Subgroups — Q₈×C₃₁×#103 #21479279

Order 7936.

1086 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	C₂×Q₈²×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1990	C₂	1	3	yes	maximal
5	C₃₁×#1987	C₂	1	3	yes	maximal
6	C₃₁×#1803	C₂	1	3	yes	maximal
7	C₄×C₃₁×#103	C₂	1	3	yes	maximal
8	Q₈²×C₃₁	—	2	4	no	—
9	Q₈²×C₃₁	C₄	1	8	yes	complemented
10	C₂×C₃₁×C₄⋊Q₈	—	2	3	no	—
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
12	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
13	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
14	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
15	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₈×Q₈×C₃₁	—	2	4	no	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	—
18	C₂×C₃₁×#103	C₂²	1	3	yes	—
19	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
21	C₃₁×#389	—	2	3	no	—
22	C₃₁×#384	C₂²	1	3	yes	—
23	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
24	C₃₁×#340	C₂²	1	6	yes	—
25	C₃₁×#340	C₂²	1	3	yes	—
26	C₃₁×#340	C₂²	1	3	yes	—
27	C₂²×Q₈×C₃₁	—	2	3	no	—
28	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
29	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
31	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
32	C₃₁×C₄⋊Q₈	—	2	24	no	—
33	C₃₁×C₄⋊Q₈	—	2	6	no	—
34	C₃₁×C₄⋊Q₈	C₂×C₄	1	12	yes	—
35	C₄×Q₈×C₃₁	—	2	8	no	—
36	C₄×Q₈×C₃₁	—	2	6	no	—
37	C₄×Q₈×C₃₁	—	4	4	no	—
38	C₄×Q₈×C₃₁	C₂×C₄	1	12	yes	—
39	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
40	C₄×Q₈×C₃₁	D₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
45	C₂×C₄⋊C₄×C₃₁	D₄	1	3	yes	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₃₁	—	2	3	no	—
50	C₂×C₄²×C₃₁	—	2	3	no	—
51	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
52	C₃₁×C₄⋊C₈	—	2	6	no	—
53	C₃₁×#103	—	2	6	no	—
54	C₃₁×#103	Q₈	1	4	yes	direct product factor
55	C₄×C₈×C₃₁	—	2	6	no	—
56	C₃₁×#95	—	2	3	no	—
57	C₂×Q₈×C₃₁	—	2	12	no	—
58	C₂×Q₈×C₃₁	—	2	1	no	—
59	C₂×Q₈×C₃₁	—	2	1	no	—
60	C₂×Q₈×C₃₁	—	2	1	no	—
61	C₂×Q₈×C₃₁	—	8	3	no	—
62	C₂×Q₈×C₃₁	—	8	3	no	—
63	C₂×Q₈×C₃₁	C₂²⋊C₄	1	4	yes	—
64	C₂×Q₈×C₃₁	SD₁₆	1	4	yes	—
65	C₂×Q₈×C₃₁	Q₁₆	1	4	yes	—
66	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
67	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
68	C₂×Q₈×C₃₁	#41	1	2	yes	—
69	C₂²×C₄×C₃₁	—	2	1	no	—
70	C₂²×C₄×C₃₁	—	2	1	no	—
71	C₂²×C₄×C₃₁	—	2	3	no	—
72	C₂²×C₄×C₃₁	—	2	3	no	—
73	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
75	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₃₁	—	2	1	no	—
77	C₂×C₈×C₃₁	—	2	1	no	—
78	C₂×C₈×C₃₁	—	2	3	no	—
79	C₂×C₈×C₃₁	—	2	3	no	—
80	C₂×C₈×C₃₁	—	2	3	no	—
81	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
82	C₄⋊C₄×C₃₁	—	2	12	no	—
83	C₄⋊C₄×C₃₁	—	2	12	no	—
84	C₄⋊C₄×C₃₁	—	2	3	no	—
85	C₄⋊C₄×C₃₁	—	2	1	no	—
86	C₄⋊C₄×C₃₁	—	2	3	no	—
87	C₄⋊C₄×C₃₁	—	2	3	no	—
88	C₄⋊C₄×C₃₁	—	4	6	no	—
89	C₄⋊C₄×C₃₁	—	8	3	no	—
90	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	12	yes	—
91	C₄⋊C₄×C₃₁	C₂²×C₄	1	6	yes	—
92	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
93	C₄²×C₃₁	—	2	12	no	—
94	C₄²×C₃₁	—	4	6	no	—
95	C₄²×C₃₁	C₂²×C₄	1	6	yes	—
96	C₄²×C₃₁	C₂×D₄	1	6	yes	—
97	Q₈×#103	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
99	Q₈×C₃₁	—	2	2	no	—
100	Q₈×C₃₁	—	2	2	no	—
101	Q₈×C₃₁	—	2	2	no	—
102	Q₈×C₃₁	—	8	12	no	—
103	Q₈×C₃₁	#103	1	16	yes	direct product factor
104	Q₈×C₃₁	C₄×Q₈	1	4	yes	complemented
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	3	no	—
109	C₂×C₄×C₃₁	—	2	3	no	—
110	C₂×C₄×C₃₁	—	2	6	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	4	1	no	—
113	C₂×C₄×C₃₁	—	4	3	no	—
114	C₂×C₄×C₃₁	—	4	1	no	—
