Subgroups — C₂×C₃₁×#1808 #21477416

Order 7936.

1790 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#107	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#380	C₂	1	2	yes	maximal
4	C₃₁×#1808	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#340	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	—	2	8	no	—
8	C₂³×C₈×C₃₁	C₄	1	8	yes	complemented
9	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₃₁	C₂²	1	4	yes	—
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
13	C₂²×C₂²⋊C₄×C₃₁	—	2	2	no	—
14	C₂×C₃₁×#107	—	2	8	no	—
15	C₂×C₃₁×#107	C₂²	1	8	yes	complemented
16	C₃₁×#380	—	2	4	no	—
17	C₃₁×#380	C₂²	1	8	yes	—
18	C₂×C₃₁×#95	—	2	2	no	—
19	C₃₁×#340	C₂²	1	8	yes	complemented
20	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	—	2	4	no	—
22	C₂³×C₄×C₃₁	—	2	4	no	—
23	C₂³×C₄×C₃₁	—	2	4	no	—
24	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₃₁	D₄	1	2	yes	—
26	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₃₁	Q₈	1	4	yes	—
28	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
29	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	—	2	4	no	—
31	C₂²×C₈×C₃₁	—	2	32	no	—
32	C₂²×C₈×C₃₁	—	2	16	no	—
33	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	16	yes	complemented
36	C₂²×C₈×C₃₁	D₄	1	8	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
44	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
45	C₃₁×#107	—	2	16	no	—
46	C₃₁×#95	—	2	8	no	—
47	C₂⁴×C₃₁	—	2	4	no	—
48	C₂⁴×C₃₁	—	2	4	no	—
49	C₂⁴×C₃₁	C₄⋊C₄	1	4	yes	—
50	C₂⁴×C₃₁	D₈	1	4	yes	—
51	C₂⁴×C₃₁	Q₁₆	1	4	yes	—
52	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
53	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	—	2	4	no	—
55	C₂²×C₄×C₃₁	—	2	8	no	—
56	C₂²×C₄×C₃₁	—	2	8	no	—
57	C₂²×C₄×C₃₁	—	2	8	no	—
58	C₂²×C₄×C₃₁	—	2	16	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	—	2	8	no	—
62	C₂²×C₄×C₃₁	—	4	8	no	—
63	C₂²×C₄×C₃₁	—	4	8	no	—
64	C₂²×C₄×C₃₁	—	4	8	no	—
65	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₃₁	#41	1	4	yes	—
75	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₃₁	—	2	8	no	—
77	C₂×C₈×C₃₁	—	2	4	no	—
78	C₂×C₈×C₃₁	—	2	32	no	—
79	C₂×C₈×C₃₁	C₂²⋊C₄	1	16	yes	complemented
80	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
83	C₄⋊C₄×C₃₁	—	2	8	no	—
84	C₄⋊C₄×C₃₁	—	2	8	no	—
85	C₄⋊C₄×C₃₁	—	4	8	no	—
86	C₂²⋊C₄×C₃₁	—	4	16	no	—
87	C₂×#1808	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	—	2	4	no	—
89	C₂³×C₃₁	—	2	4	no	—
90	C₂³×C₃₁	—	2	8	no	—
91	C₂³×C₃₁	—	2	8	no	—
92	C₂³×C₃₁	—	2	16	no	—
93	C₂³×C₃₁	—	2	16	no	—
94	C₂³×C₃₁	#107	1	16	yes	complemented
95	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
96	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₃₁	#121	1	4	yes	—
98	C₂³×C₃₁	C₂×D₈	1	4	yes	—
99	C₂³×C₃₁	C₂×D₈	1	2	yes	—
100	C₂³×C₃₁	C₂×Q₁₆	1	4	yes	—
101	C₂³×C₃₁	C₂×Q₁₆	1	2	yes	—
102	C₂³×C₃₁	#135	1	4	yes	—
103	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	8	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	8	no	—
109	C₂×C₄×C₃₁	—	2	16	no	—
110	C₂×C₄×C₃₁	—	4	8	no	—
111	C₂×C₄×C₃₁	—	4	8	no	—
112	C₂×C₄×C₃₁	—	4	8	no	—
113	C₂×C₄×C₃₁	—	8	16	no	—
114	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
115	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₃₁	#117	1	4	yes	—
118	C₂×C₄×C₃₁	#121	1	4	yes	—
119	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
120	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
122	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
124	C₈×C₃₁	—	2	4	no	—
125	C₈×C₃₁	C₂×C₂²⋊C₄	1	8	yes	complemented
126	C₂⁴×C₈	C₂×C₃₁	1	1	yes	characteristic
127	C₂²×#107	C₂×C₃₁	1	2	yes	complemented
