Subgroups — C₂×C₃₁×#1994 #21478086

Order 7936.

1422 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1994	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#339	C₂	1	4	yes	maximal
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₃₁×#533	—	2	2	no	—
8	C₃₁×#533	—	2	2	no	—
9	C₃₁×#533	C₄	1	4	yes	complemented
10	C₃₁×#533	C₂²	1	4	yes	—
11	C₂×C₃₁×C₄⋊Q₈	C₄	1	2	yes	complemented
12	C₂×C₃₁×#125	C₄	1	2	yes	complemented
13	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
14	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
15	C₂×C₃₁×#117	C₂²	1	2	yes	—
16	C₂²×C₄²×C₃₁	—	2	2	no	—
17	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
18	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#423	—	2	2	no	—
20	C₃₁×#423	C₂²	1	4	yes	—
21	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
22	C₄³×C₃₁	—	2	2	no	—
23	C₄³×C₃₁	C₂²	1	4	yes	—
24	C₃₁×#339	C₂²	1	16	yes	—
25	C₂²×Q₈×C₃₁	—	2	2	no	—
26	C₂³×C₄×C₃₁	—	2	2	no	—
27	C₂³×C₄×C₃₁	D₄	1	2	yes	—
28	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₃₁	—	2	2	no	—
30	C₂×M₁₆×C₃₁	—	2	8	no	—
31	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₃₁	—	2	2	no	—
33	C₃₁×C₄⋊Q₈	—	2	4	no	—
34	C₃₁×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
35	C₃₁×#125	—	2	4	no	—
36	C₃₁×#125	C₂×C₄	1	8	yes	complemented
37	C₃₁×C₂²⋊Q₈	—	2	8	no	—
38	C₃₁×C₂²⋊Q₈	—	2	8	no	—
39	C₄×Q₈×C₃₁	—	2	8	no	—
40	C₄×Q₈×C₃₁	—	2	8	no	—
41	C₃₁×#117	—	2	2	no	—
42	C₃₁×#117	—	2	2	no	—
43	C₃₁×#117	C₂×C₄	1	4	yes	—
44	C₃₁×#117	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
48	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄²×C₃₁	—	2	16	no	—
50	C₂×C₄²×C₃₁	—	2	8	no	—
51	C₂×C₄²×C₃₁	—	2	2	no	—
52	C₂×C₄²×C₃₁	—	2	2	no	—
53	C₂×C₄²×C₃₁	—	2	4	no	—
54	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₃₁	D₄	1	2	yes	—
58	C₂×C₄²×C₃₁	D₄	1	2	yes	—
59	C₂×C₄²×C₃₁	D₄	1	2	yes	—
60	C₂×C₄²×C₃₁	Q₈	1	8	yes	complemented
61	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
62	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
63	C₃₁×C₄⋊C₈	—	2	8	no	—
64	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂×Q₈×C₃₁	—	2	8	no	—
66	C₂×Q₈×C₃₁	—	4	8	no	—
67	C₂²×C₄×C₃₁	—	2	16	no	—
68	C₂²×C₄×C₃₁	—	2	8	no	—
69	C₂²×C₄×C₃₁	—	2	4	no	—
70	C₂²×C₄×C₃₁	—	2	8	no	—
71	C₂²×C₄×C₃₁	—	2	4	no	—
72	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
74	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
82	C₂²×C₄×C₃₁	#41	1	4	yes	—
83	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
84	M₁₆×C₃₁	—	2	8	no	—
85	M₁₆×C₃₁	—	4	4	no	—
86	C₂×C₈×C₃₁	—	2	4	no	—
87	C₂×C₈×C₃₁	—	4	4	no	—
88	C₄⋊C₄×C₃₁	—	2	16	no	—
89	C₄⋊C₄×C₃₁	—	2	8	no	—
90	C₄⋊C₄×C₃₁	—	2	4	no	—
91	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
92	C₂²⋊C₄×C₃₁	—	2	4	no	—
93	C₂²⋊C₄×C₃₁	—	2	4	no	—
94	C₄²×C₃₁	—	2	32	no	—
95	C₄²×C₃₁	—	2	4	no	—
96	C₄²×C₃₁	—	2	8	no	—
97	C₄²×C₃₁	—	2	4	no	—
98	C₄²×C₃₁	—	2	4	no	—
99	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
100	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
101	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
102	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
103	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
104	C₄²×C₃₁	C₂×D₄	1	4	yes	—
105	C₄²×C₃₁	C₂×Q₈	1	8	yes	complemented
106	C₂×#1994	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	—	2	4	no	—
108	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂³×C₃₁	C₂²⋊Q₈	1	4	yes	—
111	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₃₁	—	4	16	no	—
113	C₂×C₄×C₃₁	—	2	16	no	—
114	C₂×C₄×C₃₁	—	2	32	no	—
115	C₂×C₄×C₃₁	—	2	8	no	—
116	C₂×C₄×C₃₁	—	2	4	no	—
117	C₂×C₄×C₃₁	—	2	8	no	—
118	C₂×C₄×C₃₁	—	2	8	no	—
119	C₂×C₄×C₃₁	—	2	8	no	—
120	C₂×C₄×C₃₁	—	4	8	no	—
121	C₂×C₄×C₃₁	#104	1	8	yes	—
