Subgroups — #26188310

Order 12032.

564 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	#20253974	C₂	1	1	yes	maximal, characteristic
2	#20252489	C₂	1	2	yes	maximal
3	#20242298	C₂	1	2	yes	maximal, complemented
4	#20242240	C₂	1	4	yes	maximal, complemented
5	#20242208	C₂	1	2	yes	maximal, complemented
6	#20242136	C₂	1	1	yes	maximal, characteristic
7	#20241810	C₂	1	2	yes	maximal
8	C₄₇×#1451	C₂	1	1	yes	maximal, characteristic
9	#15689680	C₂²	1	2	yes	—
10	#15689528	C₄	1	1	yes	characteristic
11	#15689503	C₄	1	1	yes	characteristic
12	#15689474	C₄	1	2	yes	—
13	#15689364	C₄	1	2	yes	—
14	#15689364	C₄	1	2	yes	—
15	#15689253	C₂²	1	4	yes	—
16	#15689250	—	2	2	no	—
17	C₈×Dic₉₄	C₂²	1	4	yes	—
18	#15689247	—	2	2	no	—
19	C₄×Dic₁₈₈	C₂²	1	2	yes	—
20	C₄×Dic₁₈₈	C₂²	1	2	yes	—
21	#15689181	—	2	4	no	—
22	#15689176	C₂²	1	2	yes	—
23	#15688876	C₂²	1	2	yes	—
24	#15688874	C₂²	1	1	yes	characteristic
25	#15688874	C₂²	1	1	yes	characteristic
26	#15688855	C₂²	1	4	yes	—
27	#15688855	C₂²	1	4	yes	—
28	C₄₇×#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#322	C₂²	1	2	yes	—
32	C₄₇×#322	C₂²	1	2	yes	—
33	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
34	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
35	#13094024	C₂×C₄	1	2	yes	—
36	#13094024	C₂×C₄	1	2	yes	—
37	#13094016	—	2	2	no	—
38	#13094003	—	2	2	no	—
39	#13094003	C₂×C₄	1	4	yes	—
40	C₄×Dic₉₄	—	2	2	no	—
41	C₄×Dic₉₄	C₂³	1	4	yes	—
42	#13093983	—	2	2	no	—
43	#13093958	C₂×C₄	1	2	yes	—
44	#13093958	C₂×C₄	1	2	yes	—
45	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
46	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
47	Dic₄₇⋊C₈	—	2	4	no	—
48	Dic₄₇⋊C₈	—	2	4	no	—
49	#13093949	C₂×C₄	1	4	yes	—
50	#13093949	C₂×C₄	1	4	yes	—
51	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
52	#13093934	—	2	4	no	—
53	C₄×C₄₇⋊C₈	—	2	2	no	—
54	C₈×Dic₄₇	—	2	4	no	—
55	C₂×M₁₆×C₄₇	—	2	1	no	—
56	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
60	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
61	#97×C₄₇	—	2	1	no	—
62	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₄₇	C₂³	1	2	yes	—
65	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₇	C₂³	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
71	C₂×Dic₉₄	—	2	2	no	—
72	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
73	M₁₆×C₄₇	—	4	2	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	4	1	no	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
82	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₄₇	#41	1	1	yes	characteristic
85	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₂²⋊Dic₄₇	—	2	4	no	—
87	C₄⋊Dic₄₇	—	2	2	no	—
88	C₄⋊Dic₄₇	—	2	2	no	—
89	C₄⋊Dic₄₇	—	2	2	no	—
90	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
91	Dic₄₇⋊C₄	—	2	4	no	—
92	Dic₄₇⋊C₄	—	2	4	no	—
93	C₄×Dic₄₇	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	Dic₁₈₈	—	2	4	no	—
97	Dic₁₈₈	—	4	2	no	—
98	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
109	C₂×Dic₄₇	—	2	4	no	—
110	C₂×Dic₄₇	—	2	4	no	—
111	Dic₉₄	—	2	4	no	—
112	Dic₉₄	—	2	4	no	—
113	Dic₉₄	—	4	4	no	—
114	C₈×C₄₇	—	2	1	no	—
115	C₈×C₄₇	—	2	1	no	—
116	C₈×C₄₇	—	2	1	no	—
117	C₈×C₄₇	—	2	4	no	—
118	C₈×C₄₇	—	4	2	no	—
119	C₈×C₄₇	#131	1	2	yes	—
120	C₄₇⋊C₈	—	4	2	no	—
121	C₄₇⋊C₈	—	4	2	no	—
122	#18090	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	1	no	—
124	C₂²×C₄₇	—	2	1	no	—
125	C₂²×C₄₇	—	2	1	no	—
126	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₄₇	—	2	1	no	—
128	C₄×C₄₇	—	2	1	no	—
129	C₄×C₄₇	—	2	1	no	—
130	C₄×C₄₇	#444	1	2	yes	—
131	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
132	C₄×C₄₇	#568	1	1	yes	characteristic
133	C₄×C₄₇	#578	1	2	yes	—
134	Dic₄₇	—	4	4	no	—
135	Dic₄₇	—	4	4	no	—
136	#3170	—	47	1	no	—
137	#2986	—	47	2	no	—
138	#1593	—	47	2	no	—
139	#1584	—	47	4	no	—
140	#1579	—	47	2	no	—
141	#1564	—	47	1	no	—
142	#1485	—	47	2	no	—
143	#1451	D₄₇	1	1	yes	characteristic
144	C₂×C₄₇	—	4	1	no	—
