Subgroups — #26215105

Order 12032.

894 conjugacy classes of subgroups.

The subgroup lattice has 294 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	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
3	#20243257	C₂	1	2	yes	maximal, complemented
4	C₄⋊C₄×C₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	#20243245	C₂	1	4	yes	maximal, complemented
6	#20243172	C₂	1	4	yes	maximal
7	C₄₇×#2283	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	#15689176	C₄	1	8	yes	complemented
10	#15689147	C₄	1	8	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×#13093935	C₂²	1	4	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	#15688878	C₂²	1	4	yes	—
20	#15688878	C₂²	1	4	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#383	C₄	1	2	yes	—
23	C₄₇×#382	C₄	1	2	yes	—
24	C₄₇×#381	C₄	1	2	yes	—
25	C₄₇×#380	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₄₇×#376	C₂²	1	2	yes	—
29	C₂×#10625329	—	4	4	no	—
30	C₂×#10625329	C₂×C₄	1	8	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
34	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
35	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
36	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
37	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
38	#13093935	—	4	4	no	—
39	C₄×C₄₇⋊C₈	—	4	4	no	—
40	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
41	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
42	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	#95×C₄₇	C₂×C₄	1	4	yes	—
55	#95×C₄₇	C₂×C₄	1	4	yes	—
56	#95×C₄₇	C₂×C₄	1	4	yes	—
57	#95×C₄₇	C₂×C₄	1	4	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	8	yes	—
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	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₄₇	—	2	8	no	—
75	C₂²⋊C₄×C₄₇	—	4	4	no	—
76	C₄²×C₄₇	—	2	4	no	—
77	C₄²×C₄₇	—	2	4	no	—
78	C₄²×C₄₇	C₄²	1	4	yes	—
79	C₄²×C₄₇	C₄²	1	4	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	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
84	#10625329	—	4	8	no	—
85	C₂×C₄₇⋊C₈	—	2	2	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	2	2	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
92	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
93	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
94	C₂³×C₄₇	—	2	2	no	—
95	C₂³×C₄₇	—	2	2	no	—
96	C₂³×C₄₇	—	2	2	no	—
97	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
98	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	8	no	—
107	C₂×C₄×C₄₇	—	2	8	no	—
108	C₂×C₄×C₄₇	—	2	8	no	—
109	C₂×C₄×C₄₇	—	2	8	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
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	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₄₇	#131	1	4	yes	—
117	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
120	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
121	C₄₇⋊C₈	—	2	8	no	—
122	C₄₇⋊C₈	—	2	4	no	—
123	C₄₇⋊C₈	—	2	4	no	—
124	#20548	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	4	4	no	—
129	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
130	C₂²×C₄₇	#520	1	2	yes	—
131	C₂²×C₄₇	C₂×#131	1	2	yes	—
132	C₂²×C₄₇	#568	1	2	yes	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	—	2	2	no	—
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	—	4	8	no	—
138	C₄×C₄₇	#405	1	4	yes	—
139	C₄×C₄₇	#518	1	1	yes	characteristic
140	C₄×C₄₇	#519	1	1	yes	characteristic
141	C₄×C₄₇	#520	1	2	yes	—
142	C₂×#431	—	47	1	no	—
143	C₂×#423	—	47	1	no	—
144	#2283	D₄₇	1	1	yes	characteristic
145	#1770	—	47	2	no	—
146	C₈×C₄⋊C₄	—	47	2	no	—
147	#1768	—	47	4	no	—
148	#1753	—	47	4	no	—
149	C₂×C₄₇	—	4	2	no	—
150	C₂×C₄₇	#2274	1	1	yes	characteristic
151	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
152	C₂×C₄₇	#2873	1	1	yes	characteristic
153	C₂×C₄₇	#2993	1	4	yes	—
154	C₂²×M₁₆	—	47	2	no	—
155	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
156	#431	—	47	8	no	—
157	#423	—	47	8	no	—
158	C₂×C₄⋊C₈	—	47	2	no	—
159	C₂×C₄⋊C₈	—	47	2	no	—
