Subgroups — #26376331

Order 12032.

2446 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246248	C₂	1	16	yes	maximal, complemented
2	#20246002	C₂	1	2	yes	maximal
3	#20245991	C₂	1	4	yes	maximal, complemented
4	#20245786	C₂	1	4	yes	maximal, complemented
5	C₂×#15688986	C₂	1	2	yes	maximal
6	#20245741	C₂	1	2	yes	maximal
7	C₂²×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
9	#15689009	C₂²	1	32	yes	complemented
10	#15688998	C₂²	1	16	yes	—
11	#15688991	C₂²	1	8	yes	complemented
12	#15688989	C₂²	1	8	yes	complemented
13	#15688986	C₂²	1	4	yes	—
14	#15688986	C₂²	1	4	yes	—
15	#15688986	C₂²	1	8	yes	—
16	#15688983	C₂²	1	8	yes	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	8	yes	complemented
18	C₂²⋊C₄×Dic₄₇	C₂²	1	16	yes	—
19	C₄×C₂²⋊Dic₄₇	C₂²	1	8	yes	—
20	#15688969	C₂²	1	8	yes	complemented
21	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
22	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₄₇	C₄	1	16	yes	complemented
24	C₂×#117×C₄₇	C₂²	1	8	yes	—
25	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
29	C₂³×Dic₄₇	C₂³	1	8	yes	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
31	C₂×C₂²⋊Dic₄₇	—	2	16	no	—
32	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
33	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
34	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
35	C₄₇⋊#95	C₂³	1	16	yes	—
36	C₄₇⋊#95	C₂³	1	16	yes	—
37	C₄₇⋊#95	C₂³	1	16	yes	—
38	C₄₇⋊#95	C₂³	1	16	yes	—
39	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	—	2	8	no	—
41	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
46	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
47	#117×C₄₇	C₂×C₄	1	64	yes	complemented
48	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	complemented
49	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
50	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
51	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
52	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
53	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂×C₄	1	16	yes	complemented
55	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
57	C₂⁴×C₄₇	—	4	4	no	—
58	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
59	C₂⁴×C₄₇	#41	1	8	yes	—
60	C₂⁴×C₄₇	C₂⁴	1	2	yes	—
61	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	—	2	32	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	2	16	no	—
66	C₂²×C₄×C₄₇	—	4	4	no	—
67	C₂²×C₄×C₄₇	—	4	4	no	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
77	C₂²×Dic₄₇	—	2	16	no	—
78	C₂²×Dic₄₇	—	4	16	no	—
79	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
80	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
81	C₂²⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
82	C₂²⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
83	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
84	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
85	C₂²⋊Dic₄₇	—	2	16	no	—
86	C₂²⋊Dic₄₇	—	2	16	no	—
87	C₄⋊Dic₄₇	—	4	16	no	—
88	C₄×Dic₄₇	—	4	16	no	—
89	C₂³×C₄₇	—	2	8	no	—
90	C₂³×C₄₇	—	2	4	no	—
91	C₂³×C₄₇	—	4	8	no	—
92	C₂³×C₄₇	—	4	16	no	—
93	C₂³×C₄₇	#117	1	16	yes	—
94	C₂³×C₄₇	C₂³×C₄	1	4	yes	—
95	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
96	C₂³×C₄₇	C₂×#41	1	4	yes	—
97	C₂³×C₄₇	C₂×#41	1	8	yes	—
98	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	2	8	no	—
100	C₂×C₄×C₄₇	—	2	8	no	—
101	C₂×C₄×C₄₇	—	2	32	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	8	no	—
105	C₂×C₄×C₄₇	—	2	8	no	—
106	C₂×C₄×C₄₇	—	4	8	no	—
107	C₂×C₄×C₄₇	—	4	8	no	—
108	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
109	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
110	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
111	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
112	C₂×Dic₄₇	—	2	16	no	—
113	C₂×Dic₄₇	—	2	32	no	—
114	C₂×Dic₄₇	—	8	16	no	—
115	#34177	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	—	2	4	no	—
117	C₂²×C₄₇	—	2	8	no	—
118	C₂²×C₄₇	—	2	16	no	—
119	C₂²×C₄₇	—	2	8	no	—
120	C₂²×C₄₇	—	4	16	no	—
121	C₂²×C₄₇	C₂×#117	1	8	yes	—
122	C₂²×C₄₇	C₂×#117	1	4	yes	—
123	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
124	C₂²×C₄₇	C₂²×#41	1	2	yes	—
125	C₂²×C₄₇	#585	1	4	yes	—
126	C₄×C₄₇	—	2	4	no	—
127	C₄×C₄₇	—	2	4	no	—
128	C₄×C₄₇	—	2	8	no	—
129	C₄×C₄₇	—	2	8	no	—
130	Dic₄₇	—	4	16	no	—
131	C₂²×#117	D₄₇	1	1	yes	characteristic
132	#2343	—	47	16	no	—
133	#2315	—	47	2	no	—
134	#2313	—	47	4	no	—
135	#2283	—	47	4	no	—
136	C₂×#381	—	47	2	no	—
137	#2273	—	47	2	no	—
138	C₂×C₄₇	—	2	4	no	—
139	C₂×C₄₇	—	2	8	no	—
140	C₂×C₄₇	C₂²×#117	1	1	yes	characteristic
141	C₂×C₄₇	#3428	1	2	yes	—
142	C₂×C₄₇	#3491	1	4	yes	—
143	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
144	C₂×#117	Dic₄₇	1	16	yes	complemented
145	C₂×#117	C₂×D₄₇	1	8	yes	—
146	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
147	C₂²×C₂²⋊C₄	—	47	4	no	—
148	C₂²×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
149	C₂²×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
