Subgroups — #20252330

Order 6016.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689677	C₂	1	1	yes	maximal, characteristic
2	#15689254	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689253	C₂	1	2	yes	maximal, complemented
4	#15689253	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689252	C₂	1	2	yes	maximal, complemented
6	#15689247	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689148	C₂	1	2	yes	maximal
8	C₄⋊C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
9	#15689077	C₂	1	2	yes	maximal
10	C₄×#10625324	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625324	C₂²	1	1	yes	characteristic
13	C₂×#10625324	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₄₇	C₂²	1	2	yes	—
15	#13094023	C₄	1	1	yes	characteristic
16	#13094022	C₄	1	1	yes	characteristic
17	#13094014	C₄	1	2	yes	—
18	#13094004	C₄	1	1	yes	characteristic
19	#13094002	C₄	1	2	yes	—
20	#13094000	C₄	1	1	yes	characteristic
21	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
22	C₄×Dic₉₄	C₂²	1	2	yes	—
23	#13093981	C₂²	1	2	yes	—
24	C₄²×D₄₇	C₂²	1	1	yes	characteristic
25	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
27	Dic₄₇⋊C₈	C₂²	1	2	yes	—
28	Dic₄₇⋊C₈	C₂²	1	2	yes	—
29	Dic₄₇⋊C₈	C₂²	1	2	yes	—
30	#13093935	C₂²	1	2	yes	—
31	#13093934	C₂²	1	1	yes	characteristic
32	#13093934	C₂²	1	1	yes	characteristic
33	#13093934	C₂²	1	2	yes	—
34	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×Dic₄₇	C₂²	1	2	yes	—
36	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
39	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₄₇	C₂²	1	2	yes	—
41	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₄₇	C₂³	1	1	yes	characteristic
45	C₂×Dic₉₄	C₂×C₄	1	1	yes	characteristic
46	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
47	C₂×C₈×C₄₇	C₂³	1	2	yes	—
48	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
51	C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₄²×C₄₇	C₂³	1	2	yes	—
54	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
58	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
59	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
60	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
62	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
63	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₄₇	C₂³	1	2	yes	—
66	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
67	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
69	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
70	#10625324	—	2	2	no	—
71	#10625324	Q₈	1	4	yes	complemented
72	C₈×D₄₇	—	2	4	no	—
73	C₂²×D₄₇	C₂²×C₄	1	1	yes	characteristic
74	Q₈×C₄₇	—	2	2	no	—
75	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
83	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
84	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₄×D₄₇	—	2	2	no	—
86	C₄×D₄₇	—	4	1	no	—
87	C₄×D₄₇	C₂×Q₈	1	2	yes	—
88	C₄×D₄₇	#41	1	2	yes	—
89	Dic₉₄	—	2	2	no	—
90	Dic₉₄	—	4	2	no	—
91	C₈×C₄₇	—	2	2	no	—
92	C₈×C₄₇	—	2	1	no	—
93	C₈×C₄₇	C₂×Q₈	1	2	yes	—
94	C₄₇⋊C₈	—	2	2	no	—
95	C₄₇⋊C₈	—	2	1	no	—
96	C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
97	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₄₇	—	2	1	no	—
99	C₂×D₄₇	—	2	1	no	—
100	C₂×D₄₇	C₄×Q₈	1	2	yes	—
101	C₄×C₄₇	—	2	2	no	—
102	C₄×C₄₇	—	2	2	no	—
103	C₄×C₄₇	—	2	1	no	—
104	C₄×C₄₇	#131	1	2	yes	—
105	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
106	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
107	Dic₄₇	—	2	2	no	—
