Subgroups — #20257008

Order 6016.

1094 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689846	C₂	1	3	yes	maximal, complemented
2	#15689811	C₂	1	6	yes	maximal, complemented
3	Q₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689785	C₂	1	2	yes	maximal, complemented
5	#15689776	C₂	1	3	yes	maximal, complemented
6	#15689678	C₂	1	3	yes	maximal, complemented
7	#15689658	C₂	1	6	yes	maximal, complemented
8	C₄×#10625354	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#10625354	C₂	1	3	yes	maximal, complemented
10	C₄×Q₈×D₄₇	C₂	1	2	yes	maximal, complemented
11	C₄₇×#557	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625354	C₂²	1	3	yes	—
13	C₂×#10625354	C₂²	1	1	yes	characteristic
14	C₂×Q₈×D₄₇	C₂²	1	2	yes	—
15	#13094024	C₂²	1	3	yes	—
16	#13094019	C₂²	1	12	yes	—
17	#13094015	C₂²	1	6	yes	—
18	#13094012	C₂²	1	6	yes	—
19	#13094009	C₂²	1	12	yes	—
20	#13094004	C₂²	1	6	yes	—
21	#13094004	C₂²	1	6	yes	—
22	#13094000	C₂²	1	6	yes	—
23	#13093999	C₂²	1	6	yes	—
24	Q₈×Dic₄₇	C₂²	1	1	yes	characteristic
25	Q₈×Dic₄₇	C₂²	1	3	yes	—
26	C₄×Dic₉₄	C₂²	1	6	yes	—
27	#13093989	C₂²	1	6	yes	—
28	C₄×D₁₈₈	C₂²	1	3	yes	—
29	C₄×D₁₈₈	C₂²	1	3	yes	—
30	C₄×D₁₈₈	C₂²	1	6	yes	—
31	#13093986	C₂²	1	1	yes	characteristic
32	#13093986	C₂²	1	3	yes	—
33	#13093986	C₂²	1	6	yes	—
34	#13093982	C₂²	1	3	yes	—
35	#13093982	C₂²	1	6	yes	—
36	#13093982	C₂²	1	3	yes	—
37	#13093982	C₂²	1	6	yes	—
38	C₄⋊C₄×D₄₇	C₂²	1	6	yes	—
39	C₄²×D₄₇	C₂²	1	6	yes	—
40	C₄²×D₄₇	C₂²	1	3	yes	—
41	C₄⋊Q₈×C₄₇	C₂²	1	3	yes	—
42	#125×C₄₇	C₂²	1	6	yes	—
43	C₄×Q₈×C₄₇	C₂²	1	2	yes	—
44	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
45	C₄×Q₈×C₄₇	C₂²	1	3	yes	—
46	C₂×Q₈×C₄₇	C₂³	1	3	yes	—
47	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
48	#10625354	—	2	4	no	—
49	#10625354	—	2	12	no	—
50	Q₈×D₄₇	—	2	8	no	—
51	C₂×D₁₈₈	C₂³	1	6	yes	—
52	C₂×D₁₈₈	C₂³	1	3	yes	—
53	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₄₇	C₂³	1	2	yes	—
55	C₂×C₄×D₄₇	C₂³	1	6	yes	—
56	C₂×C₄×D₄₇	C₂³	1	6	yes	—
57	C₂×C₄×D₄₇	C₂³	1	3	yes	—
58	C₂×C₄×D₄₇	C₂³	1	3	yes	—
59	C₂×C₄×D₄₇	C₂³	1	3	yes	—
60	C₂×Dic₉₄	C₂³	1	6	yes	—
61	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
63	C₄⋊C₄×C₄₇	C₂³	1	6	yes	—
64	C₄⋊C₄×C₄₇	C₂³	1	6	yes	—
65	C₄⋊C₄×C₄₇	C₂³	1	6	yes	—
66	C₄²×C₄₇	C₂³	1	6	yes	—
67	C₄²×C₄₇	C₂³	1	3	yes	—
68	(C₂×D₄₇)⋊C₄	C₂³	1	2	yes	—
69	(C₂×D₄₇)⋊C₄	C₂³	1	6	yes	—
70	(C₂×D₄₇)⋊C₄	C₂³	1	6	yes	—
71	(C₂×D₄₇)⋊C₄	C₂³	1	6	yes	—
72	(C₂×D₄₇)⋊C₄	C₂³	1	12	yes	—
73	C₄⋊Dic₄₇	C₂³	1	3	yes	—
74	C₄⋊Dic₄₇	C₂³	1	3	yes	—
75	C₄⋊Dic₄₇	C₂³	1	6	yes	—
76	C₄⋊Dic₄₇	C₂³	1	6	yes	—
77	Dic₄₇⋊C₄	C₂³	1	12	yes	—
78	C₄×Dic₄₇	C₂³	1	3	yes	—
79	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
