Subgroups — #26215033

Order 12032.

894 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	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	#20245795	C₂	1	1	yes	maximal, characteristic
3	#20243260	C₂	1	4	yes	maximal, complemented
4	#20243258	C₂	1	2	yes	maximal, complemented
5	#20243253	C₂	1	2	yes	maximal, complemented
6	#20243156	C₂	1	4	yes	maximal
7	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093983	C₂²	1	2	yes	—
9	#15689176	C₄	1	8	yes	complemented
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×#13093934	C₂²	1	4	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
16	#15688993	C₄	1	2	yes	—
17	#15688989	C₄	1	2	yes	—
18	#15688983	C₄	1	2	yes	—
19	#15688983	C₄	1	2	yes	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	#15688877	C₂²	1	4	yes	—
23	#15688877	C₂²	1	4	yes	—
24	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#432	C₄	1	8	yes	complemented
26	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
28	C₂×#97×C₄₇	C₂²	1	2	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	#13093983	—	4	4	no	—
32	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
37	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
38	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
39	#13093934	—	4	4	no	—
40	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
41	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
42	C₈×Dic₄₇	—	4	4	no	—
43	C₄₇⋊#95	C₂×C₄	1	4	yes	—
44	C₄₇⋊#95	C₂×C₄	1	4	yes	—
45	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
56	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
57	#97×C₄₇	C₂×C₄	1	8	yes	—
58	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	4	2	no	—
63	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
64	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
77	C₄²×C₄₇	C₄²	1	4	yes	—
78	C₄²×C₄₇	C₄²	1	4	yes	—
79	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
81	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
83	C₂²⋊Dic₄₇	—	4	4	no	—
84	C₄⋊Dic₄₇	—	2	8	no	—
85	C₄⋊Dic₄₇	—	4	4	no	—
86	C₄⋊Dic₄₇	—	4	4	no	—
87	C₄×Dic₄₇	—	2	4	no	—
88	C₄×Dic₄₇	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	—	2	2	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
94	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
95	C₂³×C₄₇	—	2	1	no	—
96	C₂³×C₄₇	—	2	1	no	—
97	C₂³×C₄₇	—	2	2	no	—
98	C₂³×C₄₇	—	2	2	no	—
99	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
100	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	8	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₄₇	#131	1	4	yes	—
116	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
119	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
120	C₂×Dic₄₇	—	2	8	no	—
121	C₂×Dic₄₇	—	2	8	no	—
122	C₂×Dic₄₇	—	2	8	no	—
123	C₈×C₄₇	—	2	8	no	—
124	C₄₇⋊C₈	—	2	4	no	—
125	C₄₇⋊C₈	—	2	4	no	—
126	#20537	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	4	2	no	—
131	C₂²×C₄₇	—	4	2	no	—
132	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
133	C₂²×C₄₇	#518	1	1	yes	characteristic
134	C₂²×C₄₇	#519	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×#131	1	2	yes	—
136	C₂²×C₄₇	#568	1	2	yes	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	—	2	2	no	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	#405	1	4	yes	—
142	C₄×C₄₇	#520	1	2	yes	—
143	C₄×C₄₇	#520	1	2	yes	—
144	Dic₄₇	—	4	8	no	—
145	C₂×#432	D₄₇	1	1	yes	characteristic
146	C₂×#431	—	47	1	no	—
147	#2285	—	47	1	no	—
148	#1771	—	47	4	no	—
149	#1770	—	47	2	no	—
150	C₈×C₄⋊C₄	—	47	2	no	—
151	#1750	—	47	4	no	—
152	C₂×C₄₇	—	4	2	no	—
153	C₂×C₄₇	#2274	1	1	yes	characteristic
154	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
155	C₂×C₄₇	#2873	1	1	yes	characteristic
156	C₂×C₄₇	#2981	1	2	yes	—
157	C₂×C₄₇	#2985	1	2	yes	—
158	C₂×#117	—	47	2	no	—
159	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
160	#432	Dic₄₇	1	8	yes	complemented
161	#431	—	47	8	no	—
162	C₂×C₄⋊C₈	—	47	2	no	—
163	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
164	C₂×C₂²⋊C₈	—	47	2	no	—
165	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