160	C₂×C₄⋊C₈	—	47	2	no	—
161	C₂×C₂²⋊C₈	—	47	2	no	—
162	C₂×#97	—	47	4	no	—
163	C₂×C₄×C₈	—	47	1	no	—
164	C₂×C₄×C₈	—	47	1	no	—
165	C₂×C₄×C₈	—	47	4	no	—
166	#383	Dic₄₇	1	2	yes	—
167	#382	Dic₄₇	1	2	yes	—
168	#381	Dic₄₇	1	2	yes	—
169	#380	Dic₄₇	1	2	yes	—
170	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
171	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
172	#376	C₂×D₄₇	1	2	yes	—
173	#336	—	47	4	no	—
174	#336	—	47	4	no	—
175	C₄₇	#20548	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
177	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
178	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
179	C₂×M₁₆	—	47	8	no	—
180	C₂×M₁₆	—	188	4	no	—
181	C₂²×C₈	—	47	4	no	—
182	C₂²×C₈	—	47	2	no	—
183	C₂²×C₈	—	47	2	no	—
184	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
185	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
186	C₂×C₄²	C₄×D₄₇	1	4	yes	—
187	C₂×C₄²	C₄×D₄₇	1	4	yes	—
188	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
189	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
193	C₄⋊C₈	—	47	8	no	—
194	C₄⋊C₈	—	47	8	no	—
195	C₄⋊C₈	—	47	8	no	—
196	C₂²⋊C₈	—	47	8	no	—
197	#97	—	188	4	no	—
198	C₄×C₈	—	47	4	no	—
199	C₄×C₈	—	47	4	no	—
200	C₄×C₈	—	188	4	no	—
201	#95	C₂×Dic₄₇	1	4	yes	—
202	#95	C₂×Dic₄₇	1	4	yes	—
203	#95	C₂×Dic₄₇	1	4	yes	—
204	#95	C₂×Dic₄₇	1	4	yes	—
205	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
211	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
212	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
213	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
214	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
215	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
217	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
221	M₁₆	—	188	8	no	—
222	C₂×C₈	—	47	4	no	—
223	C₂×C₈	—	47	4	no	—
224	C₂×C₈	—	47	8	no	—
225	C₂×C₈	—	94	2	no	—
226	C₂×C₈	—	94	4	no	—
227	C₂×C₈	—	94	2	no	—
228	C₂×C₈	—	94	4	no	—
229	C₂×C₈	—	94	2	no	—
230	C₂×C₈	—	94	2	no	—
231	C₄⋊C₄	—	2	8	no	—
232	C₂²⋊C₄	—	4	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	C₄×Dic₄₇	1	4	yes	—
236	C₄²	C₄×Dic₄₇	1	4	yes	—
237	C₄²	C₂×C₄×D₄₇	1	2	yes	—
238	C₄²	C₂×C₄×D₄₇	1	2	yes	—
239	C₄²	C₂×C₄×D₄₇	1	2	yes	—
240	C₄²	C₂×C₄×D₄₇	1	2	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
245	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
259	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
260	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
261	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
262	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
263	C₂×C₄	#13094031	1	4	yes	—
264	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
266	C₂×C₄	#13094070	1	2	yes	—
267	C₂×C₄	#13094071	1	2	yes	—
268	C₈	—	94	8	no	—
269	C₈	—	94	4	no	—
270	C₈	—	94	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	4	4	no	—
275	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
276	C₂²	#15689606	1	2	yes	—
277	C₂²	C₂×#13094031	1	2	yes	—
278	C₂²	#15689890	1	2	yes	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	8	no	—
284	C₄	#15689079	1	4	yes	—
285	C₄	#15689597	1	1	yes	characteristic
286	C₄	#15689600	1	1	yes	characteristic
287	C₄	#15689603	1	2	yes	—
288	C₂	—	4	2	no	—
289	C₂	#20245744	1	1	yes	characteristic
290	C₂	C₂×#15689079	1	1	yes	characteristic
291	C₂	#20251807	1	1	yes	characteristic
292	C₂	#20252537	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#2283	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2283	D₄₇
hypocenter	C₄₇	#20548
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2283

Complements

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

N	H ≅ G/N	type
3. #20243257	C₂	semidirect
4. C₄⋊C₄×C₄₇⋊C₈	C₂	semidirect
5. #20243245	C₂	semidirect
9. #15689176	C₄	semidirect
10. #15689147	C₄	semidirect
175. C₄₇	#20548	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
C₂×#15689147	2
#20243257	2
C₄⋊C₄×C₄₇⋊C₈	2
#20243245	2
#20243172	2
C₄₇×#2283	2
#20548	47