108	Dic₄₇	—	2	1	no	—
109	Dic₄₇	—	2	1	no	—
110	Dic₄₇	—	2	2	no	—
111	Dic₄₇	C₄×Q₈	1	2	yes	—
112	#2962	—	47	1	no	maximal, Sylow
113	C₂×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
115	C₂×C₄₇	#568	1	1	yes	characteristic
116	D₄₇	—	2	2	no	—
117	#533	—	47	1	no	—
118	#446	—	47	1	no	—
119	#446	—	47	2	no	—
120	#446	—	47	1	no	—
121	#446	—	47	2	no	—
122	C₈×Q₈	—	47	1	no	—
123	C₈×Q₈	D₄₇	1	1	yes	characteristic, complemented
124	#424	—	47	2	no	—
125	C₂×C₄⋊C₈	—	47	1	no	—
126	#405	—	47	2	no	—
127	C₄×M₁₆	—	47	1	no	—
128	C₄₇	#2962	1	1	yes	characteristic, complemented, Sylow
129	C₂×M₁₆	—	47	1	no	—
130	C₂×M₁₆	—	47	1	no	—
131	C₂²×C₈	—	47	2	no	—
132	C₄⋊Q₈	—	47	1	no	—
133	C₄⋊Q₈	—	47	1	no	—
134	#125	—	47	2	no	—
135	C₂²⋊Q₈	—	47	1	no	—
136	C₂²⋊Q₈	—	47	2	no	—
137	C₂²⋊Q₈	—	47	1	no	—
138	C₄×Q₈	—	47	1	no	—
139	C₄×Q₈	—	47	2	no	—
140	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
141	#117	—	47	2	no	—
142	C₂×C₄²	—	47	1	no	—
143	C₄⋊C₈	—	47	2	no	—
144	C₄⋊C₈	—	47	2	no	—
145	C₄⋊C₈	—	47	2	no	—
146	C₄⋊C₈	—	47	1	no	—
147	C₄⋊C₈	—	47	2	no	—
148	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
149	C₄⋊C₈	C₂×D₄₇	1	2	yes	—
150	#97	—	47	1	no	—
151	#97	—	47	1	no	—
152	#97	—	47	2	no	—
153	#97	—	47	2	no	—
154	C₄×C₈	—	47	2	no	—
155	C₄×C₈	—	47	1	no	—
156	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
157	C₄×C₈	C₂×D₄₇	1	2	yes	—
158	C₂×Q₈	—	47	1	no	—
159	C₂×Q₈	—	47	2	no	—
160	C₂×Q₈	C₄×D₄₇	1	1	yes	characteristic
161	C₂²×C₄	—	47	1	no	—
162	C₂²×C₄	—	47	1	no	—
163	C₂²×C₄	—	47	1	no	—
164	M₁₆	—	47	4	no	—
165	M₁₆	—	94	2	no	—
166	C₂×C₈	—	47	1	no	—
167	C₂×C₈	—	47	1	no	—
168	C₂×C₈	—	47	2	no	—
169	C₂×C₈	—	94	4	no	—
170	C₂×C₈	C₂²×D₄₇	1	2	yes	—
171	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
173	C₄⋊C₄	—	47	2	no	—
174	C₄⋊C₄	—	47	2	no	—
175	C₄⋊C₄	—	47	2	no	—
176	C₄⋊C₄	—	47	2	no	—
177	C₄⋊C₄	—	47	2	no	—
178	C₄⋊C₄	—	47	2	no	—
179	C₄⋊C₄	—	47	1	no	—
180	C₄⋊C₄	C₄×D₄₇	1	2	yes	—
181	C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
182	C₂²⋊C₄	—	47	2	no	—
183	C₂²⋊C₄	—	47	2	no	—
184	C₄²	—	47	1	no	—
185	C₄²	—	47	1	no	—
186	C₄²	—	47	2	no	—
187	C₄²	—	47	1	no	—
188	C₄²	C₂²×D₄₇	1	1	yes	characteristic
189	C₄²	C₂²×D₄₇	1	2	yes	—
190	C₂³	—	47	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	94	2	no	—
193	Q₈	—	188	2	no	—
194	C₂×C₄	—	47	1	no	—
195	C₂×C₄	—	47	2	no	—
196	C₂×C₄	—	47	2	no	—
197	C₂×C₄	—	47	1	no	—
198	C₂×C₄	—	47	1	no	—
199	C₂×C₄	—	47	2	no	—
200	C₂×C₄	—	47	2	no	—
201	C₂×C₄	—	94	2	no	—
202	C₂×C₄	—	188	1	no	—
203	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	47	2	no	—
211	C₈	—	94	2	no	—
212	C₈	—	94	1	no	—
213	C₈	Q₈×D₄₇	1	2	yes	—
214	C₂²	—	47	2	no	—
215	C₂²	—	94	1	no	—
216	C₂²	—	94	1	no	—
217	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	47	2	no	—
222	C₄	—	94	1	no	—
223	C₄	—	94	2	no	—
224	C₄	—	94	1	no	—
225	C₄	—	94	2	no	—
226	C₄	#13094030	1	2	yes	—
227	C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
228	C₄	#41×D₄₇	1	1	yes	characteristic
229	C₂	—	94	2	no	—
230	C₂	C₄×Q₈×D₄₇	1	1	yes	characteristic
231	C₂	C₂×#13094030	1	1	yes	characteristic
232	C₂	#15689888	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₈×Q₈×C₄₇	C₂
hypercenter	C₈×Q₈	D₄₇
hypocenter	C₄₇	#2962
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #15689254	C₂	semidirect
3. #15689253	C₂	semidirect
4. #15689253	C₂	semidirect
5. #15689252	C₂	semidirect
6. #15689247	C₂	semidirect
11. C₈×Q₈×C₄₇	C₂	semidirect
71. #10625324	Q₈	semidirect
123. C₈×Q₈	D₄₇	semidirect
128. C₄₇	#2962	semidirect