Subgroups — #26102838

Order 12032.

560 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243368	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄₇⋊#95	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×Dic₄₇	—	2	1	no	—
5	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	2	yes	complemented
6	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂×C₄₇⋊#95	—	2	1	no	—
8	C₂×C₄₇⋊#95	C₄	1	2	yes	—
9	#15688878	—	2	1	no	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
12	C₂×C₄×C₈×C₄₇	—	2	1	no	—
13	C₄₇×#336	C₄	1	2	yes	complemented
14	C₂³×Dic₄₇	—	2	2	no	—
15	C₂²×C₄₇⋊C₈	—	2	2	no	—
16	C₂×C₄×Dic₄₇	—	2	4	no	—
17	C₂×C₄×Dic₄₇	—	4	2	no	—
18	C₂×C₄×Dic₄₇	—	4	1	no	—
19	C₂×C₄×Dic₄₇	—	4	1	no	—
20	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
21	C₂²⋊(C₄₇⋊C₈)	—	4	1	no	—
22	C₂²⋊(C₄₇⋊C₈)	—	4	1	no	—
23	C₄₇⋊#95	—	2	4	no	—
24	C₄₇⋊#95	—	2	2	no	—
25	C₄₇⋊#95	C₈	1	4	yes	complemented
26	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	—	2	2	no	—
30	C₂×C₄²×C₄₇	—	4	2	no	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
35	C₂²⋊C₈×C₄₇	—	2	4	no	—
36	C₄×C₈×C₄₇	—	4	2	no	—
37	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	—	2	2	no	—
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	—	2	2	no	—
41	C₂²×C₄×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	—	4	1	no	—
43	C₂²×C₄×C₄₇	—	4	1	no	—
44	C₂²×C₄×C₄₇	—	4	1	no	—
45	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
53	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
57	C₂²×Dic₄₇	—	2	4	no	—
58	C₂²×Dic₄₇	—	2	4	no	—
59	C₂²×Dic₄₇	—	2	4	no	—
60	C₂²×Dic₄₇	—	4	2	no	—
61	C₂²×Dic₄₇	—	4	2	no	—
62	C₂²×Dic₄₇	—	4	2	no	—
63	C₂²×Dic₄₇	—	4	2	no	—
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	—	4	2	no	—
66	C₂×C₈×C₄₇	—	4	2	no	—
67	C₄²×C₄₇	—	2	2	no	—
68	C₄²×C₄₇	—	2	2	no	—
69	C₄²×C₄₇	—	4	1	no	—
70	C₄²×C₄₇	—	4	1	no	—
71	C₄×Dic₄₇	—	4	4	no	—
72	C₄×Dic₄₇	—	4	4	no	—
73	C₂×C₄₇⋊C₈	—	4	2	no	—
74	C₂×C₄₇⋊C₈	—	4	2	no	—
75	C₂×C₄₇⋊C₈	—	4	2	no	—
76	C₂³×C₄₇	—	2	1	no	—
77	C₂³×C₄₇	—	2	1	no	—
78	C₂³×C₄₇	—	2	1	no	—
79	C₂³×C₄₇	—	2	1	no	—
80	C₂³×C₄₇	#95	1	1	yes	characteristic
81	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
82	C₂³×C₄₇	#102	1	1	yes	characteristic
83	C₂³×C₄₇	#103	1	1	yes	characteristic
84	C₂³×C₄₇	#104	1	2	yes	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	—	4	2	no	—
102	C₂×C₄×C₄₇	—	4	4	no	—
103	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
105	C₂×C₄×C₄₇	#99	1	2	yes	—
106	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#104	1	2	yes	—
109	C₂×C₄×C₄₇	#104	1	2	yes	—
110	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
111	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
113	C₂×Dic₄₇	—	2	8	no	—
114	C₂×Dic₄₇	—	4	4	no	—
115	C₂×Dic₄₇	—	4	4	no	—
116	C₂×Dic₄₇	—	4	4	no	—
117	C₂×Dic₄₇	—	4	4	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	4	4	no	—
120	C₈×C₄₇	—	4	4	no	—
121	C₄₇⋊C₈	—	8	2	no	—
122	#7028	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	2	1	no	—
125	C₂²×C₄₇	—	2	1	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	2	1	no	—
129	C₂²×C₄₇	—	2	1	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	D₄⋊C₈	1	2	yes	—
132	C₂²×C₄₇	Q₈⋊C₈	1	2	yes	—
133	C₂²×C₄₇	#336	1	1	yes	characteristic
134	C₂²×C₄₇	#339	1	1	yes	characteristic
135	C₂²×C₄₇	#339	1	1	yes	characteristic
136	C₂²×C₄₇	#339	1	1	yes	characteristic
137	C₂²×C₄₇	#340	1	1	yes	characteristic
138	C₂²×C₄₇	#342	1	1	yes	characteristic
139	C₂²×C₄₇	#344	1	1	yes	characteristic
140	C₄×C₄₇	—	2	1	no	—
141	C₄×C₄₇	—	2	1	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	2	1	no	—
144	C₄×C₄₇	—	2	1	no	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	4	1	no	—
147	C₄×C₄₇	#327	1	2	yes	—
148	C₄×C₄₇	#328	1	2	yes	—
149	Dic₄₇	—	4	8	no	—
150	#1797	—	47	1	no	—
151	C₄×C₂²⋊C₈	D₄₇	1	1	yes	characteristic
152	C₄×#95	—	47	1	no	—
153	C₂×C₄₇	—	2	2	no	—
154	C₂×C₄₇	—	2	2	no	—
155	C₂×C₄₇	#1276	1	1	yes	characteristic
156	C₂×C₄₇	#1279	1	1	yes	characteristic
157	C₂×C₄₇	#1282	1	1	yes	characteristic
158	C₂×C₄₇	#1294	1	1	yes	characteristic
159	C₂×C₄₇	#1297	1	1	yes	characteristic
160	C₂×C₄₇	#1300	1	1	yes	characteristic
161	C₂×C₄₇	#1303	1	1	yes	characteristic
162	C₂²×C₄²	—	94	1	no	—
