Subgroups — #26369315

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093963	C₂	1	1	yes	maximal, characteristic
2	#427×D₄₇	C₂	1	2	yes	maximal
3	#20245175	C₂	1	4	yes	maximal, complemented
4	#20245173	C₂	1	4	yes	maximal, complemented
5	#20245172	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245170	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
9	#117×D₄₇	—	2	2	no	—
10	C₂×#13093963	C₄	1	8	yes	complemented
11	C₂×#13093963	C₂²	1	2	yes	—
12	C₂×#13093963	C₂²	1	2	yes	—
13	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	#15689161	C₂²	1	2	yes	—
15	#15689160	C₂²	1	4	yes	—
16	#15689159	—	2	2	no	—
17	#15689158	—	2	2	no	—
18	#107×D₄₇	C₂²	1	4	yes	—
19	#106×D₄₇	—	2	2	no	—
20	#15688951	C₂²	1	2	yes	complemented
21	#15688950	C₂²	1	4	yes	complemented
22	#15688949	C₂²	1	2	yes	complemented
23	#15688948	C₂²	1	4	yes	complemented
24	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#427	C₂²	1	2	yes	—
26	C₄₇×#367	C₂²	1	2	yes	complemented
27	C₄₇×#366	C₂²	1	2	yes	complemented
28	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
30	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₈×D₄₇	D₄	1	2	yes	—
32	C₂×C₈×D₄₇	Q₈	1	4	yes	—
33	C₂×C₈×D₄₇	C₂³	1	2	yes	—
34	#13093983	—	2	2	no	—
35	#13093982	—	2	4	no	—
36	#13093981	—	4	2	no	—
37	#13093980	—	4	2	no	—
38	C₄⋊C₄×D₄₇	—	2	4	no	—
39	C₂²⋊C₄×D₄₇	—	4	2	no	—
40	C₄²×D₄₇	—	4	2	no	—
41	#13093963	C₂×C₄	1	8	yes	complemented
42	#13093963	C₂×C₄	1	8	yes	complemented
43	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
44	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
45	#13093958	C₂³	1	4	yes	—
46	#13093957	—	2	4	no	—
47	#13093956	—	2	2	no	—
48	#13093955	—	2	4	no	—
49	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	#117×C₄₇	—	2	2	no	—
51	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
52	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
54	#107×C₄₇	C₂³	1	4	yes	—
55	#106×C₄₇	—	2	2	no	—
56	C₂³×D₄₇	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
59	C₂×C₄×D₄₇	—	4	4	no	—
60	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
61	C₂×C₄×D₄₇	D₈	1	2	yes	—
62	C₂×C₄×D₄₇	Q₁₆	1	4	yes	—
63	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₄₇	—	2	4	no	—
69	C₂²⋊C₄×C₄₇	—	4	2	no	—
70	C₄²×C₄₇	—	4	2	no	—
71	C₂²⋊Dic₄₇	—	4	2	no	—
72	(C₂×D₄₇)⋊C₄	—	4	4	no	—
73	C₄⋊Dic₄₇	—	2	4	no	—
74	Dic₄₇⋊C₄	—	4	4	no	—
75	C₄×Dic₄₇	—	4	2	no	—
76	C₄×Dic₄₇	—	4	4	no	—
77	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
78	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
79	C₈×D₄₇	C₄⋊C₄	1	8	yes	—
80	C₈×D₄₇	C₂²×C₄	1	4	yes	—
81	C₈×D₄₇	C₂×Q₈	1	4	yes	—
82	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
83	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
84	C₄₇⋊C₁₆	C₂²×C₄	1	4	yes	—
85	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
86	C₂²×D₄₇	—	4	2	no	—
87	C₂²×D₄₇	#107	1	4	yes	—
88	C₂²×D₄₇	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₄₇	—	4	4	no	—
90	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×Dic₄₇	—	4	4	no	—
95	C₂×Dic₄₇	#107	1	4	yes	—
96	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
97	C₄×D₄₇	—	8	8	no	—
98	C₄×D₄₇	#107	1	8	yes	—
99	C₄×D₄₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₄×D₄₇	C₂×Q₁₆	1	4	yes	—
101	C₈×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
104	C₄₇⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
105	#33409	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	—	2	2	no	—
107	C₂²×C₄₇	C₂×#107	1	2	yes	—
108	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
109	C₂×D₄₇	—	2	4	no	—
110	C₂×D₄₇	—	4	4	no	—
111	C₂×D₄₇	C₂×#107	1	4	yes	—
112	C₄×C₄₇	—	8	4	no	—
113	C₄×C₄₇	C₂×#107	1	2	yes	—
114	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
116	Dic₄₇	—	8	4	no	—
117	Dic₄₇	C₂×#107	1	4	yes	—
118	C₂²×M₃₂	—	47	1	no	—
119	C₂×#427	—	47	2	no	—
120	#2159	—	47	4	no	—
121	#2159	—	47	4	no	—
122	#2158	—	47	1	no	—
123	#2158	—	47	2	no	—
124	#2158	D₄₇	1	1	yes	characteristic, complemented
125	C₂×C₄₇	—	2	2	no	—
126	C₂×C₄₇	C₂²×#107	1	1	yes	characteristic
127	C₂×C₄₇	#3414	1	2	yes	—
128	D₄₇	—	2	4	no	—
129	C₂³×C₈	—	47	1	no	—
130	C₂×#117	—	94	2	no	—
131	C₂×M₃₂	—	47	8	no	—
132	C₂×M₃₂	—	47	2	no	—
133	C₂×M₃₂	—	47	2	no	—
134	C₂²×C₁₆	—	47	1	no	—
135	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
