Subgroups — #26204213

Order 12032.

858 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254509	C₂	1	2	yes	maximal, complemented
2	#20253967	C₂	1	1	yes	maximal, characteristic
3	#20253845	C₂	1	2	yes	maximal, complemented
4	#20243058	C₂	1	4	yes	maximal, complemented
5	#20243005	C₂	1	2	yes	maximal, complemented
6	#20242981	C₂	1	2	yes	maximal, complemented
7	C₈×#10625329	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094055	—	2	4	no	—
10	#15689673	C₂²	1	2	yes	—
11	#15689528	C₂²	1	1	yes	characteristic
12	#15689526	C₂²	1	4	yes	—
13	#15689515	—	2	4	no	—
14	#15689502	C₂²	1	4	yes	—
15	#15689500	C₂²	1	1	yes	characteristic
16	#15689497	C₂²	1	4	yes	—
17	#15689471	C₂²	1	4	yes	—
18	#15689470	C₂²	1	2	yes	—
19	#15689436	—	2	4	no	—
20	#15689400	—	2	4	no	—
21	#15689359	—	2	2	no	—
22	C₄×D₃₇₆	—	2	2	no	—
23	#15689176	C₂²	1	1	yes	characteristic
24	C₄×#10625329	C₂²	1	1	yes	characteristic
25	#15688845	C₂²	1	2	yes	—
26	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
27	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
28	C₄₇×#500	—	2	2	no	—
29	C₄₇×#500	C₂²	1	2	yes	—
30	C₄₇×#500	C₂²	1	2	yes	—
31	C₂×#107×C₄₇	—	2	2	no	—
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×#10625350	—	2	2	no	—
34	#13094055	—	4	16	no	—
35	C₂×#10625337	—	2	8	no	—
36	C₂×#10625336	—	2	8	no	—
37	C₂×D₃₇₆	—	2	2	no	—
38	C₂×#10625329	D₄	1	2	yes	—
39	#13094024	C₂³	1	2	yes	—
40	#13094021	C₂³	1	2	yes	—
41	#13094012	—	2	2	no	—
42	#13093998	—	2	2	no	—
43	#13093998	—	2	2	no	—
44	C₄×D₁₈₈	—	2	2	no	—
45	C₄×D₁₈₈	—	2	2	no	—
46	#13093958	—	2	2	no	—
47	#13093957	—	2	8	no	—
48	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
49	#13093946	—	2	4	no	—
50	#13093944	—	2	8	no	—
51	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
52	#13093935	D₄	1	2	yes	—
53	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
54	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
55	C₂²×Q₈×C₄₇	—	2	2	no	—
56	C₂²×C₈×C₄₇	D₄	1	2	yes	—
57	C₄⋊Q₈×C₄₇	—	2	2	no	—
58	C₄⋊Q₈×C₄₇	—	2	2	no	—
59	C₄⋊Q₈×C₄₇	—	2	2	no	—
60	C₄⋊Q₈×C₄₇	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	#107×C₄₇	—	2	4	no	—
65	#107×C₄₇	—	2	4	no	—
66	C₄×C₈×C₄₇	D₄	1	2	yes	—
67	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₂×Q₈×C₄₇	—	2	4	no	—
70	C₂×Q₈×C₄₇	—	4	2	no	—
71	C₂×Q₈×C₄₇	—	4	8	no	—
72	C₂²×C₄×C₄₇	—	4	2	no	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₄₇	—	4	2	no	—
77	#10625350	—	4	8	no	—
78	C₂×D₁₈₈	—	2	2	no	—
79	C₂×D₁₈₈	—	2	2	no	—
80	C₂×D₁₈₈	—	2	2	no	—
81	C₂×C₄×D₄₇	—	4	2	no	—
82	C₂×Dic₉₄	—	2	2	no	—
83	C₂×C₈×C₄₇	—	2	4	no	—
84	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
86	C₄⋊C₄×C₄₇	—	2	4	no	—
87	C₄⋊C₄×C₄₇	—	2	4	no	—
88	C₄⋊C₄×C₄₇	—	4	2	no	—
89	C₄⋊C₄×C₄₇	—	4	2	no	—
90	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
94	#10625337	—	4	16	no	—
95	#10625336	—	4	16	no	—
96	(C₂×D₄₇)⋊C₄	—	4	2	no	—
97	(C₂×D₄₇)⋊C₄	—	4	2	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	2	2	no	—
