Subgroups — #26131523

Order 12032.

430 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689241	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688929	C₂	1	1	yes	maximal, characteristic
3	#20245093	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688899	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241483	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241481	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094046	C₂²	1	1	yes	characteristic
9	C₂×#13094030	—	2	1	no	—
10	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#15689241	—	2	2	no	—
13	#15689241	C₄	1	4	yes	complemented
14	C₂×#13093960	C₂²	1	1	yes	characteristic
15	C₂×#13093952	—	2	1	no	—
16	#15688929	C₄	1	4	yes	complemented
17	#15688926	C₄	1	2	yes	complemented
18	#15688923	C₄	1	2	yes	complemented
19	#15688899	—	2	2	no	—
20	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#427	C₂²	1	1	yes	characteristic
22	C₄₇×#405	—	2	1	no	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#339	—	2	1	no	—
25	C₂×#10625350	—	2	1	no	—
26	#13094046	—	2	1	no	—
27	#13094046	—	2	1	no	—
28	#13094046	C₂×C₄	1	2	yes	—
29	#13094046	C₂×C₄	1	2	yes	—
30	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
31	C₂×#10625325	—	2	1	no	—
32	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
33	#13094030	—	2	4	no	—
34	#13094030	—	4	2	no	—
35	C₂×#10625329	—	2	1	no	—
36	C₂×#10625324	—	4	1	no	—
37	C₂×C₈×D₄₇	—	4	1	no	—
38	C₂×C₄₇⋊C₁₆	—	2	1	no	—
39	C₂×C₄₇⋊C₁₆	—	2	1	no	—
40	C₂×C₄₇⋊C₁₆	—	2	1	no	—
41	C₂×C₄₇⋊C₁₆	—	2	1	no	—
42	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
43	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
44	#13093960	—	2	1	no	—
45	#13093960	C₂×C₄	1	2	yes	—
46	#13093952	—	2	4	no	—
47	C₂×M₁₆×C₄₇	—	2	1	no	—
48	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	#117×C₄₇	—	2	1	no	—
51	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
52	#108×C₄₇	—	2	1	no	—
53	#108×C₄₇	C₂×C₄	1	2	yes	—
54	#107×C₄₇	C₂×C₄	1	2	yes	—
55	#106×C₄₇	—	2	1	no	—
56	#97×C₄₇	—	4	1	no	—
57	C₄×C₈×C₄₇	—	4	1	no	—
58	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₄₇	D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₄₇	—	4	1	no	—
63	#10625350	—	2	2	no	—
64	#10625350	—	2	2	no	—
65	#10625350	—	4	1	no	—
66	#10625350	—	4	1	no	—
67	C₂×D₁₈₈	—	2	1	no	—
68	C₂×C₄×D₄₇	—	4	1	no	—
69	C₂×Dic₉₄	—	2	1	no	—
70	M₁₆×C₄₇	—	2	2	no	—
71	M₁₆×C₄₇	—	4	1	no	—
72	C₂×C₈×C₄₇	—	4	1	no	—
73	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
75	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
76	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
82	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
83	C₄⋊C₄×C₄₇	—	2	2	no	—
84	C₂²⋊C₄×C₄₇	—	4	1	no	—
85	C₄²×C₄₇	—	2	1	no	—
86	C₄²×C₄₇	—	4	1	no	—
87	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
88	#10625329	—	2	2	no	—
89	#10625329	—	4	1	no	—
90	C₂×C₄₇⋊C₈	—	4	1	no	—
91	Dic₁₈₈	—	2	1	no	—
92	Dic₁₈₈	C₄²	1	2	yes	—
93	D₃₇₆	—	2	1	no	—
94	D₃₇₆	C₄²	1	2	yes	—
95	#10625325	—	2	2	no	—
96	#10625325	—	2	2	no	—
97	#10625324	—	4	4	no	—
98	C₈×D₄₇	—	4	4	no	—
99	C₄₇⋊C₁₆	—	2	2	no	—
100	C₄₇⋊C₁₆	—	2	2	no	—