122	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
123	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
126	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
127	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
128	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
129	C₂×C₄×C₃₁	#124	1	2	yes	—
130	C₂×C₄×C₃₁	#124	1	2	yes	—
131	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
132	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
133	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
136	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
137	C₈×C₃₁	—	4	4	no	—
138	C₂×#533	C₂×C₃₁	1	1	yes	characteristic
139	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
140	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic
141	#1994	C₂×C₃₁	1	8	yes	direct product factor
142	C₂×#339	C₂×C₃₁	1	4	yes	—
143	C₂²×C₃₁	—	2	2	no	—
144	C₂²×C₃₁	—	2	2	no	—
145	C₂²×C₃₁	—	2	8	no	—
146	C₂²×C₃₁	#391	1	2	yes	—
147	C₂²×C₃₁	#391	1	2	yes	—
148	C₂²×C₃₁	C₂×#104	1	4	yes	—
149	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
151	C₄×C₃₁	—	2	16	no	—
152	C₄×C₃₁	—	2	4	no	—
153	C₄×C₃₁	—	4	8	no	—
154	C₄×C₃₁	#391	1	2	yes	—
155	C₄×C₃₁	#391	1	2	yes	—
156	C₄×C₃₁	C₂×#104	1	8	yes	—
157	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
158	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
159	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
160	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
161	#533	—	2	2	no	—
162	#533	—	2	2	no	—
163	#533	C₄×C₃₁	1	4	yes	complemented
164	#533	C₂²×C₃₁	1	4	yes	—
165	C₂×C₄⋊Q₈	C₄×C₃₁	1	2	yes	complemented
166	C₂×#125	C₄×C₃₁	1	2	yes	complemented
167	C₂×C₂²⋊Q₈	—	2	2	no	—
168	C₂×C₄×Q₈	—	2	2	no	—
169	C₂×#117	C₂²×C₃₁	1	2	yes	—
170	C₂²×C₄²	—	2	2	no	—
171	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
172	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
173	#423	—	2	2	no	—
174	#423	C₂²×C₃₁	1	4	yes	—
175	C₂×C₄⋊C₈	—	2	2	no	—
176	C₄³	—	2	2	no	—
177	C₄³	C₂²×C₃₁	1	4	yes	—
178	#339	C₂²×C₃₁	1	16	yes	—
179	C₂×C₃₁	—	2	2	no	—
180	C₂×C₃₁	—	2	2	no	—
181	C₂×C₃₁	#1994	1	4	yes	direct product factor
182	C₂×C₃₁	C₂×#391	1	1	yes	characteristic
183	C₂×C₃₁	C₂²×#104	1	2	yes	—
184	C₂²×Q₈	—	2	2	no	—
185	C₂³×C₄	—	2	2	no	—
186	C₂³×C₄	D₄×C₃₁	1	2	yes	—
187	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	8	no	—
190	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
191	C₂²×C₈	—	2	2	no	—
192	C₄⋊Q₈	—	2	4	no	—
193	C₄⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
194	#125	—	2	4	no	—
195	#125	C₂×C₄×C₃₁	1	8	yes	complemented
196	C₂²⋊Q₈	—	2	8	no	—
197	C₂²⋊Q₈	—	2	8	no	—
198	C₄×Q₈	—	2	8	no	—
199	C₄×Q₈	—	2	8	no	—
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	C₂×C₄×C₃₁	1	4	yes	—
203	#117	C₂³×C₃₁	1	4	yes	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
207	C₂×C₂²⋊C₄	—	2	2	no	—
208	C₂×C₄²	—	2	16	no	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	2	8	no	—
212	C₂×C₄²	—	2	4	no	—
213	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
214	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
215	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
216	C₂×C₄²	D₄×C₃₁	1	2	yes	—
217	C₂×C₄²	D₄×C₃₁	1	2	yes	—
218	C₂×C₄²	D₄×C₃₁	1	2	yes	—
219	C₂×C₄²	Q₈×C₃₁	1	8	yes	complemented
220	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
221	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
222	C₄⋊C₈	—	2	8	no	—
223	C₃₁	C₂×#1994	1	1	yes	characteristic, direct product factor, Sylow
224	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
225	C₂×Q₈	—	2	8	no	—
226	C₂×Q₈	—	4	8	no	—
227	C₂²×C₄	—	2	16	no	—
228	C₂²×C₄	—	2	8	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
233	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