100	#10625329	D₈	1	8	yes	complemented
101	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
102	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
103	D₃₇₆	—	4	4	no	—
104	C₂³×C₄₇	#127	1	1	yes	characteristic
105	C₂²×D₄₇	—	4	2	no	—
106	Q₈×C₄₇	—	4	8	no	—
107	Q₈×C₄₇	—	8	4	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	8	4	no	—
112	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
116	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂²⋊D₄₇	—	8	4	no	—
120	C₂×Dic₄₇	—	4	2	no	—
121	D₁₈₈	—	4	4	no	—
122	D₁₈₈	—	4	4	no	—
123	D₁₈₈	—	4	4	no	—
124	C₄×D₄₇	—	8	4	no	—
125	Dic₉₄	—	4	4	no	—
126	C₈×C₄₇	—	2	2	no	—
127	C₈×C₄₇	#137	1	4	yes	complemented
128	C₄₇⋊C₈	—	2	4	no	—
129	C₄₇⋊C₈	C₂×D₈	1	8	yes	—
130	#19630	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	2	1	no	—
133	C₂²×C₄₇	#493	1	2	yes	—
134	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
135	C₂×D₄₇	—	8	2	no	—
136	C₂×D₄₇	—	8	2	no	—
137	C₂×D₄₇	—	8	2	no	—
138	C₄×C₄₇	—	2	1	no	—
139	C₄×C₄₇	—	2	1	no	—
140	C₄×C₄₇	—	8	4	no	—
141	C₄×C₄₇	#493	1	2	yes	—
142	C₄×C₄₇	C₂²×D₈	1	2	yes	—
143	C₄×C₄₇	C₂×#137	1	2	yes	—
144	C₄×C₄₇	#578	1	2	yes	—
145	Dic₄₇	—	8	2	no	—
146	#3226	—	47	2	no	—
147	#3169	—	47	1	no	—
148	C₂×#500	D₄₇	1	1	yes	characteristic, complemented
149	#3152	—	47	2	no	—
150	#1722	—	47	4	no	—
151	#1713	—	47	2	no	—
152	#1706	—	47	2	no	—
153	C₈×M₁₆	—	47	1	no	—
154	C₂×C₄₇	—	2	2	no	—
155	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
156	C₂×C₄₇	#3152	1	1	yes	characteristic
157	C₂×C₄₇	#3156	1	1	yes	characteristic
158	D₄₇	—	16	2	no	—
159	C₂×#137	—	94	4	no	—
160	#532	—	47	2	no	—
161	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
162	#500	—	2	2	no	—
163	#500	—	47	4	no	—
164	#500	—	47	1	no	—
165	#500	C₂×D₄₇	1	2	yes	—
166	#500	C₂×D₄₇	1	2	yes	—
167	#497	—	94	4	no	—
168	#494	—	47	4	no	—
169	#493	—	47	1	no	—
170	#492	—	47	4	no	—
171	#486	—	47	4	no	—
172	#486	—	47	2	no	—
173	#480	—	94	4	no	—
174	#474	—	94	4	no	—
175	#466	—	94	2	no	—
176	C₄×D₈	—	94	2	no	—
177	#431	—	47	1	no	—
178	C₂×#107	—	2	2	no	—
179	C₄×M₁₆	—	47	1	no	—
180	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
181	#322	—	47	2	no	—
182	C₈²	—	47	2	no	—
183	C₄₇	#19630	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	94	2	no	—
185	C₂²×Q₈	—	2	2	no	—
186	#137	—	188	16	no	—
187	C₂×Q₁₆	—	94	8	no	—
188	C₂×SD₁₆	—	94	8	no	—
189	C₂×D₈	—	94	2	no	—
190	C₂×M₁₆	—	47	2	no	—
191	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	47	2	no	—
196	C₄⋊Q₈	C₂²×D₄₇	1	4	yes	—
197	#127	—	47	2	no	—
198	#124	—	94	2	no	—
199	#121	—	94	2	no	—
200	#121	—	94	2	no	—
201	C₄×D₄	—	94	2	no	—
202	C₄×D₄	—	94	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
206	#107	—	2	4	no	—
207	#107	—	2	4	no	—
208	#107	—	94	8	no	—
209	#107	—	94	2	no	—
210	C₄⋊C₈	—	47	2	no	—
211	#102	—	94	8	no	—
212	#102	—	94	4	no	—
213	C₂²⋊C₈	—	47	2	no	—
214	#97	—	47	2	no	—
