Subgroups — Q₈×C₄₇⋊#95 #26137764

Order 12032.

2434 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×Dic₄₇	C₂	1	2	yes	maximal
2	#20241618	C₂	1	3	yes	maximal
3	#20241617	C₂	1	6	yes	maximal
4	C₄×C₄₇⋊#95	C₂	1	3	yes	maximal
5	C₂²×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×Dic₄₇	—	2	8	no	—
7	C₂×Q₈×Dic₄₇	—	2	8	no	—
8	C₂×Q₈×Dic₄₇	C₄	1	16	yes	complemented
9	C₂²×C₄⋊Dic₄₇	C₂²	1	6	yes	—
10	C₂²×C₄×Dic₄₇	C₂²	1	6	yes	—
11	C₂×C₄₇⋊#95	C₂²	1	6	yes	—
12	C₂×C₄₇⋊#95	C₂²	1	3	yes	—
13	C₂×C₄₇⋊#95	C₂²	1	6	yes	—
14	C₂×C₄₇⋊#95	C₂²	1	1	yes	characteristic
15	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₄₇	—	2	8	no	—
17	C₂×C₄×Q₈×C₄₇	C₄	1	8	yes	complemented
18	C₂²×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
19	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
20	C₂³×Dic₄₇	C₂³	1	6	yes	—
21	C₂³×Dic₄₇	C₂³	1	2	yes	—
22	Q₈×Dic₄₇	—	2	64	no	—
23	C₂×C₄⋊Dic₄₇	—	2	12	no	—
24	C₂×C₄⋊Dic₄₇	—	2	12	no	—
25	C₂×C₄⋊Dic₄₇	C₂×C₄	1	24	yes	—
26	C₂×C₄×Dic₄₇	—	2	12	no	—
27	C₂×C₄×Dic₄₇	—	2	12	no	—
28	C₂×C₄×Dic₄₇	C₂×C₄	1	24	yes	—
29	C₄₇⋊#95	—	2	6	no	—
30	C₄₇⋊#95	—	2	12	no	—
31	C₄₇⋊#95	—	4	6	no	—
32	C₄₇⋊#95	Q₈	1	4	yes	direct product factor
33	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
34	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
35	C₂²×Q₈×C₄₇	D₄	1	4	yes	—
36	C₂²×Q₈×C₄₇	D₄	1	8	yes	—
37	C₂²×Q₈×C₄₇	Q₈	1	4	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
41	C₄×Q₈×C₄₇	—	2	32	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
43	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	12	yes	—
44	C₂×C₄²×C₄₇	—	2	12	no	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	12	yes	—
46	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
47	C₂×Q₈×C₄₇	C₄²	1	16	yes	—
48	C₂×Q₈×C₄₇	C₂²⋊C₄	1	32	yes	—
49	C₂×Q₈×C₄₇	C₂²⋊C₄	1	16	yes	—
50	C₂×Q₈×C₄₇	C₄⋊C₄	1	32	yes	—
51	C₂×Q₈×C₄₇	C₄⋊C₄	1	16	yes	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	6	no	—
55	C₂²×C₄×C₄₇	—	2	6	no	—
56	C₂²×C₄×C₄₇	—	2	3	no	—
57	C₂²×C₄×C₄₇	—	2	3	no	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	6	yes	—
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×Dic₄₇	—	2	2	no	—
68	C₂²×Dic₄₇	—	2	2	no	—
69	C₂²×Dic₄₇	—	2	2	no	—
70	C₂²×Dic₄₇	—	2	2	no	—
71	C₂²×Dic₄₇	—	2	6	no	—
72	C₂²×Dic₄₇	—	2	6	no	—
73	C₂²×Dic₄₇	—	2	6	no	—
74	C₂²×Dic₄₇	—	2	6	no	—
75	C₂²×Dic₄₇	—	2	6	no	—
76	C₂²×Dic₄₇	—	2	6	no	—
77	C₂²×Dic₄₇	C₂²×C₄	1	12	yes	—
78	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
79	C₂²×Dic₄₇	C₂×Q₈	1	4	yes	—
80	C₂²×Dic₄₇	#41	1	4	yes	—
81	C₄⋊C₄×C₄₇	—	2	24	no	—
82	C₄²×C₄₇	—	2	24	no	—
83	C₄⋊Dic₄₇	—	2	48	no	—
84	C₄×Dic₄₇	—	2	48	no	—
85	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
86	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
87	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
88	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
89	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