101	C₂³×C₄₇	#135	1	1	yes	characteristic
102	C₂²×D₄₇	—	4	1	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	4	2	no	—
105	C₂×C₄×C₄₇	—	4	2	no	—
106	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
112	C₂²⋊D₄₇	—	4	2	no	—
113	C₂²⋊D₄₇	—	4	2	no	—
114	C₂×Dic₄₇	—	4	1	no	—
115	D₁₈₈	—	2	2	no	—
116	D₁₈₈	—	4	1	no	—
117	C₄×D₄₇	—	4	2	no	—
118	C₄×D₄₇	—	4	2	no	—
119	Dic₉₄	—	2	2	no	—
120	Dic₉₄	—	4	1	no	—
121	C₈×C₄₇	—	4	2	no	—
122	C₈×C₄₇	#102	1	2	yes	—
123	C₈×C₄₇	#102	1	2	yes	—
124	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
125	C₈×C₄₇	#117	1	1	yes	characteristic
126	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
127	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
128	C₄₇⋊C₈	—	4	2	no	—
129	#12642	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	#416	1	1	yes	characteristic
132	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
133	C₂²×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
134	C₂×D₄₇	—	4	2	no	—
135	C₂×D₄₇	—	8	2	no	—
136	C₄×C₄₇	—	4	2	no	—
137	C₄×C₄₇	—	8	2	no	—
138	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
140	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
141	C₄×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
142	Dic₄₇	—	4	2	no	—
143	C₂×#443	—	47	1	no	—
144	C₂×#359	—	47	1	no	—
145	#2138	—	47	1	no	—
146	C₂×#346	—	47	1	no	—
147	#1772	D₄₇	1	1	yes	characteristic, complemented
148	#1382	—	47	1	no	—
149	#1381	—	47	1	no	—
150	C₂×C₄₇	—	2	1	no	—
151	C₂×C₄₇	—	2	1	no	—
152	C₂×C₄₇	C₄×#102	1	1	yes	characteristic
153	C₂×C₄₇	#2179	1	2	yes	—
154	D₄₇	—	8	2	no	—
155	C₂×#135	—	47	1	no	—
156	C₂×#131	—	94	1	no	—
157	C₂²×C₁₆	—	47	1	no	—
158	C₂²×C₁₆	—	47	1	no	—
159	#443	—	47	4	no	—
160	#443	—	94	2	no	—
161	C₂×#108	—	47	1	no	—
162	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
163	#427	C₂×D₄₇	1	1	yes	characteristic
164	C₂×#104	—	94	1	no	—
165	#405	—	2	1	no	—
166	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
167	#359	—	47	4	no	—
168	#358	—	47	2	no	—
169	#357	—	47	2	no	—
170	#346	—	94	2	no	—
171	#339	—	2	1	no	—
172	C₄₇	#12642	1	1	yes	characteristic, complemented, Sylow
173	C₂×#41	—	94	1	no	—
174	#135	—	47	2	no	—
175	#135	—	47	2	no	—
176	#135	—	94	1	no	—
177	#135	—	94	1	no	—
178	C₂×Q₁₆	—	47	1	no	—
179	C₂×SD₁₆	—	94	1	no	—
180	C₂×D₈	—	47	1	no	—
181	#131	—	94	4	no	—
182	#131	—	188	2	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	94	1	no	—
185	C₂×M₁₆	—	188	1	no	—
186	C₂²×C₈	—	188	1	no	—
187	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
189	#117	—	2	1	no	—
190	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂×C₁₆	—	47	2	no	—
192	C₂×C₁₆	—	47	2	no	—
193	C₂×C₁₆	—	94	1	no	—
194	C₂×C₁₆	—	94	1	no	—
195	C₂×C₁₆	—	94	1	no	—
196	C₂×C₁₆	—	94	1	no	—
197	#108	—	2	1	no	—
198	#108	—	47	2	no	—
199	#108	—	94	1	no	—
200	#108	C₄×D₄₇	1	2	yes	—
201	#107	C₄×D₄₇	1	2	yes	—
202	#106	—	2	1	no	—
203	#104	—	94	4	no	—
204	#97	—	4	1	no	—
205	C₄×C₈	—	4	1	no	—
206	C₄×C₈	C₄×D₄₇	1	2	yes	—
207	C₄×C₈	D₁₈₈	1	2	yes	—
208	#41	—	94	2	no	—
209	#41	—	94	2	no	—
210	#41	—	188	1	no	—
211	#41	—	188	1	no	—
212	C₂×Q₈	—	94	1	no	—
213	C₂×D₄	—	94	1	no	—