234	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
235	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
238	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
241	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
242	C₂²×C₄	#41×C₃₁	1	4	yes	—
243	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
244	M₁₆	—	2	8	no	—
245	M₁₆	—	4	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	4	no	—
248	C₄⋊C₄	—	2	16	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₄²	—	2	32	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	8	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	4	no	—
259	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
260	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
261	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
262	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
263	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
264	C₄²	C₂×D₄×C₃₁	1	4	yes	—
265	C₄²	C₂×Q₈×C₃₁	1	8	yes	complemented
266	C₂³	—	2	4	no	—
267	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
268	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
269	C₂³	C₃₁×C₂²⋊Q₈	1	4	yes	—
270	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
271	Q₈	—	4	16	no	—
272	C₂×C₄	—	2	16	no	—
273	C₂×C₄	—	2	32	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	4	8	no	—
280	C₂×C₄	C₃₁×#104	1	8	yes	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
282	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
283	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
285	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
286	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
287	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
288	C₂×C₄	C₃₁×#124	1	2	yes	—
289	C₂×C₄	C₃₁×#124	1	2	yes	—
290	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
291	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
292	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
294	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
295	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
296	C₈	—	4	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	8	no	—
300	C₂²	C₃₁×#391	1	2	yes	—
301	C₂²	C₃₁×#391	1	2	yes	—
302	C₂²	C₂×C₃₁×#104	1	4	yes	—
303	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
304	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
305	C₄	—	2	16	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	8	no	—
308	C₄	C₃₁×#391	1	2	yes	—
309	C₄	C₃₁×#391	1	2	yes	—
310	C₄	C₂×C₃₁×#104	1	8	yes	—
311	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
312	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
313	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	C₃₁×#1994	1	4	yes	direct product factor
317	C₂	C₂×C₃₁×#391	1	1	yes	characteristic
318	C₂	C₂²×C₃₁×#104	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1994	C₂	direct (N × H)
9. C₃₁×#533	C₄	semidirect
11. C₂×C₃₁×C₄⋊Q₈	C₄	semidirect
12. C₂×C₃₁×#125	C₄	semidirect
34. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
36. C₃₁×#125	C₂×C₄	semidirect
60. C₂×C₄²×C₃₁	Q₈	semidirect
105. C₄²×C₃₁	C₂×Q₈	semidirect
106. C₂×#1994	C₃₁	direct (N × H)
141. #1994	C₂×C₃₁	direct (N × H)
163. #533	C₄×C₃₁	semidirect
165. C₂×C₄⋊Q₈	C₄×C₃₁	semidirect
166. C₂×#125	C₄×C₃₁	semidirect
181. C₂×C₃₁	#1994	direct (N × H)
193. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
195. #125	C₂×C₄×C₃₁	semidirect
219. C₂×C₄²	Q₈×C₃₁	semidirect
223. C₃₁	C₂×#1994	direct (N × H)
265. C₄²	C₂×Q₈×C₃₁	semidirect
316. C₂	C₃₁×#1994	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#533	2
C₂×C₃₁×#423	2
C₂×C₄³×C₃₁	2
C₃₁×#1994	2
C₂×C₃₁×#339	2
C₂×#1994	31

Subgroups — C2×C31×#1994 #21478086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1994 #21478086