Subgroups — C₂²⋊Q₈×Dic₄₇ #20245924

Order 6016.

656 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#15689030	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689022	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689009	C₂	1	2	yes	maximal, complemented
6	#15688998	C₂	1	2	yes	maximal
7	#15688983	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×Dic₄₇	C₂	1	2	yes	maximal, complemented
9	C₄⋊C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊C₄×Dic₄₇	C₂	1	2	yes	maximal
11	C₂×C₂²⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂³×Dic₄₇	C₂²	1	1	yes	characteristic
14	Q₈×Dic₄₇	—	2	4	no	—
15	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
20	C₂×C₄×Dic₄₇	—	2	4	no	—
21	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
22	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
23	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
27	C₄₇⋊#95	C₂²	1	2	yes	—
28	C₄₇⋊#95	C₂²	1	2	yes	—
29	C₄₇⋊#95	C₂²	1	2	yes	—
30	C₄₇⋊#95	C₂²	1	2	yes	—
31	C₄₇⋊#95	C₂²	1	2	yes	—
32	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
34	C₂²⋊Q₈×C₄₇	C₄	1	8	yes	complemented
35	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
36	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
38	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₄₇	—	2	2	no	—
40	C₂×Q₈×C₄₇	—	2	2	no	—
41	C₂×Q₈×C₄₇	C₂×C₄	1	4	yes	complemented
42	C₂²×C₄×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	—	2	2	no	—
44	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
45	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×Dic₄₇	—	2	2	no	—
50	C₂²×Dic₄₇	—	2	2	no	—
51	C₂²×Dic₄₇	—	2	2	no	—
52	C₂²×Dic₄₇	—	2	2	no	—
53	C₂²×Dic₄₇	—	2	2	no	—
54	C₂²×Dic₄₇	Q₈	1	4	yes	complemented
55	C₂²×Dic₄₇	C₂³	1	2	yes	—
56	C₂²×Dic₄₇	C₂³	1	2	yes	—
57	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
58	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
59	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
60	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	complemented
62	C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	complemented
63	C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
64	C₂²⋊Dic₄₇	—	2	4	no	—
65	C₄⋊Dic₄₇	—	2	2	no	—
66	C₄⋊Dic₄₇	—	2	2	no	—
67	C₄⋊Dic₄₇	—	2	4	no	—
68	C₄⋊Dic₄₇	—	2	4	no	—
69	C₄×Dic₄₇	—	2	2	no	—
70	C₄×Dic₄₇	—	2	2	no	—
71	C₄×Dic₄₇	—	2	2	no	—
72	C₄×Dic₄₇	—	2	4	no	—
73	C₄×Dic₄₇	—	2	4	no	—
74	C₄×Dic₄₇	D₄	1	4	yes	—
75	C₂³×C₄₇	—	2	1	no	—
76	C₂³×C₄₇	—	2	1	no	—
77	C₂³×C₄₇	—	2	1	no	—
78	C₂³×C₄₇	—	2	1	no	—
79	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂³×C₄₇	C₂×Q₈	1	2	yes	—
81	C₂³×C₄₇	#41	1	2	yes	—
82	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
83	Q₈×C₄₇	—	2	8	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	8	no	—
91	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
92	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
93	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₄₇	#41	1	2	yes	—
97	C₂×Dic₄₇	—	2	8	no	—
98	C₂×Dic₄₇	—	2	1	no	—
99	C₂×Dic₄₇	—	2	1	no	—
100	C₂×Dic₄₇	—	2	1	no	—
101	C₂×Dic₄₇	—	2	1	no	—
102	C₂×Dic₄₇	—	2	1	no	—
