Subgroups — #26131492

Order 12032.

526 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688924	C₂	1	2	yes	maximal
2	C₂×#15688899	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13093958	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241479	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#13093958	C₂²	1	2	yes	—
8	C₄×C₄⋊Dic₄₇	—	2	1	no	—
9	#15688924	C₄	1	8	yes	complemented
10	#15688899	—	2	2	no	—
11	#15688886	—	2	1	no	—
12	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₄₇	—	2	1	no	—
14	C₄×D₈×C₄₇	—	2	2	no	—
15	C₄×D₈×C₄₇	C₄	1	4	yes	complemented
16	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#102×C₄₇	—	2	1	no	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	C₂×C₄×Dic₄₇	—	4	1	no	—
21	C₂×C₄₇⋊C₁₆	—	2	2	no	—
22	C₂×C₄₇⋊C₁₆	—	2	2	no	—
23	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
24	#13093958	—	2	2	no	—
25	#13093958	C₂×C₄	1	4	yes	complemented
26	C₄₇⋊#95	—	4	1	no	—
27	C₂²×D₄×C₄₇	—	2	1	no	—
28	C₂³×C₄×C₄₇	—	4	1	no	—
29	C₂×D₈×C₄₇	—	2	1	no	—
30	C₂×D₈×C₄₇	—	2	2	no	—
31	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
32	C₂×D₈×C₄₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₄×D₄×C₄₇	—	2	4	no	—
36	C₄×D₄×C₄₇	—	4	2	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	#107×C₄₇	—	2	1	no	—
41	#107×C₄₇	C₂×C₄	1	2	yes	—
42	#102×C₄₇	—	2	4	no	—
43	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₄₇	D₄	1	2	yes	—
45	C₂⁴×C₄₇	—	4	1	no	—
46	C₂×D₄×C₄₇	—	2	2	no	—
47	C₂×D₄×C₄₇	—	2	4	no	—
48	C₂×D₄×C₄₇	—	4	1	no	—
49	C₂×D₄×C₄₇	—	4	2	no	—
50	C₂²×C₄×C₄₇	—	4	1	no	—
51	C₂²×C₄×C₄₇	—	4	4	no	—
52	C₂²×C₄×C₄₇	—	8	4	no	—
53	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₄₇	—	4	2	no	—
57	D₈×C₄₇	—	2	2	no	—
58	D₈×C₄₇	—	2	4	no	—
59	D₈×C₄₇	C₄²	1	4	yes	complemented
60	C₂×C₈×C₄₇	—	2	2	no	—
61	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₄₇	#41	1	2	yes	—
66	C₄⋊C₄×C₄₇	—	2	2	no	—
67	C₄⋊C₄×C₄₇	—	4	1	no	—
68	C₂²⋊C₄×C₄₇	—	4	4	no	—
69	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₄₇	SD₁₆	1	2	yes	—
71	C₄⋊Dic₄₇	—	2	4	no	—
72	C₄⋊Dic₄₇	—	4	2	no	—
73	C₄×Dic₄₇	—	8	2	no	—
74	C₄₇⋊C₁₆	—	2	4	no	—
75	C₂³×C₄₇	—	4	2	no	—
76	C₂³×C₄₇	—	4	4	no	—
77	C₂³×C₄₇	—	8	2	no	—
78	C₂³×C₄₇	—	8	1	no	—
79	C₂³×C₄₇	—	8	1	no	—
80	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
81	D₄×C₄₇	—	2	4	no	—
82	D₄×C₄₇	—	4	2	no	—
83	D₄×C₄₇	—	4	4	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	4	2	no	—
88	C₂×C₄×C₄₇	—	8	8	no	—
89	C₂×C₄×C₄₇	—	8	2	no	—
90	C₂×C₄×C₄₇	#102	1	2	yes	—
91	C₂×C₄×C₄₇	#102	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₄₇	#135	1	2	yes	—
96	C₂×Dic₄₇	—	4	4	no	—
97	C₂×Dic₄₇	—	8	2	no	—
98	C₂×Dic₄₇	—	8	2	no	—
99	C₈×C₄₇	—	2	2	no	—
100	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
101	C₈×C₄₇	#117	1	1	yes	characteristic
102	C₈×C₄₇	C₄×D₄	1	2	yes	—
103	#12637	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	—	4	4	no	—
105	C₂²×C₄₇	—	8	2	no	—
106	C₂²×C₄₇	—	8	2	no	—
107	C₂²×C₄₇	—	8	4	no	—
108	C₂²×C₄₇	—	8	4	no	—
109	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
110	C₂²×C₄₇	C₄×D₈	1	2	yes	—
111	C₂²×C₄₇	#509	1	1	yes	characteristic
112	C₂²×C₄₇	#509	1	2	yes	—
113	C₂²×C₄₇	#510	1	1	yes	characteristic
114	C₄×C₄₇	—	2	2	no	—
115	C₄×C₄₇	—	2	2	no	—
116	C₄×C₄₇	—	8	2	no	—
117	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
118	C₄×C₄₇	#416	1	1	yes	characteristic
119	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
120	Dic₄₇	—	8	4	no	—
121	C₂×C₄×D₈	D₄₇	1	1	yes	characteristic
122	C₂×#357	—	47	2	no	—
123	C₂×#346	—	47	1	no	—
124	C₄×#107	—	47	1	no	—
125	#1380	—	47	2	no	—
126	C₂×C₄₇	—	8	4	no	—
127	C₂×C₄₇	C₄×#102	1	1	yes	characteristic
128	C₂×C₄₇	#2139	1	1	yes	characteristic
129	C₂×C₄₇	#2140	1	1	yes	characteristic
130	C₂×C₄₇	#2175	1	2	yes	—
131	C₂×C₄₇	#2177	1	2	yes	—
132	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
133	C₂×C₄×D₄	—	2	1	no	—
134	C₂²×C₁₆	—	47	2	no	—
135	C₄×D₈	—	2	2	no	—
136	C₄×D₈	Dic₄₇	1	4	yes	complemented
137	C₂×#107	—	47	2	no	—
138	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