90	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
91	C₂³×C₄₇	C₂×#41	1	2	yes	—
92	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
93	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
94	C₂³×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
95	C₂³×C₄₇	2₋¹⁺⁴	1	2	yes	—
96	Q₈×C₄₇	#95	1	64	yes	complemented
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	6	no	—
102	C₂×C₄×C₄₇	—	2	6	no	—
103	C₂×C₄×C₄₇	—	2	6	no	—
104	C₂×C₄×C₄₇	—	4	6	no	—
105	C₂×C₄×C₄₇	—	4	6	no	—
106	C₂×C₄×C₄₇	C₂×C₄²	1	12	yes	—
107	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	24	yes	—
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	24	yes	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
111	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	complemented
112	C₂×Dic₄₇	—	2	4	no	—
113	C₂×Dic₄₇	—	2	4	no	—
114	C₂×Dic₄₇	—	2	4	no	—
115	C₂×Dic₄₇	—	2	4	no	—
116	C₂×Dic₄₇	—	2	4	no	—
117	C₂×Dic₄₇	—	2	4	no	—
118	C₂×Dic₄₇	—	2	12	no	—
119	C₂×Dic₄₇	—	2	12	no	—
120	C₂×Dic₄₇	—	2	12	no	—
121	C₂×Dic₄₇	—	4	6	no	—
122	C₂×Dic₄₇	—	4	6	no	—
123	C₂×Dic₄₇	—	4	6	no	—
124	C₂×Dic₄₇	—	4	6	no	—
125	C₂×Dic₄₇	C₄×Q₈	1	8	yes	complemented
126	Q₈×#95	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
128	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	2	yes	—
129	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
131	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
133	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
134	C₂²×C₄₇	C₄×#41	1	2	yes	—
135	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
136	C₂²×C₄₇	#519	1	2	yes	—
137	C₂²×C₄₇	#519	1	1	yes	characteristic
138	C₂²×C₄₇	#520	1	2	yes	—
139	C₂²×C₄₇	#520	1	1	yes	characteristic
140	C₂²×C₄₇	#548	1	2	yes	—
141	C₂²×C₄₇	#548	1	2	yes	—
142	C₂²×C₄₇	#548	1	2	yes	—
143	C₂²×C₄₇	D₄×Q₈	1	2	yes	—
144	C₂²×C₄₇	D₄×Q₈	1	1	yes	characteristic
145	C₂²×C₄₇	#550	1	2	yes	—
146	C₂²×C₄₇	#550	1	1	yes	characteristic
147	C₂²×C₄₇	#557	1	2	yes	—
148	C₂²×C₄₇	#557	1	1	yes	characteristic
149	C₂²×C₄₇	Q₈²	1	1	yes	characteristic
150	C₄×C₄₇	—	2	4	no	—
151	C₄×C₄₇	—	4	6	no	—
152	C₄×C₄₇	C₂×#95	1	24	yes	—
153	Dic₄₇	—	2	8	no	—
154	Dic₄₇	—	4	12	no	—
155	C₂²×C₄×Q₈	—	47	2	no	—
156	C₂²×C₄×Q₈	D₄₇	1	1	yes	characteristic
157	C₄⋊#95	—	47	3	no	—
158	C₄⋊#95	—	47	6	no	—
159	C₄×#95	—	47	3	no	—
160	C₂×C₄₇	C₂²×#95	1	1	yes	characteristic
161	C₂×C₄₇	C₄²×Q₈	1	1	yes	characteristic
162	C₂×C₄₇	#2276	1	1	yes	characteristic
163	C₂×C₄₇	Q₈×C₂²⋊C₄	1	2	yes	—
164	C₂×C₄₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₄₇	#2341	1	2	yes	—
166	C₂×C₄₇	#2341	1	1	yes	characteristic
167	C₂×C₄₇	Q₈×C₄⋊C₄	1	2	yes	—
168	C₂×C₄₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₄₇	#2351	1	2	yes	—
170	C₂×C₄₇	#2351	1	1	yes	characteristic
171	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
172	C₂×C₄×Q₈	—	2	8	no	—
173	C₂×C₄×Q₈	—	47	16	no	—
174	C₂×C₄×Q₈	—	94	8	no	—
175	C₂×C₄×Q₈	—	94	8	no	—