214	C₂×D₄	—	188	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	188	1	no	—
217	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
218	Q₁₆	—	47	2	no	—
219	Q₁₆	—	94	1	no	—
220	SD₁₆	—	94	2	no	—
221	SD₁₆	—	94	2	no	—
222	D₈	—	47	2	no	—
223	D₈	—	94	1	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	4	1	no	—
226	M₁₆	—	94	2	no	—
227	M₁₆	—	188	1	no	—
228	M₁₆	—	188	4	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	188	1	no	—
231	C₂×C₈	—	188	4	no	—
232	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
233	C₂×C₈	#10625334	1	1	yes	characteristic
234	C₂×C₈	#10625334	1	1	yes	characteristic
235	C₂×C₈	#10625336	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
239	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
240	C₂×C₈	#10625350	1	1	yes	characteristic
241	C₂×C₈	#10625350	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	4	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
247	C₁₆	—	94	2	no	—
248	C₁₆	—	94	2	no	—
249	C₂³	—	188	1	no	—
250	C₂³	#13094044	1	1	yes	characteristic
251	Q₈	—	94	2	no	—
252	Q₈	—	188	1	no	—
253	D₄	—	94	2	no	—
254	D₄	—	188	2	no	—
255	D₄	—	188	2	no	—
256	D₄	—	188	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	188	1	no	—
261	C₂×C₄	—	188	2	no	—
262	C₂×C₄	—	188	2	no	—
263	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
264	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
265	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
266	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
267	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
268	C₂×C₄	#13094044	1	1	yes	characteristic
269	C₈	—	4	2	no	—
270	C₈	—	188	2	no	—
271	C₈	#13093944	1	2	yes	—
272	C₈	#13093944	1	2	yes	—
273	C₈	C₄²×D₄₇	1	1	yes	characteristic
274	C₈	#13093981	1	1	yes	characteristic
275	C₈	C₄×D₁₈₈	1	1	yes	characteristic
276	C₈	C₄×D₁₈₈	1	1	yes	characteristic
277	C₂²	—	2	2	no	—
278	C₂²	—	376	2	no	—
279	C₂²	—	188	2	no	—
280	C₂²	#15689117	1	1	yes	characteristic
281	C₂²	C₄×#10625334	1	1	yes	characteristic
282	C₂²	C₄×#10625336	1	1	yes	characteristic
283	C₄	—	4	2	no	—
284	C₄	—	8	2	no	—
285	C₄	—	188	2	no	—
286	C₄	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
287	C₄	C₂×#13093944	1	1	yes	characteristic
288	C₄	C₄×#10625334	1	1	yes	characteristic
289	C₄	C₄×#10625336	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	376	2	no	—
293	C₂	C₄×#13093944	1	1	yes	characteristic
294	C₂	#20245272	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#10625334
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1772	C₂
socle	C₂²×C₄₇	C₄×D₈
prime radical	C₂×#13094046	C₂²
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1772	D₄₇
hypocenter	C₄₇	#12642
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
4. C₂×#15688899	C₂	semidirect
5. #20241483	C₂	semidirect
6. #20241481	C₂	semidirect
7. C₄₇×#1772	C₂	semidirect
13. #15689241	C₄	semidirect
16. #15688929	C₄	semidirect
17. #15688926	C₄	semidirect
18. #15688923	C₄	semidirect
147. #1772	D₄₇	semidirect
172. C₄₇	#12642	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689241	2
C₂×#15688929	2
#20245093	2
C₂×#15688899	2
#20241483	2
#20241481	2
C₄₇×#1772	2
#12642	47