Subgroups — #20252448

Order 6016.

402 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689634	C₂	1	1	yes	maximal, characteristic
2	#15689199	C₂	1	2	yes	maximal, complemented
3	#15689198	C₂	1	2	yes	maximal, complemented
4	#15689197	C₂	1	2	yes	maximal, complemented
5	#15689192	C₂	1	2	yes	maximal, complemented
6	#15689181	C₂	1	2	yes	maximal, complemented
7	#15689088	C₂	1	2	yes	maximal, complemented
8	#15689087	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#407	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10625350	C₂²	1	1	yes	characteristic
11	C₂×#10625329	C₂²	1	2	yes	—
12	C₂×#10625324	C₂²	1	2	yes	complemented
13	C₂×C₈×D₄₇	C₂²	1	2	yes	complemented
14	#13094007	C₄	1	2	yes	—
15	#13094003	C₄	1	1	yes	characteristic
16	#13094003	C₄	1	1	yes	characteristic
17	#13093998	C₄	1	1	yes	characteristic
18	#13093998	C₄	1	1	yes	characteristic
19	#13093993	C₄	1	2	yes	—
20	C₄×C₂²⋊D₄₇	C₂²	1	2	yes	—
21	C₄×C₂²⋊D₄₇	C₂²	1	2	yes	—
22	#13093983	C₂²	1	1	yes	characteristic
23	Dic₄₇⋊C₈	C₂²	1	2	yes	complemented
24	Dic₄₇⋊C₈	C₂²	1	2	yes	complemented
25	C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
26	C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
27	C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
28	(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	complemented
29	(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	complemented
30	#13093934	C₂²	1	2	yes	complemented
31	C₈×Dic₄₇	C₂²	1	2	yes	complemented
32	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
34	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
35	C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
37	C₂⁴×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	—	4	2	no	—
39	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
48	#10625350	—	4	2	no	—
49	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₄₇	C₂³	1	2	yes	—
51	C₂×Dic₉₄	C₂×C₄	1	1	yes	characteristic
52	M₁₆×C₄₇	—	4	2	no	—
53	C₂×C₈×C₄₇	C₂³	1	2	yes	—
54	C₂×C₈×C₄₇	C₂³	1	2	yes	—
55	C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
56	C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
58	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
62	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
63	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
64	C₄×Dic₄₇	C₂³	1	2	yes	—
65	#10625329	—	4	2	no	—
66	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
67	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
68	#10625324	—	4	2	no	—
69	C₈×D₄₇	—	4	2	no	—
70	C₂³×C₄₇	—	4	1	no	—
71	C₂³×C₄₇	—	4	1	no	—
72	C₂³×C₄₇	—	4	1	no	—
73	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	4	1	no	—
81	C₂×C₄×C₄₇	—	4	1	no	—
82	C₂×C₄×C₄₇	—	4	1	no	—
83	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₂²⋊D₄₇	—	4	2	no	—
88	C₂²⋊D₄₇	—	4	2	no	—
89	C₂²⋊D₄₇	—	4	2	no	—
90	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
91	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
92	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
93	D₁₈₈	—	4	1	no	—
94	C₄×D₄₇	—	2	2	no	—
95	C₄×D₄₇	—	2	2	no	—
96	Dic₉₄	—	4	1	no	—
97	C₈×C₄₇	—	2	2	no	—
98	C₈×C₄₇	—	2	2	no	—
99	C₄₇⋊C₈	—	2	2	no	—
100	C₄₇⋊C₈	—	2	2	no	—
101	C₂²×C₄₇	—	2	1	no	—
102	C₂²×C₄₇	—	2	1	no	—
103	C₂²×C₄₇	—	2	1	no	—
104	C₂²×C₄₇	—	2	1	no	—
105	C₂²×C₄₇	—	2	1	no	—
106	C₂²×C₄₇	—	2	1	no	—
107	C₂²×C₄₇	—	2	1	no	—
108	C₂²×C₄₇	—	2	1	no	—
109	C₂²×C₄₇	—	2	1	no	—
110	C₂²×C₄₇	—	4	4	no	—
111	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₄₇	—	2	2	no	—
113	C₂×D₄₇	—	2	2	no	—
114	C₂×D₄₇	—	2	2	no	—
115	C₄×C₄₇	—	2	1	no	—
116	C₄×C₄₇	—	2	1	no	—
117	C₄×C₄₇	—	2	1	no	—
118	C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
119	Dic₄₇	—	2	2	no	—
120	Dic₄₇	—	2	2	no	—
121	Dic₄₇	—	2	2	no	—
122	#2978	—	47	1	no	maximal, Sylow
123	C₂×C₄₇	—	4	1	no	—
124	C₂×C₄₇	—	4	1	no	—
125	C₂×C₄₇	—	4	1	no	—
