Subgroups — #26147442

Order 12032.

6350 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688986	C₂	1	4	yes	maximal, complemented
2	C₂×#15688979	C₂	1	2	yes	maximal
3	#20241611	C₂	1	8	yes	maximal, complemented
4	C₂⁵×C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
6	C₂²×C₂²⋊Dic₄₇	C₂²	1	8	yes	—
7	#15688986	—	2	16	no	—
8	#15688979	—	2	8	no	—
9	C₂×C₄₇⋊#95	C₂²	1	16	yes	complemented
10	C₂⁶×C₄₇	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₄₇	—	2	8	no	—
12	C₂⁴×C₄×C₄₇	—	2	16	no	—
13	C₂⁴×C₄×C₄₇	C₄	1	8	yes	complemented
14	C₂⁴×C₄×C₄₇	C₂²	1	4	yes	—
15	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
16	C₂³×Dic₄₇	C₂³	1	8	yes	—
17	C₂×C₂²⋊Dic₄₇	—	2	8	no	—
18	C₂×C₂²⋊Dic₄₇	—	2	16	no	—
19	C₂×C₂²⋊Dic₄₇	—	2	16	no	—
20	C₂×C₂²⋊Dic₄₇	—	2	32	no	—
21	C₄₇⋊#95	—	2	32	no	—
22	C₂⁵×C₄₇	—	2	8	no	—
23	C₂⁵×C₄₇	—	2	16	no	—
24	C₂⁵×C₄₇	C₂×C₄	1	4	yes	—
25	C₂⁵×C₄₇	D₄	1	8	yes	—
26	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
27	C₂⁵×C₄₇	C₂³	1	2	yes	—
28	C₂³×C₄×C₄₇	—	2	128	no	—
29	C₂³×C₄×C₄₇	—	2	32	no	—
30	C₂³×C₄×C₄₇	—	2	16	no	—
31	C₂³×C₄×C₄₇	—	2	8	no	—
32	C₂³×C₄×C₄₇	—	2	8	no	—
33	C₂³×C₄×C₄₇	—	2	64	no	—
34	C₂³×C₄×C₄₇	—	2	32	no	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	16	yes	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
37	C₂³×C₄×C₄₇	D₄	1	16	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
39	C₂⁴×C₄₇	—	2	128	no	—
40	C₂⁴×C₄₇	—	2	16	no	—
41	C₂⁴×C₄₇	—	2	32	no	—
42	C₂⁴×C₄₇	—	2	32	no	—
43	C₂⁴×C₄₇	—	2	64	no	—
44	C₂⁴×C₄₇	—	2	8	no	—
45	C₂⁴×C₄₇	—	2	8	no	—
46	C₂⁴×C₄₇	—	2	8	no	—
47	C₂⁴×C₄₇	C₂²⋊C₄	1	16	yes	—
48	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
50	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
51	C₂⁴×C₄₇	C₂×D₄	1	8	yes	—
52	C₂⁴×C₄₇	C₂×D₄	1	8	yes	—
53	C₂⁴×C₄₇	#41	1	16	yes	—
54	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	128	no	—
56	C₂²×C₄×C₄₇	—	2	256	no	—
57	C₂²×C₄×C₄₇	—	2	32	no	—
58	C₂²×C₄×C₄₇	—	2	16	no	—
59	C₂²×C₄×C₄₇	—	2	16	no	—
60	C₂²×C₄×C₄₇	—	2	64	no	—
61	C₂²×C₄×C₄₇	—	2	8	no	—
62	C₂²×C₄×C₄₇	—	2	8	no	—
63	C₂²×C₄×C₄₇	—	2	64	no	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	32	yes	complemented
65	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	16	yes	—
67	C₂²×Dic₄₇	—	2	16	no	—
68	C₂²×Dic₄₇	—	2	8	no	—
69	C₂²×Dic₄₇	—	2	32	no	—
70	C₂²⋊Dic₄₇	—	4	16	no	—
71	C₂²⋊Dic₄₇	—	4	32	no	—
72	C₂³×C₄₇	—	2	256	no	—
73	C₂³×C₄₇	—	2	128	no	—
74	C₂³×C₄₇	—	2	8	no	—
75	C₂³×C₄₇	—	2	8	no	—
76	C₂³×C₄₇	—	2	8	no	—
77	C₂³×C₄₇	—	2	16	no	—
78	C₂³×C₄₇	—	2	16	no	—
79	C₂³×C₄₇	—	2	16	no	—
80	C₂³×C₄₇	—	2	32	no	—
81	C₂³×C₄₇	—	2	64	no	—
82	C₂³×C₄₇	—	2	32	no	—
83	C₂³×C₄₇	—	2	64	no	—
84	C₂³×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
85	C₂³×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
86	C₂³×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
