Subgroups — #26108397

Order 12032.

804 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244191	C₂	1	1	yes	maximal, characteristic
2	#20243393	C₂	1	2	yes	maximal
3	C₂×#15688872	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
5	C₂×Q₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×#13093948	—	2	2	no	—
8	C₂×#13093948	—	2	2	no	—
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	#15688993	C₂²	1	1	yes	characteristic
11	#15688885	—	2	2	no	—
12	#15688872	—	2	2	no	—
13	Q₈⋊(C₄₇⋊C₈)	—	2	4	no	—
14	Q₈²×C₄₇	—	2	2	no	—
15	Q₈²×C₄₇	—	2	4	no	—
16	Q₈²×C₄₇	C₄	1	4	yes	complemented
17	C₂×C₄⋊Q₈×C₄₇	—	2	2	no	—
18	C₂×C₄⋊Q₈×C₄₇	—	2	2	no	—
19	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₄₇	—	2	2	no	—
21	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄₇⋊C₈	—	2	1	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂×C₄⋊Dic₄₇	—	2	1	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
28	#13093948	—	4	4	no	—
29	#13093948	—	4	4	no	—
30	C₄×C₄₇⋊C₈	—	2	2	no	—
31	C₂²×Q₈×C₄₇	—	2	2	no	—
32	C₂²×Q₈×C₄₇	D₄	1	2	yes	—
33	C₂²×Q₈×C₄₇	D₄	1	2	yes	—
34	C₄⋊Q₈×C₄₇	—	2	8	no	—
35	C₄⋊Q₈×C₄₇	—	2	8	no	—
36	C₄⋊Q₈×C₄₇	—	2	8	no	—
37	C₄⋊Q₈×C₄₇	—	2	8	no	—
38	C₄⋊Q₈×C₄₇	—	2	2	no	—
39	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
40	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
41	C₄×Q₈×C₄₇	—	2	8	no	—
42	C₄×Q₈×C₄₇	—	2	8	no	—
43	C₄×Q₈×C₄₇	—	2	2	no	—
44	C₄×Q₈×C₄₇	—	2	2	no	—
45	C₄×Q₈×C₄₇	C₂×C₄	1	4	yes	—
46	C₄×Q₈×C₄₇	D₄	1	4	yes	—
47	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
51	C₂×C₄²×C₄₇	—	2	2	no	—
52	C₂×C₄²×C₄₇	—	2	2	no	—
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×Q₈×C₄₇	—	2	8	no	—
55	C₂×Q₈×C₄₇	—	2	2	no	—
56	C₂×Q₈×C₄₇	—	2	2	no	—
57	C₂×Q₈×C₄₇	—	2	2	no	—
58	C₂×Q₈×C₄₇	—	2	2	no	—
59	C₂×Q₈×C₄₇	—	4	2	no	—
60	C₂×Q₈×C₄₇	—	4	2	no	—
61	C₂×Q₈×C₄₇	—	8	2	no	—
62	C₂×Q₈×C₄₇	—	8	2	no	—
63	C₂×Q₈×C₄₇	C₂²⋊C₄	1	4	yes	—
64	C₂×Q₈×C₄₇	C₂²⋊C₄	1	4	yes	—
65	C₂×Q₈×C₄₇	SD₁₆	1	4	yes	—
66	C₂×Q₈×C₄₇	Q₁₆	1	4	yes	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
72	C₂²×Dic₄₇	—	4	1	no	—
73	C₄⋊C₄×C₄₇	—	2	16	no	—
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₄⋊C₄×C₄₇	—	2	4	no	—
78	C₄⋊C₄×C₄₇	—	2	2	no	—
79	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
80	C₄²×C₄₇	—	2	8	no	—
81	C₄²×C₄₇	—	2	4	no	—
82	C₄²×C₄₇	—	2	4	no	—
83	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₄₇	C₂×D₄	1	2	yes	—
86	C₄⋊Dic₄₇	—	4	2	no	—
87	C₄⋊Dic₄₇	—	4	2	no	—
88	C₄⋊Dic₄₇	—	4	2	no	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂×C₄₇⋊C₈	—	4	1	no	—
91	C₂×C₄₇⋊C₈	—	4	1	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂³×C₄₇	#120	1	1	yes	characteristic
96	Q₈×C₄₇	—	2	4	no	—