80	C₄×Dic₄₇	C₂³	1	3	yes	—
81	C₄×Dic₄₇	C₂³	1	6	yes	—
82	C₄×Dic₄₇	C₂³	1	6	yes	—
83	C₂²×D₄₇	C₂⁴	1	3	yes	—
84	C₂²×D₄₇	C₂⁴	1	2	yes	—
85	Q₈×C₄₇	—	2	6	no	—
86	Q₈×C₄₇	#41	1	4	yes	complemented
87	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
88	C₂×C₄×C₄₇	C₂⁴	1	6	yes	—
89	C₂×C₄×C₄₇	C₂⁴	1	3	yes	—
90	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂⁴	1	3	yes	—
92	C₂×Dic₄₇	C₂⁴	1	3	yes	—
93	C₂×Dic₄₇	C₂⁴	1	1	yes	characteristic
94	C₂×Dic₄₇	C₂⁴	1	6	yes	—
95	C₂×Dic₄₇	C₂⁴	1	1	yes	characteristic
96	D₁₈₈	—	2	12	no	—
97	D₁₈₈	—	2	6	no	—
98	D₁₈₈	#41	1	4	yes	complemented
99	C₄×D₄₇	—	2	12	no	—
100	C₄×D₄₇	—	2	12	no	—
101	C₄×D₄₇	—	2	6	no	—
102	C₄×D₄₇	—	2	6	no	—
103	C₄×D₄₇	—	2	4	no	—
104	C₄×D₄₇	—	2	6	no	—
105	Dic₉₄	—	2	12	no	—
106	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
107	C₂×D₄₇	—	2	2	no	—
108	C₂×D₄₇	—	2	2	no	—
109	C₂×D₄₇	—	2	3	no	—
110	C₂×D₄₇	—	2	3	no	—
111	C₂×D₄₇	—	2	3	no	—
112	C₂×D₄₇	C₂×#41	1	4	yes	—
113	C₄×C₄₇	—	2	6	no	—
114	C₄×C₄₇	—	2	2	no	—
115	C₄×C₄₇	—	2	3	no	—
116	C₄×C₄₇	C₂×#41	1	2	yes	—
117	C₄×C₄₇	C₂×#41	1	6	yes	—
118	Dic₄₇	—	2	6	no	—
119	Dic₄₇	—	2	3	no	—
120	Dic₄₇	—	2	1	no	—
121	Dic₄₇	2₋¹⁺⁴	1	2	yes	—
122	#3482	—	47	1	no	maximal, Sylow
123	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
124	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
125	C₂×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
126	D₄₇	—	2	4	no	—
127	D₄₇	—	4	3	no	—
128	#557	D₄₇	1	1	yes	characteristic, complemented
129	#556	—	47	3	no	—
130	#552	—	47	6	no	—
131	D₄×Q₈	—	47	1	no	—
132	#548	—	47	2	no	—
133	#547	—	47	3	no	—
134	#533	—	47	3	no	—
135	#529	—	47	6	no	—
136	C₄×#41	—	47	3	no	—
137	C₄×#41	—	47	1	no	—
138	C₂×C₄×Q₈	—	47	2	no	—
139	C₄₇	#3482	1	1	yes	characteristic, complemented, Sylow
140	C₂×#41	—	47	3	no	—
141	C₂×#41	—	47	1	no	—
142	C₂²×Q₈	—	47	2	no	—
143	C₄⋊Q₈	—	47	3	no	—
144	C₄⋊Q₈	C₂×D₄₇	1	3	yes	—
145	#126	—	47	12	no	—
146	#125	—	47	6	no	—
147	#125	C₂×D₄₇	1	6	yes	—
148	#124	—	47	6	no	—
149	#123	—	47	12	no	—
150	C₂²⋊Q₈	—	47	6	no	—
151	C₂²⋊Q₈	—	47	6	no	—
152	C₂²⋊Q₈	—	47	6	no	—
153	#121	—	47	6	no	—
154	C₄×Q₈	—	47	1	no	—
155	C₄×Q₈	—	47	3	no	—
156	C₄×Q₈	—	47	6	no	—
157	C₄×Q₈	—	47	6	no	—
158	C₄×Q₈	C₂×D₄₇	1	2	yes	—
159	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
160	C₄×Q₈	C₂×D₄₇	1	3	yes	—
161	C₄×D₄	—	47	1	no	—
162	C₄×D₄	—	47	3	no	—
163	C₄×D₄	—	47	6	no	—
164	C₄×D₄	—	47	3	no	—
165	C₄×D₄	—	47	3	no	—
166	C₄×D₄	—	47	6	no	—
167	#117	—	47	3	no	—
168	#117	—	47	6	no	—
169	#117	—	47	3	no	—
170	#117	—	47	6	no	—
171	C₂×C₄⋊C₄	—	47	6	no	—
172	C₂×C₄²	—	47	6	no	—
173	C₂×C₄²	—	47	3	no	—
174	#41	—	94	12	no	—
175	#41	—	94	4	no	—
176	C₂×Q₈	—	47	6	no	—
177	C₂×Q₈	—	94	8	no	—
178	C₂×Q₈	C₂²×D₄₇	1	3	yes	—