215	C₄×C₈	—	47	2	no	—
216	C₄×C₈	—	47	2	no	—
217	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
218	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
220	#41	—	188	8	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	4	8	no	—
224	C₂×Q₈	—	94	2	no	—
225	C₂×D₄	—	94	2	no	—
226	C₂×D₄	—	94	2	no	—
227	C₂×D₄	—	94	2	no	—
228	C₂×D₄	—	188	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	188	2	no	—
231	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
234	Q₁₆	—	188	16	no	—
235	SD₁₆	—	188	16	no	—
236	D₈	—	188	4	no	—
237	M₁₆	—	47	8	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	47	4	no	—
240	C₂×C₈	—	47	4	no	—
241	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	94	2	no	—
248	C₄⋊C₄	—	94	2	no	—
249	C₂²⋊C₄	—	188	2	no	—
250	C₂²⋊C₄	—	188	2	no	—
251	C₄²	D₄×D₄₇	1	1	yes	characteristic
252	C₄²	D₄×D₄₇	1	1	yes	characteristic
253	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
254	C₄²	C₂³×D₄₇	1	1	yes	characteristic
255	C₂³	—	188	2	no	—
256	C₂³	#13094020	1	1	yes	characteristic
257	Q₈	—	4	8	no	—
258	Q₈	—	8	4	no	—
259	Q₈	—	188	4	no	—
260	D₄	—	376	4	no	—
261	D₄	—	188	4	no	—
262	D₄	—	188	4	no	—
263	D₄	—	188	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	—	376	4	no	—
269	C₂×C₄	—	188	2	no	—
270	C₂×C₄	#13094020	1	1	yes	characteristic
271	C₂×C₄	#13094020	1	1	yes	characteristic
272	C₂×C₄	#13094020	1	1	yes	characteristic
273	C₂×C₄	D₈×D₄₇	1	4	yes	—
274	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	47	8	no	—
279	C₈	—	94	4	no	—
280	C₈	#13094055	1	4	yes	complemented
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	376	2	no	—
284	C₂²	—	376	2	no	—
285	C₂²	—	376	2	no	—
286	C₂²	#15689498	1	2	yes	—
287	C₂²	C₂×#13094020	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	8	4	no	—
291	C₄	—	376	2	no	—
292	C₄	#15689498	1	2	yes	—
293	C₄	C₂×D₈×D₄₇	1	2	yes	—
294	C₄	C₂×#13094055	1	2	yes	—
295	C₄	#15689936	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	752	2	no	—
298	C₂	C₂×#15689498	1	1	yes	characteristic
299	C₂	#20253845	1	1	yes	characteristic
300	C₂	#20253877	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13094020
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#500	C₂
socle	C₂²×C₄₇	C₂×#127
prime radical	#20253967	C₂
hypercenter	C₂×#500	D₄₇
hypocenter	C₄₇	#19630
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
1. #20254509	C₂	semidirect
3. #20253845	C₂	semidirect
4. #20243058	C₂	semidirect
5. #20243005	C₂	semidirect
6. #20242981	C₂	semidirect
7. C₈×#10625329	C₂	semidirect
8. C₂×C₄₇×#500	C₂	semidirect
100. #10625329	D₈	semidirect
127. C₈×C₄₇	#137	semidirect
148. C₂×#500	D₄₇	semidirect
183. C₄₇	#19630	semidirect
280. C₈	#13094055	semidirect

Maximal subgroups

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

subgroup	index
#20254509	2
#20253967	2
#20253845	2
#20243058	2
#20243005	2
#20242981	2
C₈×#10625329	2
C₂×C₄₇×#500	2
#19630	47