139	C₂×#102	—	2	1	no	—
140	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
141	C₄×C₄⋊C₄	—	94	1	no	—
142	#357	—	47	8	no	—
143	#346	—	94	2	no	—
144	#340	—	94	1	no	—
145	C₄₇	#12637	1	1	yes	characteristic, complemented, Sylow
146	C₂²×D₄	—	2	1	no	—
147	C₂³×C₄	—	4	1	no	—
148	C₂×D₈	—	2	2	no	—
149	C₂×D₈	—	2	1	no	—
150	C₂×D₈	C₄×D₄₇	1	4	yes	—
151	C₂×D₈	C₂×Dic₄₇	1	2	yes	—
152	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
153	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
154	C₄×D₄	—	2	4	no	—
155	C₄×D₄	—	4	2	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	94	2	no	—
158	C₂×C₂²⋊C₄	—	4	1	no	—
159	C₂×C₄²	—	188	1	no	—
160	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
161	C₂×C₁₆	—	47	4	no	—
162	C₂×C₁₆	—	94	2	no	—
163	C₂×C₁₆	—	94	2	no	—
164	#107	—	2	1	no	—
165	#107	—	47	4	no	—
166	#107	—	94	2	no	—
167	#107	C₂×Dic₄₇	1	2	yes	—
168	#102	—	2	4	no	—
169	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
170	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
171	#95	—	188	1	no	—
172	C₂⁴	—	4	1	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂²×C₄	—	4	1	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	—	8	4	no	—
180	C₂²×C₄	—	188	2	no	—
181	C₂²×C₄	#10625334	1	1	yes	characteristic
182	C₂²×C₄	#10625334	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
184	D₈	—	2	2	no	—
185	D₈	—	2	4	no	—
186	D₈	C₄×Dic₄₇	1	4	yes	complemented
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
189	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
190	C₂×C₈	#10625350	1	2	yes	—
191	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
192	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	94	4	no	—
196	C₄⋊C₄	—	188	2	no	—
197	C₂²⋊C₄	—	4	4	no	—
198	C₄²	—	376	2	no	—
199	C₄²	#10625335	1	2	yes	—
200	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
201	C₁₆	—	94	4	no	—
202	C₂³	—	4	2	no	—
203	C₂³	—	4	4	no	—
204	C₂³	—	8	1	no	—
205	C₂³	—	8	2	no	—
206	C₂³	—	8	1	no	—
207	C₂³	C₂×#10625334	1	1	yes	characteristic
208	D₄	—	2	4	no	—
209	D₄	—	4	2	no	—
210	D₄	—	4	4	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	8	8	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	376	2	no	—
218	C₂×C₄	—	376	2	no	—
219	C₂×C₄	—	188	4	no	—
220	C₂×C₄	#13093945	1	2	yes	—
221	C₂×C₄	#13093945	1	2	yes	—
222	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
223	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
224	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
225	C₂×C₄	#13094044	1	2	yes	—
226	C₈	—	2	2	no	—
227	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
228	C₈	#13093983	1	1	yes	characteristic
229	C₈	C₄×C₂²⋊D₄₇	1	2	yes	—
230	C₂²	—	4	4	no	—
231	C₂²	—	8	4	no	—
232	C₂²	—	8	4	no	—
233	C₂²	—	8	2	no	—
234	C₂²	—	8	2	no	—
235	C₂²	C₂×#13093945	1	1	yes	characteristic
236	C₂²	C₄×#10625334	1	2	yes	—
237	C₂²	#15689558	1	1	yes	characteristic
238	C₂²	#15689559	1	2	yes	—
239	C₂²	#15689563	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	376	4	no	—
244	C₄	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
245	C₄	#15689119	1	1	yes	characteristic
246	C₄	C₄×#10625335	1	2	yes	—
247	C₂	—	8	4	no	—
248	C₂	C₄×#13093945	1	1	yes	characteristic
249	C₂	#20245100	1	1	yes	characteristic
250	C₂	#20245104	1	1	yes	characteristic
251	C₂	#20245252	1	2	yes	—
252	C₂	#20245262	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄×D₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂²×D₈×C₄₇	C₂²
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₄×D₈	D₄₇
hypocenter	C₄₇	#12637
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₂×#15688899	C₂	semidirect
3. C₄×#13093958	C₂	semidirect
4. #20241479	C₂	semidirect
9. #15688924	C₄	semidirect
15. C₄×D₈×C₄₇	C₄	semidirect
23. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
25. #13093958	C₂×C₄	semidirect
59. D₈×C₄₇	C₄²	semidirect
136. C₄×D₈	Dic₄₇	semidirect
145. C₄₇	#12637	semidirect
186. D₈	C₄×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688924	2
C₂×#15688899	2
C₄×#13093958	2
#20241479	2
C₂×C₄×D₈×C₄₇	2
#12637	47