Subgroups — #26109357

Order 12032.

438 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243516	C₂	1	1	yes	maximal, characteristic
2	#20243393	C₂	1	2	yes	maximal
3	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
4	#20241272	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×#13093948	—	2	2	no	—
8	#15688993	C₂²	1	2	yes	—
9	#15688886	—	2	1	no	—
10	#15688886	—	2	1	no	—
11	#15688885	—	2	2	no	—
12	#15688876	—	2	2	no	—
13	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₄₇	—	2	2	no	—
15	C₈×Q₈×C₄₇	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
17	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₇	—	2	1	no	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂×C₄⋊Dic₄₇	—	2	2	no	—
22	C₂×C₄⋊Dic₄₇	—	2	2	no	—
23	C₂×C₄⋊Dic₄₇	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	#13093948	—	4	4	no	—
27	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	—	2	1	no	—
29	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₄×Q₈×C₄₇	—	2	1	no	—
32	C₄×Q₈×C₄₇	—	2	1	no	—
33	C₄×Q₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₄×Q₈×C₄₇	Q₈	1	2	yes	—
35	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	—	2	1	no	—
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₄₇	—	2	4	no	—
40	C₄⋊C₈×C₄₇	—	2	1	no	—
41	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₄₇	—	2	4	no	—
43	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₄₇	D₄	1	2	yes	—
45	C₂×Q₈×C₄₇	—	2	1	no	—
46	C₂×Q₈×C₄₇	—	2	1	no	—
47	C₂×Q₈×C₄₇	—	2	1	no	—
48	C₂×Q₈×C₄₇	C₄⋊C₄	1	2	yes	—
49	C₂×Q₈×C₄₇	D₈	1	2	yes	—
50	C₂×Q₈×C₄₇	Q₁₆	1	2	yes	—
51	C₂²×C₄×C₄₇	—	2	1	no	—
52	C₂²×C₄×C₄₇	—	2	1	no	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
56	C₂²×Dic₄₇	—	4	2	no	—
57	C₂×C₈×C₄₇	—	2	2	no	—
58	C₂×C₈×C₄₇	—	2	2	no	—
59	C₂×C₈×C₄₇	—	2	2	no	—
60	C₂×C₈×C₄₇	—	4	2	no	—
61	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₄⋊C₄×C₄₇	—	2	2	no	—
64	C₄⋊C₄×C₄₇	—	2	1	no	—
65	C₄⋊C₄×C₄₇	—	2	2	no	—
66	C₄⋊C₄×C₄₇	C₄⋊C₄	1	2	yes	—
67	C₄²×C₄₇	—	2	2	no	—
68	C₄²×C₄₇	—	2	2	no	—
69	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₄₇	#41	1	1	yes	characteristic
73	C₄⋊Dic₄₇	—	4	2	no	—
74	C₄⋊Dic₄₇	—	4	2	no	—
75	C₄⋊Dic₄₇	—	4	2	no	—
76	C₄⋊Dic₄₇	—	4	2	no	—
77	C₄⋊Dic₄₇	—	8	2	no	—
78	C₂×C₄₇⋊C₈	—	4	2	no	—
79	C₂×C₄₇⋊C₈	—	4	2	no	—
80	C₂×C₄₇⋊C₈	—	4	2	no	—
81	C₂³×C₄₇	#121	1	1	yes	characteristic
82	Q₈×C₄₇	—	2	2	no	—
83	Q₈×C₄₇	—	2	2	no	—
84	Q₈×C₄₇	—	2	2	no	—
85	Q₈×C₄₇	#107	1	4	yes	—
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	4	2	no	—
92	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
95	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₄×C₄₇	#135	1	2	yes	—
101	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#137	1	2	yes	—
104	C₂×Dic₄₇	—	8	2	no	—
105	C₂×Dic₄₇	—	8	2	no	—
106	C₂×Dic₄₇	—	8	2	no	—
107	C₈×C₄₇	—	2	2	no	—
108	C₈×C₄₇	—	2	2	no	—
109	C₈×C₄₇	—	4	2	no	—
110	C₄₇⋊C₈	—	8	2	no	—
111	#8649	—	47	1	no	maximal, Sylow
112	C₂²×C₄₇	#380	1	1	yes	characteristic
113	C₂²×C₄₇	#464	1	2	yes	—
114	C₂²×C₄₇	#466	1	1	yes	characteristic
115	C₂²×C₄₇	#467	1	1	yes	characteristic
116	C₂²×C₄₇	#471	1	2	yes	—
117	C₄×C₄₇	—	2	4	no	—
118	C₄×C₄₇	—	2	1	no	—
119	C₄×C₄₇	—	2	1	no	—
120	C₄×C₄₇	—	4	2	no	—
121	C₄×C₄₇	#416	1	1	yes	characteristic
122	C₄×C₄₇	#419	1	1	yes	characteristic
123	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
124	C₄×C₄₇	#427	1	1	yes	characteristic
125	C₄×C₄₇	#466	1	1	yes	characteristic
126	C₄×C₄₇	#467	1	1	yes	characteristic
127	C₄×C₄₇	#479	1	2	yes	—
128	Dic₄₇	—	16	2	no	—
129	C₂×C₈×Q₈	D₄₇	1	1	yes	characteristic
130	#1834	—	47	1	no	—
131	#1803	—	47	2	no	—
132	C₂×#335	—	47	1	no	—
133	#1296	—	47	2	no	—
134	C₂×C₄₇	#1682	1	2	yes	—
135	C₂×C₄₇	#1684	1	2	yes	—
136	C₂×C₄₇	#1805	1	1	yes	characteristic
137	C₂×C₄₇	#1808	1	1	yes	characteristic
138	C₂×C₄₇	#1815	1	1	yes	characteristic
139	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
140	C₈×Q₈	—	2	2	no	—
141	C₈×Q₈	Dic₄₇	1	4	yes	complemented
