Subgroups — #26379096

Order 12032.

3390 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246565	C₂	1	4	yes	maximal, complemented
2	#20246453	C₂	1	4	yes	maximal, complemented
3	D₄×C₂²⋊Dic₄₇	C₂	1	8	yes	maximal, complemented
4	#20246114	C₂	1	8	yes	maximal, complemented
5	#20245966	C₂	1	4	yes	maximal, complemented
6	#20245786	C₂	1	2	yes	maximal, complemented
7	C₂²×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×D₄×Dic₄₇	C₂²	1	16	yes	complemented
9	C₂²×C₂²⋊Dic₄₇	C₂²	1	8	yes	—
10	#15689026	C₂²	1	16	yes	complemented
11	#15689003	C₂²	1	32	yes	complemented
12	#15689003	C₂²	1	16	yes	complemented
13	#15689003	C₂²	1	16	yes	complemented
14	#15688991	C₂²	1	4	yes	complemented
15	#15688989	C₂²	1	4	yes	complemented
16	#15688986	C₂²	1	4	yes	—
17	#15688983	C₂²	1	4	yes	—
18	#15688979	C₂²	1	8	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	complemented
20	C₄×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
21	#15688969	C₂²	1	4	yes	complemented
22	C₂³×D₄×C₄₇	C₂²	1	4	yes	—
23	C₂³×D₄×C₄₇	C₂²	1	2	yes	—
24	C₂×#127×C₄₇	C₄	1	16	yes	complemented
25	C₂×#127×C₄₇	C₂²	1	2	yes	—
26	C₂×#127×C₄₇	C₂²	1	4	yes	—
27	C₂×#127×C₄₇	C₂²	1	2	yes	—
28	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
29	C₂³×Dic₄₇	C₂³	1	16	yes	—
30	D₄×Dic₄₇	—	4	32	no	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
32	C₂×C₂²⋊Dic₄₇	—	4	16	no	—
33	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
34	C₂×C₂²⋊Dic₄₇	C₂³	1	32	yes	—
35	C₂×C₂²⋊Dic₄₇	C₂³	1	16	yes	—
36	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
37	C₄₇⋊#95	C₂³	1	8	yes	—
38	C₄₇⋊#95	C₂³	1	8	yes	—
39	C₄₇⋊#95	C₂³	1	8	yes	—
40	C₄₇⋊#95	C₂³	1	8	yes	—
41	C₂⁵×C₄₇	C₂³	1	4	yes	—
42	C₂²×D₄×C₄₇	—	4	16	no	—
43	C₂²×D₄×C₄₇	—	4	8	no	—
44	C₂²×D₄×C₄₇	C₂×C₄	1	32	yes	complemented
45	C₂²×D₄×C₄₇	C₂×C₄	1	16	yes	complemented
46	C₂²×D₄×C₄₇	C₂³	1	8	yes	—
47	C₂²×D₄×C₄₇	C₂³	1	8	yes	—
48	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
49	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
50	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
51	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	#127×C₄₇	C₂×C₄	1	16	yes	complemented
53	#127×C₄₇	C₂×C₄	1	32	yes	complemented
54	#127×C₄₇	C₂×C₄	1	16	yes	complemented
55	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	complemented
56	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂⁴×C₄₇	—	2	16	no	—
60	C₂⁴×C₄₇	—	4	16	no	—
61	C₂⁴×C₄₇	C₂²×C₄	1	16	yes	—
62	C₂⁴×C₄₇	C₂⁴	1	8	yes	—
63	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₂×D₄×C₄₇	—	2	32	no	—
65	C₂×D₄×C₄₇	—	4	64	no	—
66	C₂×D₄×C₄₇	—	4	32	no	—
67	C₂×D₄×C₄₇	—	4	4	no	—
68	C₂×D₄×C₄₇	—	4	8	no	—
69	C₂×D₄×C₄₇	—	4	16	no	—
70	C₂×D₄×C₄₇	—	4	4	no	—
71	C₂×D₄×C₄₇	C₂²×C₄	1	32	yes	—
72	C₂×D₄×C₄₇	C₂²×C₄	1	32	yes	—
73	C₂×D₄×C₄₇	C₂²×C₄	1	16	yes	—
74	C₂×D₄×C₄₇	C₂²×C₄	1	16	yes	—
75	C₂²×C₄×C₄₇	—	2	4	no	—
76	C₂²×C₄×C₄₇	—	2	8	no	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
79	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
83	C₂²×Dic₄₇	—	4	16	no	—
84	C₂²×Dic₄₇	—	4	32	no	—
85	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
86	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
87	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
89	C₂²⋊Dic₄₇	—	2	16	no	—
90	C₂²⋊Dic₄₇	—	4	16	no	—
91	C₂²⋊Dic₄₇	—	4	32	no	—
92	C₄⋊Dic₄₇	—	2	16	no	—
93	C₄×Dic₄₇	—	2	16	no	—
94	C₂³×C₄₇	—	2	4	no	—
95	C₂³×C₄₇	—	2	8	no	—
96	C₂³×C₄₇	—	2	32	no	—
