Subgroups — #26115325

Order 12032.

882 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251783	C₂	1	1	yes	maximal, characteristic
2	#20243540	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15688866	C₂	1	4	yes	maximal, complemented
4	C₄×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13094021	—	2	1	no	—
6	C₂×#13094021	—	2	2	no	—
7	C₂×#13094021	—	2	1	no	—
8	C₂×#13094021	C₄	1	4	yes	—
9	C₂×#13094021	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	#15688866	—	2	8	no	—
13	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	—	2	1	no	—
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#336	—	2	1	no	—
17	C₂²×D₁₈₈	—	2	4	no	—
18	C₂²×D₁₈₈	—	2	4	no	—
19	C₂²×D₁₈₈	—	2	2	no	—
20	C₂²×D₁₈₈	—	2	2	no	—
21	C₂²×D₁₈₈	C₂×C₄	1	4	yes	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	#13094021	—	2	4	no	—
24	#13094021	—	2	4	no	—
25	#13094021	—	4	4	no	—
26	#13094021	—	4	8	no	—
27	#13094021	—	4	4	no	—
28	#13094021	—	4	2	no	—
29	#13094021	—	4	4	no	—
30	#13094021	—	4	2	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
32	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
33	C₂²×C₈×C₄₇	—	2	2	no	—
34	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	D₄	1	2	yes	—
37	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₄₇	—	2	2	no	—
40	C₄⋊C₈×C₄₇	D₄	1	4	yes	complemented
41	C₄×C₈×C₄₇	—	4	2	no	—
42	C₂³×D₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂×D₁₈₈	—	2	8	no	—
49	C₂×D₁₈₈	—	4	8	no	—
50	C₂×D₁₈₈	—	4	8	no	—
51	C₂×D₁₈₈	—	4	8	no	—
52	C₂×D₁₈₈	—	4	8	no	—
53	C₂×D₁₈₈	—	4	8	no	—
54	C₂×D₁₈₈	—	4	8	no	—
55	C₂×D₁₈₈	—	4	8	no	—
56	C₂×D₁₈₈	—	4	4	no	—
57	C₂×D₁₈₈	—	4	2	no	—
58	C₂×D₁₈₈	—	4	2	no	—
59	C₂×D₁₈₈	—	4	2	no	—
60	C₂×D₁₈₈	—	4	4	no	—
61	C₂×D₁₈₈	—	4	2	no	—
62	C₂×D₁₈₈	—	4	4	no	—
63	C₂×D₁₈₈	—	4	4	no	—
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	—	4	2	no	—
66	C₂×C₈×C₄₇	—	4	2	no	—
67	C₄²×C₄₇	—	2	4	no	—
68	C₄²×C₄₇	—	2	2	no	—
69	C₄²×C₄₇	D₈	1	4	yes	—
70	C₄²×C₄₇	D₈	1	2	yes	—
71	C₄²×C₄₇	SD₁₆	1	4	yes	—
72	C₄²×C₄₇	SD₁₆	1	2	yes	—
73	C₄²×C₄₇	C₂×D₄	1	2	yes	—
74	C₄²×C₄₇	#41	1	2	yes	—
75	C₂×C₄₇⋊C₈	—	4	2	no	—
76	C₂×C₄₇⋊C₈	—	4	4	no	—
77	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×D₄₇	—	4	4	no	—
79	C₂²×D₄₇	—	4	8	no	—
80	C₂²×D₄₇	—	4	4	no	—
81	C₂²×D₄₇	—	4	8	no	—
82	C₂²×D₄₇	—	4	4	no	—
83	C₂×C₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	#100	1	4	yes	—
87	C₂×C₄×C₄₇	#102	1	4	yes	—
88	C₂×C₄×C₄₇	#102	1	4	yes	—
89	C₂×C₄×C₄₇	#102	1	4	yes	—
90	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₄₇	#121	1	2	yes	—
92	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
93	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
