Subgroups — #26226655

Order 12032.

818 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245786	C₂	1	2	yes	maximal
2	#20244158	C₂	1	8	yes	maximal, complemented
3	#20243417	C₂	1	2	yes	maximal
4	#20243413	C₂	1	2	yes	maximal
5	C₂²×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093958	C₂²	1	4	yes	—
7	C₂×#13093956	C₂²	1	4	yes	—
8	#15688991	—	2	2	no	—
9	#15688989	C₂²	1	4	yes	—
10	#15688986	—	2	2	no	—
11	#15688983	C₂²	1	4	yes	—
12	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
13	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
14	#15688969	—	2	2	no	—
15	#15688886	C₂²	1	4	yes	—
16	#15688886	C₂²	1	4	yes	—
17	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₄₇	C₄	1	8	yes	complemented
21	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
22	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
23	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
24	C₂×C₄×Dic₄₇	—	2	4	no	—
25	#13093958	—	4	4	no	—
26	#13093956	—	4	4	no	—
27	C₄₇⋊#95	—	2	4	no	—
28	C₄₇⋊#95	—	2	4	no	—
29	C₄₇⋊#95	—	2	4	no	—
30	C₄₇⋊#95	—	2	4	no	—
31	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	—	4	2	no	—
35	C₂²×C₈×C₄₇	—	4	2	no	—
36	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
38	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₄₇	D₄	1	2	yes	—
42	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
44	C₄⋊C₈×C₄₇	C₂×C₄	1	16	yes	—
45	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	—	2	4	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	—	4	2	no	—
50	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₄₇	—	2	8	no	—
59	C₂×C₈×C₄₇	—	2	4	no	—
60	C₂×C₈×C₄₇	—	2	4	no	—
61	C₂×C₈×C₄₇	—	4	4	no	—
62	C₂×C₈×C₄₇	—	4	4	no	—
63	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
65	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
66	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
67	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
68	C₂²⋊Dic₄₇	—	4	8	no	—
69	C₄⋊Dic₄₇	—	2	8	no	—
70	C₄⋊Dic₄₇	—	2	8	no	—
71	C₄×Dic₄₇	—	4	8	no	—
72	C₂³×C₄₇	—	2	2	no	—
73	C₂³×C₄₇	—	2	2	no	—
74	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂³×C₄₇	#135	1	4	yes	—
76	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	—	2	8	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	4	4	no	—
85	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
87	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
91	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
92	C₂×Dic₄₇	—	4	8	no	—
93	C₂×Dic₄₇	—	4	8	no	—
94	C₂×Dic₄₇	—	4	8	no	—
95	C₈×C₄₇	—	2	4	no	—
96	C₈×C₄₇	—	2	4	no	—
97	#21960	—	47	1	no	maximal, Sylow
98	C₂²×C₄₇	—	2	2	no	—
99	C₂²×C₄₇	—	2	2	no	—
100	C₂²×C₄₇	—	2	8	no	—
101	C₂²×C₄₇	#427	1	4	yes	—
102	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₄₇	#537	1	2	yes	—
104	C₂²×C₄₇	C₂×#135	1	2	yes	—
105	C₂²×C₄₇	#575	1	2	yes	—
106	C₄×C₄₇	—	2	2	no	—
107	C₄×C₄₇	—	2	4	no	—
108	C₄×C₄₇	—	2	4	no	—
109	C₄×C₄₇	#518	1	1	yes	characteristic
110	C₄×C₄₇	#519	1	1	yes	characteristic
111	C₄×C₄₇	#544	1	2	yes	—
112	Dic₄₇	—	8	8	no	—
113	C₂²×C₄⋊C₈	D₄₇	1	1	yes	characteristic
114	#2283	—	47	2	no	—
115	#1949	—	47	8	no	—
116	#1808	—	47	2	no	—
117	#1807	—	47	2	no	—
118	C₂×C₄₇	—	2	4	no	—
119	C₂×C₄₇	#2316	1	1	yes	characteristic
120	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
121	C₂×C₄₇	#2911	1	1	yes	characteristic
122	C₂×C₄₇	#3250	1	4	yes	—
123	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
124	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
