Subgroups — C₂×C₃₁×#1993 #21478079

Order 7936.

1422 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#382	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1993	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#340	C₂	1	4	yes	maximal
6	C₂³×Q₈×C₃₁	—	2	1	no	—
7	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
8	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₃₁×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₃₁×C₄⋊Q₈	C₄	1	8	yes	complemented
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
13	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
15	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#97	—	2	4	no	—
17	C₂×C₃₁×#97	C₂²	1	4	yes	—
18	C₃₁×#382	—	2	2	no	—
19	C₃₁×#382	C₂²	1	4	yes	—
20	C₂×C₃₁×#95	—	2	1	no	—
21	C₂×C₃₁×#95	—	2	1	no	—
22	C₃₁×#340	C₂²	1	16	yes	—
23	C₂²×Q₈×C₃₁	—	2	8	no	—
24	C₂²×Q₈×C₃₁	—	2	4	no	—
25	C₂²×Q₈×C₃₁	—	4	8	no	—
26	C₂³×C₄×C₃₁	—	2	2	no	—
27	C₂³×C₄×C₃₁	—	2	2	no	—
28	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	—	2	4	no	—
32	C₂²×C₈×C₃₁	—	2	4	no	—
33	C₂²×C₈×C₃₁	Q₈	1	8	yes	complemented
34	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
35	C₃₁×C₄⋊Q₈	—	2	16	no	—
36	C₃₁×C₄⋊Q₈	—	2	8	no	—
37	C₃₁×C₄⋊Q₈	C₂×C₄	1	16	yes	complemented
38	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
47	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₃₁	D₄	1	2	yes	—
49	C₂×C₄²×C₃₁	D₄	1	4	yes	—
50	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
51	C₃₁×#97	—	2	8	no	—
52	C₃₁×#95	—	2	4	no	—
53	C₃₁×#95	—	2	4	no	—
54	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂×Q₈×C₃₁	—	2	16	no	—
56	C₂×Q₈×C₃₁	—	4	16	no	—
57	C₂×Q₈×C₃₁	—	4	32	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	—	2	8	no	—
61	C₂²×C₄×C₃₁	—	2	4	no	—
62	C₂²×C₄×C₃₁	—	2	4	no	—
63	C₂²×C₄×C₃₁	—	4	8	no	—
64	C₂²×C₄×C₃₁	—	4	4	no	—
65	C₂²×C₄×C₃₁	—	4	4	no	—
66	C₂²×C₄×C₃₁	—	4	4	no	—
67	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
68	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
69	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
75	C₂²×C₄×C₃₁	#41	1	4	yes	—
76	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₃₁	—	2	16	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	4	no	—
80	C₂×C₈×C₃₁	C₂×Q₈	1	8	yes	complemented
81	C₄⋊C₄×C₃₁	—	2	16	no	—
82	C₄⋊C₄×C₃₁	—	2	8	no	—
83	C₄⋊C₄×C₃₁	—	2	4	no	—
84	C₄⋊C₄×C₃₁	—	2	8	no	—
85	C₄⋊C₄×C₃₁	—	4	4	no	—
86	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
87	C₄²×C₃₁	C₂²⋊C₄	1	8	yes	—
88	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
89	C₄²×C₃₁	C₂×D₄	1	4	yes	—
90	C₂×#1993	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂³×C₃₁	#124	1	2	yes	—
93	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
94	C₂³×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
95	C₂³×C₃₁	#137	1	4	yes	—
96	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₃₁	—	4	32	no	—
98	C₂×C₄×C₃₁	—	2	4	no	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	2	8	no	—
101	C₂×C₄×C₃₁	—	2	8	no	—
102	C₂×C₄×C₃₁	—	2	8	no	—
103	C₂×C₄×C₃₁	—	4	8	no	—
104	C₂×C₄×C₃₁	—	4	16	no	—
105	C₂×C₄×C₃₁	—	4	4	no	—
106	C₂×C₄×C₃₁	—	4	4	no	—
107	C₂×C₄×C₃₁	—	4	4	no	—
108	C₂×C₄×C₃₁	—	8	8	no	—
109	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
111	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₃₁	C₄×Q₈	1	8	yes	—
113	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
114	C₂×C₄×C₃₁	#124	1	2	yes	—
115	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
119	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
120	C₈×C₃₁	—	2	8	no	—
121	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
122	C₂²×#97	C₂×C₃₁	1	1	yes	characteristic
123	C₂×#382	C₂×C₃₁	1	1	yes	characteristic
124	#1993	C₂×C₃₁	1	8	yes	direct product factor
125	C₂×#340	C₂×C₃₁	1	4	yes	—
126	C₂²×C₃₁	#391	1	4	yes	—
127	C₂²×C₃₁	#418	1	4	yes	—
128	C₂²×C₃₁	#419	1	4	yes	—
129	C₂²×C₃₁	#489	1	8	yes	—
130	C₂²×C₃₁	#501	1	8	yes	—
131	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
133	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
135	C₂²×C₃₁	C₂×#137	1	2	yes	—
136	C₄×C₃₁	—	2	4	no	—
137	C₄×C₃₁	—	2	4	no	—
138	C₄×C₃₁	—	4	8	no	—
139	C₄×C₃₁	—	8	4	no	—
140	C₄×C₃₁	#391	1	4	yes	—
141	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
142	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
143	C₂³×Q₈	—	2	1	no	—
144	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
145	C₂×C₄⋊Q₈	—	2	2	no	—
146	C₂×C₄⋊Q₈	—	2	4	no	—
147	C₂×C₄⋊Q₈	C₄×C₃₁	1	8	yes	complemented
148	C₂×C₄⋊Q₈	C₂²×C₃₁	1	4	yes	—
149	C₂²×C₄⋊C₄	—	2	1	no	—
150	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
151	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
152	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
153	C₂×#97	—	2	4	no	—
154	C₂×#97	C₂²×C₃₁	1	4	yes	—
155	#382	—	2	2	no	—
156	#382	C₂²×C₃₁	1	4	yes	—
