Subgroups — #26188096

Order 12032.

578 conjugacy classes of subgroups.

The subgroup lattice has 282 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	#20252335	C₂	1	2	yes	maximal
3	#20242298	C₂	1	2	yes	maximal, complemented
4	#20242261	C₂	1	4	yes	maximal, complemented
5	#20242208	C₂	1	2	yes	maximal, complemented
6	C₂×#15688874	C₂	1	1	yes	maximal, characteristic
7	#20241792	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	#15689251	—	2	4	no	—
17	C₈×Dic₉₄	C₂²	1	4	yes	—
18	C₄×Dic₁₈₈	C₂²	1	2	yes	—
19	C₄×Dic₁₈₈	C₂²	1	2	yes	—
20	#15689151	—	2	2	no	—
21	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
22	#15689081	—	2	2	no	—
23	#15688874	C₂²	1	1	yes	characteristic
24	#15688874	C₂²	1	2	yes	—
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	C₂²×C₄₇⋊C₈	—	2	2	no	—
36	#13094024	C₂×C₄	1	2	yes	—
37	#13094024	C₂×C₄	1	2	yes	—
38	#13094016	—	2	2	no	—
39	#13094003	—	2	2	no	—
40	#13094003	C₂×C₄	1	4	yes	—
41	C₄×Dic₉₄	—	2	2	no	—
42	C₄×Dic₉₄	C₂³	1	4	yes	—
43	#13093983	—	2	2	no	—
44	#13093958	C₂×C₄	1	2	yes	—
45	#13093958	C₂×C₄	1	2	yes	—
46	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
47	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
48	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
49	Dic₄₇⋊C₈	—	2	4	no	—
50	Dic₄₇⋊C₈	—	2	4	no	—
51	#13093949	C₂×C₄	1	4	yes	—
52	#13093949	C₂×C₄	1	4	yes	—
53	#13093934	—	2	4	no	—
54	C₈×Dic₄₇	—	2	4	no	—
55	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
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	—	2	1	no	—
60	C₂²⋊C₈×C₄₇	—	2	1	no	—
61	#97×C₄₇	D₄	1	2	yes	—
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	—	2	4	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	C₂×C₄₇⋊C₈	—	4	4	no	—
97	Dic₁₈₈	—	2	4	no	—
98	Dic₁₈₈	—	4	2	no	—
99	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	4	1	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	#137	1	4	yes	—
107	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₄₇	—	2	4	no	—
111	C₂×Dic₄₇	—	2	4	no	—
112	Dic₉₄	—	2	4	no	—
113	Dic₉₄	—	2	4	no	—
114	Dic₉₄	—	8	2	no	—
115	C₈×C₄₇	—	2	1	no	—
116	C₈×C₄₇	—	2	1	no	—
117	C₈×C₄₇	—	2	1	no	—
118	C₈×C₄₇	—	2	4	no	—
119	C₈×C₄₇	—	4	2	no	—
120	C₈×C₄₇	#131	1	2	yes	—
121	C₄₇⋊C₈	—	4	2	no	—
122	C₄₇⋊C₈	—	4	2	no	—
123	#18072	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	1	no	—
125	C₂²×C₄₇	—	2	1	no	—
126	C₂²×C₄₇	#441	1	2	yes	—
127	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₄₇	—	2	1	no	—
129	C₄×C₄₇	—	2	1	no	—
130	C₄×C₄₇	—	2	1	no	—
131	C₄×C₄₇	#441	1	2	yes	—
132	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
133	C₄×C₄₇	#568	1	1	yes	characteristic
134	C₄×C₄₇	C₂×#137	1	2	yes	—
135	Dic₄₇	—	4	4	no	—
136	Dic₄₇	—	4	4	no	—
137	#3170	—	47	1	no	—
138	#2962	—	47	2	no	—
139	#1593	—	47	2	no	—
140	#1587	—	47	4	no	—
141	#1579	—	47	2	no	—
142	C₂×#334	—	47	1	no	—
143	#1481	—	47	2	no	—
144	#1451	D₄₇	1	1	yes	characteristic
145	C₂×C₄₇	—	2	2	no	—
146	C₂×C₄₇	#2935	1	1	yes	characteristic
147	C₂×C₄₇	C₂×#441	1	1	yes	characteristic
148	C₂×C₄₇	#2957	1	1	yes	characteristic
149	#533	—	47	2	no	—
150	#500	—	47	1	no	—
151	#494	—	47	1	no	—
152	#487	—	47	2	no	—
153	#467	—	47	2	no	—
154	#467	—	47	2	no	—
155	#446	—	47	4	no	—
156	#446	—	94	4	no	—
157	C₈×Q₈	—	47	4	no	—
158	C₄×Q₁₆	—	47	2	no	—
159	C₄×Q₁₆	—	47	2	no	—
160	#431	C₂×D₄₇	1	1	yes	characteristic
161	#424	—	94	2	no	—
162	C₂×C₄⋊C₈	—	47	2	no	—
163	#405	—	94	2	no	—
164	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
166	#334	—	47	1	no	—
