Subgroups — #26233994

Order 12032.

766 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
2	#20246275	C₂	1	2	yes	maximal
3	#20244124	C₂	1	2	yes	maximal, complemented
4	#20243965	C₂	1	4	yes	maximal, complemented
5	#20243480	C₂	1	2	yes	maximal
6	#20243288	C₂	1	2	yes	maximal, complemented
7	#20243164	C₂	1	1	yes	maximal, characteristic
8	C₄₇×#1807	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093983	C₂²	1	2	yes	—
10	#15689161	C₄	1	8	yes	complemented
11	C₂×#13093958	C₂²	1	1	yes	characteristic
12	C₂×#13093958	C₂²	1	1	yes	characteristic
13	C₂×#13093956	C₂²	1	2	yes	—
14	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×#13093934	C₂²	1	2	yes	—
16	#15689019	C₂²	1	4	yes	—
17	#15689017	—	2	4	no	—
18	#15688987	—	2	2	no	—
19	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
20	C₄×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
21	#15688968	—	2	2	no	—
22	#15688885	C₂²	1	4	yes	—
23	#15688885	C₂²	1	4	yes	—
24	#15688877	C₂²	1	2	yes	—
25	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#106×C₄₇	C₂²	1	2	yes	—
27	C₄₇×#380	C₂²	1	2	yes	—
28	C₄₇×#340	C₂²	1	2	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	#13093983	—	2	4	no	—
31	#13093983	C₂×C₄	1	8	yes	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
34	C₂×Dic₄₇⋊C₄	—	2	4	no	—
35	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
36	C₂×C₂²⋊Dic₄₇	D₄	1	2	yes	—
37	C₂×C₄×Dic₄₇	—	2	2	no	—
38	C₂×C₄×Dic₄₇	—	2	4	no	—
39	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
40	#13093958	C₂×C₄	1	4	yes	—
41	#13093958	C₂×C₄	1	4	yes	—
42	#13093956	C₂×C₄	1	8	yes	—
43	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
44	#13093934	—	2	4	no	—
45	C₄₇⋊#95	—	2	4	no	—
46	C₄₇⋊#95	—	2	4	no	—
47	C₄₇⋊#95	D₄	1	2	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	—	2	4	no	—
50	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
54	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
55	#106×C₄₇	—	2	4	no	—
56	#95×C₄₇	—	2	2	no	—
57	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₄₇	—	2	2	no	—
66	C₂²×Dic₄₇	—	2	2	no	—
67	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	8	no	—
70	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₄₇	#41	1	4	yes	—
73	C₄⋊C₄×C₄₇	—	2	4	no	—
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₂²⋊C₄×C₄₇	—	8	2	no	—
76	C₂²⋊Dic₄₇	—	2	8	no	—
77	C₂²⋊Dic₄₇	—	4	2	no	—
78	C₄⋊Dic₄₇	—	2	2	no	—
79	C₄⋊Dic₄₇	—	2	2	no	—
80	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
81	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
82	Dic₄₇⋊C₄	—	8	4	no	—
83	C₄×Dic₄₇	—	2	8	no	—
84	C₄×Dic₄₇	—	2	2	no	—
85	C₄×Dic₄₇	—	2	2	no	—
86	C₄×Dic₄₇	—	4	8	no	—
87	C₂×C₄₇⋊C₈	—	2	2	no	—
88	C₂×C₄₇⋊C₈	—	2	2	no	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	C₄×D₄	1	2	yes	—
94	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	4	4	no	—
100	C₂×C₄×C₄₇	—	4	4	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
107	C₂×Dic₄₇	—	2	4	no	—
108	C₂×Dic₄₇	—	2	4	no	—
109	C₂×Dic₄₇	—	2	4	no	—
110	C₂×Dic₄₇	—	2	4	no	—
111	C₂×Dic₄₇	—	2	4	no	—
112	C₂×Dic₄₇	—	4	4	no	—
113	C₂×Dic₄₇	—	4	2	no	—
114	C₂×Dic₄₇	—	4	2	no	—
115	C₈×C₄₇	—	2	2	no	—
116	C₈×C₄₇	#117	1	4	yes	—
117	C₄₇⋊C₈	—	4	2	no	—
118	#22619	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	4	4	no	—
123	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂²×C₄₇	#547	1	2	yes	—
125	C₂²×C₄₇	#575	1	2	yes	—
126	C₂²×C₄₇	#577	1	2	yes	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	—	8	4	no	—
129	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
130	C₄×C₄₇	#519	1	1	yes	characteristic
131	C₄×C₄₇	#552	1	2	yes	—
132	Dic₄₇	—	4	4	no	—
133	Dic₄₇	—	4	4	no	—
134	Dic₄₇	—	4	4	no	—
135	C₂×#427	—	47	1	no	—
136	#2346	—	47	2	no	—
137	#1943	—	47	2	no	—
138	#1919	—	47	4	no	—
139	#1825	—	47	2	no	—
140	#1807	D₄₇	1	1	yes	characteristic
141	#1776	—	47	2	no	—
142	#1752	—	47	1	no	—
143	C₂×C₄₇	—	4	2	no	—
144	C₂×C₄₇	#2344	1	1	yes	characteristic
145	C₂×C₄₇	#2943	1	1	yes	characteristic