97	Q₈×C₄₇	—	2	4	no	—
98	Q₈×C₄₇	—	2	4	no	—
99	Q₈×C₄₇	—	4	4	no	—
100	Q₈×C₄₇	—	8	2	no	—
101	Q₈×C₄₇	—	8	8	no	—
102	Q₈×C₄₇	#103	1	8	yes	complemented
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	4	2	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
116	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#120	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
120	C₂×C₄×C₄₇	#137	1	2	yes	—
121	C₂×C₄×C₄₇	#137	1	2	yes	—
122	C₂×Dic₄₇	—	8	2	no	—
123	C₂×Dic₄₇	—	8	1	no	—
124	C₄₇⋊C₈	—	4	2	no	—
125	C₄₇⋊C₈	—	8	2	no	—
126	#8451	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	#379	1	1	yes	characteristic
128	C₂²×C₄₇	#448	1	2	yes	—
129	C₂²×C₄₇	#451	1	2	yes	—
130	C₂²×C₄₇	#453	1	1	yes	characteristic
131	C₂²×C₄₇	#455	1	1	yes	characteristic
132	C₄×C₄₇	—	2	1	no	—
133	C₄×C₄₇	—	2	1	no	—
134	C₄×C₄₇	—	2	8	no	—
135	C₄×C₄₇	—	4	4	no	—
136	C₄×C₄₇	—	4	4	no	—
137	C₄×C₄₇	C₂×#103	1	2	yes	—
138	C₄×C₄₇	#419	1	2	yes	—
139	C₄×C₄₇	#448	1	2	yes	—
140	C₄×C₄₇	#451	1	2	yes	—
141	Dic₄₇	—	16	1	no	—
142	C₂×Q₈²	D₄₇	1	1	yes	characteristic
143	#1956	—	47	1	no	—
144	#1803	—	47	2	no	—
145	C₂×#333	—	47	1	no	—
146	C₂×Q₈⋊C₈	—	47	2	no	—
147	C₂×C₄₇	#1623	1	1	yes	characteristic
148	C₂×C₄₇	#1633	1	1	yes	characteristic
149	C₂×C₄₇	#1645	1	2	yes	—
150	C₂×C₄₇	#1787	1	2	yes	—
151	C₂×C₄₇	#1792	1	1	yes	characteristic
152	Q₈²	—	2	2	no	—
153	Q₈²	—	2	4	no	—
154	Q₈²	Dic₄₇	1	4	yes	complemented
155	C₂×C₄⋊Q₈	—	2	2	no	—
156	C₂×C₄⋊Q₈	—	2	2	no	—
157	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
158	C₂×C₄×Q₈	—	2	2	no	—
159	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
160	C₂×C₄⋊C₈	—	47	2	no	—
161	C₂×#103	—	94	2	no	—
162	C₂×#103	—	94	2	no	—
163	C₂×C₄×C₈	—	47	1	no	—
164	#384	—	47	1	no	—
165	#340	—	94	2	no	—
166	#333	—	94	2	no	—
167	Q₈⋊C₈	—	94	4	no	—
168	C₄₇	#8451	1	1	yes	characteristic, complemented, Sylow
169	C₂²×Q₈	—	2	2	no	—
170	C₂²×Q₈	C₂²⋊D₄₇	1	2	yes	—
171	C₂²×Q₈	C₂²⋊D₄₇	1	2	yes	—
172	C₂²×C₈	—	94	1	no	—
173	C₂²×C₈	—	94	2	no	—
174	C₄⋊Q₈	—	2	8	no	—
175	C₄⋊Q₈	—	2	8	no	—
176	C₄⋊Q₈	—	2	8	no	—
177	C₄⋊Q₈	—	2	8	no	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
180	C₄⋊Q₈	C₂²⋊D₄₇	1	2	yes	—
181	C₄×Q₈	—	2	8	no	—
182	C₄×Q₈	—	2	8	no	—
183	C₄×Q₈	—	2	2	no	—
184	C₄×Q₈	—	2	2	no	—
185	C₄×Q₈	C₂×Dic₄₇	1	4	yes	—
186	C₄×Q₈	C₂²⋊D₄₇	1	4	yes	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	94	2	no	—
191	C₂×C₄⋊C₄	—	94	1	no	—
192	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
196	C₄⋊C₈	—	94	2	no	—
197	C₄⋊C₈	—	94	2	no	—
198	#103	—	188	4	no	—
199	#103	—	188	4	no	—
200	C₄×C₈	—	94	2	no	—
201	C₂×Q₈	—	2	8	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	—	4	2	no	—
208	C₂×Q₈	—	8	2	no	—
209	C₂×Q₈	—	8	2	no	—