179	C₂×Q₈	C₂²×D₄₇	1	1	yes	characteristic
180	C₂×D₄	—	47	1	no	—
181	C₂×D₄	—	47	6	no	—
182	C₂×D₄	—	47	3	no	—
183	C₂²×C₄	—	47	2	no	—
184	C₂²×C₄	—	47	6	no	—
185	C₂²×C₄	—	47	6	no	—
186	C₂²×C₄	—	47	3	no	—
187	C₂²×C₄	—	47	3	no	—
188	C₂²×C₄	—	47	3	no	—
189	C₄⋊C₄	—	47	12	no	—
190	C₄⋊C₄	—	47	3	no	—
191	C₄⋊C₄	—	47	6	no	—
192	C₄⋊C₄	—	47	3	no	—
193	C₄⋊C₄	—	47	6	no	—
194	C₄⋊C₄	C₂²×D₄₇	1	6	yes	—
195	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
196	C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
197	C₄⋊C₄	C₂²×D₄₇	1	6	yes	—
198	C₄⋊C₄	C₂²×D₄₇	1	6	yes	—
199	C₂²⋊C₄	—	47	2	no	—
200	C₂²⋊C₄	—	47	6	no	—
201	C₂²⋊C₄	—	47	6	no	—
202	C₂²⋊C₄	—	47	6	no	—
203	C₂²⋊C₄	—	47	12	no	—
204	C₄²	—	47	1	no	—
205	C₄²	—	47	3	no	—
206	C₄²	—	47	3	no	—
207	C₄²	—	47	6	no	—
208	C₄²	—	47	6	no	—
209	C₄²	C₂²×D₄₇	1	3	yes	—
210	C₄²	C₂²×D₄₇	1	6	yes	—
211	C₂³	—	47	2	no	—
212	C₂³	—	47	3	no	—
213	Q₈	—	2	6	no	—
214	Q₈	—	94	12	no	—
215	Q₈	#10625354	1	4	yes	complemented
216	D₄	—	47	4	no	—
217	D₄	—	94	12	no	—
218	D₄	—	94	6	no	—
219	C₂×C₄	—	47	1	no	—
220	C₂×C₄	—	47	3	no	—
221	C₂×C₄	—	47	1	no	—
222	C₂×C₄	—	47	6	no	—
223	C₂×C₄	—	94	4	no	—
224	C₂×C₄	—	94	6	no	—
225	C₂×C₄	—	94	12	no	—
226	C₂×C₄	—	94	6	no	—
227	C₂×C₄	—	94	12	no	—
228	C₂×C₄	—	94	6	no	—
229	C₂×C₄	C₂³×D₄₇	1	2	yes	—
230	C₂×C₄	C₂³×D₄₇	1	3	yes	—
231	C₂×C₄	C₂³×D₄₇	1	6	yes	—
232	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
233	C₂×C₄	C₂³×D₄₇	1	3	yes	—
234	C₂²	—	47	4	no	—
235	C₂²	—	94	2	no	—
236	C₂²	—	94	2	no	—
237	C₂²	—	94	3	no	—
238	C₂²	—	94	3	no	—
239	C₂²	—	94	3	no	—
240	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
241	C₄	—	2	6	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	3	no	—
244	C₄	—	47	2	no	—
245	C₄	—	94	3	no	—
246	C₄	—	94	1	no	—
247	C₄	—	94	6	no	—
248	C₄	#41×D₄₇	1	2	yes	—
249	C₄	C₂×#10625354	1	6	yes	—
250	C₂	—	94	4	no	—
251	C₂	—	188	3	no	—
252	C₂	C₂×#41×D₄₇	1	1	yes	characteristic
253	C₂	C₂²×#10625354	1	1	yes	characteristic
254	C₂	2₋¹⁺⁴×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₄₇×#557	C₂
hypercenter	#557	D₄₇
hypocenter	C₄₇	#3482
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #557

Complements

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

N	H ≅ G/N	type
1. #15689846	C₂	semidirect
2. #15689811	C₂	semidirect
3. Q₈×D₁₈₈	C₂	semidirect
4. #15689785	C₂	semidirect
5. #15689776	C₂	semidirect
6. #15689678	C₂	semidirect
7. #15689658	C₂	semidirect
8. C₄×#10625354	C₂	semidirect
9. C₄×#10625354	C₂	semidirect
10. C₄×Q₈×D₄₇	C₂	semidirect
11. C₄₇×#557	C₂	semidirect
86. Q₈×C₄₇	#41	semidirect
98. D₁₈₈	#41	semidirect
128. #557	D₄₇	semidirect
139. C₄₇	#3482	semidirect
215. Q₈	#10625354	semidirect