142	C₂×C₄⋊C₈	—	2	1	no	—
143	C₂×C₄⋊C₈	—	47	2	no	—
144	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
145	C₂×#103	—	94	2	no	—
146	C₂×C₄×C₈	—	2	1	no	—
147	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
148	#384	—	47	2	no	—
149	#340	—	94	1	no	—
150	#340	—	94	1	no	—
151	#340	—	94	2	no	—
152	#335	—	94	2	no	—
153	C₄₇	#8649	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	94	2	no	—
157	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	Dic₉₄	1	2	yes	—
162	C₄×Q₈	C₂×Dic₄₇	1	2	yes	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	94	2	no	—
165	C₂×C₄⋊C₄	—	94	2	no	—
166	C₂×C₄⋊C₄	—	94	2	no	—
167	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
168	C₂×C₄²	—	2	1	no	—
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	94	2	no	—
173	C₄⋊C₈	—	94	2	no	—
174	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
175	#103	—	188	4	no	—
176	C₄×C₈	—	2	4	no	—
177	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
178	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
179	C₂×Q₈	—	2	1	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	D₃₇₆	1	2	yes	—
183	C₂×Q₈	Dic₁₈₈	1	2	yes	—
184	C₂×Q₈	C₄⋊Dic₄₇	1	2	yes	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	188	2	no	—
188	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
189	C₂²×C₄	#10625350	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	188	2	no	—
196	C₂×C₈	—	188	2	no	—
197	C₂×C₈	—	188	2	no	—
198	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
199	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	376	2	no	—
204	C₄⋊C₄	—	188	2	no	—
205	C₄⋊C₄	—	188	2	no	—
206	C₄⋊C₄	—	188	2	no	—
207	C₄⋊C₄	—	188	2	no	—
208	C₄⋊C₄	C₄⋊Dic₄₇	1	2	yes	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
212	C₄²	#10625350	1	1	yes	characteristic
213	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
215	C₂³	#13093998	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	Q₈	—	2	2	no	—
218	Q₈	—	2	2	no	—
219	Q₈	#13093958	1	4	yes	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	376	2	no	—
227	C₂×C₄	—	376	2	no	—
228	C₂×C₄	—	376	2	no	—
229	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
231	C₂×C₄	#13093983	1	1	yes	characteristic
232	C₂×C₄	#13093998	1	1	yes	characteristic
233	C₂×C₄	#13094003	1	1	yes	characteristic
234	C₂×C₄	#13094003	1	1	yes	characteristic
235	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
236	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
237	C₂×C₄	#13094044	1	2	yes	—
238	C₂×C₄	#13094046	1	1	yes	characteristic
239	C₂×C₄	#13094046	1	1	yes	characteristic
240	C₂×C₄	#13094055	1	2	yes	—
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	376	2	no	—
245	C₂²	#15688983	1	1	yes	characteristic
246	C₂²	#15689345	1	2	yes	—
247	C₂²	#15689359	1	1	yes	characteristic
248	C₂²	#15689364	1	1	yes	characteristic
249	C₂²	#15689386	1	2	yes	—
250	C₄	—	2	4	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	752	2	no	—
255	C₄	#15689119	1	1	yes	characteristic
256	C₄	#15689132	1	1	yes	characteristic
257	C₄	C₂×#13093958	1	1	yes	characteristic
258	C₄	#15689161	1	1	yes	characteristic
259	C₄	#15689359	1	1	yes	characteristic
260	C₄	#15689364	1	1	yes	characteristic
261	C₄	#15689430	1	2	yes	—
262	C₂	#20242832	1	2	yes	—
263	C₂	#20242846	1	2	yes	—
264	C₂	#20243403	1	1	yes	characteristic
265	C₂	#20243417	1	1	yes	characteristic
266	C₂	#20243447	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093998
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₈×Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#121
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₈×Q₈	D₄₇
hypocenter	C₄₇	#8649
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₄₇	C₄	semidirect
141. C₈×Q₈	Dic₄₇	semidirect
153. C₄₇	#8649	semidirect

Maximal subgroups

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

subgroup	index
#20243516	2
#20243393	2
C₂×#15688876	2
#20241272	2
C₂×C₈×Q₈×C₄₇	2
#8649	47