Subgroups — #26134910

Order 12032.

374 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	C₂×#15689528	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093964	C₂	1	1	yes	maximal, characteristic
3	#20241498	C₂	1	4	yes	maximal
4	C₄₇×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	—	2	1	no	—
6	C₂×#13093964	C₂²	1	2	yes	—
7	#15689528	—	2	1	no	—
8	#15689528	—	2	1	no	—
9	#15689528	C₄	1	2	yes	—
10	#15689528	C₄	1	2	yes	—
11	C₂×#13093958	—	2	1	no	—
12	C₂×#13093958	C₂²	1	2	yes	—
13	#15688899	—	2	1	no	—
14	C₄×C₄₇⋊C₁₆	—	2	1	no	—
15	C₂×#108×C₄₇	—	2	1	no	—
16	C₂×#108×C₄₇	C₂²	1	2	yes	—
17	C₄₇×#423	—	2	1	no	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×Dic₉₄	—	4	1	no	—
20	#13094024	—	2	2	no	—
21	#13094024	—	2	2	no	—
22	#13094024	—	4	1	no	—
23	#13094024	—	4	1	no	—
24	C₂×C₄⋊Dic₄₇	—	2	2	no	—
25	C₂×C₄⋊Dic₄₇	—	4	1	no	—
26	#13093964	—	2	2	no	—
27	#13093964	Q₈	1	4	yes	complemented
28	C₂×C₄₇⋊C₁₆	—	2	2	no	—
29	#13093958	—	2	2	no	—
30	#13093958	—	2	2	no	—
31	#13093958	—	2	2	no	—
32	#13093958	C₂×C₄	1	4	yes	—
33	C₂×M₁₆×C₄₇	—	2	2	no	—
34	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
37	#108×C₄₇	—	2	4	no	—
38	#108×C₄₇	—	4	2	no	—
39	C₄⋊C₈×C₄₇	—	4	1	no	—
40	C₄⋊C₈×C₄₇	—	4	1	no	—
41	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₄₇	—	4	2	no	—
49	C₂×Dic₉₄	—	4	2	no	—
50	C₂×Dic₉₄	—	4	2	no	—
51	C₂×Dic₉₄	—	8	4	no	—
52	M₁₆×C₄₇	—	4	2	no	—
53	M₁₆×C₄₇	—	4	2	no	—
54	C₂×C₈×C₄₇	—	4	2	no	—
55	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
58	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
59	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₄₇	#41	1	2	yes	—
63	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₄₇	D₈	1	1	yes	characteristic
66	C₄²×C₄₇	D₈	1	1	yes	characteristic
67	C₄⋊Dic₄₇	—	2	4	no	—
68	C₄⋊Dic₄₇	—	4	2	no	—
69	C₄⋊Dic₄₇	—	4	2	no	—
70	C₄⋊Dic₄₇	—	4	2	no	—
71	C₄₇⋊C₁₆	—	2	4	no	—
72	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	—	2	2	no	—
75	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
77	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
85	C₂×Dic₄₇	—	4	4	no	—
86	C₂×Dic₄₇	—	8	4	no	—
87	Dic₉₄	—	8	4	no	—
88	Dic₉₄	—	8	4	no	—
89	C₈×C₄₇	—	8	2	no	—
90	C₈×C₄₇	#103	1	2	yes	—
91	C₈×C₄₇	#103	1	2	yes	—
92	C₈×C₄₇	C₄×Q₈	1	2	yes	—
93	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
94	#13290	—	47	1	no	maximal, Sylow
95	C₂²×C₄₇	—	2	1	no	—
96	C₂²×C₄₇	—	2	1	no	—
97	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
98	C₂²×C₄₇	#487	1	1	yes	characteristic
99	C₂²×C₄₇	#498	1	1	yes	characteristic
100	C₄×C₄₇	—	2	1	no	—
101	C₄×C₄₇	—	2	1	no	—
102	C₄×C₄₇	#361	1	2	yes	—
103	C₄×C₄₇	#362	1	2	yes	—
104	C₄×C₄₇	#391	1	1	yes	characteristic
105	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
106	C₄×C₄₇	#486	1	1	yes	characteristic
107	C₄×C₄₇	#500	1	1	yes	characteristic
108	Dic₄₇	—	8	4	no	—
109	C₂×#500	—	47	1	no	—
110	C₄×M₃₂	—	47	1	no	—
111	#1848	D₄₇	1	1	yes	characteristic
112	#1387	—	47	4	no	—
113	C₂×C₄₇	—	2	1	no	—
114	C₂×C₄₇	—	2	1	no	—
115	C₂×C₄₇	#1990	1	1	yes	characteristic
116	C₂×C₄₇	C₂×#361	1	1	yes	characteristic
117	C₂×C₄₇	C₂×#362	1	1	yes	characteristic
118	C₂×C₄⋊Q₈	—	94	1	no	—
119	C₂×M₃₂	—	47	2	no	—
120	#500	—	47	2	no	—
121	#500	—	47	2	no	—
122	#500	—	94	1	no	—
123	#500	—	94	1	no	—
124	C₂×#108	—	2	1	no	—
125	C₂×#108	C₂×D₄₇	1	2	yes	—
126	C₂×#107	—	47	2	no	—
127	C₂×#107	—	94	1	no	—
128	#423	—	2	1	no	—
129	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
130	#346	—	94	1	no	—
131	C₄×C₁₆	—	94	1	no	—
