Subgroups — #20247985

Order 6016.

440 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689052	C₂	1	2	yes	maximal, complemented
2	#15689049	C₂	1	2	yes	maximal, complemented
3	#15689025	C₂	1	1	yes	maximal, characteristic
4	#15689025	C₂	1	1	yes	maximal, characteristic
5	#15689025	C₂	1	1	yes	maximal, characteristic
6	#15689021	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689013	C₂	1	2	yes	maximal, complemented
8	#15689012	C₂	1	2	yes	maximal, complemented
9	#15688989	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
17	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
18	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
19	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
20	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
21	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
22	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
23	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
24	C₄₇⋊#95	C₂²	1	2	yes	—
25	C₄₇⋊#95	C₂²	1	2	yes	—
26	C₄₇⋊#95	C₂²	1	2	yes	—
27	C₄₇⋊#95	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
33	#95×C₄₇	C₂²	1	1	yes	characteristic
34	#95×C₄₇	C₂²	1	1	yes	characteristic
35	C₂³×D₄₇	C₂³	1	2	yes	—
36	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂²×Dic₄₇	C₂³	1	2	yes	—
44	C₂²×Dic₄₇	C₂³	1	2	yes	—
45	C₂²×Dic₄₇	C₂³	1	2	yes	—
46	C₂×D₁₈₈	—	2	2	no	—
47	C₂×D₁₈₈	—	2	2	no	—
48	C₂×D₁₈₈	—	2	2	no	—
49	C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₄²×C₄₇	—	2	2	no	—
52	C₄²×C₄₇	—	2	2	no	—
53	C₄²×C₄₇	—	2	2	no	—
54	(C₂×D₄₇)⋊C₄	—	2	4	no	—
55	(C₂×D₄₇)⋊C₄	—	2	4	no	—
56	(C₂×D₄₇)⋊C₄	—	4	2	no	—
57	(C₂×D₄₇)⋊C₄	—	4	2	no	—
58	(C₂×D₄₇)⋊C₄	—	4	2	no	—
59	(C₂×D₄₇)⋊C₄	—	4	2	no	—
60	C₄⋊Dic₄₇	—	2	2	no	—
61	C₄⋊Dic₄₇	—	2	2	no	—
62	C₄⋊Dic₄₇	—	2	2	no	—
63	Dic₄₇⋊C₄	—	4	2	no	—
64	C₄×Dic₄₇	—	4	2	no	—
65	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
66	C₂²×D₄₇	—	2	2	no	—
67	C₂²×D₄₇	—	2	2	no	—
68	C₂²×D₄₇	—	2	2	no	—
69	C₂²×D₄₇	—	2	2	no	—
70	C₂²×D₄₇	—	2	2	no	—
71	C₂²×D₄₇	—	2	2	no	—
72	C₂²×D₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	—	2	1	no	—
74	C₂×C₄×C₄₇	—	2	1	no	—
75	C₂×C₄×C₄₇	—	2	1	no	—
76	C₂×C₄×C₄₇	—	2	1	no	—
77	C₂×C₄×C₄₇	—	2	1	no	—
78	C₂×C₄×C₄₇	—	2	1	no	—
79	C₂×C₄×C₄₇	—	2	1	no	—
80	C₂×C₄×C₄₇	—	2	1	no	—
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	1	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	#41	1	2	yes	—
90	C₂×C₄×C₄₇	#41	1	2	yes	—
91	C₂×C₄×C₄₇	#41	1	2	yes	—
92	C₂×C₄×C₄₇	#41	1	2	yes	—
93	C₂×C₄×C₄₇	#41	1	2	yes	—
94	C₂×Dic₄₇	—	2	2	no	—
95	C₂×Dic₄₇	—	2	2	no	—
96	C₂×Dic₄₇	—	2	2	no	—
97	C₂×Dic₄₇	—	2	2	no	—
98	C₂×Dic₄₇	—	2	2	no	—
99	C₂×Dic₄₇	—	2	2	no	—
100	C₂×Dic₄₇	—	2	2	no	—
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	2	no	—
103	D₁₈₈	—	4	4	no	—
104	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
105	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
106	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
107	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
108	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
109	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂×D₄₇	—	4	2	no	—
112	C₂×D₄₇	—	4	2	no	—
113	C₂×D₄₇	—	4	2	no	—
114	C₂×D₄₇	—	4	2	no	—
115	C₂×D₄₇	—	4	2	no	—
116	C₂×D₄₇	—	4	2	no	—
117	C₂×D₄₇	—	4	2	no	—
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	2	2	no	—
120	C₄×C₄₇	—	4	1	no	—
121	C₄×C₄₇	—	4	1	no	—
122	C₄×C₄₇	—	4	1	no	—
123	C₄×C₄₇	—	4	1	no	—
124	C₄×C₄₇	#126	1	4	yes	—
125	Dic₄₇	—	4	2	no	—
126	Dic₄₇	—	4	2	no	—
127	Dic₄₇	—	4	2	no	—
128	#2513	—	47	1	no	maximal, Sylow
129	C₂×C₄₇	C₂×#126	1	1	yes	characteristic
130	C₂×C₄₇	#529	1	1	yes	characteristic
131	C₂×C₄₇	#540	1	1	yes	characteristic
132	C₂×C₄₇	#553	1	1	yes	characteristic
133	C₂×C₄₇	#555	1	1	yes	characteristic
