Subgroups — #26106365

Order 12032.

650 conjugacy classes of subgroups.

The subgroup lattice has 300 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	#20241272	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	C₄	1	2	yes	—
6	#15689176	—	2	2	no	—
7	#15689176	C₄	1	4	yes	complemented
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
10	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
11	#15688993	C₂²	1	2	yes	—
12	#15688989	C₄	1	2	yes	—
13	#15688983	—	2	1	no	—
14	#15688983	—	2	1	no	—
15	C₄×C₄⋊Dic₄₇	—	2	1	no	—
16	C₄×C₄⋊Dic₄₇	—	2	1	no	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#431	—	2	2	no	—
19	C₄₇×#431	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
22	C₂×C₄×C₈×C₄₇	—	2	1	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	#13093983	—	4	4	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄×Dic₄₇	—	2	2	no	—
30	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
31	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
32	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
33	C₄×C₄₇⋊C₈	—	2	4	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	—	2	2	no	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	2	yes	—
46	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₄₇	—	2	2	no	—
49	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
50	C₂²⋊C₈×C₄₇	—	2	4	no	—
51	C₄×C₈×C₄₇	—	2	4	no	—
52	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	—	4	4	no	—
72	C₄²×C₄₇	C₄²	1	2	yes	—
73	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
76	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
77	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
82	C₂²⋊Dic₄₇	—	4	4	no	—
83	C₄⋊Dic₄₇	—	4	4	no	—
84	C₄⋊Dic₄₇	—	4	4	no	—
85	C₄⋊Dic₄₇	—	4	4	no	—
86	C₄×Dic₄₇	—	4	4	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	4	4	no	—
89	C₂³×C₄₇	—	2	1	no	—
90	C₂³×C₄₇	—	2	1	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	#95	1	2	yes	—
94	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	#95	1	2	yes	—
106	C₂×C₄×C₄₇	#95	1	2	yes	—
107	C₂×C₄×C₄₇	#95	1	2	yes	—
108	C₂×C₄×C₄₇	#100	1	2	yes	—
109	C₂×C₄×C₄₇	#101	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#135	1	4	yes	—
117	C₂×C₄×C₄₇	#135	1	4	yes	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	4	4	no	—
120	C₂×Dic₄₇	—	4	4	no	—
121	C₈×C₄₇	—	4	4	no	—
122	C₄₇⋊C₈	—	4	4	no	—
123	#8038	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	4	2	no	—
128	C₂²×C₄₇	—	4	2	no	—
129	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
132	C₂²×C₄₇	#416	1	2	yes	—
133	C₂²×C₄₇	#416	1	2	yes	—
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	—	2	2	no	—
136	C₄×C₄₇	—	2	1	no	—
137	C₄×C₄₇	—	2	1	no	—
138	C₄×C₄₇	#343	1	4	yes	—
139	C₄×C₄₇	#416	1	2	yes	—
140	C₄×C₄₇	#416	1	2	yes	—
141	C₄×C₄₇	#427	1	2	yes	—
142	C₄×C₄₇	#427	1	2	yes	—
143	Dic₄₇	—	8	4	no	—
144	C₂×#431	—	47	1	no	—
145	C₂×#431	D₄₇	1	1	yes	characteristic
146	#2285	—	47	1	no	—
147	#1296	—	47	4	no	—
148	C₂×C₄₇	—	4	2	no	—
149	C₂×C₄₇	#1543	1	2	yes	—
150	C₂×C₄₇	#1544	1	2	yes	—
151	C₂×C₄₇	#1735	1	1	yes	characteristic
152	C₂×C₄₇	#1735	1	1	yes	characteristic
153	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
154	C₂×#117	—	47	2	no	—
155	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
156	#431	—	2	2	no	—
157	#431	—	47	4	no	—
158	#431	—	94	2	no	—
159	#431	Dic₄₇	1	4	yes	complemented
160	C₂×C₄⋊C₈	—	47	2	no	—
161	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
162	C₂×C₂²⋊C₈	—	2	1	no	—
163	C₂×C₂²⋊C₈	—	94	1	no	—
164	C₂×C₄×C₈	—	2	1	no	—
165	C₂×C₄×C₈	—	94	1	no	—
166	#384	—	47	2	no	—
167	#382	—	47	2	no	—
168	#380	—	94	1	no	—
