Subgroups — #26102862

Order 12032.

476 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243456	C₂	1	1	yes	maximal, characteristic
2	#20241615	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1728	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
5	C₂×C₄₇⋊#95	—	2	1	no	—
6	C₂×C₄₇⋊#95	—	2	1	no	—
7	C₂×C₄₇⋊#95	C₄	1	2	yes	—
8	#15688878	—	2	1	no	—
9	#15688878	C₄	1	2	yes	complemented
10	C₂²×M₁₆×C₄₇	—	2	1	no	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#336	—	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	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
17	C₄₇⋊#95	—	2	4	no	—
18	C₄₇⋊#95	—	4	1	no	—
19	C₄₇⋊#95	—	4	1	no	—
20	C₄₇⋊#95	—	4	2	no	—
21	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₄₇	—	2	4	no	—
25	C₂×M₁₆×C₄₇	—	4	2	no	—
26	C₂×M₁₆×C₄₇	—	4	1	no	—
27	C₂×M₁₆×C₄₇	—	4	1	no	—
28	C₂²×C₈×C₄₇	—	2	2	no	—
29	C₂×C₄²×C₄₇	—	4	2	no	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
34	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	—	2	2	no	—
36	C₂²×C₄×C₄₇	—	2	2	no	—
37	C₂²×C₄×C₄₇	—	2	2	no	—
38	C₂²×C₄×C₄₇	—	2	2	no	—
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	—	4	1	no	—
41	C₂²×C₄×C₄₇	—	4	1	no	—
42	C₂²×C₄×C₄₇	—	4	1	no	—
43	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
53	C₂²×Dic₄₇	—	2	4	no	—
54	C₂²×Dic₄₇	—	4	2	no	—
55	C₂²×Dic₄₇	—	4	2	no	—
56	C₂²×Dic₄₇	—	4	2	no	—
57	C₂²×Dic₄₇	—	4	2	no	—
58	C₂²×Dic₄₇	—	4	2	no	—
59	C₂²×Dic₄₇	—	4	2	no	—
60	M₁₆×C₄₇	—	4	4	no	—
61	M₁₆×C₄₇	—	4	4	no	—
62	C₂×C₈×C₄₇	—	2	4	no	—
63	C₂×C₈×C₄₇	—	4	2	no	—
64	C₂×C₈×C₄₇	—	4	2	no	—
65	C₄²×C₄₇	—	4	1	no	—
66	C₄²×C₄₇	—	4	1	no	—
67	C₄²×C₄₇	—	4	1	no	—
68	C₄²×C₄₇	—	4	1	no	—
69	C₂×C₄₇⋊C₈	—	4	2	no	—
70	C₂×C₄₇⋊C₈	—	4	2	no	—
71	C₂×C₄₇⋊C₈	—	4	2	no	—
72	C₂³×C₄₇	—	2	1	no	—
73	C₂³×C₄₇	—	2	1	no	—
74	C₂³×C₄₇	—	2	1	no	—
75	C₂³×C₄₇	—	2	1	no	—
76	C₂³×C₄₇	#95	1	1	yes	characteristic
77	C₂³×C₄₇	#99	1	2	yes	—
78	C₂³×C₄₇	#102	1	1	yes	characteristic
79	C₂³×C₄₇	#103	1	1	yes	characteristic
80	C₂³×C₄₇	#104	1	2	yes	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
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	1	no	—
94	C₂×C₄×C₄₇	—	2	1	no	—
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	—	2	1	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	4	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#104	1	2	yes	—
105	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
107	C₂×Dic₄₇	—	4	4	no	—
108	C₂×Dic₄₇	—	4	4	no	—
109	C₂×Dic₄₇	—	4	4	no	—
110	C₂×Dic₄₇	—	8	2	no	—
111	C₂×Dic₄₇	—	8	2	no	—
112	C₂×Dic₄₇	—	8	2	no	—
113	C₂×Dic₄₇	—	8	2	no	—
114	C₈×C₄₇	—	4	4	no	—
115	C₄₇⋊C₈	—	8	2	no	—
116	#7037	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	2	1	no	—
119	C₂²×C₄₇	—	2	1	no	—
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	2	1	no	—
123	C₂²×C₄₇	—	2	1	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	#327	1	2	yes	—
126	C₂²×C₄₇	#328	1	2	yes	—
127	C₂²×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
128	C₂²×C₄₇	#339	1	1	yes	characteristic
129	C₂²×C₄₇	#339	1	1	yes	characteristic
130	C₂²×C₄₇	#340	1	1	yes	characteristic
131	C₂²×C₄₇	#342	1	1	yes	characteristic
132	C₂²×C₄₇	#344	1	1	yes	characteristic
133	C₂²×C₄₇	#344	1	1	yes	characteristic
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	—	2	1	no	—
136	C₄×C₄₇	—	2	1	no	—
137	C₄×C₄₇	—	2	1	no	—
138	C₄×C₄₇	—	2	1	no	—
139	C₄×C₄₇	—	4	1	no	—
140	C₄×C₄₇	—	4	1	no	—
141	C₄×C₄₇	—	4	1	no	—
142	Dic₄₇	—	8	4	no	—
143	#1817	—	47	1	no	—
144	#1728	D₄₇	1	1	yes	characteristic
145	#1434	—	47	1	no	—
146	C₂×C₄₇	—	2	2	no	—
147	C₂×C₄₇	—	2	2	no	—
148	C₂×C₄₇	#1284	1	1	yes	characteristic
149	C₂×C₄₇	#1286	1	1	yes	characteristic
150	C₂×C₄₇	#1288	1	1	yes	characteristic
151	C₂×C₄₇	#1294	1	1	yes	characteristic
152	C₂×C₄₇	#1300	1	1	yes	characteristic
153	C₂×C₄₇	#1303	1	1	yes	characteristic
154	C₂×C₄₇	#1306	1	1	yes	characteristic
155	C₂²×M₁₆	—	2	1	no	—
