁	—	4	4	no	—
146	C₂²×C₃₁	—	8	2	no	—
147	C₂²×C₃₁	—	8	2	no	—
148	C₂²×C₃₁	—	8	8	no	—
149	C₂²×C₃₁	C₄×D₈	1	2	yes	—
150	C₂²×C₃₁	#509	1	2	yes	—
151	C₂²×C₃₁	#510	1	2	yes	—
152	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
153	C₄×C₃₁	—	2	2	no	—
154	C₄×C₃₁	—	2	2	no	—
155	C₄×C₃₁	—	4	4	no	—
156	C₄×C₃₁	—	8	2	no	—
157	C₄×C₃₁	—	8	2	no	—
158	C₄×C₃₁	C₄×D₈	1	2	yes	—
159	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
160	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
161	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
162	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×C₄×Q₈	—	2	1	no	—
164	C₂×C₄×D₄	—	2	1	no	—
165	C₂×SD₃₂	C₄×C₃₁	1	8	yes	complemented
166	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	—
167	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
168	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
169	C₄×Q₁₆	—	2	2	no	—
170	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	complemented
171	C₄×D₈	—	2	2	no	—
172	C₄×D₈	C₂²×C₃₁	1	4	yes	—
173	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×#103	—	2	1	no	—
176	C₂×#102	—	2	1	no	—
177	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
178	#366	C₂²×C₃₁	1	4	yes	complemented
179	#358	C₂²×C₃₁	1	4	yes	complemented
180	#357	C₂²×C₃₁	1	4	yes	—
181	#346	C₂²×C₃₁	1	4	yes	complemented
182	C₂×C₃₁	—	8	4	no	—
183	C₂×C₃₁	#2175	1	4	yes	direct product factor
184	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
185	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
186	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
187	C₂²×Q₈	—	2	1	no	—
188	C₂²×D₄	—	2	1	no	—
189	C₂³×C₄	—	4	1	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	—	2	2	no	—
192	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
193	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	2	2	no	—
196	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
197	C₂×D₈	C₂³×C₃₁	1	2	yes	—
198	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
200	C₄×Q₈	—	2	4	no	—
201	C₄×Q₈	—	4	2	no	—
202	C₄×D₄	—	2	4	no	—
203	C₄×D₄	—	4	2	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	4	1	no	—
207	C₂×C₂²⋊C₄	—	4	1	no	—
208	C₂×C₄²	—	4	1	no	—
209	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
210	SD₃₂	C₂×C₄×C₃₁	1	16	yes	complemented
211	C₂×C₁₆	—	2	2	no	—
212	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
213	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
214	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
215	#107	—	2	1	no	—
216	#107	—	2	1	no	—
217	#107	C₂³×C₃₁	1	2	yes	—
218	#107	C₂³×C₃₁	1	2	yes	—
219	#103	—	2	4	no	—
220	#102	—	2	4	no	—
221	C₄×C₈	D₄×C₃₁	1	2	yes	—
222	C₄×C₈	C₂³×C₃₁	1	2	yes	—
223	C₃₁	C₂×#2175	1	1	yes	characteristic, direct product factor, Sylow
224	C₂⁴	—	4	1	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	—	8	4	no	—
238	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
239	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	2	2	no	—
243	Q₁₆	C₂²×C₄×C₃₁	1	4	yes	complemented
244	D₈	—	2	4	no	—
245	D₈	—	2	2	no	—
246	D₈	C₂²×C₄×C₃₁	1	4	yes	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
249	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
250	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂×C₈	#41×C₃₁	1	2	yes	—
252	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₂²⋊C₄	—	4	4	no	—
259	C₄²	—	4	4	no	—
260	C₄²	D₈×C₃₁	1	2	yes	—
261	C₄²	C₂×D₄×C₃₁	1	2	yes	—
262	C₁₆	—	2	2	no	—
263	C₁₆	C₂²×C₄×C₃₁	1	4	yes	complemented
264	C₂³	—	4	4	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	8	2	no	—
267	C₂³	—	8	2	no	—
268	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
269	Q₈	—	2	4	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	4	2	no	—
272	D₄	—	2	4	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	8	8	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
285	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
286	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
287	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
288	C₂×C₄	C₃₁×#135	1	2	yes	—
289	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	C₄×D₄×C₃₁	1	2	yes	—
292	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
293	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
294	C₂²	—	4	4	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	2	no	—
297	C₂²	—	8	8	no	—
298	C₂²	C₄×D₈×C₃₁	1	2	yes	—
299	C₂²	C₃₁×#509	1	2	yes	—
300	C₂²	C₃₁×#510	1	2	yes	—
301	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	4	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₄	C₄×D₈×C₃₁	1	2	yes	—
308	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
309	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
310	C₂	—	8	4	no	—
311	C₂	C₃₁×#2175	1	4	yes	direct product factor
312	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
313	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
314	C₂	C₂×C₃₁×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
4. C₃₁×#2175	C₂	direct (N × H)
5. C₂×C₃₁×#366	C₂	semidirect
6. C₂×C₃₁×#358	C₂	semidirect
8. C₂×C₃₁×#346	C₂	semidirect
13. C₂×C₃₁×SD₃₂	C₄	semidirect
18. C₄×Q₁₆×C₃₁	C₂²	semidirect
26. C₃₁×#366	C₂²	semidirect
27. C₃₁×#358	C₂²	semidirect
29. C₃₁×#346	C₂²	semidirect
35. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
53. C₃₁×SD₃₂	C₂×C₄	semidirect
55. C₂×C₁₆×C₃₁	C₂×C₄	semidirect
85. Q₁₆×C₃₁	C₂²×C₄	semidirect
105. C₁₆×C₃₁	C₂²×C₄	semidirect
106. C₂×#2175	C₃₁	direct (N × H)
138. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
140. #2175	C₂×C₃₁	direct (N × H)
141. C₂×#366	C₂×C₃₁	semidirect
142. C₂×#358	C₂×C₃₁	semidirect
144. C₂×#346	C₂×C₃₁	semidirect
165. C₂×SD₃₂	C₄×C₃₁	semidirect
170. C₄×Q₁₆	C₂²×C₃₁	semidirect
178. #366	C₂²×C₃₁	semidirect
179. #358	C₂²×C₃₁	semidirect
181. #346	C₂²×C₃₁	semidirect
183. C₂×C₃₁	#2175	direct (N × H)
192. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
210. SD₃₂	C₂×C₄×C₃₁	semidirect
212. C₂×C₁₆	C₂×C₄×C₃₁	semidirect
223. C₃₁	C₂×#2175	direct (N × H)
243. Q₁₆	C₂²×C₄×C₃₁	semidirect
263. C₁₆	C₂²×C₄×C₃₁	semidirect
311. C₂	C₃₁×#2175	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×C₄×Q₁₆×C₃₁	2
C₂×C₄×D₈×C₃₁	2
C₃₁×#2175	2
C₂×C₃₁×#366	2
C₂×C₃₁×#358	2
C₂×C₃₁×#357	2
C₂×C₃₁×#346	2
C₂×#2175	31

Subgroups — C2×C31×#2175 #21490207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2175 #21490207