87	C₂³×C₄₇	#117	1	16	yes	—
88	C₂³×C₄₇	#120	1	16	yes	—
89	C₂³×C₄₇	#123	1	32	yes	—
90	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
91	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
92	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
93	C₂³×C₄₇	C₂×#41	1	8	yes	—
94	C₂×C₄×C₄₇	—	2	128	no	—
95	C₂×C₄×C₄₇	—	2	8	no	—
96	C₂×C₄×C₄₇	—	2	8	no	—
97	C₂×C₄×C₄₇	—	2	32	no	—
98	C₂×C₄×C₄₇	—	2	64	no	—
99	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	16	yes	—
100	C₂×Dic₄₇	—	4	8	no	—
101	C₂×Dic₄₇	—	4	16	no	—
102	C₂×Dic₄₇	—	4	32	no	—
103	#14681	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	—	2	128	no	—
105	C₂²×C₄₇	—	2	32	no	—
106	C₂²×C₄₇	—	2	64	no	—
107	C₂²×C₄₇	—	2	8	no	—
108	C₂²×C₄₇	—	2	8	no	—
109	C₂²×C₄₇	—	2	8	no	—
110	C₂²×C₄₇	—	2	16	no	—
111	C₂²×C₄₇	—	2	32	no	—
112	C₂²×C₄₇	#379	1	8	yes	—
113	C₂²×C₄₇	#381	1	16	yes	complemented
114	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	2	yes	—
116	C₂²×C₄₇	C₂×#117	1	4	yes	—
117	C₂²×C₄₇	C₂×#120	1	4	yes	—
118	C₂²×C₄₇	C₂×#123	1	8	yes	—
119	C₂²×C₄₇	#525	1	16	yes	—
120	C₄×C₄₇	—	2	16	no	—
121	Dic₄₇	—	8	8	no	—
122	C₂⁵×C₄	D₄₇	1	1	yes	characteristic
123	C₂×#381	—	47	4	no	—
124	C₂×#379	—	47	2	no	—
125	C₂²⋊#95	—	47	8	no	—
126	C₂×C₄₇	—	2	8	no	—
127	C₂×C₄₇	—	2	16	no	—
128	C₂×C₄₇	C₂×#379	1	1	yes	characteristic
129	C₂×C₄₇	C₂×#381	1	2	yes	—
130	C₂×C₄₇	#2280	1	4	yes	—
131	C₂×C₄₇	#2849	1	8	yes	—
132	C₂⁶	C₂×D₄₇	1	1	yes	characteristic
133	C₂⁴×C₄	—	2	8	no	—
134	C₂⁴×C₄	—	2	16	no	—
135	C₂⁴×C₄	Dic₄₇	1	8	yes	complemented
136	C₂⁴×C₄	C₂×D₄₇	1	4	yes	—
137	C₂⁴×C₄	C₂×D₄₇	1	2	yes	—
138	C₂²×C₂²⋊C₄	—	47	8	no	—
139	C₂²×C₂²⋊C₄	—	47	4	no	—
140	#381	—	94	16	no	—
141	#379	—	94	8	no	—
142	C₂×#95	—	47	16	no	—
143	C₄₇	#14681	1	1	yes	characteristic, complemented, Sylow
144	C₂⁵	—	2	16	no	—
145	C₂⁵	—	2	8	no	—
146	C₂⁵	C₂×Dic₄₇	1	4	yes	—
147	C₂⁵	C₂²⋊D₄₇	1	8	yes	—
148	C₂⁵	C₂²×D₄₇	1	2	yes	—
149	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
150	C₂³×C₄	—	2	128	no	—
151	C₂³×C₄	—	2	32	no	—
152	C₂³×C₄	—	2	16	no	—
153	C₂³×C₄	—	2	8	no	—
154	C₂³×C₄	—	2	8	no	—
155	C₂³×C₄	—	2	64	no	—
156	C₂³×C₄	—	2	32	no	—
157	C₂³×C₄	—	47	8	no	—
158	C₂³×C₄	C₂×Dic₄₇	1	16	yes	—
159	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
160	C₂³×C₄	C₂²⋊D₄₇	1	16	yes	—
161	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
162	C₂×C₂²⋊C₄	—	94	32	no	—
163	C₂×C₂²⋊C₄	—	94	8	no	—
164	C₂×C₂²⋊C₄	—	94	16	no	—
165	C₂×C₂²⋊C₄	—	94	16	no	—
166	#95	—	94	32	no	—
167	C₂⁴	—	2	128	no	—
168	C₂⁴	—	2	16	no	—
169	C₂⁴	—	2	32	no	—
170	C₂⁴	—	2	32	no	—
171	C₂⁴	—	2	64	no	—
172	C₂⁴	—	2	8	no	—
173	C₂⁴	—	2	8	no	—
174	C₂⁴	—	2	8	no	—
175	C₂⁴	C₂²⋊Dic₄₇	1	16	yes	—
176	C₂⁴	#10625350	1	16	yes	—
177	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
178	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