146	C₂×C₄₇	#2952	1	1	yes	characteristic
147	C₂×C₄₇	#3348	1	4	yes	—
148	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
149	C₂×#117	—	47	2	no	—
150	#427	—	47	8	no	—
151	C₂×#107	—	47	1	no	—
152	C₂×#107	—	47	1	no	—
153	C₂×#106	—	47	2	no	—
154	C₂×#106	C₂×D₄₇	1	2	yes	—
155	C₂×C₂²⋊C₈	—	47	1	no	—
156	C₂×#97	—	47	2	no	—
157	#389	—	47	4	no	—
158	#388	—	94	4	no	—
159	#382	—	94	2	no	—
160	#380	C₂×D₄₇	1	2	yes	—
161	C₄×C₄⋊C₄	—	47	4	no	—
162	C₄×C₂²⋊C₄	—	47	1	no	—
163	#376	—	94	2	no	—
164	#340	—	47	4	no	—
165	#340	—	47	4	no	—
166	#340	C₂×D₄₇	1	2	yes	—
167	#336	—	47	2	no	—
168	C₄₇	#22619	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	—	47	2	no	—
172	C₂²×C₈	C₄×D₄₇	1	4	yes	—
173	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
175	#117	—	47	8	no	—
176	#117	—	94	4	no	—
177	C₂×C₄⋊C₄	—	47	2	no	—
178	C₂×C₄⋊C₄	—	47	2	no	—
179	C₂×C₄⋊C₄	—	94	4	no	—
180	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
181	C₂×C₂²⋊C₄	—	2	2	no	—
182	C₂×C₂²⋊C₄	—	47	2	no	—
183	C₂×C₂²⋊C₄	—	94	2	no	—
184	C₂×C₄²	—	47	2	no	—
185	C₂×C₄²	—	94	2	no	—
186	C₂×C₄²	—	94	4	no	—
187	#107	—	47	4	no	—
188	#107	—	47	4	no	—
189	#106	—	2	4	no	—
190	#106	—	47	8	no	—
191	C₂²⋊C₈	—	94	2	no	—
192	#97	—	94	4	no	—
193	#95	—	2	2	no	—
194	#95	—	47	2	no	—
195	#95	—	94	4	no	—
196	#95	—	94	4	no	—
197	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	47	4	no	—
202	C₂²×C₄	—	94	2	no	—
203	C₂²×C₄	—	94	2	no	—
204	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
205	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
206	C₂²×C₄	#10625354	1	2	yes	—
207	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	8	no	—
210	C₂×C₈	—	94	2	no	—
211	C₂×C₈	—	94	2	no	—
212	C₂×C₈	—	94	2	no	—
213	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
214	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
215	C₂×C₈	#10625352	1	4	yes	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	47	4	no	—
219	C₄⋊C₄	—	47	4	no	—
220	C₄⋊C₄	—	94	2	no	—
221	C₄⋊C₄	—	94	2	no	—
222	C₄⋊C₄	—	376	4	no	—
223	C₂²⋊C₄	—	8	2	no	—
224	C₂²⋊C₄	—	94	8	no	—
225	C₂²⋊C₄	—	188	2	no	—
226	C₄²	—	94	8	no	—
227	C₄²	—	94	2	no	—
228	C₄²	—	94	2	no	—
229	C₄²	—	188	8	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	#13093986	1	2	yes	—
234	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	94	4	no	—
243	C₂×C₄	—	94	4	no	—
244	C₂×C₄	—	94	4	no	—
245	C₂×C₄	—	94	4	no	—
246	C₂×C₄	—	94	4	no	—
247	C₂×C₄	—	188	4	no	—
248	C₂×C₄	—	188	2	no	—
249	C₂×C₄	—	188	2	no	—
250	C₂×C₄	#13093986	1	2	yes	—
251	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
252	C₂×C₄	C₂×#10625352	1	2	yes	—
253	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
254	C₂×C₄	#13094071	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	188	2	no	—
257	C₈	#13093980	1	4	yes	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂×#13093986	1	1	yes	characteristic
263	C₂²	#15689773	1	2	yes	—
264	C₂²	#15689920	1	2	yes	—
265	C₂²	#15689929	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	8	4	no	—
268	C₄	—	188	4	no	—
269	C₄	—	188	4	no	—
270	C₄	—	188	4	no	—
271	C₄	C₂×#13093980	1	1	yes	characteristic
272	C₄	#15689600	1	1	yes	characteristic
273	C₄	#15689815	1	2	yes	—
274	C₂	—	4	2	no	—
275	C₂	#20246259	1	1	yes	characteristic
276	C₂	#20252188	1	1	yes	characteristic
277	C₂	#20252271	1	1	yes	characteristic
278	C₂	#20255909	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1807	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1807	D₄₇
hypocenter	C₄₇	#22619
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1807

Complements

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

N	H ≅ G/N	type
3. #20244124	C₂	semidirect
4. #20243965	C₂	semidirect
6. #20243288	C₂	semidirect
10. #15689161	C₄	semidirect
168. C₄₇	#22619	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689161	2
#20246275	2
#20244124	2
#20243965	2
#20243480	2
#20243288	2
#20243164	2
C₄₇×#1807	2
#22619	47