103	C₂×Dic₄₇	—	2	1	no	—
104	C₂×Dic₄₇	—	2	8	no	—
105	C₂×Dic₄₇	—	2	2	no	—
106	C₂×Dic₄₇	—	2	2	no	—
107	C₂×Dic₄₇	—	2	2	no	—
108	C₂×Dic₄₇	—	2	2	no	—
109	C₂×Dic₄₇	—	2	2	no	—
110	C₂×Dic₄₇	—	2	2	no	—
111	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
112	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
113	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
114	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
115	C₂×Dic₄₇	#41	1	2	yes	—
116	C₂×Dic₄₇	#41	1	2	yes	—
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	C₄×Q₈	1	4	yes	complemented
124	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
127	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
128	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
129	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	2	4	no	—
133	C₄×C₄₇	—	2	4	no	—
134	C₄×C₄₇	C₄×D₄	1	4	yes	—
135	Dic₄₇	—	2	2	no	—
136	Dic₄₇	—	2	2	no	—
137	Dic₄₇	—	2	2	no	—
138	Dic₄₇	—	4	2	no	—
139	Dic₄₇	—	4	2	no	—
140	Dic₄₇	C₂²⋊Q₈	1	4	yes	direct product factor
141	C₄×C₂²⋊Q₈	—	47	1	no	maximal, Sylow
142	C₂×C₄₇	—	2	4	no	—
143	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
145	C₂×C₄₇	C₄×#41	1	1	yes	characteristic
146	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
147	C₂×C₄₇	#525	1	1	yes	characteristic
148	C₂×C₄₇	#529	1	1	yes	characteristic
149	C₂×C₄₇	#533	1	1	yes	characteristic
150	C₂×C₂²⋊Q₈	D₄₇	1	1	yes	characteristic
151	C₂×C₄×Q₈	—	47	1	no	—
152	C₂²×C₄²	—	47	1	no	—
153	#391	—	47	1	no	—
154	#389	—	47	1	no	—
155	#387	—	47	2	no	—
156	#385	—	47	2	no	—
157	#380	—	47	1	no	—
158	C₄×C₄⋊C₄	—	47	2	no	—
159	C₄×C₄⋊C₄	—	47	1	no	—
160	C₄×C₂²⋊C₄	—	47	2	no	—
161	C₄₇	C₄×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
163	C₂³×C₄	—	47	1	no	—
164	C₂³×C₄	—	47	1	no	—
165	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
166	C₂²⋊Q₈	Dic₄₇	1	8	yes	direct product factor
167	C₄×Q₈	—	94	4	no	—
168	C₂×C₄⋊C₄	—	47	1	no	—
169	C₂×C₄⋊C₄	—	47	2	no	—
170	C₂×C₄⋊C₄	—	47	1	no	—
171	C₂×C₄⋊C₄	—	47	2	no	—
172	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	—	47	2	no	—
175	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
176	C₂×C₄²	—	47	1	no	—
177	C₂×C₄²	—	47	2	no	—
178	C₂×C₄²	—	47	2	no	—
179	C₂×C₄²	—	47	1	no	—
180	C₂×C₄²	—	47	1	no	—
181	C₂×C₄²	—	47	1	no	—
182	C₂×C₄²	—	94	4	no	—
183	#95	—	47	2	no	—
184	#95	—	47	2	no	—
185	#95	—	47	2	no	—
186	#95	—	47	2	no	—
187	#95	—	47	2	no	—
188	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	C₂×Dic₄₇	1	4	yes	complemented
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	47	1	no	—
195	C₂²×C₄	—	47	4	no	—
196	C₂²×C₄	—	47	1	no	—
197	C₂²×C₄	—	47	1	no	—
198	C₂²×C₄	—	47	1	no	—
199	C₂²×C₄	—	47	2	no	—
200	C₂²×C₄	—	47	2	no	—
201	C₂²×C₄	—	94	2	no	—
202	C₂²×C₄	—	94	2	no	—
203	C₂²×C₄	—	94	2	no	—
204	C₂²×C₄	—	94	2	no	—
205	C₂²×C₄	—	94	2	no	—
206	C₂²×C₄	C₂×Dic₄₇	1	4	yes	—
207	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
209	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
210	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
211	C₄⋊C₄	—	94	4	no	—
212	C₄⋊C₄	—	94	4	no	—