98	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
99	D₁₈₈	—	8	8	no	—
100	D₁₈₈	—	8	16	no	—
101	D₁₈₈	—	8	16	no	—
102	D₁₈₈	—	8	4	no	—
103	D₁₈₈	—	8	4	no	—
104	D₁₈₈	—	8	4	no	—
105	D₁₈₈	—	8	2	no	—
106	D₁₈₈	—	8	2	no	—
107	C₈×C₄₇	—	4	4	no	—
108	C₄₇⋊C₈	—	8	2	no	—
109	#9695	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	#390	1	1	yes	characteristic
111	C₂²×C₄₇	C₂×#100	1	2	yes	—
112	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
113	C₂²×C₄₇	C₂×#102	1	2	yes	—
114	C₂²×C₄₇	#418	1	1	yes	characteristic
115	C₂×D₄₇	—	8	8	no	—
116	C₂×D₄₇	—	8	8	no	—
117	C₂×D₄₇	—	8	4	no	—
118	C₂×D₄₇	—	8	4	no	—
119	C₂×D₄₇	—	8	4	no	—
120	C₄×C₄₇	—	2	4	no	—
121	C₄×C₄₇	—	2	2	no	—
122	C₄×C₄₇	#331	1	8	yes	complemented
123	C₄×C₄₇	#459	1	2	yes	—
124	C₄×C₄₇	#460	1	2	yes	—
125	C₄×C₄₇	#486	1	2	yes	—
126	C₄×C₄₇	#492	1	1	yes	characteristic
127	C₄×C₄₇	#493	1	1	yes	characteristic
128	#2867	—	47	1	no	—
129	#1842	—	47	1	no	—
130	C₄×C₄⋊C₈	D₄₇	1	1	yes	characteristic, complemented
131	C₂×#331	—	47	4	no	—
132	C₂×C₄₇	C₂×#331	1	2	yes	—
133	C₂×C₄₇	#1545	1	2	yes	—
134	C₂×C₄₇	#1957	1	1	yes	characteristic
135	C₂×C₄₇	#1965	1	1	yes	characteristic
136	C₂×C₄₇	#1982	1	1	yes	characteristic
137	D₄₇	—	16	4	no	—
138	C₂×#127	—	47	4	no	—
139	C₂×#127	—	47	2	no	—
140	C₂×#127	—	94	2	no	—
141	C₂×#127	—	94	1	no	—
142	C₂×#127	—	94	1	no	—
143	C₂×C₄⋊C₈	—	47	2	no	—
144	C₂×C₄⋊C₈	—	94	2	no	—
145	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
146	C₂×C₄×C₈	—	2	1	no	—
147	C₄³	C₂×D₄₇	1	1	yes	characteristic
148	#336	—	2	1	no	—
149	#331	—	94	8	no	—
150	C₄₇	#9695	1	1	yes	characteristic, complemented, Sylow
151	C₂²×D₄	—	47	4	no	—
152	C₂²×D₄	—	94	4	no	—
153	C₂²×D₄	—	94	4	no	—
154	C₂²×D₄	—	94	2	no	—
155	C₂²×D₄	—	94	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	94	2	no	—
158	#127	—	94	4	no	—
159	#127	—	94	4	no	—
160	#127	—	188	8	no	—
161	#127	—	188	4	no	—
162	#127	—	188	4	no	—
163	#127	—	188	4	no	—
164	#127	—	188	2	no	—
165	#127	—	188	2	no	—
166	C₂×C₄²	C₄×D₄₇	1	2	yes	—
167	C₂×C₄²	D₁₈₈	1	2	yes	—
168	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	94	4	no	—
173	C₄⋊C₈	—	188	4	no	—
174	C₄⋊C₈	D₁₈₈	1	4	yes	complemented
175	C₄×C₈	—	4	2	no	—
176	C₂⁴	—	94	4	no	—
177	C₂×D₄	—	94	8	no	—
178	C₂×D₄	—	188	8	no	—
179	C₂×D₄	—	188	8	no	—
180	C₂×D₄	—	188	8	no	—
181	C₂×D₄	—	188	8	no	—
182	C₂×D₄	—	188	8	no	—
183	C₂×D₄	—	188	8	no	—
184	C₂×D₄	—	188	8	no	—
185	C₂×D₄	—	188	4	no	—
186	C₂×D₄	—	188	2	no	—
187	C₂×D₄	—	188	2	no	—
188	C₂×D₄	—	188	2	no	—
189	C₂×D₄	—	188	4	no	—
190	C₂×D₄	—	188	2	no	—
191	C₂×D₄	—	188	4	no	—
192	C₂×D₄	—	188	4	no	—
193	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
194	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