97	C₂³×C₄₇	—	2	8	no	—
98	C₂³×C₄₇	—	2	32	no	—
99	C₂³×C₄₇	—	4	32	no	—
100	C₂³×C₄₇	—	8	32	no	—
101	C₂³×C₄₇	C₂³×C₄	1	16	yes	—
102	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
103	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
104	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
105	D₄×C₄₇	—	2	64	no	—
106	D₄×C₄₇	—	4	32	no	—
107	D₄×C₄₇	—	4	16	no	—
108	D₄×C₄₇	—	4	16	no	—
109	C₂×C₄×C₄₇	—	2	8	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	16	no	—
114	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
117	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
118	C₂×C₄×C₄₇	H(5,𝔽₂)	1	8	yes	—
119	C₂×Dic₄₇	—	2	16	no	—
120	C₂×Dic₄₇	—	2	16	no	—
121	C₂×Dic₄₇	—	2	16	no	—
122	C₂×Dic₄₇	—	8	32	no	—
123	#34364	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	4	no	—
125	C₂²×C₄₇	—	2	8	no	—
126	C₂²×C₄₇	—	2	32	no	—
127	C₂²×C₄₇	—	2	16	no	—
128	C₂²×C₄₇	—	4	16	no	—
129	C₂²×C₄₇	—	4	16	no	—
130	C₂²×C₄₇	—	8	32	no	—
131	C₂²×C₄₇	#518	1	4	yes	—
132	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	2	yes	—
134	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	4	yes	—
135	C₄×C₄₇	—	2	2	no	—
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	—	2	4	no	—
138	C₄×C₄₇	#518	1	8	yes	—
139	Dic₄₇	—	4	16	no	—
140	C₂²×#127	D₄₇	1	1	yes	characteristic
141	#2380	—	47	4	no	—
142	#2366	—	47	4	no	—
143	D₄×C₂²⋊C₄	—	47	8	no	—
144	#2328	—	47	8	no	—
145	#2308	—	47	4	no	—
146	#2283	—	47	2	no	—
147	C₂×C₄₇	—	2	4	no	—
148	C₂×C₄₇	—	4	16	no	—
149	C₂×C₄₇	C₂×#518	1	1	yes	characteristic
150	C₂×C₄₇	C₂×#518	1	2	yes	—
151	C₂×C₄₇	#3543	1	4	yes	—
152	C₂³×D₄	C₂×D₄₇	1	2	yes	—
153	C₂³×D₄	C₂×D₄₇	1	4	yes	—
154	C₂×#127	Dic₄₇	1	16	yes	complemented
155	C₂×#127	C₂×D₄₇	1	4	yes	—
156	C₂×#127	C₂×D₄₇	1	2	yes	—
157	C₂×#127	C₂×D₄₇	1	2	yes	—
158	C₂×C₄×D₄	—	47	16	no	—
159	C₂²×C₂²⋊C₄	—	47	8	no	—
160	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
161	#390	—	47	16	no	—
162	#386	—	47	16	no	—
163	#386	—	47	16	no	—
164	#386	—	47	32	no	—
165	#383	—	47	4	no	—
166	#382	—	47	4	no	—
167	#381	—	47	4	no	—
168	#380	—	47	4	no	—
169	#379	—	47	8	no	—
170	C₄×C₄⋊C₄	—	47	4	no	—
171	C₄×C₂²⋊C₄	—	47	4	no	—
172	#376	—	47	4	no	—
173	C₄₇	#34364	1	1	yes	characteristic, complemented, Sylow
174	C₂⁵	C₂²×D₄₇	1	4	yes	—
175	C₂²×D₄	—	4	16	no	—
176	C₂²×D₄	—	4	8	no	—
177	C₂²×D₄	C₂×Dic₄₇	1	32	yes	complemented
178	C₂²×D₄	C₂×Dic₄₇	1	16	yes	complemented
179	C₂²×D₄	C₂²×D₄₇	1	8	yes	—
180	C₂²×D₄	C₂²×D₄₇	1	4	yes	—
181	C₂²×D₄	C₂²×D₄₇	1	8	yes	—
182	C₂²×D₄	C₂²×D₄₇	1	4	yes	—
183	C₂³×C₄	—	47	16	no	—
184	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
185	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
186	#127	C₂×Dic₄₇	1	16	yes	complemented
187	#127	C₂×Dic₄₇	1	32	yes	complemented
188	#127	C₂×Dic₄₇	1	16	yes	complemented
189	C₄×D₄	—	188	32	no	—
190	C₂×C₄⋊C₄	—	47	8	no	—
191	C₂×C₂²⋊C₄	—	47	8	no	—
192	C₂×C₂²⋊C₄	—	47	16	no	—
193	C₂×C₂²⋊C₄	—	47	32	no	—
194	C₂×C₂²⋊C₄	—	188	16	no	—
195	C₂×C₄²	—	47	8	no	—
196	C₂×C₄²	C₂×Dic₄₇	1	8	yes	complemented
197	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
198	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
200	#95	—	47	8	no	—
201	#95	—	47	8	no	—
202	#95	—	47	8	no	—
203	#95	—	47	8	no	—
204	C₂⁴	—	2	16	no	—
205	C₂⁴	—	4	16	no	—
206	C₂⁴	C₂²×Dic₄₇	1	16	yes	—
207	C₂⁴	C₂³×D₄₇	1	8	yes	—
208	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
209	C₂×D₄	—	2	32	no	—
210	C₂×D₄	—	4	64	no	—
211	C₂×D₄	—	4	32	no	—