136	#427	—	47	2	no	—
137	#427	—	47	4	no	—
138	#427	—	94	2	no	—
139	#427	—	94	2	no	—
140	#427	C₂×D₄₇	1	2	yes	—
141	C₂×#107	—	47	4	no	—
142	C₂×#106	—	94	2	no	—
143	#367	—	47	2	no	—
144	#367	—	47	4	no	—
145	#367	C₂×D₄₇	1	2	yes	complemented
146	#366	—	47	2	no	—
147	#366	—	47	4	no	—
148	#366	C₂×D₄₇	1	2	yes	complemented
149	C₄₇	#33409	1	1	yes	characteristic, complemented, Sylow
150	C₂³×C₄	—	47	1	no	—
151	C₂²×C₈	—	47	1	no	—
152	C₂²×C₈	—	47	4	no	—
153	C₂²×C₈	—	47	2	no	—
154	C₂²×C₈	—	47	4	no	—
155	C₂²×C₈	—	47	2	no	—
156	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
157	#117	—	2	2	no	—
158	#117	—	94	4	no	—
159	#117	—	94	2	no	—
160	#117	—	188	2	no	—
161	#117	—	188	2	no	—
162	C₂×C₄⋊C₄	—	94	4	no	—
163	C₂×C₂²⋊C₄	—	188	2	no	—
164	C₂×C₄²	—	188	2	no	—
165	M₃₂	—	47	8	no	—
166	M₃₂	—	47	8	no	—
167	C₂×C₁₆	—	47	4	no	—
168	C₂×C₁₆	—	47	2	no	—
169	C₂×C₁₆	C₄×D₄₇	1	4	yes	complemented
170	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
171	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
172	#107	—	47	4	no	—
173	#107	—	94	4	no	—
174	#107	C₂²×D₄₇	1	4	yes	—
175	#106	—	2	2	no	—
176	#106	—	94	4	no	—
177	#106	—	94	2	no	—
178	C₂⁴	—	47	1	no	—
179	C₂²×C₄	—	47	1	no	—
180	C₂²×C₄	—	47	4	no	—
181	C₂²×C₄	—	47	2	no	—
182	C₂²×C₄	—	47	4	no	—
183	C₂²×C₄	—	47	2	no	—
184	C₂²×C₄	—	188	4	no	—
185	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
186	C₂×C₈	—	47	4	no	—
187	C₂×C₈	—	47	2	no	—
188	C₂×C₈	—	47	8	no	—
189	C₂×C₈	—	47	4	no	—
190	C₂×C₈	—	47	4	no	—
191	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
192	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
193	C₂×C₈	Q₈×D₄₇	1	2	yes	—
194	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	94	4	no	—
197	C₄⋊C₄	—	188	4	no	—
198	C₂²⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	188	4	no	—
200	C₂²⋊C₄	—	188	2	no	—
201	C₄²	—	4	2	no	—
202	C₄²	—	188	4	no	—
203	C₄²	—	188	2	no	—
204	C₁₆	—	47	4	no	—
205	C₁₆	C₂×C₄×D₄₇	1	2	yes	—
206	C₁₆	C₂×C₄×D₄₇	1	2	yes	—
207	C₂³	—	47	2	no	—
208	C₂³	—	47	4	no	—
209	C₂³	—	188	2	no	—
210	C₂³	D₈×D₄₇	1	1	yes	characteristic
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	47	2	no	—
213	C₂×C₄	—	47	4	no	—
214	C₂×C₄	—	47	4	no	—
215	C₂×C₄	—	47	4	no	—
216	C₂×C₄	—	47	8	no	—
217	C₂×C₄	—	376	8	no	—
218	C₂×C₄	—	188	4	no	—
219	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
220	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
221	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
222	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
223	C₈	—	47	4	no	—
224	C₈	C₄⋊C₄×D₄₇	1	2	yes	—
225	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
226	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
227	C₂²	—	2	2	no	—
228	C₂²	—	47	4	no	—
229	C₂²	—	94	4	no	—
230	C₂²	—	188	4	no	—
231	C₂²	#107×D₄₇	1	2	yes	—
232	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
233	C₄	—	8	4	no	—
234	C₄	—	47	4	no	—
235	C₄	—	376	4	no	—
236	C₄	#107×D₄₇	1	2	yes	—
237	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
238	C₄	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
239	C₂	—	2	2	no	—
240	C₂	—	94	4	no	—
241	C₂	C₂×#107×D₄₇	1	1	yes	characteristic
242	C₂	#20256521	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2158	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂³×D₄₇	D₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2158	D₄₇
hypocenter	C₄₇	#33409
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #20245175	C₂	semidirect
4. #20245173	C₂	semidirect
5. #20245172	C₂	semidirect
6. #20245170	C₂	semidirect
7. C₄₇×#2158	C₂	semidirect
10. C₂×#13093963	C₄	semidirect
20. #15688951	C₂²	semidirect
21. #15688950	C₂²	semidirect
22. #15688949	C₂²	semidirect
23. #15688948	C₂²	semidirect
26. C₄₇×#367	C₂²	semidirect
27. C₄₇×#366	C₂²	semidirect
41. #13093963	C₂×C₄	semidirect
42. #13093963	C₂×C₄	semidirect
43. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
51. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
124. #2158	D₄₇	semidirect
145. #367	C₂×D₄₇	semidirect
148. #366	C₂×D₄₇	semidirect
149. C₄₇	#33409	semidirect
169. C₂×C₁₆	C₄×D₄₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13093963	2
#427×D₄₇	2
#20245175	2
#20245173	2
#20245172	2
#20245170	2
C₄₇×#2158	2
#33409	47