Subgroups — #20257272

Order 6016.

992 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689848	C₂	1	4	yes	maximal, complemented
2	#15689847	C₂	1	4	yes	maximal, complemented
3	#15689808	C₂	1	8	yes	maximal, complemented
4	#15689734	C₂	1	4	yes	maximal, complemented
5	#15689697	C₂	1	2	yes	maximal
6	#15689680	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689678	C₂	1	4	yes	maximal, complemented
8	#15689601	C₂	1	2	yes	maximal, complemented
9	C₄×#10625350	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#10625350	C₂²	1	1	yes	characteristic
12	#13094024	C₂²	1	2	yes	—
13	#13094022	C₂²	1	4	yes	—
14	#13094019	C₂²	1	16	yes	—
15	#13094016	C₂²	1	2	yes	—
16	#13094015	C₂²	1	8	yes	—
17	#13094014	C₂²	1	16	yes	—
18	#13094013	C₂²	1	4	yes	—
19	#13094007	C₂²	1	8	yes	—
20	#13094004	C₂²	1	8	yes	—
21	#13094003	C₂²	1	4	yes	—
22	#13094000	C₂²	1	8	yes	—
23	Q₈×Dic₄₇	C₂²	1	8	yes	—
24	Q₈×Dic₄₇	C₂²	1	8	yes	—
25	C₄×Dic₉₄	C₂²	1	4	yes	—
26	C₄×Dic₉₄	C₂²	1	2	yes	—
27	#13093989	C₂²	1	8	yes	—
28	C₄×D₁₈₈	C₂²	1	2	yes	—
29	C₄×C₂²⋊D₄₇	C₂²	1	4	yes	—
30	#13093983	C₂²	1	8	yes	—
31	#13093983	C₂²	1	1	yes	characteristic
32	#13093983	C₂²	1	2	yes	—
33	#13093982	C₂²	1	8	yes	—
34	#13093981	C₂²	1	2	yes	—
35	C₄²×D₄₇	C₂²	1	2	yes	—
36	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
37	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
38	C₄⋊Q₈×C₄₇	C₂²	1	4	yes	—
39	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
40	C₂×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
41	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
42	C₂×Q₈×C₄₇	—	2	8	no	—
43	C₂×Q₈×C₄₇	C₂³	1	4	yes	—
44	C₂×Q₈×C₄₇	C₂³	1	4	yes	—
45	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
47	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₄₇	C₂³	1	2	yes	—
49	#10625350	—	2	4	no	—
50	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₄₇	C₂³	1	2	yes	—
52	C₂×C₄×D₄₇	C₂³	1	4	yes	—
53	C₂×Dic₉₄	C₂³	1	4	yes	—
54	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
56	C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
57	C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₄²×C₄₇	C₂³	1	2	yes	—
60	C₂²⋊Dic₄₇	C₂³	1	2	yes	—
61	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
62	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
63	(C₂×D₄₇)⋊C₄	C₂³	1	16	yes	—
64	C₄⋊Dic₄₇	C₂³	1	4	yes	—
65	C₄⋊Dic₄₇	C₂³	1	2	yes	—
66	C₄⋊Dic₄₇	C₂³	1	4	yes	—
67	C₄⋊Dic₄₇	C₂³	1	16	yes	—
68	Dic₄₇⋊C₄	C₂³	1	4	yes	—
69	Dic₄₇⋊C₄	C₂³	1	16	yes	—
70	Dic₄₇⋊C₄	C₂³	1	8	yes	—
71	C₄×Dic₄₇	C₂³	1	2	yes	—
72	C₄×Dic₄₇	C₂³	1	4	yes	—
73	C₄×Dic₄₇	C₂³	1	16	yes	—
74	C₄×Dic₄₇	C₂³	1	4	yes	—
75	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×D₄₇	C₂⁴	1	2	yes	—
77	Q₈×C₄₇	—	2	8	no	—
78	Q₈×C₄₇	—	2	8	no	—
79	C₂×C₄×C₄₇	—	2	8	no	—
80	C₂×C₄×C₄₇	—	2	4	no	—
81	C₂×C₄×C₄₇	#41	1	4	yes	—
82	C₂×C₄×C₄₇	C₂⁴	1	8	yes	—
83	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
84	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
87	C₂²⋊D₄₇	—	2	4	no	—
88	C₂×Dic₄₇	C₂⁴	1	4	yes	—
89	C₂×Dic₄₇	C₂⁴	1	8	yes	—
90	C₂×Dic₄₇	C₂⁴	1	2	yes	—
91	D₁₈₈	—	2	2	no	—
92	C₄×D₄₇	—	2	4	no	—
93	C₄×D₄₇	—	2	8	no	—
94	Dic₉₄	—	2	2	no	—
95	Dic₉₄	—	2	8	no	—
96	C₂²×C₄₇	—	2	2	no	—
97	C₂²×C₄₇	C₂×#41	1	2	yes	—
98	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
99	C₂×D₄₇	—	2	2	no	—
100	C₂×D₄₇	—	2	4	no	—
101	C₄×C₄₇	—	2	8	no	—
102	C₄×C₄₇	—	2	2	no	—
103	C₄×C₄₇	C₂×#41	1	2	yes	—
104	C₄×C₄₇	C₂×#41	1	2	yes	—
105	C₄×C₄₇	2₋¹⁺⁴	1	4	yes	—
106	Dic₄₇	—	2	2	no	—
107	Dic₄₇	—	2	8	no	—
108	Dic₄₇	—	2	4	no	—
109	#3509	—	47	1	no	maximal, Sylow
110	C₂×C₄₇	—	2	2	no	—
111	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