212	C₂×D₄	—	4	4	no	—
213	C₂×D₄	—	4	8	no	—
214	C₂×D₄	—	4	16	no	—
215	C₂×D₄	—	4	4	no	—
216	C₂×D₄	C₂²×Dic₄₇	1	32	yes	—
217	C₂×D₄	C₂²×Dic₄₇	1	32	yes	—
218	C₂×D₄	C₂²×Dic₄₇	1	16	yes	—
219	C₂×D₄	C₂²×Dic₄₇	1	16	yes	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	47	16	no	—
223	C₂²×C₄	—	188	16	no	—
224	C₂²×C₄	—	188	32	no	—
225	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
226	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
227	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
228	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
231	C₄⋊C₄	—	94	16	no	—
232	C₂²⋊C₄	—	94	16	no	—
233	C₂²⋊C₄	—	188	16	no	—
234	C₂²⋊C₄	—	188	32	no	—
235	C₄²	—	94	16	no	—
236	C₄²	C₂²×Dic₄₇	1	4	yes	—
237	C₄²	C₂²×Dic₄₇	1	8	yes	—
238	C₄²	C₂²×Dic₄₇	1	4	yes	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	8	no	—
241	C₂³	—	2	32	no	—
242	C₂³	—	2	8	no	—
243	C₂³	—	2	32	no	—
244	C₂³	—	4	32	no	—
245	C₂³	—	8	32	no	—
246	C₂³	C₂³×Dic₄₇	1	16	yes	—
247	C₂³	C₂³×Dic₄₇	1	2	yes	—
248	C₂³	#13094069	1	4	yes	—
249	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
250	D₄	—	2	64	no	—
251	D₄	—	4	32	no	—
252	D₄	—	4	16	no	—
253	D₄	—	4	16	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	94	16	no	—
260	C₂×C₄	—	94	16	no	—
261	C₂×C₄	—	94	16	no	—
262	C₂×C₄	—	376	32	no	—
263	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
264	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
265	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
266	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
267	C₂×C₄	#13094069	1	8	yes	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	8	no	—
270	C₂²	—	2	32	no	—
271	C₂²	—	2	16	no	—
272	C₂²	—	4	16	no	—
273	C₂²	—	4	16	no	—
274	C₂²	—	8	32	no	—
275	C₂²	#15689598	1	4	yes	—
276	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
277	C₂²	C₂×#13094069	1	2	yes	—
278	C₂²	C₂×#13094069	1	4	yes	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	4	no	—
282	C₄	—	188	16	no	—
283	C₄	#15689598	1	8	yes	—
284	C₂	—	2	4	no	—
285	C₂	—	4	16	no	—
286	C₂	C₂×#15689598	1	1	yes	characteristic
287	C₂	C₂×#15689598	1	2	yes	—
288	C₂	#20257721	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Fitting, prime radical	C₂²×#127×C₄₇	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂²×#127	D₄₇
hypocenter	C₄₇	#34364
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. #20246565	C₂	semidirect
2. #20246453	C₂	semidirect
3. D₄×C₂²⋊Dic₄₇	C₂	semidirect
4. #20246114	C₂	semidirect
5. #20245966	C₂	semidirect
6. #20245786	C₂	semidirect
8. C₂×D₄×Dic₄₇	C₂²	semidirect
10. #15689026	C₂²	semidirect
11. #15689003	C₂²	semidirect
12. #15689003	C₂²	semidirect
13. #15689003	C₂²	semidirect
14. #15688991	C₂²	semidirect
15. #15688989	C₂²	semidirect
19. C₄×C₄⋊Dic₄₇	C₂²	semidirect
21. #15688969	C₂²	semidirect
24. C₂×#127×C₄₇	C₄	semidirect
44. C₂²×D₄×C₄₇	C₂×C₄	semidirect
45. C₂²×D₄×C₄₇	C₂×C₄	semidirect
52. #127×C₄₇	C₂×C₄	semidirect
53. #127×C₄₇	C₂×C₄	semidirect
54. #127×C₄₇	C₂×C₄	semidirect
55. C₂×C₄²×C₄₇	C₂×C₄	semidirect
154. C₂×#127	Dic₄₇	semidirect
173. C₄₇	#34364	semidirect
177. C₂²×D₄	C₂×Dic₄₇	semidirect
178. C₂²×D₄	C₂×Dic₄₇	semidirect
186. #127	C₂×Dic₄₇	semidirect
187. #127	C₂×Dic₄₇	semidirect
188. #127	C₂×Dic₄₇	semidirect
196. C₂×C₄²	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20246565	2
#20246453	2
D₄×C₂²⋊Dic₄₇	2
#20246114	2
#20245966	2
#20245786	2
C₂²×#127×C₄₇	2
#34364	47