156	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
157	C₂×C₂²⋊C₈	—	94	1	no	—
158	C₂×#95	—	47	2	no	—
159	C₂×#95	—	94	1	no	—
160	C₂×#95	—	94	1	no	—
161	#336	—	2	1	no	—
162	#336	—	47	2	no	—
163	#336	—	94	1	no	—
164	#336	Dic₄₇	1	2	yes	complemented
165	C₄₇	#7037	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	—	94	2	no	—
167	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
168	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
169	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	4	2	no	—
172	C₂×M₁₆	—	4	1	no	—
173	C₂×M₁₆	—	4	1	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	94	2	no	—
176	C₂×C₄²	—	4	2	no	—
177	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂²⋊C₈	—	188	2	no	—
182	#95	—	94	4	no	—
183	#95	—	188	2	no	—
184	#95	—	188	1	no	—
185	#95	—	188	1	no	—
186	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	94	4	no	—
196	C₂²×C₄	—	188	2	no	—
197	C₂²×C₄	—	188	2	no	—
198	C₂²×C₄	—	188	2	no	—
199	C₂²×C₄	—	188	2	no	—
200	C₂²×C₄	—	188	2	no	—
201	C₂²×C₄	—	188	2	no	—
202	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
203	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
204	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
205	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
206	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	#10625334	1	1	yes	characteristic
208	C₂²×C₄	#10625335	1	1	yes	characteristic
209	C₂²×C₄	#10625336	1	1	yes	characteristic
210	C₂²×C₄	#10625337	1	1	yes	characteristic
211	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
212	M₁₆	—	4	4	no	—
213	M₁₆	—	4	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	188	2	no	—
218	C₂×C₈	—	188	2	no	—
219	C₂×C₈	—	188	2	no	—
220	C₄²	—	4	1	no	—
221	C₄²	—	4	1	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	C₄₇⋊#95	1	1	yes	characteristic
229	C₂³	#13093938	1	2	yes	—
230	C₂³	#13093944	1	1	yes	characteristic
231	C₂³	#13093947	1	1	yes	characteristic
232	C₂³	#13093952	1	2	yes	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
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	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	376	2	no	—
255	C₂×C₄	—	376	2	no	—
256	C₂×C₄	—	376	2	no	—
257	C₂×C₄	—	376	2	no	—
258	C₂×C₄	—	188	4	no	—
259	C₂×C₄	—	188	4	no	—
260	C₂×C₄	—	188	4	no	—
261	C₂×C₄	#13093945	1	1	yes	characteristic
262	C₂×C₄	#13093948	1	1	yes	characteristic
263	C₂×C₄	#13093950	1	2	yes	—
264	C₂×C₄	#13093955	1	1	yes	characteristic
265	C₂×C₄	#13093957	1	1	yes	characteristic
266	C₈	—	4	4	no	—
267	C₈	—	376	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	2	no	—
276	C₂²	#15688856	1	2	yes	—
277	C₂²	#15688859	1	2	yes	—
278	C₂²	#15688880	1	1	yes	characteristic
279	C₂²	#15688883	1	1	yes	characteristic
280	C₂²	#15688884	1	1	yes	characteristic
281	C₂²	#15688885	1	1	yes	characteristic
282	C₂²	#15688889	1	1	yes	characteristic
283	C₂²	#15688894	1	1	yes	characteristic
284	C₂²	#15688895	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	376	4	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	#20241242	1	1	yes	characteristic
297	C₂	#20241248	1	1	yes	characteristic
298	C₂	#20241253	1	1	yes	characteristic
299	C₂	#20241267	1	1	yes	characteristic
300	C₂	#20241282	1	1	yes	characteristic
301	C₂	#20241289	1	1	yes	characteristic
302	C₂	#20241297	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₄₇⋊#95
commutator	C₂²×C₄×C₄₇	C₄²
Frattini	C₂²×C₄²	C₂×D₄₇
Fitting	C₄₇×#1728	C₂
socle	C₂³×C₄₇	#95
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂²×C₄²×C₄₇	C₂²
hypercenter	#1728	D₄₇
hypocenter	C₄₇	#7037
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
9. #15688878	C₄	semidirect
13. C₄₇×#336	C₄	semidirect
164. #336	Dic₄₇	semidirect
165. C₄₇	#7037	semidirect

Maximal subgroups

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

subgroup	index
#20243456	2
#20241615	2
C₄₇×#1728	2
#7037	47