166	C₂×#97	—	47	4	no	—
167	C₂×#97	C₂×D₄₇	1	2	yes	—
168	C₂×C₄×C₈	—	47	1	no	—
169	C₂×C₄×C₈	—	47	1	no	—
170	C₂×C₄×C₈	—	47	4	no	—
171	#384	—	47	2	no	—
172	#382	—	47	2	no	—
173	#380	—	47	2	no	—
174	#380	—	47	2	no	—
175	C₄×C₄⋊C₄	—	47	2	no	—
176	C₄×C₄⋊C₄	—	47	2	no	—
177	#336	—	47	4	no	—
178	#336	—	47	4	no	—
179	C₄₇	#20537	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
181	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
182	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
183	C₂²×C₈	—	47	2	no	—
184	C₂²×C₈	—	47	2	no	—
185	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
186	#117	—	188	4	no	—
187	C₂×C₄⋊C₄	—	47	4	no	—
188	C₂×C₄⋊C₄	—	47	4	no	—
189	C₂×C₄⋊C₄	—	47	4	no	—
190	C₂×C₂²⋊C₄	—	47	4	no	—
191	C₂×C₄²	—	47	4	no	—
192	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
193	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
194	C₂×C₄²	C₄×D₄₇	1	4	yes	—
195	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
196	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
197	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
198	C₄⋊C₈	—	47	8	no	—
199	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
200	C₂²⋊C₈	—	47	8	no	—
201	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
202	#97	—	188	4	no	—
203	#97	C₂×Dic₄₇	1	8	yes	—
204	C₄×C₈	—	47	4	no	—
205	C₄×C₈	—	47	4	no	—
206	C₄×C₈	—	188	4	no	—
207	#95	—	47	4	no	—
208	#95	—	47	4	no	—
209	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	47	8	no	—
215	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
216	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
217	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
222	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
224	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	47	4	no	—
228	C₂×C₈	—	47	4	no	—
229	C₂×C₈	—	94	2	no	—
230	C₂×C₈	—	94	2	no	—
231	C₂×C₈	—	94	2	no	—
232	C₂×C₈	—	94	2	no	—
233	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
234	C₄⋊C₄	—	94	8	no	—
235	C₄⋊C₄	—	188	4	no	—
236	C₄⋊C₄	—	188	4	no	—
237	C₂²⋊C₄	—	188	4	no	—
238	C₄²	—	94	4	no	—
239	C₄²	—	94	4	no	—
240	C₄²	C₄×Dic₄₇	1	4	yes	—
241	C₄²	C₄×Dic₄₇	1	4	yes	—
242	C₄²	C₂×C₄×D₄₇	1	2	yes	—
243	C₄²	C₂×C₄×D₄₇	1	2	yes	—
244	C₄²	C₂²×Dic₄₇	1	2	yes	—
245	C₄²	C₂²×Dic₄₇	1	2	yes	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
251	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	94	8	no	—
262	C₂×C₄	—	94	8	no	—
263	C₂×C₄	—	94	8	no	—
264	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
265	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
266	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
267	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
268	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
269	C₂×C₄	#13094032	1	4	yes	—
270	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
271	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
272	C₂×C₄	#13094070	1	2	yes	—
273	C₂×C₄	#13094071	1	2	yes	—
274	C₈	—	2	8	no	—
275	C₈	—	94	4	no	—
276	C₈	—	94	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
283	C₂²	#15689597	1	1	yes	characteristic
284	C₂²	#15689600	1	1	yes	characteristic
285	C₂²	C₂×#13094032	1	2	yes	—
286	C₂²	#15689889	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	188	8	no	—
292	C₄	#15689081	1	4	yes	—
293	C₄	#15689603	1	2	yes	—
294	C₄	#15689606	1	2	yes	—
295	C₂	—	4	2	no	—
296	C₂	#20245744	1	1	yes	characteristic
297	C₂	C₂×#15689081	1	1	yes	characteristic
298	C₂	#20251806	1	1	yes	characteristic
299	C₂	#20252466	1	2	yes	—
300	C₂	#20252485	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄₇×#432	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#20537
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #20243260	C₂	semidirect
4. #20243258	C₂	semidirect
5. #20243253	C₂	semidirect
9. #15689176	C₄	semidirect
25. C₄₇×#432	C₄	semidirect
160. #432	Dic₄₇	semidirect
179. C₄₇	#20537	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
#20245795	2
#20243260	2
#20243258	2
#20243253	2
#20243156	2
C₂×C₄₇×#432	2
#20537	47