115	C₂×C₄×C₃₁	—	4	3	no	—
116	C₂×C₄×C₃₁	—	4	3	no	—
117	C₂×C₄×C₃₁	—	4	3	no	—
118	C₂×C₄×C₃₁	—	4	3	no	—
119	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
120	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
121	C₂×C₄×C₃₁	C₂×SD₁₆	1	6	yes	—
122	C₂×C₄×C₃₁	C₂×Q₁₆	1	6	yes	—
123	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
129	C₈×C₃₁	—	2	2	no	—
130	C₈×C₃₁	—	4	3	no	—
131	C₂×Q₈²	C₂×C₃₁	1	1	yes	characteristic
132	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
133	Q₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
134	#1990	C₂×C₃₁	1	3	yes	—
135	#1987	C₂×C₃₁	1	3	yes	—
136	#1803	C₂×C₃₁	1	3	yes	—
137	C₄×#103	C₂×C₃₁	1	3	yes	—
138	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₃₁	#548	1	1	yes	characteristic
140	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
141	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
142	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
143	C₂²×C₃₁	#576	1	1	yes	characteristic
144	C₂²×C₃₁	#577	1	1	yes	characteristic
145	C₄×C₃₁	—	2	4	no	—
146	C₄×C₃₁	—	2	6	no	—
147	C₄×C₃₁	—	4	2	no	—
148	C₄×C₃₁	—	4	6	no	—
149	C₄×C₃₁	—	8	3	no	—
150	C₄×C₃₁	C₂×#103	1	12	yes	—
151	C₄×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₄×C₃₁	#519	1	1	yes	characteristic
153	C₄×C₃₁	#548	1	1	yes	characteristic
154	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
155	Q₈²	—	2	4	no	—
156	Q₈²	C₄×C₃₁	1	8	yes	complemented
157	C₂×C₄⋊Q₈	—	2	3	no	—
158	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
159	C₂×C₄×Q₈	—	2	1	no	—
160	C₂×C₄×Q₈	—	2	1	no	—
161	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
162	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
163	C₈×Q₈	—	2	4	no	—
164	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	—
165	C₂×#103	C₂²×C₃₁	1	3	yes	—
166	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
168	#389	—	2	3	no	—
169	#384	C₂²×C₃₁	1	3	yes	—
170	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
171	#340	C₂²×C₃₁	1	6	yes	—
172	#340	C₂²×C₃₁	1	3	yes	—
173	#340	C₂²×C₃₁	1	3	yes	—
174	C₂×C₃₁	Q₈×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₃₁	C₂²×#103	1	1	yes	characteristic
176	C₂×C₃₁	#2897	1	1	yes	characteristic
177	C₂×C₃₁	#3362	1	1	yes	characteristic
178	C₂×C₃₁	#3363	1	1	yes	characteristic
179	C₂×C₃₁	Q₈×SD₁₆	1	1	yes	characteristic
180	C₂×C₃₁	Q₈×Q₁₆	1	1	yes	characteristic
181	C₂²×Q₈	—	2	3	no	—
182	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
183	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
185	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
186	C₄⋊Q₈	—	2	24	no	—
187	C₄⋊Q₈	—	2	6	no	—
188	C₄⋊Q₈	C₂×C₄×C₃₁	1	12	yes	—
189	C₄×Q₈	—	2	8	no	—
190	C₄×Q₈	—	2	6	no	—
191	C₄×Q₈	—	4	4	no	—
192	C₄×Q₈	C₂×C₄×C₃₁	1	12	yes	—
193	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
194	C₄×Q₈	D₄×C₃₁	1	4	yes	—
195	C₂×C₄⋊C₄	—	2	3	no	—
196	C₂×C₄⋊C₄	—	2	3	no	—
197	C₂×C₄⋊C₄	—	2	3	no	—
198	C₂×C₄⋊C₄	—	2	3	no	—
199	C₂×C₄⋊C₄	D₄×C₃₁	1	3	yes	—
200	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
201	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
202	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
203	C₂×C₄²	—	2	3	no	—
204	C₂×C₄²	—	2	3	no	—
205	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
206	C₄⋊C₈	—	2	6	no	—
207	#103	—	2	6	no	—
208	#103	Q₈×C₃₁	1	4	yes	direct product factor
209	C₄×C₈	—	2	6	no	—
210	#95	—	2	3	no	—
211	C₃₁	Q₈×#103	1	1	yes	characteristic, direct product factor, Sylow
212	C₂×Q₈	—	2	12	no	—
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	2	1	no	—
216	C₂×Q₈	—	8	3	no	—
217	C₂×Q₈	—	8	3	no	—
218	C₂×Q₈	C₂²⋊C₄×C₃₁	1	4	yes	—
