Subgroups — #26104698

Order 12032.

384 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251770	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20241213	C₂	1	2	yes	maximal
4	#20241212	C₂	1	2	yes	maximal
5	C₄₇×#1839	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
8	#15689029	C₄	1	1	yes	characteristic
9	#15689029	C₄	1	1	yes	characteristic
10	#15689029	C₂²	1	1	yes	characteristic
11	#15689021	C₄	1	1	yes	characteristic
12	#15689021	C₄	1	1	yes	characteristic
13	#15689021	C₂²	1	1	yes	characteristic
14	#15689007	—	2	1	no	—
15	#15689007	—	2	1	no	—
16	#15689007	—	2	1	no	—
17	#15688991	—	2	1	no	—
18	#15688878	C₂²	1	2	yes	—
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#336	—	2	1	no	—
21	C₄₇×#336	—	2	1	no	—
22	C₄₇×#336	C₂²	1	2	yes	—
23	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×Dic₄₇⋊C₄	—	2	1	no	—
29	C₂×Dic₄₇⋊C₄	—	2	1	no	—
30	C₂×Dic₄₇⋊C₄	—	2	1	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
32	C₄×C₄₇⋊C₈	—	4	2	no	—
33	C₄₇⋊#95	—	2	1	no	—
34	C₄₇⋊#95	—	2	1	no	—
35	C₄₇⋊#95	—	2	1	no	—
36	C₄₇⋊#95	—	2	1	no	—
37	C₄₇⋊#95	—	2	1	no	—
38	C₄₇⋊#95	—	2	1	no	—
39	C₄₇⋊#95	—	2	1	no	—
40	C₄₇⋊#95	—	2	1	no	—
41	C₄₇⋊#95	—	2	1	no	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₄₇	—	2	1	no	—
58	C₂²×Dic₄₇	—	2	1	no	—
59	C₂²×Dic₄₇	—	2	1	no	—
60	C₂²×Dic₄₇	—	2	1	no	—
61	C₂×Dic₉₄	—	2	2	no	—
62	C₂×Dic₉₄	—	4	1	no	—
63	C₂×Dic₉₄	—	4	1	no	—
64	C₂×C₈×C₄₇	—	4	2	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	—	2	2	no	—
70	C₄²×C₄₇	—	4	1	no	—
71	C₄²×C₄₇	—	4	1	no	—
72	C₄²×C₄₇	—	4	1	no	—
73	C₄²×C₄₇	SD₁₆	1	2	yes	—
74	C₄²×C₄₇	Q₁₆	1	2	yes	—
75	C₄⋊Dic₄₇	—	2	2	no	—
76	C₄⋊Dic₄₇	—	4	1	no	—
77	C₄⋊Dic₄₇	—	4	1	no	—
78	Dic₄₇⋊C₄	—	4	2	no	—
79	Dic₄₇⋊C₄	—	4	2	no	—
80	Dic₄₇⋊C₄	—	4	2	no	—
81	C₂×C₄₇⋊C₈	—	2	4	no	—
82	C₂×C₄₇⋊C₈	—	4	2	no	—
83	C₂×C₄₇⋊C₈	—	4	2	no	—
84	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	#103	1	2	yes	—
101	C₂×C₄×C₄₇	#103	1	2	yes	—
102	C₂×C₄×C₄₇	#104	1	2	yes	—
103	C₂×C₄×C₄₇	#131	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#135	1	2	yes	—
107	C₂×Dic₄₇	—	4	1	no	—
108	C₂×Dic₄₇	—	4	1	no	—
109	C₂×Dic₄₇	—	4	1	no	—
110	C₂×Dic₄₇	—	4	1	no	—
111	C₂×Dic₄₇	—	4	1	no	—
112	C₂×Dic₄₇	—	4	1	no	—
113	C₂×Dic₄₇	—	4	1	no	—
114	C₂×Dic₄₇	—	4	1	no	—
115	C₂×Dic₄₇	—	4	1	no	—
116	C₂×Dic₄₇	—	4	1	no	—
117	C₂×Dic₄₇	—	4	1	no	—
118	C₂×Dic₄₇	—	4	1	no	—
119	Dic₉₄	—	8	2	no	—
120	C₈×C₄₇	—	8	2	no	—
121	C₄₇⋊C₈	—	4	4	no	—
122	#7673	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	#408	1	1	yes	characteristic
124	C₂²×C₄₇	#410	1	1	yes	characteristic
125	C₂²×C₄₇	#413	1	1	yes	characteristic
126	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
127	C₂²×C₄₇	#416	1	1	yes	characteristic
128	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
129	C₂²×C₄₇	#421	1	1	yes	characteristic
130	C₄×C₄₇	—	2	2	no	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	2	1	no	—
135	C₄×C₄₇	—	2	1	no	—
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	—	4	1	no	—
138	Dic₄₇	—	8	1	no	—
139	Dic₄₇	—	8	1	no	—
140	Dic₄₇	—	8	1	no	—
141	Dic₄₇	—	8	1	no	—
142	#2863	—	47	1	no	—
143	#1839	D₄₇	1	1	yes	characteristic
144	C₄×C₄⋊C₈	—	47	1	no	—
145	#1272	—	47	2	no	—
146	#1271	—	47	2	no	—
147	C₂×C₄₇	#1473	1	1	yes	characteristic
148	C₂×C₄₇	#1481	1	1	yes	characteristic
149	C₂×C₄₇	#1494	1	1	yes	characteristic
150	C₂×C₄₇	#1503	1	1	yes	characteristic
151	C₂×C₄₇	#1521	1	1	yes	characteristic
152	C₂×C₄₇	#1529	1	1	yes	characteristic
153	C₂×C₄₇	#1542	1	1	yes	characteristic
154	C₂×C₄⋊C₈	—	94	1	no	—
155	C₂×C₄×C₈	—	94	1	no	—
156	#391	—	47	1	no	—
157	#391	—	47	1	no	—
158	#391	—	47	1	no	—
159	#389	—	47	1	no	—
160	#389	—	47	1	no	—