128	C₂×#380	C₂×C₃₁	1	2	yes	—
129	#1808	C₂×C₃₁	1	8	yes	direct product factor
130	C₂×#340	C₂×C₃₁	1	2	yes	complemented
131	C₂²×C₃₁	—	2	32	no	—
132	C₂²×C₃₁	—	2	4	no	—
133	C₂²×C₃₁	—	2	4	no	—
134	C₂²×C₃₁	—	2	8	no	—
135	C₂²×C₃₁	—	2	8	no	—
136	C₂²×C₃₁	#380	1	4	yes	—
137	C₂²×C₃₁	C₂×#107	1	4	yes	—
138	C₂²×C₃₁	C₂×#107	1	8	yes	complemented
139	C₂²×C₃₁	#427	1	4	yes	—
140	C₂²×C₃₁	#466	1	8	yes	—
141	C₂²×C₃₁	#467	1	8	yes	—
142	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₃₁	C₂×#121	1	2	yes	—
144	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
145	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
146	C₂²×C₃₁	C₂×#135	1	2	yes	—
147	C₄×C₃₁	—	2	4	no	—
148	C₄×C₃₁	—	8	8	no	—
149	C₄×C₃₁	#380	1	4	yes	—
150	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
151	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
152	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
153	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
154	C₂³×C₈	—	2	8	no	—
155	C₂³×C₈	C₄×C₃₁	1	8	yes	complemented
156	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
157	C₂³×C₈	C₂²×C₃₁	1	4	yes	—
158	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
160	C₂²×C₂²⋊C₄	—	2	2	no	—
161	C₂×#107	—	2	8	no	—
162	C₂×#107	C₂²×C₃₁	1	8	yes	complemented
163	#380	—	2	4	no	—
164	#380	C₂²×C₃₁	1	8	yes	—
165	C₂×#95	—	2	2	no	—
166	#340	C₂²×C₃₁	1	8	yes	complemented
167	C₂×C₃₁	—	2	8	no	—
168	C₂×C₃₁	#1808	1	8	yes	direct product factor
169	C₂×C₃₁	C₂×#380	1	1	yes	characteristic
170	C₂×C₃₁	C₂²×#107	1	1	yes	characteristic
171	C₂×C₃₁	C₂×#427	1	1	yes	characteristic
172	C₂×C₃₁	C₂×#466	1	2	yes	—
173	C₂×C₃₁	C₂×#467	1	2	yes	—
174	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
175	C₂³×C₄	—	2	4	no	—
176	C₂³×C₄	—	2	4	no	—
177	C₂³×C₄	—	2	4	no	—
178	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
179	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
180	C₂³×C₄	D₄×C₃₁	1	2	yes	—
181	C₂³×C₄	Q₈×C₃₁	1	4	yes	—
182	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
183	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
184	C₂²×C₈	—	2	4	no	—
185	C₂²×C₈	—	2	32	no	—
186	C₂²×C₈	—	2	16	no	—
187	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
188	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
189	C₂²×C₈	C₂×C₄×C₃₁	1	16	yes	complemented
190	C₂²×C₈	D₄×C₃₁	1	8	yes	—
191	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
192	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
193	C₂×C₄⋊C₄	—	2	8	no	—
194	C₂×C₄⋊C₄	—	2	8	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
197	C₂×C₂²⋊C₄	—	2	8	no	—
198	C₂×C₂²⋊C₄	—	4	8	no	—
199	#107	—	2	16	no	—
200	#95	—	2	8	no	—
201	C₃₁	C₂×#1808	1	1	yes	characteristic, direct product factor, Sylow
202	C₂⁴	—	2	4	no	—
203	C₂⁴	—	2	4	no	—
204	C₂⁴	C₄⋊C₄×C₃₁	1	4	yes	—
205	C₂⁴	D₈×C₃₁	1	4	yes	—
206	C₂⁴	Q₁₆×C₃₁	1	4	yes	—
207	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
208	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	—	2	16	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	4	8	no	—
218	C₂²×C₄	—	4	8	no	—
219	C₂²×C₄	—	4	8	no	—
220	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
221	C₂²×C₄	C₄⋊C₄×C₃₁	1	8	yes	—
222	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
223	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
226	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
228	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
229	C₂²×C₄	#41×C₃₁	1	4	yes	—
230	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
231	C₂×C₈	—	2	8	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	32	no	—
234	C₂×C₈	C₂²⋊C₄×C₃₁	1	16	yes	complemented