134	C₂×C₄₇	#559	1	1	yes	characteristic
135	C₂×C₄₇	#559	1	1	yes	characteristic
136	D₄₇	—	8	2	no	—
137	#397	—	47	2	no	—
138	#397	—	47	2	no	—
139	#390	—	47	1	no	—
140	#390	—	47	1	no	—
141	#390	—	47	1	no	—
142	#389	—	47	1	no	—
143	#388	—	47	2	no	—
144	#388	—	47	2	no	—
145	#382	—	47	1	no	—
146	C₄×C₄⋊C₄	—	47	1	no	—
147	C₄×C₄⋊C₄	D₄₇	1	1	yes	characteristic, complemented
148	C₄₇	#2513	1	1	yes	characteristic, complemented, Sylow
149	C₂²×D₄	—	47	1	no	—
150	C₂×C₄⋊C₄	—	47	1	no	—
151	C₂×C₄⋊C₄	—	47	1	no	—
152	C₂×C₄⋊C₄	—	47	1	no	—
153	C₂×C₄⋊C₄	—	47	2	no	—
154	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	—	47	2	no	—
157	C₂×C₂²⋊C₄	—	47	2	no	—
158	C₂×C₂²⋊C₄	—	47	2	no	—
159	C₂×C₂²⋊C₄	—	47	2	no	—
160	C₂×C₂²⋊C₄	—	47	2	no	—
161	C₂×C₂²⋊C₄	—	47	2	no	—
162	C₂×C₄²	—	47	2	no	—
163	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
164	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
165	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
166	#95	—	47	2	no	—
167	#95	—	47	2	no	—
168	#95	—	47	2	no	—
169	#95	—	47	2	no	—
170	#95	C₂×D₄₇	1	1	yes	characteristic
171	#95	C₂×D₄₇	1	1	yes	characteristic
172	C₂⁴	—	47	2	no	—
173	C₂×D₄	—	94	2	no	—
174	C₂×D₄	—	94	2	no	—
175	C₂×D₄	—	94	2	no	—
176	C₂²×C₄	—	47	2	no	—
177	C₂²×C₄	—	47	2	no	—
178	C₂²×C₄	—	47	2	no	—
179	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
182	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
184	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	94	2	no	—
189	C₄⋊C₄	—	94	2	no	—
190	C₄⋊C₄	—	94	2	no	—
191	C₄⋊C₄	—	188	2	no	—
192	C₂²⋊C₄	—	94	4	no	—
193	C₂²⋊C₄	—	94	4	no	—
194	C₂²⋊C₄	—	188	2	no	—
195	C₂²⋊C₄	—	188	2	no	—
196	C₂²⋊C₄	—	188	2	no	—
197	C₂²⋊C₄	—	188	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	188	2	no	—
202	C₂³	—	94	2	no	—
203	C₂³	—	94	2	no	—
204	C₂³	—	94	2	no	—
205	C₂³	—	94	2	no	—
206	C₂³	—	94	2	no	—
207	C₂³	—	94	2	no	—
208	C₂³	—	94	2	no	—
209	C₂³	C₂³×D₄₇	1	1	yes	characteristic
210	D₄	—	188	4	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	94	2	no	—
228	C₂×C₄	—	94	2	no	—
229	C₂×C₄	—	94	2	no	—
230	C₂×C₄	—	94	2	no	—
231	C₂×C₄	—	94	2	no	—
232	C₂×C₄	—	94	2	no	—
233	C₂×C₄	—	94	2	no	—
234	C₂×C₄	—	94	2	no	—
235	C₂×C₄	—	94	2	no	—
236	C₂×C₄	#10625350	1	2	yes	—
237	C₂×C₄	#10625350	1	2	yes	—
238	C₂×C₄	#10625350	1	2	yes	—
239	C₂×C₄	#10625352	1	2	yes	—
240	C₂×C₄	#10625354	1	2	yes	—
241	C₂²	—	188	2	no	—
242	C₂²	—	188	2	no	—
243	C₂²	—	188	2	no	—
244	C₂²	—	188	2	no	—
245	C₂²	—	188	2	no	—
246	C₂²	—	188	2	no	—
247	C₂²	—	188	2	no	—
248	C₂²	C₂×#10625352	1	1	yes	characteristic
249	C₂²	C₂×#10625350	1	1	yes	characteristic
250	C₂²	C₂×#10625350	1	1	yes	characteristic
251	C₂²	C₂×#10625350	1	1	yes	characteristic
252	C₂²	C₂×#10625354	1	1	yes	characteristic
253	C₂²	#13094070	1	1	yes	characteristic
254	C₂²	#13094070	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	1	no	—
261	C₄	—	188	2	no	—
262	C₄	—	188	2	no	—
263	C₄	—	188	2	no	—
264	C₄	#13094019	1	4	yes	—
265	C₂	—	376	2	no	—
266	C₂	C₂×#13094019	1	1	yes	characteristic
267	C₂	#15689657	1	1	yes	characteristic
268	C₂	#15689728	1	1	yes	characteristic
269	C₂	#15689819	1	1	yes	characteristic
270	C₂	#15689838	1	1	yes	characteristic
271	C₂	#15689858	1	1	yes	characteristic
272	C₂	#15689858	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄×C₄⋊C₄×C₄₇	C₂
hypercenter	C₄×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#2513
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #15689052	C₂	semidirect
2. #15689049	C₂	semidirect
6. #15689021	C₂	semidirect
7. #15689013	C₂	semidirect
8. #15689012	C₂	semidirect
9. #15688989	C₂	semidirect
10. C₄×C₄⋊Dic₄₇	C₂	semidirect
11. C₄×C₄⋊C₄×C₄₇	C₂	semidirect
147. C₄×C₄⋊C₄	D₄₇	semidirect
148. C₄₇	#2513	semidirect