161	#389	—	47	1	no	—
162	#387	—	94	1	no	—
163	#387	—	94	1	no	—
164	#387	—	94	1	no	—
165	#383	—	94	1	no	—
166	C₄³	C₂×D₄₇	1	1	yes	characteristic
167	#336	—	2	1	no	—
168	#336	—	2	1	no	—
169	#336	—	47	2	no	—
170	#336	C₂×D₄₇	1	2	yes	—
171	C₄₇	#7673	1	1	yes	characteristic, complemented, Sylow
172	C₂²×Q₈	—	47	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	94	2	no	—
175	C₂×C₄⋊C₄	—	47	1	no	—
176	C₂×C₄⋊C₄	—	47	1	no	—
177	C₂×C₄⋊C₄	—	47	1	no	—
178	C₂×C₄⋊C₄	—	94	1	no	—
179	C₂×C₄⋊C₄	—	94	1	no	—
180	C₂×C₄⋊C₄	—	94	1	no	—
181	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
182	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
183	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
184	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
186	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
187	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
188	C₄⋊C₈	—	94	4	no	—
189	C₄×C₈	—	188	2	no	—
190	#95	—	94	1	no	—
191	#95	—	94	1	no	—
192	#95	—	94	1	no	—
193	#95	—	94	1	no	—
194	#95	—	94	1	no	—
195	#95	—	94	1	no	—
196	#95	—	94	1	no	—
197	#95	—	94	1	no	—
198	#95	—	94	1	no	—
199	C₂×Q₈	—	94	2	no	—
200	C₂×Q₈	—	188	1	no	—
201	C₂×Q₈	—	188	1	no	—
202	C₂²×C₄	—	94	1	no	—
203	C₂²×C₄	—	94	1	no	—
204	C₂²×C₄	—	94	1	no	—
205	C₂²×C₄	—	94	1	no	—
206	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	94	4	no	—
217	C₂×C₈	—	188	2	no	—
218	C₂×C₈	—	188	2	no	—
219	C₄⋊C₄	—	94	2	no	—
220	C₄⋊C₄	—	188	1	no	—
221	C₄⋊C₄	—	188	1	no	—
222	C₄⋊C₄	—	188	2	no	—
223	C₄⋊C₄	—	188	2	no	—
224	C₄⋊C₄	—	188	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	4	1	no	—
229	C₄²	—	4	1	no	—
230	C₄²	—	4	1	no	—
231	C₄²	#10625325	1	2	yes	—
232	C₄²	Dic₁₈₈	1	2	yes	—
233	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
234	Q₈	—	376	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	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	—	188	1	no	—
251	C₂×C₄	—	188	1	no	—
252	C₂×C₄	—	188	1	no	—
253	C₂×C₄	—	188	1	no	—
254	C₂×C₄	—	188	1	no	—
255	C₂×C₄	—	188	1	no	—
256	C₂×C₄	—	188	1	no	—
257	C₂×C₄	—	188	1	no	—
258	C₂×C₄	—	188	1	no	—
259	C₂×C₄	—	188	1	no	—
260	C₂×C₄	—	188	1	no	—
261	C₂×C₄	—	188	1	no	—
262	C₂×C₄	#13093949	1	2	yes	—
263	C₂×C₄	#13093949	1	2	yes	—
264	C₂×C₄	#13093950	1	2	yes	—
265	C₂×C₄	#13094032	1	2	yes	—
266	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
267	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
268	C₂×C₄	#13094044	1	2	yes	—
269	C₈	—	8	2	no	—
270	C₈	—	188	4	no	—
271	C₂²	#15689091	1	1	yes	characteristic
272	C₂²	#15689096	1	1	yes	characteristic
273	C₂²	#15689105	1	1	yes	characteristic
274	C₂²	C₂×#13093949	1	1	yes	characteristic
275	C₂²	#15689117	1	1	yes	characteristic
276	C₂²	C₂×#13093950	1	1	yes	characteristic
277	C₂²	#15689141	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	376	1	no	—
287	C₄	—	376	1	no	—
288	C₄	—	376	1	no	—
289	C₄	—	376	1	no	—
290	C₂	#20241758	1	1	yes	characteristic
291	C₂	#20241792	1	1	yes	characteristic
292	C₂	#20241846	1	1	yes	characteristic
293	C₂	#20241888	1	1	yes	characteristic
294	C₂	#20241963	1	1	yes	characteristic
295	C₂	#20241996	1	1	yes	characteristic
296	C₂	#20242043	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1839	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1839	D₄₇
hypocenter	C₄₇	#7673
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1839

Complements

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

N	H ≅ G/N	type
171. C₄₇	#7673	semidirect

Maximal subgroups

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

subgroup	index
#20251770	2
C₄×C₄⋊(C₄₇⋊C₈)	2
#20241213	2
#20241212	2
C₄₇×#1839	2
#7673	47