163	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
164	C₂×C₄⋊C₈	—	47	2	no	—
165	C₂×C₂²⋊C₈	—	2	1	no	—
166	C₂×C₂²⋊C₈	—	94	1	no	—
167	C₂×C₄×C₈	—	2	1	no	—
168	C₂×#95	—	47	2	no	—
169	C₂×#95	—	94	1	no	—
170	#336	—	94	1	no	—
171	#336	Dic₄₇	1	2	yes	complemented
172	C₄₇	#7028	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	—	94	2	no	—
174	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
175	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
176	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	94	2	no	—
179	C₂×C₄²	—	4	2	no	—
180	C₂×C₄²	—	94	4	no	—
181	C₂×C₄²	—	188	2	no	—
182	C₂×C₄²	—	188	1	no	—
183	C₂×C₄²	—	188	1	no	—
184	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
185	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
186	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
188	C₄⋊C₈	—	188	2	no	—
189	C₂²⋊C₈	—	2	4	no	—
190	C₂²⋊C₈	—	188	1	no	—
191	C₂²⋊C₈	—	188	1	no	—
192	C₄×C₈	—	4	2	no	—
193	#95	—	47	4	no	—
194	#95	—	94	4	no	—
195	#95	—	94	2	no	—
196	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	—	94	4	no	—
205	C₂²×C₄	—	94	4	no	—
206	C₂²×C₄	—	94	4	no	—
207	C₂²×C₄	—	188	2	no	—
208	C₂²×C₄	—	188	2	no	—
209	C₂²×C₄	—	188	2	no	—
210	C₂²×C₄	—	188	2	no	—
211	C₂²×C₄	C₈×D₄₇	1	2	yes	—
212	C₂²×C₄	#10625324	1	2	yes	—
213	C₂²×C₄	#10625325	1	1	yes	characteristic
214	C₂²×C₄	#10625325	1	1	yes	characteristic
215	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
216	C₂²×C₄	Dic₁₈₈	1	1	yes	characteristic
217	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
218	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
221	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	188	2	no	—
227	C₂×C₈	—	188	2	no	—
228	C₂×C₈	—	188	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	4	1	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	188	4	no	—
234	C₄²	—	188	4	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	C₄₇⋊#95	1	1	yes	characteristic
240	C₂³	(C₂×D₄₇)⋊C₈	1	2	yes	—
241	C₂³	#13093946	1	1	yes	characteristic
242	C₂³	#13093949	1	1	yes	characteristic
243	C₂³	#13093952	1	2	yes	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	94	8	no	—
263	C₂×C₄	—	188	4	no	—
264	C₂×C₄	—	188	4	no	—
265	C₂×C₄	—	188	4	no	—
266	C₂×C₄	—	188	4	no	—
267	C₂×C₄	—	188	4	no	—
268	C₂×C₄	—	188	4	no	—
269	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
270	C₂×C₄	#13093934	1	1	yes	characteristic
271	C₂×C₄	#13093938	1	2	yes	—
272	C₂×C₄	#13093946	1	1	yes	characteristic
273	C₂×C₄	#13093949	1	1	yes	characteristic
274	C₂×C₄	#13093950	1	2	yes	—
275	C₂×C₄	#13093951	1	2	yes	—
276	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
277	C₂×C₄	#13093956	1	1	yes	characteristic
278	C₂×C₄	#13093958	1	1	yes	characteristic
279	C₈	—	4	4	no	—
280	C₈	—	376	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	#15688852	1	2	yes	—
290	C₂²	#15688855	1	2	yes	—
291	C₂²	#15688877	1	1	yes	characteristic
292	C₂²	#15688882	1	1	yes	characteristic
293	C₂²	#15688883	1	1	yes	characteristic
294	C₂²	#15688884	1	1	yes	characteristic
295	C₂²	#15688886	1	1	yes	characteristic
296	C₂²	#15688889	1	1	yes	characteristic
297	C₂²	#15688895	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	1	no	—
305	C₄	—	188	8	no	—
306	C₄	#15688858	1	2	yes	—
307	C₄	#15688861	1	2	yes	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	#20241220	1	1	yes	characteristic
311	C₂	#20241228	1	1	yes	characteristic
312	C₂	#20241234	1	1	yes	characteristic
313	C₂	#20241268	1	1	yes	characteristic
314	C₂	#20241274	1	1	yes	characteristic
315	C₂	#20241282	1	1	yes	characteristic
316	C₂	#20241290	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₄₇
Fitting	C₄×C₂²⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	#95
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂²×C₄²×C₄₇	C₂²
hypercenter	C₄×C₂²⋊C₈	D₄₇
hypocenter	C₄₇	#7028
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
5. C₂×C₄⋊(C₄₇⋊C₈)	C₄	semidirect
13. C₄₇×#336	C₄	semidirect
25. C₄₇⋊#95	C₈	semidirect
171. #336	Dic₄₇	semidirect
172. C₄₇	#7028	semidirect

Maximal subgroups

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

subgroup	index
#20243368	2
C₄×C₄₇⋊#95	2
C₄×C₂²⋊C₈×C₄₇	2
#7028	47