125	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
126	C₂×#107	—	47	4	no	—
127	C₂×#106	—	47	4	no	—
128	C₂×C₄⋊C₈	Dic₄₇	1	8	yes	complemented
129	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
130	#383	—	94	2	no	—
131	#382	—	47	4	no	—
132	#381	—	94	2	no	—
133	#380	—	47	4	no	—
134	C₄×C₄⋊C₄	—	47	4	no	—
135	C₄×C₂²⋊C₄	—	94	2	no	—
136	#376	—	94	2	no	—
137	#340	—	47	4	no	—
138	#340	—	47	4	no	—
139	C₄₇	#21960	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
141	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
142	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
143	C₂²×C₈	—	4	2	no	—
144	C₂²×C₈	—	4	2	no	—
145	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
146	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
147	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
148	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
149	C₂×C₄⋊C₄	—	47	8	no	—
150	C₂×C₂²⋊C₄	—	94	4	no	—
151	C₂×C₄²	—	94	4	no	—
152	C₂×C₄²	Dic₉₄	1	4	yes	—
153	C₂×C₄²	D₁₈₈	1	2	yes	—
154	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
155	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
156	#107	—	188	4	no	—
157	#106	—	188	4	no	—
158	C₄⋊C₈	C₂×Dic₄₇	1	16	yes	—
159	#95	—	94	4	no	—
160	#95	—	94	4	no	—
161	#95	—	94	4	no	—
162	#95	—	94	4	no	—
163	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	94	8	no	—
169	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
170	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
171	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
172	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
174	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
175	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
176	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	4	4	no	—
180	C₂×C₈	—	4	4	no	—
181	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
182	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
183	C₄⋊C₄	—	94	8	no	—
184	C₄⋊C₄	—	94	8	no	—
185	C₂²⋊C₄	—	188	8	no	—
186	C₄²	—	188	8	no	—
187	C₄²	C₄⋊Dic₄₇	1	8	yes	—
188	C₄²	C₂×Dic₉₄	1	4	yes	—
189	C₄²	C₂²×Dic₄₇	1	4	yes	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
193	C₂³	#13094046	1	4	yes	—
194	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	188	8	no	—
204	C₂×C₄	—	188	8	no	—
205	C₂×C₄	—	188	8	no	—
206	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
209	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
211	C₂×C₄	#13094069	1	2	yes	—
212	C₂×C₄	#13094072	1	2	yes	—
213	C₈	—	2	4	no	—
214	C₈	—	2	4	no	—
215	C₂²	—	2	8	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	#15689161	1	4	yes	—
219	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
220	C₂²	#15689704	1	2	yes	—
221	C₂²	C₂×#13094046	1	2	yes	—
222	C₂²	#15689923	1	2	yes	—
223	C₄	—	2	4	no	—
224	C₄	—	2	4	no	—
225	C₄	—	2	2	no	—
226	C₄	—	376	8	no	—
227	C₄	#15689598	1	1	yes	characteristic
228	C₄	#15689601	1	1	yes	characteristic
229	C₄	#15689754	1	2	yes	—
230	C₂	—	2	4	no	—
231	C₂	#20246006	1	1	yes	characteristic
232	C₂	C₂×#15689161	1	1	yes	characteristic
233	C₂	#20251978	1	1	yes	characteristic
234	C₂	#20254740	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#21960
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #20244158	C₂	semidirect
20. C₂×C₄⋊C₈×C₄₇	C₄	semidirect
128. C₂×C₄⋊C₈	Dic₄₇	semidirect
139. C₄₇	#21960	semidirect

Maximal subgroups

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

subgroup	index
#20245786	2
#20244158	2
#20243417	2
#20243413	2
C₂²×C₄⋊C₈×C₄₇	2
#21960	47