195	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	188	2	no	—
202	C₂×C₈	—	188	4	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	#10625325	1	4	yes	—
206	C₄²	D₃₇₆	1	4	yes	—
207	C₄²	#10625334	1	2	yes	—
208	C₄²	#10625336	1	2	yes	—
209	C₄²	C₂×D₁₈₈	1	2	yes	—
210	C₄²	#10625350	1	2	yes	—
211	C₂³	—	188	4	no	—
212	C₂³	—	188	8	no	—
213	C₂³	—	188	4	no	—
214	C₂³	—	188	8	no	—
215	C₂³	—	188	4	no	—
216	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
217	D₄	—	376	8	no	—
218	D₄	—	376	16	no	—
219	D₄	—	376	2	no	—
220	D₄	—	376	2	no	—
221	D₄	—	376	4	no	—
222	D₄	—	376	4	no	—
223	D₄	—	376	16	no	—
224	D₄	—	376	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	#13093940	1	4	yes	—
229	C₂×C₄	#13093944	1	4	yes	—
230	C₂×C₄	#13093946	1	4	yes	—
231	C₂×C₄	#13093946	1	4	yes	—
232	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
233	C₂×C₄	#13093998	1	2	yes	—
234	C₂×C₄	#13094012	1	1	yes	characteristic
235	C₂×C₄	#13094021	1	1	yes	characteristic
236	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
237	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
238	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
239	C₂×C₄	C₂×#10625325	1	2	yes	—
240	C₂×C₄	#13094049	1	2	yes	—
241	C₈	—	4	4	no	—
242	C₈	—	376	2	no	—
243	C₂²	—	376	4	no	—
244	C₂²	—	376	4	no	—
245	C₂²	—	376	8	no	—
246	C₂²	—	376	8	no	—
247	C₂²	—	376	4	no	—
248	C₂²	#15689025	1	1	yes	characteristic
249	C₂²	C₂×#13093940	1	2	yes	—
250	C₂²	C₂×#13093944	1	1	yes	characteristic
251	C₂²	C₂×#13093946	1	2	yes	—
252	C₂²	#15689127	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	4	no	—
255	C₄	#15688866	1	8	yes	complemented
256	C₄	#15689318	1	2	yes	—
257	C₄	#15689323	1	2	yes	—
258	C₄	#15689470	1	2	yes	—
259	C₄	#15689496	1	1	yes	characteristic
260	C₄	#15689500	1	1	yes	characteristic
261	C₂	—	752	4	no	—
262	C₂	C₂×#15688866	1	2	yes	—
263	C₂	#20242055	1	2	yes	—
264	C₂	#20244194	1	1	yes	characteristic
265	C₂	#20244228	1	1	yes	characteristic
266	C₂	#20244315	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×C₄⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	#20251783	C₂
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#9695
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #20243540	C₂	semidirect
3. C₂×#15688866	C₂	semidirect
4. C₄×C₄⋊C₈×C₄₇	C₂	semidirect
40. C₄⋊C₈×C₄₇	D₄	semidirect
122. C₄×C₄₇	#331	semidirect
130. C₄×C₄⋊C₈	D₄₇	semidirect
150. C₄₇	#9695	semidirect
174. C₄⋊C₈	D₁₈₈	semidirect
255. C₄	#15688866	semidirect

Maximal subgroups

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

subgroup	index
#20251783	2
#20243540	2
C₂×#15688866	2
C₄×C₄⋊C₈×C₄₇	2
#9695	47