126	C₂×C₄₇	#518	1	1	yes	characteristic
127	C₂×C₄₇	#568	1	1	yes	characteristic
128	C₂×C₄₇	#568	1	1	yes	characteristic
129	D₄₇	—	4	2	no	—
130	#525	—	47	1	no	—
131	#436	—	47	2	no	—
132	#436	—	47	2	no	—
133	#435	—	47	2	no	—
134	#435	—	47	2	no	—
135	#433	—	47	2	no	—
136	#408	—	47	2	no	—
137	#407	—	47	1	no	—
138	#407	D₄₇	1	1	yes	characteristic, complemented
139	C₄₇	#2978	1	1	yes	characteristic, complemented, Sylow
140	C₂×#41	—	47	1	no	—
141	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
142	C₂×M₁₆	—	47	2	no	—
143	C₂×M₁₆	—	47	2	no	—
144	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
145	C₂²×C₈	—	47	2	no	—
146	#123	—	47	2	no	—
147	C₂²⋊Q₈	—	47	1	no	—
148	C₂²⋊Q₈	—	47	1	no	—
149	#121	—	47	1	no	—
150	#121	—	47	1	no	—
151	#120	—	47	2	no	—
152	C₄×D₄	—	47	2	no	—
153	C₄×D₄	—	47	2	no	—
154	#117	—	47	1	no	—
155	C₄⋊C₈	—	47	2	no	—
156	C₄⋊C₈	—	47	2	no	—
157	C₂²⋊C₈	—	47	2	no	—
158	C₂²⋊C₈	—	47	1	no	—
159	C₂²⋊C₈	—	47	1	no	—
160	C₂²⋊C₈	—	47	2	no	—
161	C₂²⋊C₈	—	47	2	no	—
162	C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
163	C₂²⋊C₈	C₂×D₄₇	1	1	yes	characteristic
164	C₂²⋊C₈	C₂×D₄₇	1	1	yes	characteristic
165	#97	—	47	2	no	—
166	C₄×C₈	—	47	2	no	—
167	C₂⁴	C₄×D₄₇	1	1	yes	characteristic
168	#41	—	188	2	no	—
169	C₂×Q₈	—	47	1	no	—
170	C₂×D₄	—	47	2	no	—
171	C₂×D₄	—	47	2	no	—
172	C₂×D₄	—	47	2	no	—
173	C₂×D₄	—	47	1	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	47	2	no	—
176	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
177	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
178	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
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	M₁₆	—	4	2	no	—
183	M₁₆	—	188	2	no	—
184	M₁₆	—	188	2	no	—
185	C₂×C₈	—	47	2	no	—
186	C₂×C₈	—	47	2	no	—
187	C₂×C₈	—	188	2	no	—
188	C₂×C₈	C₂²×D₄₇	1	2	yes	—
189	C₂×C₈	C₂²×D₄₇	1	2	yes	—
190	C₄⋊C₄	—	47	2	no	—
191	C₄⋊C₄	—	47	2	no	—
192	C₄⋊C₄	—	47	1	no	—
193	C₄⋊C₄	—	47	1	no	—
194	C₂²⋊C₄	—	47	2	no	—
195	C₂²⋊C₄	—	47	2	no	—
196	C₂²⋊C₄	—	47	2	no	—
197	C₂²⋊C₄	—	47	2	no	—
198	C₂²⋊C₄	—	47	2	no	—
199	C₄²	—	47	2	no	—
200	C₂³	—	4	1	no	—
201	C₂³	—	4	1	no	—
202	C₂³	—	4	1	no	—
203	C₂³	—	47	2	no	—
204	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
205	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
206	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
207	Q₈	—	188	1	no	—
208	D₄	—	188	1	no	—
209	D₄	—	188	2	no	—
210	D₄	—	188	2	no	—
211	D₄	—	188	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	47	2	no	—
219	C₂×C₄	—	47	2	no	—
220	C₂×C₄	—	47	2	no	—
221	C₂×C₄	—	94	2	no	—
222	C₂×C₄	—	94	2	no	—
223	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
224	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	94	2	no	—
230	C₈	—	94	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	4	4	no	—
241	C₂²	—	94	2	no	—
242	C₂²	—	94	2	no	—
243	C₂²	—	94	2	no	—
244	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	94	2	no	—
249	C₄	—	94	2	no	—
250	C₄	—	94	2	no	—
251	C₄	#13094069	1	2	yes	—
252	C₂	—	4	1	no	—
253	C₂	—	4	1	no	—
254	C₂	—	4	1	no	—
255	C₂	—	188	2	no	—
256	C₂	#15689596	1	1	yes	characteristic
257	C₂	#15689889	1	1	yes	characteristic
258	C₂	#15689889	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#407	C₂
hypercenter	#407	D₄₇
hypocenter	C₄₇	#2978
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
2. #15689199	C₂	semidirect
3. #15689198	C₂	semidirect
4. #15689197	C₂	semidirect
5. #15689192	C₂	semidirect
6. #15689181	C₂	semidirect
7. #15689088	C₂	semidirect
8. #15689087	C₂	semidirect
9. C₄₇×#407	C₂	semidirect
12. C₂×#10625324	C₂²	semidirect
13. C₂×C₈×D₄₇	C₂²	semidirect
23. Dic₄₇⋊C₈	C₂²	semidirect
24. Dic₄₇⋊C₈	C₂²	semidirect
28. (C₂×D₄₇)⋊C₈	C₂²	semidirect
29. (C₂×D₄₇)⋊C₈	C₂²	semidirect
30. #13093934	C₂²	semidirect
31. C₈×Dic₄₇	C₂²	semidirect
138. #407	D₄₇	semidirect
139. C₄₇	#2978	semidirect