210	C₂×Q₈	#10625336	1	4	yes	—
211	C₂×Q₈	#10625337	1	4	yes	—
212	C₂×Q₈	C₂²⋊Dic₄₇	1	4	yes	—
213	C₂×Q₈	C₂²⋊Dic₄₇	1	4	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	188	1	no	—
218	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
220	C₂×C₈	—	94	2	no	—
221	C₂×C₈	—	188	2	no	—
222	C₂×C₈	—	188	2	no	—
223	C₂×C₈	—	188	2	no	—
224	C₂×C₈	—	188	1	no	—
225	C₂×C₈	—	188	1	no	—
226	C₄⋊C₄	—	2	16	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	188	2	no	—
233	C₄⋊C₄	—	188	2	no	—
234	C₄⋊C₄	—	188	2	no	—
235	C₄⋊C₄	C₂²⋊Dic₄₇	1	4	yes	—
236	C₄²	—	2	8	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
240	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
241	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
242	C₂³	#13093993	1	1	yes	characteristic
243	Q₈	—	2	4	no	—
244	Q₈	—	2	4	no	—
245	Q₈	—	2	4	no	—
246	Q₈	—	4	4	no	—
247	Q₈	—	8	2	no	—
248	Q₈	—	8	8	no	—
249	Q₈	#13093948	1	8	yes	complemented
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	376	2	no	—
262	C₂×C₄	—	376	1	no	—
263	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
264	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
265	C₂×C₄	#13093993	1	1	yes	characteristic
266	C₂×C₄	#13093993	1	2	yes	—
267	C₂×C₄	C₂×#10625336	1	2	yes	—
268	C₂×C₄	C₂×#10625337	1	2	yes	—
269	C₂×C₄	#13094055	1	2	yes	—
270	C₂×C₄	#13094055	1	2	yes	—
271	C₈	—	376	2	no	—
272	C₈	—	188	2	no	—
273	C₂²	#15688979	1	1	yes	characteristic
274	C₂²	#15689262	1	2	yes	—
275	C₂²	#15689279	1	2	yes	—
276	C₂²	#15689291	1	1	yes	characteristic
277	C₂²	#15689301	1	1	yes	characteristic
278	C₄	—	2	8	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	4	4	no	—
282	C₄	—	4	4	no	—
283	C₄	—	752	1	no	—
284	C₄	C₂×#13093948	1	2	yes	—
285	C₄	#15689132	1	2	yes	—
286	C₄	#15689262	1	2	yes	—
287	C₄	#15689279	1	2	yes	—
288	C₂	#20242466	1	1	yes	characteristic
289	C₂	#20242528	1	1	yes	characteristic
290	C₂	#20242609	1	2	yes	—
291	C₂	#20243323	1	2	yes	—
292	C₂	#20243342	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093993
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×Q₈²×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#120
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×Q₈²	D₄₇
hypocenter	C₄₇	#8451
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×Q₈²

Complements

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

N	H ≅ G/N	type
16. Q₈²×C₄₇	C₄	semidirect
102. Q₈×C₄₇	#103	semidirect
154. Q₈²	Dic₄₇	semidirect
168. C₄₇	#8451	semidirect
249. Q₈	#13093948	semidirect

Maximal subgroups

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

subgroup	index
#20244191	2
#20243393	2
C₂×#15688872	2
C₂×Q₈⋊(C₄₇⋊C₈)	2
C₂×Q₈²×C₄₇	2
#8451	47