235	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
236	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
237	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
238	C₄⋊C₄	—	2	8	no	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	4	8	no	—
241	C₂²⋊C₄	—	4	16	no	—
242	C₂³	—	2	8	no	—
243	C₂³	—	2	16	no	—
244	C₂³	—	2	16	no	—
245	C₂³	—	2	4	no	—
246	C₂³	—	2	4	no	—
247	C₂³	—	2	8	no	—
248	C₂³	C₃₁×#107	1	16	yes	complemented
249	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
250	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
251	C₂³	C₃₁×#121	1	4	yes	—
252	C₂³	C₂×D₈×C₃₁	1	4	yes	—
253	C₂³	C₂×D₈×C₃₁	1	2	yes	—
254	C₂³	C₂×Q₁₆×C₃₁	1	4	yes	—
255	C₂³	C₂×Q₁₆×C₃₁	1	2	yes	—
256	C₂³	C₃₁×#135	1	4	yes	—
257	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	16	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	—	4	8	no	—
267	C₂×C₄	—	8	16	no	—
268	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
269	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
270	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
271	C₂×C₄	C₃₁×#117	1	4	yes	—
272	C₂×C₄	C₃₁×#121	1	4	yes	—
273	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
274	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
276	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
277	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
278	C₈	—	2	4	no	—
279	C₈	C₂×C₂²⋊C₄×C₃₁	1	8	yes	complemented
280	C₂²	—	2	32	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	2	8	no	—
285	C₂²	C₃₁×#380	1	4	yes	—
286	C₂²	C₂×C₃₁×#107	1	4	yes	—
287	C₂²	C₂×C₃₁×#107	1	8	yes	complemented
288	C₂²	C₃₁×#427	1	4	yes	—
289	C₂²	C₃₁×#466	1	8	yes	—
290	C₂²	C₃₁×#467	1	8	yes	—
291	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
292	C₂²	C₂×C₃₁×#121	1	2	yes	—
293	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
294	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
295	C₂²	C₂×C₃₁×#135	1	2	yes	—
296	C₄	—	2	4	no	—
297	C₄	—	8	8	no	—
298	C₄	C₃₁×#380	1	4	yes	—
299	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
300	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
301	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
302	C₂	—	2	8	no	—
303	C₂	C₃₁×#1808	1	8	yes	direct product factor
304	C₂	C₂×C₃₁×#380	1	1	yes	characteristic
305	C₂	C₂²×C₃₁×#107	1	1	yes	characteristic
306	C₂	C₂×C₃₁×#427	1	1	yes	characteristic
307	C₂	C₂×C₃₁×#466	1	2	yes	—
308	C₂	C₂×C₃₁×#467	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁵×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×#107	C₂	semidirect
4. C₃₁×#1808	C₂	direct (N × H)
5. C₂×C₃₁×#340	C₂	semidirect
8. C₂³×C₈×C₃₁	C₄	semidirect
15. C₂×C₃₁×#107	C₂²	semidirect
19. C₃₁×#340	C₂²	semidirect
35. C₂²×C₈×C₃₁	C₂×C₄	semidirect
79. C₂×C₈×C₃₁	C₂²⋊C₄	semidirect
87. C₂×#1808	C₃₁	direct (N × H)
94. C₂³×C₃₁	#107	semidirect
125. C₈×C₃₁	C₂×C₂²⋊C₄	semidirect
127. C₂²×#107	C₂×C₃₁	semidirect
129. #1808	C₂×C₃₁	direct (N × H)
130. C₂×#340	C₂×C₃₁	semidirect
138. C₂²×C₃₁	C₂×#107	semidirect
155. C₂³×C₈	C₄×C₃₁	semidirect
162. C₂×#107	C₂²×C₃₁	semidirect
166. #340	C₂²×C₃₁	semidirect
168. C₂×C₃₁	#1808	direct (N × H)
189. C₂²×C₈	C₂×C₄×C₃₁	semidirect
201. C₃₁	C₂×#1808	direct (N × H)
234. C₂×C₈	C₂²⋊C₄×C₃₁	semidirect
248. C₂³	C₃₁×#107	semidirect
279. C₈	C₂×C₂²⋊C₄×C₃₁	semidirect
287. C₂²	C₂×C₃₁×#107	semidirect
303. C₂	C₃₁×#1808	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₃₁	2
C₂²×C₃₁×#107	2
C₂×C₃₁×#380	2
C₃₁×#1808	2
C₂×C₃₁×#340	2
C₂×#1808	31

Subgroups — C2×C31×#1808 #21477416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1808 #21477416