Subgroups — #26102839

Order 12032.

448 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1797	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄₇⋊C₈	—	2	1	no	—
5	C₂×C₂²⋊(C₄₇⋊C₈)	C₄	1	2	yes	—
6	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
7	#15688878	—	2	1	no	—
8	#15688878	—	2	1	no	—
9	#15688878	C₄	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₄₇	C₄	1	2	yes	—
12	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
13	C₄₇×#336	—	2	1	no	—
14	C₂²×C₄₇⋊C₈	—	2	4	no	—
15	C₂²×C₄₇⋊C₈	—	2	2	no	—
16	C₂²×C₄₇⋊C₈	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	—	4	2	no	—
18	C₂²×C₄₇⋊C₈	—	4	1	no	—
19	C₂²×C₄₇⋊C₈	—	4	1	no	—
20	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
21	C₂²⋊(C₄₇⋊C₈)	C₈	1	4	yes	complemented
22	C₄×C₄₇⋊C₈	—	4	2	no	—
23	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	—	2	2	no	—
27	C₂×C₄²×C₄₇	—	4	2	no	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
32	C₄⋊C₈×C₄₇	—	4	2	no	—
33	C₂²⋊C₈×C₄₇	—	4	1	no	—
34	C₂²⋊C₈×C₄₇	—	4	1	no	—
35	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
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	C₂²⋊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	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
51	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
55	C₂×C₈×C₄₇	—	4	2	no	—
56	C₂×C₈×C₄₇	—	4	2	no	—
57	C₂×C₈×C₄₇	—	4	2	no	—
58	C₄²×C₄₇	—	2	2	no	—
59	C₄²×C₄₇	—	2	2	no	—
60	C₄²×C₄₇	—	4	1	no	—
61	C₄²×C₄₇	—	4	1	no	—
62	C₂×C₄₇⋊C₈	—	2	4	no	—
63	C₂×C₄₇⋊C₈	—	2	4	no	—
64	C₂×C₄₇⋊C₈	—	4	2	no	—
65	C₂×C₄₇⋊C₈	—	4	2	no	—
66	C₂×C₄₇⋊C₈	—	4	2	no	—
67	C₂×C₄₇⋊C₈	—	4	4	no	—
68	C₂×C₄₇⋊C₈	—	4	4	no	—
69	C₂×C₄₇⋊C₈	—	4	2	no	—
70	C₂³×C₄₇	—	2	1	no	—
71	C₂³×C₄₇	—	2	1	no	—
72	C₂³×C₄₇	—	2	1	no	—
73	C₂³×C₄₇	—	2	1	no	—
74	C₂³×C₄₇	#95	1	1	yes	characteristic
75	C₂³×C₄₇	C₄⋊C₈	1	2	yes	—
76	C₂³×C₄₇	#106	1	1	yes	characteristic
77	C₂³×C₄₇	#107	1	1	yes	characteristic
78	C₂³×C₄₇	#108	1	2	yes	—
79	C₂×C₄×C₄₇	—	2	1	no	—
80	C₂×C₄×C₄₇	—	2	1	no	—
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	1	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	2	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	—	4	2	no	—
95	C₂×C₄×C₄₇	—	4	2	no	—
96	C₂×C₄×C₄₇	—	4	4	no	—
97	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
100	C₂×C₄×C₄₇	#99	1	2	yes	—
101	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
105	C₂×C₄×C₄₇	#104	1	2	yes	—
106	C₂×C₄×C₄₇	#108	1	2	yes	—
107	C₈×C₄₇	—	8	2	no	—
108	C₄₇⋊C₈	—	4	4	no	—
109	C₄₇⋊C₈	—	4	4	no	—
110	#7029	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	—	2	2	no	—
112	C₂²×C₄₇	—	2	1	no	—
113	C₂²×C₄₇	—	2	1	no	—
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	2	1	no	—
116	C₂²×C₄₇	—	2	1	no	—
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	#334	1	2	yes	—
120	C₂²×C₄₇	#335	1	2	yes	—
121	C₂²×C₄₇	#336	1	1	yes	characteristic
122	C₂²×C₄₇	#339	1	1	yes	characteristic
123	C₂²×C₄₇	#340	1	1	yes	characteristic
124	C₂²×C₄₇	#341	1	1	yes	characteristic
125	C₂²×C₄₇	#341	1	1	yes	characteristic
126	C₂²×C₄₇	#342	1	1	yes	characteristic
127	C₂²×C₄₇	#344	1	1	yes	characteristic
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	—	2	1	no	—
130	C₄×C₄₇	—	2	1	no	—
131	C₄×C₄₇	—	2	1	no	—
132	C₄×C₄₇	—	2	1	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	4	1	no	—
135	C₄×C₄₇	#327	1	2	yes	—
136	C₄×C₄₇	#328	1	2	yes	—
137	#1797	D₄₇	1	1	yes	characteristic
138	C₄×C₂²⋊C₈	—	47	1	no	—
139	C₂×#336	—	47	1	no	—
140	C₂×C₄₇	—	2	2	no	—
141	C₂×C₄₇	—	2	2	no	—
142	C₂×C₄₇	#1277	1	1	yes	characteristic
143	C₂×C₄₇	#1278	1	1	yes	characteristic
144	C₂×C₄₇	#1282	1	1	yes	characteristic
145	C₂×C₄₇	#1294	1	1	yes	characteristic
146	C₂×C₄₇	#1298	1	1	yes	characteristic
147	C₂×C₄₇	#1301	1	1	yes	characteristic
148	C₂×C₄₇	#1302	1	1	yes	characteristic
149	C₂³×C₈	—	94	1	no	—
150	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