219	C₂×Q₈	SD₁₆×C₃₁	1	4	yes	—
220	C₂×Q₈	Q₁₆×C₃₁	1	4	yes	—
221	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
222	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
223	C₂×Q₈	#41×C₃₁	1	2	yes	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	3	no	—
227	C₂²×C₄	—	2	3	no	—
228	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
229	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
230	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	3	no	—
234	C₂×C₈	—	2	3	no	—
235	C₂×C₈	—	2	3	no	—
236	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
237	C₄⋊C₄	—	2	12	no	—
238	C₄⋊C₄	—	2	12	no	—
239	C₄⋊C₄	—	2	3	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	3	no	—
242	C₄⋊C₄	—	2	3	no	—
243	C₄⋊C₄	—	4	6	no	—
244	C₄⋊C₄	—	8	3	no	—
245	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	12	yes	—
246	C₄⋊C₄	C₂²×C₄×C₃₁	1	6	yes	—
247	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
248	C₄²	—	2	12	no	—
249	C₄²	—	4	6	no	—
250	C₄²	C₂²×C₄×C₃₁	1	6	yes	—
251	C₄²	C₂×D₄×C₃₁	1	6	yes	—
252	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
253	Q₈	—	2	2	no	—
254	Q₈	—	2	2	no	—
255	Q₈	—	2	2	no	—
256	Q₈	—	8	12	no	—
257	Q₈	C₃₁×#103	1	16	yes	direct product factor
258	Q₈	C₄×Q₈×C₃₁	1	4	yes	complemented
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	3	no	—
263	C₂×C₄	—	2	3	no	—
264	C₂×C₄	—	2	6	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	3	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	3	no	—
270	C₂×C₄	—	4	3	no	—
271	C₂×C₄	—	4	3	no	—
272	C₂×C₄	—	4	3	no	—
273	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
274	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
275	C₂×C₄	C₂×SD₁₆×C₃₁	1	6	yes	—
276	C₂×C₄	C₂×Q₁₆×C₃₁	1	6	yes	—
277	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
280	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
281	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
282	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	4	3	no	—
285	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
286	C₂²	C₃₁×#548	1	1	yes	characteristic
287	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
288	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
289	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
290	C₂²	C₃₁×#576	1	1	yes	characteristic
291	C₂²	C₃₁×#577	1	1	yes	characteristic
292	C₄	—	2	4	no	—
293	C₄	—	2	6	no	—
294	C₄	—	4	2	no	—
295	C₄	—	4	6	no	—
296	C₄	—	8	3	no	—
297	C₄	C₂×C₃₁×#103	1	12	yes	—
298	C₄	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
299	C₄	C₃₁×#519	1	1	yes	characteristic
300	C₄	C₃₁×#548	1	1	yes	characteristic
301	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
302	C₂	Q₈×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
303	C₂	C₂²×C₃₁×#103	1	1	yes	characteristic
304	C₂	C₃₁×#2897	1	1	yes	characteristic
305	C₂	C₃₁×#3362	1	1	yes	characteristic
306	C₂	C₃₁×#3363	1	1	yes	characteristic
307	C₂	Q₈×SD₁₆×C₃₁	1	1	yes	characteristic
308	C₂	Q₈×Q₁₆×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. Q₈²×C₃₁	C₄	semidirect
54. C₃₁×#103	Q₈	direct (N × H)
97. Q₈×#103	C₃₁	direct (N × H)
103. Q₈×C₃₁	#103	direct (N × H)
104. Q₈×C₃₁	C₄×Q₈	semidirect
156. Q₈²	C₄×C₃₁	semidirect
208. #103	Q₈×C₃₁	direct (N × H)
211. C₃₁	Q₈×#103	direct (N × H)
257. Q₈	C₃₁×#103	direct (N × H)
258. Q₈	C₄×Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈²×C₃₁	2
C₂×C₈×Q₈×C₃₁	2
Q₈×C₄⋊C₄×C₃₁	2
C₃₁×#1990	2
C₃₁×#1987	2
C₃₁×#1803	2
C₄×C₃₁×#103	2
Q₈×#103	31

Subgroups — Q8×C31×#103 #21479279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₃₁×#103 #21479279