157	C₂×#95	—	2	1	no	—
158	C₂×#95	—	2	1	no	—
159	#340	C₂²×C₃₁	1	16	yes	—
160	C₂×C₃₁	#1993	1	8	yes	direct product factor
161	C₂×C₃₁	C₂×#391	1	1	yes	characteristic
162	C₂×C₃₁	C₂×#418	1	1	yes	characteristic
163	C₂×C₃₁	C₂×#419	1	1	yes	characteristic
164	C₂×C₃₁	C₂×#489	1	2	yes	—
165	C₂×C₃₁	C₂×#501	1	2	yes	—
166	C₂²×Q₈	—	2	8	no	—
167	C₂²×Q₈	—	2	4	no	—
168	C₂²×Q₈	—	4	8	no	—
169	C₂³×C₄	—	2	2	no	—
170	C₂³×C₄	—	2	2	no	—
171	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
172	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
173	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
174	C₂²×C₈	—	2	4	no	—
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	Q₈×C₃₁	1	8	yes	complemented
177	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
178	C₄⋊Q₈	—	2	16	no	—
179	C₄⋊Q₈	—	2	8	no	—
180	C₄⋊Q₈	C₂×C₄×C₃₁	1	16	yes	complemented
181	C₂×C₄⋊C₄	—	2	8	no	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	8	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
188	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
189	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
190	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
191	C₂×C₄²	D₄×C₃₁	1	2	yes	—
192	C₂×C₄²	D₄×C₃₁	1	4	yes	—
193	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
194	#97	—	2	8	no	—
195	#95	—	2	4	no	—
196	#95	—	2	4	no	—
197	C₃₁	C₂×#1993	1	1	yes	characteristic, direct product factor, Sylow
198	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
199	C₂×Q₈	—	2	16	no	—
200	C₂×Q₈	—	4	32	no	—
201	C₂×Q₈	—	4	16	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	8	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	8	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
212	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
213	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
218	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
219	C₂²×C₄	#41×C₃₁	1	4	yes	—
220	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
221	C₂×C₈	—	2	16	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	C₂×Q₈×C₃₁	1	8	yes	complemented
225	C₄⋊C₄	—	2	16	no	—
226	C₄⋊C₄	—	2	8	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
231	C₄²	C₂²⋊C₄×C₃₁	1	8	yes	—
232	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
233	C₄²	C₂×D₄×C₃₁	1	4	yes	—
234	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
235	C₂³	C₃₁×#124	1	2	yes	—
236	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
237	C₂³	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
238	C₂³	C₃₁×#137	1	4	yes	—
239	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
240	Q₈	—	4	32	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	4	16	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	8	8	no	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
254	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
255	C₂×C₄	C₄×Q₈×C₃₁	1	8	yes	—
256	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
257	C₂×C₄	C₃₁×#124	1	2	yes	—
258	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
259	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
262	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
263	C₈	—	2	8	no	—
264	C₂²	C₃₁×#391	1	4	yes	—
265	C₂²	C₃₁×#418	1	4	yes	—
266	C₂²	C₃₁×#419	1	4	yes	—
267	C₂²	C₃₁×#489	1	8	yes	—
268	C₂²	C₃₁×#501	1	8	yes	—
269	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
270	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
271	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
272	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
273	C₂²	C₂×C₃₁×#137	1	2	yes	—
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	4	8	no	—
277	C₄	—	8	4	no	—
278	C₄	C₃₁×#391	1	4	yes	—
279	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
280	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
281	C₂	C₃₁×#1993	1	8	yes	direct product factor
282	C₂	C₂×C₃₁×#391	1	1	yes	characteristic
283	C₂	C₂×C₃₁×#418	1	1	yes	characteristic
284	C₂	C₂×C₃₁×#419	1	1	yes	characteristic
285	C₂	C₂×C₃₁×#489	1	2	yes	—
286	C₂	C₂×C₃₁×#501	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1993	C₂	direct (N × H)
10. C₂×C₃₁×C₄⋊Q₈	C₄	semidirect
33. C₂²×C₈×C₃₁	Q₈	semidirect
37. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
80. C₂×C₈×C₃₁	C₂×Q₈	semidirect
90. C₂×#1993	C₃₁	direct (N × H)
124. #1993	C₂×C₃₁	direct (N × H)
147. C₂×C₄⋊Q₈	C₄×C₃₁	semidirect
160. C₂×C₃₁	#1993	direct (N × H)
176. C₂²×C₈	Q₈×C₃₁	semidirect
180. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
197. C₃₁	C₂×#1993	direct (N × H)
224. C₂×C₈	C₂×Q₈×C₃₁	semidirect
281. C₂	C₃₁×#1993	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×C₄⋊Q₈	2
C₂²×C₃₁×#97	2
C₂×C₃₁×#382	2
C₃₁×#1993	2
C₂×C₃₁×#340	2
C₂×#1993	31

Subgroups — C2×C31×#1993 #21478079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1993 #21478079