145	C₂×C₄₇	#2935	1	1	yes	characteristic
146	C₂×C₄₇	#2950	1	1	yes	characteristic
147	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
148	#533	—	47	2	no	—
149	#500	—	47	1	no	—
150	#494	—	47	1	no	—
151	#487	—	47	2	no	—
152	#467	—	47	2	no	—
153	#467	—	47	2	no	—
154	#446	—	47	4	no	—
155	#446	—	94	2	no	—
156	C₈×Q₈	—	47	4	no	—
157	C₈×Q₈	—	94	2	no	—
158	C₄×Q₁₆	—	47	2	no	—
159	C₄×Q₁₆	—	47	2	no	—
160	#433	—	94	4	no	—
161	#431	—	47	2	no	—
162	#431	C₂×D₄₇	1	1	yes	characteristic
163	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
165	#335	—	47	2	no	—
166	#334	—	47	1	no	—
167	#334	—	47	1	no	—
168	Q₈⋊C₈	—	47	4	no	—
169	Q₈⋊C₈	—	47	4	no	—
170	#322	C₂×D₄₇	1	2	yes	—
171	#322	C₂×D₄₇	1	2	yes	—
172	C₄₇	#18090	1	1	yes	characteristic, complemented, Sylow
173	C₂×Q₁₆	—	47	2	no	—
174	C₂×Q₁₆	—	47	2	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
177	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
178	C₄⋊Q₈	—	47	2	no	—
179	C₄⋊Q₈	—	47	2	no	—
180	#125	—	94	2	no	—
181	C₂²⋊Q₈	—	47	4	no	—
182	C₂²⋊Q₈	—	94	2	no	—
183	C₄×Q₈	—	47	4	no	—
184	C₄×Q₈	—	94	2	no	—
185	#117	—	94	2	no	—
186	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
187	#107	—	47	2	no	—
188	#107	—	47	2	no	—
189	C₄⋊C₈	—	47	2	no	—
190	C₄⋊C₈	—	47	2	no	—
191	C₄⋊C₈	—	94	4	no	—
192	C₄⋊C₈	—	94	4	no	—
193	C₄⋊C₈	D₁₈₈	1	2	yes	—
194	#103	—	47	4	no	—
195	#103	—	47	4	no	—
196	C₂²⋊C₈	—	94	2	no	—
197	C₂²⋊C₈	D₁₈₈	1	2	yes	—
198	#97	—	2	1	no	—
199	#97	—	94	4	no	—
200	C₄×C₈	—	94	4	no	—
201	C₄×C₈	—	94	2	no	—
202	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
203	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
204	C₄×C₈	C₂²×D₄₇	1	2	yes	—
205	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₄₇	1	2	yes	—
208	C₂×Q₈	—	47	4	no	—
209	C₂×Q₈	—	94	2	no	—
210	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
212	C₂²×C₄	#10625350	1	1	yes	characteristic
213	Q₁₆	—	94	4	no	—
214	Q₁₆	—	188	2	no	—
215	M₁₆	—	4	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	94	2	no	—
220	C₂×C₈	—	94	2	no	—
221	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
225	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
226	C₄⋊C₄	—	47	4	no	—
227	C₄⋊C₄	—	94	4	no	—
228	C₄⋊C₄	—	94	2	no	—
229	C₄⋊C₄	—	94	2	no	—
230	C₄⋊C₄	—	94	2	no	—
231	C₄⋊C₄	—	94	4	no	—
232	C₂²⋊C₄	—	94	4	no	—
233	C₄²	—	94	4	no	—
234	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
235	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
236	C₄²	#10625350	1	1	yes	characteristic
237	C₄²	C₂³×D₄₇	1	1	yes	characteristic
238	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
239	Q₈	—	94	4	no	—
240	Q₈	—	94	4	no	—
241	Q₈	—	188	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	94	4	no	—
247	C₂×C₄	—	94	4	no	—
248	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
249	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
250	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
253	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	4	no	—
258	C₈	—	4	2	no	—
259	C₈	—	188	2	no	—
260	C₈	—	188	2	no	—
261	C₈	#13094032	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	188	4	no	—
270	C₄	—	188	4	no	—
271	C₄	#15689246	1	2	yes	—
272	C₄	C₂×#13094032	1	1	yes	characteristic
273	C₄	#15689889	1	1	yes	characteristic
274	C₄	#15689940	1	2	yes	—
275	C₂	—	4	1	no	—
276	C₂	#20252134	1	1	yes	characteristic
277	C₂	#20252257	1	1	yes	characteristic
278	C₂	C₂×#15689246	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1451	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1451	D₄₇
hypocenter	C₄₇	#18090
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #20242298	C₂	semidirect
4. #20242240	C₂	semidirect
5. #20242208	C₂	semidirect
172. C₄₇	#18090	semidirect

Maximal subgroups

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

subgroup	index
#20253974	2
#20252489	2
#20242298	2
#20242240	2
#20242208	2
#20242136	2
#20241810	2
C₄₇×#1451	2
#18090	47