150	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
151	#387	—	47	32	no	—
152	#385	—	47	16	no	—
153	#383	—	47	8	no	—
154	#382	—	47	8	no	—
155	#381	—	47	4	no	—
156	#381	—	47	8	no	—
157	#381	—	47	4	no	—
158	#380	—	47	8	no	—
159	C₄×C₄⋊C₄	—	47	8	no	—
160	C₄×C₂²⋊C₄	—	47	8	no	—
161	C₄×C₂²⋊C₄	—	47	16	no	—
162	#376	—	47	8	no	—
163	C₂×#95	—	47	8	no	—
164	C₄₇	#34177	1	1	yes	characteristic, complemented, Sylow
165	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
166	C₂³×C₄	—	2	8	no	—
167	C₂³×C₄	—	47	8	no	—
168	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
169	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
170	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
171	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
172	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
173	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
174	#117	C₂×Dic₄₇	1	64	yes	complemented
175	C₂×C₄⋊C₄	—	47	16	no	—
176	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	complemented
177	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
178	C₂×C₂²⋊C₄	—	47	8	no	—
179	C₂×C₂²⋊C₄	—	47	8	no	—
180	C₂×C₂²⋊C₄	—	94	16	no	—
181	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
182	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
183	C₂×C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
184	C₂×C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
185	C₂×C₄²	—	47	16	no	—
186	C₂×C₄²	C₂×Dic₄₇	1	16	yes	complemented
187	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
188	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
189	#95	—	47	16	no	—
190	#95	—	47	16	no	—
191	#95	—	47	16	no	—
192	#95	—	47	16	no	—
193	C₂⁴	—	4	4	no	—
194	C₂⁴	#10625350	1	8	yes	—
195	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
196	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
197	C₂⁴	C₂³×D₄₇	1	2	yes	—
198	C₂²×C₄	—	2	32	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	16	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	—	47	16	no	—
205	C₂²×C₄	—	94	16	no	—
206	C₂²×C₄	—	188	16	no	—
207	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
208	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
209	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
210	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
211	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
212	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
213	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
214	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
215	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
216	C₄⋊C₄	—	188	16	no	—
217	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
218	C₂²⋊C₄	—	94	16	no	—
219	C₂²⋊C₄	—	94	16	no	—
220	C₂²⋊C₄	C₂²×Dic₄₇	1	16	yes	—
221	C₂²⋊C₄	C₂²×Dic₄₇	1	16	yes	—
222	C₄²	—	188	16	no	—
223	C₄²	C₂²×Dic₄₇	1	16	yes	—
224	C₄²	C₂²×Dic₄₇	1	16	yes	—
225	C₂³	—	2	4	no	—
226	C₂³	—	2	8	no	—
227	C₂³	—	4	16	no	—
228	C₂³	—	4	8	no	—
229	C₂³	#13093983	1	16	yes	—
230	C₂³	C₂³×Dic₄₇	1	4	yes	—
231	C₂³	C₂³×Dic₄₇	1	2	yes	—
232	C₂³	C₂×#10625350	1	4	yes	—
233	C₂³	C₂×#10625350	1	8	yes	—
234	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	32	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	4	8	no	—
244	C₂×C₄	—	94	16	no	—
245	C₂×C₄	—	94	32	no	—
246	C₂×C₄	—	376	16	no	—
247	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
248	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
249	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
250	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
251	C₂²	—	2	16	no	—
252	C₂²	—	2	8	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	4	16	no	—
256	C₂²	C₂×#13093983	1	8	yes	—
257	C₂²	C₂×#13093983	1	4	yes	—
258	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
259	C₂²	C₂²×#10625350	1	2	yes	—
260	C₂²	#15689959	1	4	yes	—
261	C₄	—	2	4	no	—
262	C₄	—	2	4	no	—
263	C₄	—	2	8	no	—
264	C₄	—	2	8	no	—
265	C₄	—	188	16	no	—
266	C₂	—	2	8	no	—
267	C₂	—	2	4	no	—
268	C₂	C₂²×#13093983	1	1	yes	characteristic
269	C₂	#20256591	1	2	yes	—
270	C₂	#20257086	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Fitting	C₂²×#117×C₄₇	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁵×C₄₇	C₂³
hypercenter	C₂²×#117	D₄₇
hypocenter	C₄₇	#34177
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
1. #20246248	C₂	semidirect
3. #20245991	C₂	semidirect
4. #20245786	C₂	semidirect
9. #15689009	C₂²	semidirect
11. #15688991	C₂²	semidirect
12. #15688989	C₂²	semidirect
17. C₄×C₄⋊Dic₄₇	C₂²	semidirect
20. #15688969	C₂²	semidirect
23. C₂×#117×C₄₇	C₄	semidirect
47. #117×C₄₇	C₂×C₄	semidirect
48. C₂×C₄⋊C₄×C₄₇	C₂×C₄	semidirect
54. C₂×C₄²×C₄₇	C₂×C₄	semidirect
144. C₂×#117	Dic₄₇	semidirect
164. C₄₇	#34177	semidirect
174. #117	C₂×Dic₄₇	semidirect
176. C₂×C₄⋊C₄	C₂×Dic₄₇	semidirect
186. 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
#20246248	2
#20246002	2
#20245991	2
#20245786	2
C₂×#15688986	2
#20245741	2
C₂²×#117×C₄₇	2
#34177	47