Subgroups — #20245863

Order 6016.

500 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#15689030	C₂	1	1	yes	maximal, characteristic
3	#15689021	C₂	1	1	yes	maximal, characteristic
4	#15689019	C₂	1	1	yes	maximal, characteristic
5	#15689008	C₂	1	2	yes	maximal
6	#15689006	C₂	1	2	yes	maximal
7	C₄⋊C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
8	#15688974	C₂	1	2	yes	maximal
9	#15688968	C₂	1	2	yes	maximal
10	C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#391	C₂	1	1	yes	maximal, characteristic
12	Q₈×Dic₄₇	C₄	1	8	yes	complemented
13	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
17	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
23	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
25	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
26	C₄₇⋊#95	C₂²	1	2	yes	—
27	C₄₇⋊#95	C₂²	1	2	yes	—
28	C₄₇⋊#95	C₂²	1	2	yes	—
29	C₄₇⋊#95	C₂²	1	2	yes	—
30	C₄₇⋊#95	C₂²	1	2	yes	—
31	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
34	#95×C₄₇	C₂²	1	2	yes	—
35	#95×C₄₇	C₂²	1	2	yes	—
36	C₂×Q₈×C₄₇	—	2	2	no	—
37	C₂×Q₈×C₄₇	—	2	2	no	—
38	C₂×Q₈×C₄₇	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂²×Dic₄₇	C₂³	1	2	yes	—
45	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
46	C₂²×Dic₄₇	C₂³	1	2	yes	—
47	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
48	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
49	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₄²×C₄₇	—	2	2	no	—
52	C₄⋊Dic₄₇	—	2	2	no	—
53	C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
54	C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
55	Dic₄₇⋊C₄	—	2	4	no	—
56	Dic₄₇⋊C₄	—	2	4	no	—
57	C₄×Dic₄₇	—	2	2	no	—
58	C₄×Dic₄₇	—	2	2	no	—
59	C₄×Dic₄₇	—	2	2	no	—
60	C₄×Dic₄₇	—	2	2	no	—
61	C₄×Dic₄₇	—	2	4	no	—
62	C₄×Dic₄₇	C₂×C₄	1	8	yes	—
63	C₄×Dic₄₇	C₂×C₄	1	4	yes	—
64	C₄×Dic₄₇	Q₈	1	4	yes	complemented
65	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
66	Q₈×C₄₇	—	2	8	no	—
67	C₂×C₄×C₄₇	—	2	1	no	—
68	C₂×C₄×C₄₇	—	2	1	no	—
69	C₂×C₄×C₄₇	—	2	1	no	—
70	C₂×C₄×C₄₇	—	2	1	no	—
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	—	2	2	no	—
75	C₂×C₄×C₄₇	—	2	2	no	—
76	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
79	C₂×C₄×C₄₇	#41	1	2	yes	—
80	C₂×C₄×C₄₇	#41	1	2	yes	—
81	C₂×C₄×C₄₇	#41	1	2	yes	—
82	C₂×Dic₄₇	—	2	1	no	—
83	C₂×Dic₄₇	—	2	1	no	—
84	C₂×Dic₄₇	—	2	1	no	—
85	C₂×Dic₄₇	—	2	1	no	—
86	C₂×Dic₄₇	—	2	1	no	—
87	C₂×Dic₄₇	—	2	1	no	—
88	C₂×Dic₄₇	—	2	2	no	—
89	C₂×Dic₄₇	—	2	2	no	—
90	C₂×Dic₄₇	—	2	2	no	—
91	C₂×Dic₄₇	—	2	2	no	—
92	C₂×Dic₄₇	—	2	2	no	—
93	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
94	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
95	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
96	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
97	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
98	C₂×Dic₄₇	#41	1	2	yes	—
99	C₂×Dic₄₇	#41	1	2	yes	—
100	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
101	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
103	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
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	C₄×C₄₇	—	2	4	no	—
108	C₄×C₄₇	—	2	2	no	—
109	C₄×C₄₇	—	2	2	no	—
110	C₄×C₄₇	—	4	2	no	—
111	C₄×C₄₇	#117	1	4	yes	—
112	Dic₄₇	—	2	2	no	—
113	Dic₄₇	—	2	4	no	—
114	Dic₄₇	—	2	2	no	—
115	Dic₄₇	—	2	2	no	—
116	Dic₄₇	—	4	2	no	—
117	Dic₄₇	C₄×Q₈	1	4	yes	—
118	#2295	—	47	1	no	maximal, Sylow
119	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
120	C₂×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₂×C₄₇	C₄×#41	1	1	yes	characteristic
122	C₂×C₄₇	#529	1	1	yes	characteristic
123	C₂×C₄₇	#529	1	1	yes	characteristic