169	#380	—	94	1	no	—
170	C₄×C₄⋊C₄	—	94	1	no	—
171	C₄×C₄⋊C₄	—	94	1	no	—
172	C₄₇	#8038	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
174	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
175	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	94	2	no	—
178	#117	—	188	4	no	—
179	C₂×C₄⋊C₄	—	47	4	no	—
180	C₂×C₄⋊C₄	—	94	2	no	—
181	C₂×C₄⋊C₄	—	94	2	no	—
182	C₂×C₂²⋊C₄	—	94	2	no	—
183	C₂×C₄²	—	94	2	no	—
184	C₂×C₄²	Dic₉₄	1	2	yes	—
185	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
186	C₂×C₄²	C₄×D₄₇	1	2	yes	—
187	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
188	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
189	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	47	4	no	—
194	C₄⋊C₈	—	94	2	no	—
195	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
196	C₂²⋊C₈	—	2	4	no	—
197	C₂²⋊C₈	—	94	4	no	—
198	C₄×C₈	—	2	4	no	—
199	C₄×C₈	—	94	4	no	—
200	#95	—	94	2	no	—
201	#95	—	94	2	no	—
202	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	94	4	no	—
208	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
209	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
210	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
211	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
212	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
214	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
215	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	—	94	4	no	—
223	C₂×C₈	—	188	4	no	—
224	C₄⋊C₄	—	188	4	no	—
225	C₄⋊C₄	—	188	4	no	—
226	C₄⋊C₄	—	188	4	no	—
227	C₂²⋊C₄	—	188	4	no	—
228	C₄²	—	188	4	no	—
229	C₄²	C₄×Dic₄₇	1	2	yes	—
230	C₄²	Dic₄₇⋊C₄	1	2	yes	—
231	C₄²	Dic₄₇⋊C₄	1	2	yes	—
232	C₄²	C₄⋊Dic₄₇	1	2	yes	—
233	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
234	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
235	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
236	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
237	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
238	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₄₇⋊#95	1	2	yes	—
244	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	188	4	no	—
256	C₂×C₄	—	188	4	no	—
257	C₂×C₄	—	188	4	no	—
258	C₂×C₄	C₄₇⋊#95	1	2	yes	—
259	C₂×C₄	C₄₇⋊#95	1	2	yes	—
260	C₂×C₄	C₄₇⋊#95	1	2	yes	—
261	C₂×C₄	#13093940	1	2	yes	—
262	C₂×C₄	#13093942	1	2	yes	—
263	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
264	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
266	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
269	C₂×C₄	#13094044	1	4	yes	—
270	C₂×C₄	#13094046	1	4	yes	—
271	C₈	—	4	4	no	—
272	C₈	—	188	4	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
279	C₂²	C₂×#13093940	1	1	yes	characteristic
280	C₂²	C₂×#13093942	1	1	yes	characteristic
281	C₂²	#15689117	1	2	yes	—
282	C₂²	#15689120	1	2	yes	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	376	4	no	—
288	C₄	#15688891	1	4	yes	—
289	C₄	#15689119	1	2	yes	—
290	C₄	#15689120	1	2	yes	—
291	C₄	#15689159	1	2	yes	—
292	C₄	#15689161	1	2	yes	—
293	C₂	—	4	2	no	—
294	C₂	#20242048	1	2	yes	—
295	C₂	#20242052	1	2	yes	—
296	C₂	#20243104	1	1	yes	characteristic
297	C₂	#20243105	1	1	yes	characteristic
298	C₂	C₂×#15688891	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#431	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#431	D₄₇
hypocenter	C₄₇	#8038
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20241272	C₂	semidirect
7. #15689176	C₄	semidirect
19. C₄₇×#431	C₄	semidirect
159. #431	Dic₄₇	semidirect
172. C₄₇	#8038	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
#20241272	2
C₂×C₄₇×#431	2
#8038	47