132	C₄₇	#13290	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	188	1	no	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
137	C₄⋊Q₈	—	94	2	no	—
138	C₄⋊Q₈	—	94	2	no	—
139	C₄⋊Q₈	—	188	1	no	—
140	C₄⋊Q₈	—	188	1	no	—
141	C₂×C₄⋊C₄	—	94	2	no	—
142	C₂×C₄⋊C₄	—	188	1	no	—
143	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
144	M₃₂	—	47	4	no	—
145	M₃₂	—	94	2	no	—
146	C₂×C₁₆	—	94	2	no	—
147	#108	—	2	4	no	—
148	#108	—	4	2	no	—
149	#107	—	47	4	no	—
150	#107	—	94	2	no	—
151	#107	—	94	2	no	—
152	#107	—	94	2	no	—
153	C₄⋊C₈	—	4	1	no	—
154	C₄⋊C₈	—	4	1	no	—
155	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
156	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
157	C₄×C₈	D₁₈₈	1	1	yes	characteristic
158	C₄×C₈	D₁₈₈	1	1	yes	characteristic
159	C₂×Q₈	—	376	4	no	—
160	C₂×Q₈	—	188	2	no	—
161	C₂×Q₈	—	188	2	no	—
162	C₂²×C₄	—	188	2	no	—
163	C₂²×C₄	#10625336	1	1	yes	characteristic
164	C₂²×C₄	#10625336	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
166	M₁₆	—	4	2	no	—
167	M₁₆	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
170	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
171	C₂×C₈	#10625335	1	2	yes	—
172	C₂×C₈	#10625337	1	2	yes	—
173	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
174	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
175	C₂×C₈	#10625352	1	2	yes	—
176	C₂×C₈	Q₈×D₄₇	1	2	yes	—
177	C₄⋊C₄	—	94	4	no	—
178	C₄⋊C₄	—	188	2	no	—
179	C₄⋊C₄	—	188	2	no	—
180	C₄⋊C₄	—	188	2	no	—
181	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
182	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
183	C₄²	#10625334	1	1	yes	characteristic
184	C₄²	#10625334	1	1	yes	characteristic
185	C₁₆	—	94	4	no	—
186	C₂³	C₂×#10625336	1	1	yes	characteristic
187	Q₈	—	376	4	no	—
188	Q₈	—	376	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	376	4	no	—
192	C₂×C₄	—	188	4	no	—
193	C₂×C₄	#13093944	1	1	yes	characteristic
194	C₂×C₄	#13093944	1	1	yes	characteristic
195	C₂×C₄	#13093944	1	1	yes	characteristic
196	C₂×C₄	#13093944	1	1	yes	characteristic
197	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
198	C₂×C₄	#13094011	1	1	yes	characteristic
199	C₂×C₄	#13094022	1	1	yes	characteristic
200	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
201	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
202	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
203	C₈	—	8	2	no	—
204	C₈	#13093947	1	2	yes	—
205	C₈	#13093947	1	2	yes	—
206	C₈	#13093989	1	2	yes	—
207	C₈	#13094004	1	2	yes	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	C₂×#13093944	1	1	yes	characteristic
211	C₂²	#15689473	1	1	yes	characteristic
212	C₂²	#15689519	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	376	4	no	—
216	C₄	#15688935	1	2	yes	—
217	C₄	#15688938	1	2	yes	—
218	C₄	#15689028	1	1	yes	characteristic
219	C₄	C₂×#13093947	1	1	yes	characteristic
220	C₄	#15689469	1	1	yes	characteristic
221	C₄	#15689526	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	#20244348	1	1	yes	characteristic
225	C₂	C₂×#15688935	1	1	yes	characteristic
226	C₂	C₂×#15688938	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093944
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1848	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂³×C₄₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1848	D₄₇
hypocenter	C₄₇	#13290
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
27. #13093964	Q₈	semidirect
132. C₄₇	#13290	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689528	2
C₄×#13093964	2
#20241498	2
C₄₇×#1848	2
#13290	47