112	C₂×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
113	D₄₇	—	4	2	no	—
114	#557	—	47	4	no	—
115	#556	—	47	4	no	—
116	#552	—	47	8	no	—
117	#541	—	47	4	no	—
118	#536	—	47	2	no	—
119	#533	—	47	4	no	—
120	#533	—	47	1	no	—
121	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
122	#519	—	47	2	no	—
123	C₄×#41	—	47	1	no	—
124	C₄₇	#3509	1	1	yes	characteristic, complemented, Sylow
125	C₂×#41	—	47	1	no	—
126	C₂²×Q₈	C₂×D₄₇	1	2	yes	—
127	C₄⋊Q₈	—	47	4	no	—
128	C₄⋊Q₈	—	47	2	no	—
129	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
130	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
131	C₄⋊Q₈	C₂×D₄₇	1	4	yes	—
132	#126	—	47	16	no	—
133	#125	—	47	8	no	—
134	#125	—	47	16	no	—
135	#125	—	47	2	no	—
136	#124	—	47	4	no	—
137	#123	—	47	8	no	—
138	C₂²⋊Q₈	—	47	8	no	—
139	C₂²⋊Q₈	—	47	4	no	—
140	C₂²⋊Q₈	—	47	8	no	—
141	C₄×Q₈	—	47	8	no	—
142	C₄×Q₈	—	47	8	no	—
143	C₄×Q₈	—	47	8	no	—
144	C₄×Q₈	—	47	2	no	—
145	C₄×Q₈	—	47	4	no	—
146	C₄×D₄	—	47	4	no	—
147	C₄×D₄	—	47	2	no	—
148	#117	—	47	1	no	—
149	#117	—	47	2	no	—
150	#117	—	47	8	no	—
151	#117	—	47	8	no	—
152	#117	—	47	2	no	—
153	C₂×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
154	C₂×C₄²	—	47	2	no	—
155	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
156	#41	—	94	4	no	—
157	C₂×Q₈	—	2	8	no	—
158	C₂×Q₈	—	47	4	no	—
159	C₂×Q₈	—	47	1	no	—
160	C₂×Q₈	C₂²×D₄₇	1	4	yes	—
161	C₂×Q₈	C₂²×D₄₇	1	4	yes	—
162	C₂×D₄	—	47	2	no	—
163	C₂×D₄	—	47	1	no	—
164	C₂²×C₄	—	47	4	no	—
165	C₂²×C₄	—	47	2	no	—
166	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
167	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
168	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
169	C₄⋊C₄	—	47	4	no	—
170	C₄⋊C₄	—	47	16	no	—
171	C₄⋊C₄	—	47	8	no	—
172	C₄⋊C₄	—	47	4	no	—
173	C₄⋊C₄	—	47	2	no	—
174	C₄⋊C₄	—	47	4	no	—
175	C₄⋊C₄	—	47	16	no	—
176	C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
177	C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
178	C₂²⋊C₄	—	47	2	no	—
179	C₂²⋊C₄	—	47	4	no	—
180	C₂²⋊C₄	—	47	4	no	—
181	C₂²⋊C₄	—	47	16	no	—
182	C₄²	—	47	2	no	—
183	C₄²	—	47	4	no	—
184	C₄²	—	47	16	no	—
185	C₄²	—	47	4	no	—
186	C₄²	C₂²×D₄₇	1	1	yes	characteristic
187	C₄²	C₂²×D₄₇	1	1	yes	characteristic
188	C₄²	C₂²×D₄₇	1	2	yes	—
189	C₂³	—	47	2	no	—
190	C₂³	C₂³×D₄₇	1	1	yes	characteristic
191	Q₈	—	2	8	no	—
192	Q₈	—	2	8	no	—
193	Q₈	—	94	2	no	—
194	Q₈	—	94	8	no	—
195	D₄	—	94	4	no	—
196	D₄	—	94	2	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	47	4	no	—
200	C₂×C₄	—	47	8	no	—
201	C₂×C₄	—	47	2	no	—
202	C₂×C₄	—	94	4	no	—
203	C₂×C₄	—	94	8	no	—
204	C₂×C₄	#10625352	1	4	yes	—
205	C₂×C₄	C₂³×D₄₇	1	8	yes	—
206	C₂×C₄	C₂³×D₄₇	1	2	yes	—
207	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
209	C₂×C₄	C₂³×D₄₇	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	—	94	2	no	—
212	C₂²	—	94	4	no	—
213	C₂²	C₂×#10625352	1	2	yes	—
214	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
215	C₄	—	2	8	no	—
216	C₄	—	2	2	no	—
217	C₄	—	94	2	no	—
218	C₄	—	94	4	no	—
219	C₄	—	94	8	no	—
220	C₄	C₂×#10625352	1	2	yes	—
221	C₄	C₂×#10625352	1	2	yes	—
222	C₄	#13094072	1	4	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	188	2	no	—
225	C₂	C₂²×#10625352	1	1	yes	characteristic
226	C₂	C₂×#13094072	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#3509
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #15689848	C₂	semidirect
2. #15689847	C₂	semidirect
3. #15689808	C₂	semidirect
4. #15689734	C₂	semidirect
6. #15689680	C₂	semidirect
7. #15689678	C₂	semidirect
8. #15689601	C₂	semidirect
10. C₂×C₄⋊Q₈×C₄₇	C₂	semidirect
121. C₂×C₄⋊Q₈	D₄₇	semidirect
124. C₄₇	#3509	semidirect