176	C₂×C₄×Q₈	Dic₄₇	1	8	yes	complemented
177	C₂²×C₄⋊C₄	—	47	6	no	—
178	C₂²×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
179	C₂²×C₄²	—	47	6	no	—
180	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
181	C₂×#95	—	47	1	no	—
182	C₂×#95	—	47	6	no	—
183	C₂×#95	—	47	3	no	—
184	C₂×#95	—	47	6	no	—
185	C₄₇	Q₈×#95	1	1	yes	characteristic, complemented, Sylow
186	C₂²×Q₈	Dic₉₄	1	4	yes	—
187	C₂²×Q₈	C₄×D₄₇	1	8	yes	—
188	C₂²×Q₈	D₁₈₈	1	4	yes	—
189	C₂²×Q₈	C₂×Dic₄₇	1	4	yes	—
190	C₂²×Q₈	C₂²⋊D₄₇	1	8	yes	—
191	C₂³×C₄	—	47	2	no	—
192	C₂³×C₄	—	47	6	no	—
193	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
194	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
195	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
196	C₄×Q₈	—	2	32	no	—
197	C₄×Q₈	—	94	64	no	—
198	C₂×C₄⋊C₄	—	2	12	no	—
199	C₂×C₄⋊C₄	—	47	24	no	—
200	C₂×C₄⋊C₄	—	94	12	no	—
201	C₂×C₄⋊C₄	—	94	12	no	—
202	C₂×C₄⋊C₄	C₂×Dic₄₇	1	12	yes	—
203	C₂×C₄²	—	2	12	no	—
204	C₂×C₄²	—	47	24	no	—
205	C₂×C₄²	—	94	12	no	—
206	C₂×C₄²	—	94	12	no	—
207	C₂×C₄²	C₂×Dic₄₇	1	12	yes	—
208	#95	—	47	4	no	—
209	#95	—	94	12	no	—
210	#95	—	94	6	no	—
211	#95	—	188	6	no	—
212	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
213	C₂×Q₈	C₄×Dic₄₇	1	16	yes	—
214	C₂×Q₈	Dic₄₇⋊C₄	1	32	yes	—
215	C₂×Q₈	C₄⋊Dic₄₇	1	16	yes	—
216	C₂×Q₈	(C₂×D₄₇)⋊C₄	1	32	yes	—
217	C₂×Q₈	C₂²⋊Dic₄₇	1	16	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	6	no	—
221	C₂²×C₄	—	2	6	no	—
222	C₂²×C₄	—	2	3	no	—
223	C₂²×C₄	—	2	3	no	—
224	C₂²×C₄	—	47	4	no	—
225	C₂²×C₄	—	47	4	no	—
226	C₂²×C₄	—	47	4	no	—
227	C₂²×C₄	—	47	12	no	—
228	C₂²×C₄	—	94	2	no	—
229	C₂²×C₄	—	94	2	no	—
230	C₂²×C₄	—	94	2	no	—
231	C₂²×C₄	—	94	2	no	—
232	C₂²×C₄	—	94	6	no	—
233	C₂²×C₄	—	94	6	no	—
234	C₂²×C₄	—	94	6	no	—
235	C₂²×C₄	—	94	6	no	—
236	C₂²×C₄	—	94	6	no	—
237	C₂²×C₄	—	94	6	no	—
238	C₂²×C₄	C₂×Dic₉₄	1	6	yes	—
239	C₂²×C₄	C₂×C₄×D₄₇	1	12	yes	—
240	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
241	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
242	C₂²×C₄	#10625354	1	2	yes	—
243	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
244	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
245	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
246	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
247	C₄⋊C₄	—	2	24	no	—
248	C₄⋊C₄	—	94	48	no	—
249	C₄²	—	2	24	no	—
250	C₄²	—	94	48	no	—
251	C₂³	C₂²×C₄×D₄₇	1	2	yes	—
252	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
253	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
254	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
255	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
256	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
257	C₂³	#41×D₄₇	1	2	yes	—
258	C₂³	C₂×#10625354	1	1	yes	characteristic
259	C₂³	#13094070	1	1	yes	characteristic
260	C₂³	#13094071	1	1	yes	characteristic
261	C₂³	#13094072	1	2	yes	—
262	Q₈	C₄₇⋊#95	1	64	yes	direct product factor