151	C₂×C₄⋊C₈	Dic₄₇	1	2	yes	—
152	C₂×C₂²⋊C₈	—	2	1	no	—
153	C₂×C₂²⋊C₈	—	47	2	no	—
154	C₂×C₄×C₈	—	94	1	no	—
155	#336	—	2	1	no	—
156	#336	—	47	2	no	—
157	#336	—	94	1	no	—
158	#336	—	94	1	no	—
159	C₄₇	#7029	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
161	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
162	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	94	4	no	—
165	C₂²×C₈	—	94	2	no	—
166	C₂²×C₈	—	94	2	no	—
167	C₂²×C₈	—	188	2	no	—
168	C₂²×C₈	—	188	1	no	—
169	C₂²×C₈	—	188	1	no	—
170	C₂×C₄²	—	4	2	no	—
171	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
172	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
173	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
175	C₄⋊C₈	—	4	2	no	—
176	C₂²⋊C₈	—	4	1	no	—
177	C₂²⋊C₈	—	4	1	no	—
178	C₂²⋊C₈	—	47	4	no	—
179	C₂²⋊C₈	—	94	2	no	—
180	C₄×C₈	—	188	2	no	—
181	C₂⁴	Dic₄₇⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	4	1	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	C₈×D₄₇	1	2	yes	—
190	C₂²×C₄	#10625324	1	2	yes	—
191	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
192	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
193	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
194	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
195	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	#10625334	1	1	yes	characteristic
197	C₂²×C₄	#10625335	1	1	yes	characteristic
198	C₂²×C₄	#10625336	1	1	yes	characteristic
199	C₂²×C₄	#10625337	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	94	4	no	—
205	C₂×C₈	—	94	4	no	—
206	C₂×C₈	—	188	2	no	—
207	C₂×C₈	—	188	2	no	—
208	C₂×C₈	—	188	2	no	—
209	C₂×C₈	—	188	4	no	—
210	C₂×C₈	—	188	4	no	—
211	C₂×C₈	—	188	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	4	1	no	—
215	C₄²	—	4	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	C₄₇⋊#95	1	1	yes	characteristic
221	C₂³	Dic₄₇⋊C₈	1	2	yes	—
222	C₂³	#13093955	1	1	yes	characteristic
223	C₂³	#13093957	1	1	yes	characteristic
224	C₂³	#13093959	1	2	yes	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
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	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
244	C₂×C₄	#13093934	1	1	yes	characteristic
245	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
246	C₂×C₄	#13093939	1	2	yes	—
247	C₂×C₄	#13093944	1	1	yes	characteristic
248	C₂×C₄	#13093945	1	1	yes	characteristic
249	C₂×C₄	#13093947	1	1	yes	characteristic
250	C₂×C₄	#13093948	1	1	yes	characteristic
251	C₂×C₄	#13093951	1	2	yes	—
252	C₂×C₄	#13093959	1	2	yes	—
253	C₈	—	8	2	no	—
254	C₈	—	188	4	no	—
255	C₈	—	188	4	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	#15688873	1	2	yes	—
265	C₂²	#15688875	1	2	yes	—
266	C₂²	#15688877	1	1	yes	characteristic
267	C₂²	#15688882	1	1	yes	characteristic
268	C₂²	#15688885	1	1	yes	characteristic
269	C₂²	#15688887	1	1	yes	characteristic
270	C₂²	#15688887	1	1	yes	characteristic
271	C₂²	#15688890	1	1	yes	characteristic
272	C₂²	#15688894	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	—	4	1	no	—
280	C₄	#15688857	1	2	yes	—
281	C₄	#15688860	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	2	2	no	—
284	C₂	#20241222	1	1	yes	characteristic
285	C₂	#20241224	1	1	yes	characteristic
286	C₂	#20241235	1	1	yes	characteristic
287	C₂	#20241266	1	1	yes	characteristic
288	C₂	#20241275	1	1	yes	characteristic
289	C₂	#20241283	1	1	yes	characteristic
290	C₂	#20241286	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₄₇
Fitting	C₄₇×#1797	C₂
socle	C₂³×C₄₇	#95
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂²×C₄²×C₄₇	C₂²
hypercenter	#1797	D₄₇
hypocenter	C₄₇	#7029
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
21. C₂²⋊(C₄₇⋊C₈)	C₈	semidirect
159. C₄₇	#7029	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊(C₄₇⋊C₈)	2
C₂×#15688878	2
C₄₇×#1797	2
#7029	47