167	#334	—	47	2	no	—
168	#334	—	47	1	no	—
169	Q₈⋊C₈	—	47	4	no	—
170	Q₈⋊C₈	—	47	4	no	—
171	#322	C₂×D₄₇	1	2	yes	—
172	#322	C₂×D₄₇	1	2	yes	—
173	C₄₇	#18072	1	1	yes	characteristic, complemented, Sylow
174	C₂×Q₁₆	—	47	2	no	—
175	C₂×Q₁₆	—	47	2	no	—
176	C₂×M₁₆	D₁₈₈	1	2	yes	—
177	C₂²×C₈	—	94	2	no	—
178	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
180	C₄⋊Q₈	—	47	2	no	—
181	C₄⋊Q₈	—	47	2	no	—
182	#125	—	94	2	no	—
183	C₂²⋊Q₈	—	47	4	no	—
184	C₂²⋊Q₈	—	94	2	no	—
185	C₄×Q₈	—	47	4	no	—
186	C₄×Q₈	—	94	2	no	—
187	#117	—	94	2	no	—
188	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
189	#107	—	47	2	no	—
190	#107	—	47	2	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	47	2	no	—
193	C₄⋊C₈	—	47	2	no	—
194	C₄⋊C₈	—	94	4	no	—
195	C₄⋊C₈	—	94	4	no	—
196	C₄⋊C₈	—	94	2	no	—
197	#103	—	47	4	no	—
198	#103	—	47	4	no	—
199	C₂²⋊C₈	—	2	1	no	—
200	#97	—	94	4	no	—
201	#97	D₁₈₈	1	2	yes	—
202	C₄×C₈	—	94	4	no	—
203	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
204	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
205	C₄×C₈	C₂²×D₄₇	1	2	yes	—
206	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₄₇	1	2	yes	—
209	C₂×Q₈	—	47	4	no	—
210	C₂×Q₈	—	94	2	no	—
211	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
213	C₂²×C₄	#10625350	1	1	yes	characteristic
214	Q₁₆	—	94	4	no	—
215	Q₁₆	—	188	2	no	—
216	M₁₆	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	94	2	no	—
221	C₂×C₈	—	94	2	no	—
222	C₂×C₈	—	188	4	no	—
223	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
227	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
228	C₄⋊C₄	—	47	4	no	—
229	C₄⋊C₄	—	94	4	no	—
230	C₄⋊C₄	—	94	2	no	—
231	C₄⋊C₄	—	94	2	no	—
232	C₄⋊C₄	—	94	2	no	—
233	C₄⋊C₄	—	94	4	no	—
234	C₂²⋊C₄	—	94	4	no	—
235	C₄²	—	94	4	no	—
236	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
237	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
238	C₄²	#10625350	1	1	yes	characteristic
239	C₄²	C₂³×D₄₇	1	1	yes	characteristic
240	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
241	Q₈	—	94	4	no	—
242	Q₈	—	94	4	no	—
243	Q₈	—	376	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	94	4	no	—
248	C₂×C₄	—	94	4	no	—
249	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
250	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
251	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
252	C₂×C₄	#13094057	1	4	yes	—
253	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
255	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	4	no	—
260	C₈	—	4	2	no	—
261	C₈	—	188	2	no	—
262	C₈	—	188	2	no	—
263	C₈	#13094032	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#15689231	1	2	yes	—
267	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	188	4	no	—
272	C₄	—	188	4	no	—
273	C₄	#15689231	1	2	yes	—
274	C₄	C₂×#13094032	1	1	yes	characteristic
275	C₄	#15689889	1	1	yes	characteristic
276	C₄	C₂×#13094057	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	#20252134	1	1	yes	characteristic
279	C₂	C₂×#15689231	1	1	yes	characteristic
280	C₂	#20252306	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	#18072
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #20242298	C₂	semidirect
4. #20242261	C₂	semidirect
5. #20242208	C₂	semidirect
173. C₄₇	#18072	semidirect

Maximal subgroups

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

subgroup	index
#20253974	2
#20252335	2
#20242298	2
#20242261	2
#20242208	2
C₂×#15688874	2
#20241792	2
C₄₇×#1451	2
#18072	47