213	C₄⋊C₄	—	94	2	no	—
214	C₄⋊C₄	—	94	2	no	—
215	C₄⋊C₄	C₂×Dic₄₇	1	4	yes	complemented
216	C₄⋊C₄	C₂×Dic₄₇	1	8	yes	complemented
217	C₂²⋊C₄	—	94	4	no	—
218	C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
219	C₄²	—	47	4	no	—
220	C₄²	—	94	4	no	—
221	C₄²	—	94	2	no	—
222	C₄²	—	94	2	no	—
223	C₄²	—	94	4	no	—
224	C₄²	—	94	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	Q₈×D₄₇	1	2	yes	—
230	C₂³	#10625354	1	2	yes	—
231	C₂³	C₂²×Dic₄₇	1	2	yes	—
232	C₂³	C₂³×D₄₇	1	1	yes	characteristic
233	Q₈	—	2	8	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	47	2	no	—
242	C₂×C₄	—	47	2	no	—
243	C₂×C₄	—	47	2	no	—
244	C₂×C₄	—	47	2	no	—
245	C₂×C₄	—	47	2	no	—
246	C₂×C₄	—	47	2	no	—
247	C₂×C₄	—	94	8	no	—
248	C₂×C₄	—	94	8	no	—
249	C₂×C₄	—	94	1	no	—
250	C₂×C₄	—	94	1	no	—
251	C₂×C₄	—	94	1	no	—
252	C₂×C₄	—	94	1	no	—
253	C₂×C₄	—	94	1	no	—
254	C₂×C₄	—	94	1	no	—
255	C₂×C₄	—	94	2	no	—
256	C₂×C₄	—	94	2	no	—
257	C₂×C₄	—	94	2	no	—
258	C₂×C₄	—	94	2	no	—
259	C₂×C₄	—	94	2	no	—
260	C₂×C₄	—	94	2	no	—
261	C₂×C₄	D₄×D₄₇	1	2	yes	—
262	C₂×C₄	#10625352	1	2	yes	—
263	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
264	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
265	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
266	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	Q₈×Dic₄₇	1	4	yes	complemented
274	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
275	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
276	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
277	C₂²	#41×D₄₇	1	1	yes	characteristic
278	C₂²	#41×D₄₇	1	1	yes	characteristic
279	C₂²	C₂×#10625352	1	1	yes	characteristic
280	C₂²	C₂×#10625354	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	4	no	—
283	C₄	—	2	4	no	—
284	C₄	—	47	4	no	—
285	C₄	—	94	2	no	—
286	C₄	—	94	2	no	—
287	C₄	—	94	2	no	—
288	C₄	—	188	2	no	—
289	C₄	—	188	2	no	—
290	C₄	D₄×Dic₄₇	1	4	yes	—
291	C₂	—	2	4	no	—
292	C₂	C₂×D₄×Dic₄₇	1	1	yes	characteristic
293	C₂	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
294	C₂	#41×Dic₄₇	1	1	yes	characteristic
295	C₂	C₂²⋊Q₈×D₄₇	1	1	yes	characteristic
296	C₂	#15689636	1	1	yes	characteristic
297	C₂	#15689659	1	1	yes	characteristic
298	C₂	#15689675	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₂×C₂²⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×C₂²⋊Q₈	D₄₇
hypocenter	C₄₇	C₄×C₂²⋊Q₈
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₄₇	C₂	semidirect
3. #15689030	C₂	semidirect
4. #15689022	C₂	semidirect
5. #15689009	C₂	semidirect
8. C₄⋊C₄×Dic₄₇	C₂	semidirect
9. C₄⋊C₄×Dic₄₇	C₂	semidirect
34. C₂²⋊Q₈×C₄₇	C₄	semidirect
41. C₂×Q₈×C₄₇	C₂×C₄	semidirect
54. C₂²×Dic₄₇	Q₈	semidirect
61. C₄⋊C₄×C₄₇	C₂×C₄	semidirect
62. C₄⋊C₄×C₄₇	C₂×C₄	semidirect
123. C₂²×C₄₇	C₄×Q₈	semidirect
140. Dic₄₇	C₂²⋊Q₈	direct (N × H)
161. C₄₇	C₄×C₂²⋊Q₈	semidirect
166. C₂²⋊Q₈	Dic₄₇	direct (N × H)
191. C₂×Q₈	C₂×Dic₄₇	semidirect
215. C₄⋊C₄	C₂×Dic₄₇	semidirect
216. C₄⋊C₄	C₂×Dic₄₇	semidirect
273. C₂²	Q₈×Dic₄₇	semidirect

Subgroups — C22⋊Q8×Dic47 #20245924

Conjugacy classes of subgroups

Complements

Subgroups — C₂²⋊Q₈×Dic₄₇ #20245924