124	C₂×C₄₇	#533	1	1	yes	characteristic
125	C₂×C₄₇	#533	1	1	yes	characteristic
126	C₂×C₄×Q₈	—	47	1	no	—
127	#391	—	47	1	no	—
128	#391	D₄₇	1	1	yes	characteristic
129	#389	—	47	1	no	—
130	#389	—	47	1	no	—
131	#387	—	47	2	no	—
132	#387	—	47	2	no	—
133	C₄×C₄⋊C₄	—	47	1	no	—
134	C₄×C₄⋊C₄	—	47	2	no	—
135	#376	—	47	2	no	—
136	C₄³	—	47	1	no	—
137	C₄₇	#2295	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
139	C₄×Q₈	—	47	8	no	—
140	C₂×C₄⋊C₄	—	47	2	no	—
141	C₂×C₄⋊C₄	—	47	1	no	—
142	C₂×C₄⋊C₄	—	47	2	no	—
143	C₂×C₄⋊C₄	—	47	2	no	—
144	C₂×C₄⋊C₄	—	47	1	no	—
145	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
146	C₂×C₄²	—	47	1	no	—
147	C₂×C₄²	—	47	2	no	—
148	C₂×C₄²	—	47	1	no	—
149	C₂×C₄²	—	47	1	no	—
150	C₂×C₄²	—	47	1	no	—
151	C₂×C₄²	—	47	1	no	—
152	C₂×C₄²	—	47	1	no	—
153	C₂×C₄²	—	47	2	no	—
154	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
155	#95	—	47	2	no	—
156	#95	—	47	2	no	—
157	#95	—	47	2	no	—
158	#95	—	47	2	no	—
159	#95	—	47	2	no	—
160	#95	C₂×D₄₇	1	2	yes	—
161	#95	C₂×D₄₇	1	2	yes	—
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	C₄×D₄₇	1	4	yes	—
165	C₂²×C₄	—	47	1	no	—
166	C₂²×C₄	—	47	1	no	—
167	C₂²×C₄	—	47	1	no	—
168	C₂²×C₄	—	47	1	no	—
169	C₂²×C₄	—	47	2	no	—
170	C₂²×C₄	—	47	2	no	—
171	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
172	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
173	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
175	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	47	8	no	—
178	C₄⋊C₄	—	47	4	no	—
179	C₄⋊C₄	—	94	4	no	—
180	C₄⋊C₄	—	94	2	no	—
181	C₄⋊C₄	—	94	4	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	47	8	no	—
184	C₄²	—	47	4	no	—
185	C₄²	—	47	4	no	—
186	C₄²	—	94	2	no	—
187	C₄²	—	94	2	no	—
188	C₄²	—	94	4	no	—
189	C₄²	—	94	2	no	—
190	C₄²	—	94	2	no	—
191	C₂³	C₂³×D₄₇	1	1	yes	characteristic
192	Q₈	—	2	8	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	47	2	no	—
203	C₂×C₄	—	47	2	no	—
204	C₂×C₄	—	47	2	no	—
205	C₂×C₄	—	47	2	no	—
206	C₂×C₄	—	47	4	no	—
207	C₂×C₄	—	47	2	no	—
208	C₂×C₄	—	47	2	no	—
209	C₂×C₄	—	94	1	no	—
210	C₂×C₄	—	94	1	no	—
211	C₂×C₄	—	94	2	no	—
212	C₂×C₄	—	94	1	no	—
213	C₂×C₄	—	94	1	no	—
214	C₂×C₄	—	94	1	no	—
215	C₂×C₄	—	94	1	no	—
216	C₂×C₄	—	94	2	no	—
217	C₂×C₄	—	94	2	no	—
218	C₂×C₄	—	94	2	no	—
219	C₂×C₄	—	94	2	no	—
220	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
221	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
222	C₂×C₄	#10625352	1	2	yes	—
223	C₂×C₄	#10625352	1	2	yes	—
224	C₂×C₄	Q₈×D₄₇	1	2	yes	—
225	C₂×C₄	#10625354	1	2	yes	—
226	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
227	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
228	C₂²	#41×D₄₇	1	1	yes	characteristic
229	C₂²	#41×D₄₇	1	1	yes	characteristic
230	C₂²	C₂×#10625352	1	1	yes	characteristic
231	C₂²	C₂×#10625352	1	1	yes	characteristic
232	C₂²	C₂×#10625354	1	1	yes	characteristic
233	C₄	—	2	4	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	4	2	no	—
237	C₄	—	47	4	no	—
238	C₄	—	94	2	no	—
239	C₄	—	94	4	no	—
240	C₄	—	94	2	no	—
241	C₄	—	94	2	no	—
242	C₄	—	188	2	no	—
243	C₄	#13093980	1	4	yes	—
244	C₂	C₂×#13093980	1	1	yes	characteristic
245	C₂	C₄×Q₈×D₄₇	1	1	yes	characteristic
246	C₂	C₄×#10625354	1	1	yes	characteristic
247	C₂	#15689652	1	1	yes	characteristic
248	C₂	#15689659	1	1	yes	characteristic
249	C₂	#15689675	1	1	yes	characteristic
250	C₂	#15689678	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#391	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	#391	D₄₇
hypocenter	C₄₇	#2295
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
12. Q₈×Dic₄₇	C₄	semidirect
64. C₄×Dic₄₇	Q₈	semidirect
137. C₄₇	#2295	semidirect