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	6	no	—
268	C₂×C₄	—	2	6	no	—
269	C₂×C₄	—	2	6	no	—
270	C₂×C₄	—	4	6	no	—
271	C₂×C₄	—	4	6	no	—
272	C₂×C₄	—	47	8	no	—
273	C₂×C₄	—	94	4	no	—
274	C₂×C₄	—	94	4	no	—
275	C₂×C₄	—	94	4	no	—
276	C₂×C₄	—	94	4	no	—
277	C₂×C₄	—	94	4	no	—
278	C₂×C₄	—	94	4	no	—
279	C₂×C₄	—	94	12	no	—
280	C₂×C₄	—	94	12	no	—
281	C₂×C₄	—	94	12	no	—
282	C₂×C₄	—	188	6	no	—
283	C₂×C₄	—	188	6	no	—
284	C₂×C₄	—	188	6	no	—
285	C₂×C₄	—	188	6	no	—
286	C₂×C₄	C₂×C₄×Dic₄₇	1	12	yes	—
287	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	24	yes	—
288	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	12	yes	—
289	C₂×C₄	C₂×Dic₄₇⋊C₄	1	24	yes	—
290	C₂×C₄	C₂×C₄⋊Dic₄₇	1	12	yes	—
291	C₂×C₄	Q₈×Dic₄₇	1	4	yes	complemented
292	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
293	C₂²	C₂²×(C₂×D₄₇)⋊C₄	1	2	yes	—
294	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
295	C₂²	C₂²×Dic₄₇⋊C₄	1	2	yes	—
296	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
297	C₂²	C₄×Q₈×D₄₇	1	2	yes	—
298	C₂²	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
299	C₂²	C₄×#10625354	1	2	yes	—
300	C₂²	#41×Dic₄₇	1	1	yes	characteristic
301	C₂²	#15689599	1	2	yes	—
302	C₂²	#15689601	1	1	yes	characteristic
303	C₂²	#15689605	1	2	yes	—
304	C₂²	#15689606	1	1	yes	characteristic
305	C₂²	#15689780	1	2	yes	—
306	C₂²	#15689784	1	2	yes	—
307	C₂²	#15689785	1	2	yes	—
308	C₂²	Q₈×C₂²⋊D₄₇	1	2	yes	—
309	C₂²	Q₈×D₁₈₈	1	1	yes	characteristic
310	C₂²	#15689792	1	2	yes	—
311	C₂²	#15689795	1	1	yes	characteristic
312	C₂²	#15689849	1	2	yes	—
313	C₂²	#15689852	1	1	yes	characteristic
314	C₂²	Q₈×Dic₉₄	1	1	yes	characteristic
315	C₄	—	2	4	no	—
316	C₄	—	4	6	no	—
317	C₄	—	94	8	no	—
318	C₄	—	188	12	no	—
319	C₄	C₂×C₄₇⋊#95	1	24	yes	—
320	C₂	C₂²×C₄₇⋊#95	1	1	yes	characteristic
321	C₂	C₄×Q₈×Dic₄₇	1	1	yes	characteristic
322	C₂	#20245753	1	1	yes	characteristic
323	C₂	Q₈×(C₂×D₄₇)⋊C₄	1	2	yes	—
324	C₂	Q₈×C₂²⋊Dic₄₇	1	1	yes	characteristic
325	C₂	#20246226	1	2	yes	—
326	C₂	#20246229	1	1	yes	characteristic
327	C₂	Q₈×Dic₄₇⋊C₄	1	2	yes	—
328	C₂	Q₈×C₄⋊Dic₄₇	1	1	yes	characteristic
329	C₂	#20246306	1	2	yes	—
330	C₂	#20246309	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂²×C₄×Q₈×C₄₇	C₂
socle, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₄₇
hypocenter	C₄₇	Q₈×#95
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
8. C₂×Q₈×Dic₄₇	C₄	semidirect
17. C₂×C₄×Q₈×C₄₇	C₄	semidirect
32. C₄₇⋊#95	Q₈	direct (N × H)
96. Q₈×C₄₇	#95	semidirect
111. C₂×C₄×C₄₇	C₄×Q₈	semidirect
125. C₂×Dic₄₇	C₄×Q₈	semidirect
176. C₂×C₄×Q₈	Dic₄₇	semidirect
185. C₄₇	Q₈×#95	semidirect
262. Q₈	C₄₇⋊#95	direct (N × H)
291. C₂×C₄	Q₈×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×Dic₄₇	2
#20241618	2
#20241617	2
C₄×C₄₇⋊#95	2
C₂²×C₄×Q₈×C₄₇	2
Q₈×#95	47

Subgroups — Q8×C47⋊#95 #26137764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₄₇⋊#95 #26137764