179	C₂⁴	C₂×C₂²⋊D₄₇	1	4	yes	—
180	C₂⁴	C₂×C₂²⋊D₄₇	1	8	yes	—
181	C₂⁴	C₂×C₂²⋊D₄₇	1	8	yes	—
182	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
183	C₂²×C₄	—	2	256	no	—
184	C₂²×C₄	—	2	128	no	—
185	C₂²×C₄	—	2	32	no	—
186	C₂²×C₄	—	2	16	no	—
187	C₂²×C₄	—	2	16	no	—
188	C₂²×C₄	—	2	64	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	64	no	—
192	C₂²×C₄	—	94	16	no	—
193	C₂²×C₄	—	94	8	no	—
194	C₂²×C₄	—	94	32	no	—
195	C₂²×C₄	C₂²⋊Dic₄₇	1	32	yes	complemented
196	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
197	C₂²×C₄	C₂×C₂²⋊D₄₇	1	16	yes	—
198	C₂²⋊C₄	—	188	16	no	—
199	C₂²⋊C₄	—	188	32	no	—
200	C₂³	—	2	128	no	—
201	C₂³	—	2	256	no	—
202	C₂³	—	2	32	no	—
203	C₂³	—	2	64	no	—
204	C₂³	—	2	8	no	—
205	C₂³	—	2	8	no	—
206	C₂³	—	2	8	no	—
207	C₂³	—	2	16	no	—
208	C₂³	—	2	16	no	—
209	C₂³	—	2	16	no	—
210	C₂³	—	2	64	no	—
211	C₂³	—	2	32	no	—
212	C₂³	C₂×C₂²⋊Dic₄₇	1	4	yes	—
213	C₂³	C₂×C₂²⋊Dic₄₇	1	8	yes	—
214	C₂³	C₂×C₂²⋊Dic₄₇	1	8	yes	—
215	C₂³	#13093983	1	16	yes	—
216	C₂³	#13093993	1	16	yes	—
217	C₂³	#13094007	1	32	yes	—
218	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
219	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
220	C₂³	C₂²×C₂²⋊D₄₇	1	4	yes	—
221	C₂³	C₂×#10625350	1	8	yes	—
222	C₂×C₄	—	2	128	no	—
223	C₂×C₄	—	2	64	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	32	no	—
227	C₂×C₄	—	188	8	no	—
228	C₂×C₄	—	188	16	no	—
229	C₂×C₄	—	188	32	no	—
230	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	16	yes	—
231	C₂²	—	2	128	no	—
232	C₂²	—	2	8	no	—
233	C₂²	—	2	8	no	—
234	C₂²	—	2	16	no	—
235	C₂²	—	2	32	no	—
236	C₂²	—	2	32	no	—
237	C₂²	—	2	64	no	—
238	C₂²	—	2	8	no	—
239	C₂²	#15688979	1	8	yes	—
240	C₂²	#15688986	1	16	yes	complemented
241	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
242	C₂²	C₂²×C₂²⋊Dic₄₇	1	2	yes	—
243	C₂²	C₂×#13093983	1	4	yes	—
244	C₂²	C₂×#13093993	1	4	yes	—
245	C₂²	C₂×#13094007	1	8	yes	—
246	C₂²	#15689634	1	16	yes	—
247	C₄	—	2	16	no	—
248	C₄	—	376	8	no	—
249	C₂	—	2	8	no	—
250	C₂	—	2	16	no	—
251	C₂	C₂×#15688979	1	1	yes	characteristic
252	C₂	C₂×#15688986	1	2	yes	—
253	C₂	#20245771	1	4	yes	—
254	C₂	#20251680	1	8	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Fitting	C₂⁵×C₄×C₄₇	C₂
socle, cosocle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂⁶×C₄₇	C₂²
hypercenter	C₂⁵×C₄	D₄₇
hypocenter	C₄₇	#14681
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#15688986	C₂	semidirect
3. #20241611	C₂	semidirect
9. C₂×C₄₇⋊#95	C₂²	semidirect
13. C₂⁴×C₄×C₄₇	C₄	semidirect
64. C₂²×C₄×C₄₇	C₂²⋊C₄	semidirect
113. C₂²×C₄₇	#381	semidirect
135. C₂⁴×C₄	Dic₄₇	semidirect
143. C₄₇	#14681	semidirect
195. C₂²×C₄	C₂²⋊Dic₄₇	semidirect
240. C₂²	#15688986	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688986	2
C₂×#15688979	2
#20241611	2
C₂⁵×C₄×C₄₇	2
#14681	47