Subgroups — #26131157

Order 12032.

606 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
3	#20241479	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#427	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	—	2	1	no	—
6	C₂²×C₄₇⋊C₁₆	—	2	2	no	—
7	C₂²×C₄₇⋊C₁₆	—	2	2	no	—
8	C₂²×C₄₇⋊C₁₆	C₄	1	4	yes	complemented
9	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
10	#15689161	—	2	2	no	—
11	#15689161	C₄	1	4	yes	complemented
12	C₂×#13093958	C₂²	1	2	yes	—
13	C₂×#13093956	—	2	1	no	—
14	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₄₇	—	2	1	no	—
16	C₄₇×#427	—	2	2	no	—
17	C₄₇×#427	C₄	1	4	yes	complemented
18	C₂×#107×C₄₇	C₂²	1	2	yes	—
19	C₂×#106×C₄₇	—	2	1	no	—
20	#13093983	—	2	4	no	—
21	#13093983	—	4	2	no	—
22	C₂×C₄⋊Dic₄₇	—	2	2	no	—
23	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
24	C₂×C₄×Dic₄₇	—	4	1	no	—
25	C₂×C₄₇⋊C₁₆	—	2	8	no	—
26	C₂×C₄₇⋊C₁₆	—	2	2	no	—
27	C₂×C₄₇⋊C₁₆	—	2	2	no	—
28	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
29	#13093958	—	2	2	no	—
30	#13093958	C₂×C₄	1	4	yes	—
31	#13093956	—	2	4	no	—
32	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₄₇	D₄	1	2	yes	—
38	C₂²×C₈×C₄₇	D₄	1	2	yes	—
39	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
40	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
41	#117×C₄₇	—	2	4	no	—
42	#117×C₄₇	—	4	2	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
45	C₂×C₄²×C₄₇	—	4	1	no	—
46	#107×C₄₇	—	2	2	no	—
47	#107×C₄₇	C₂×C₄	1	4	yes	—
48	#106×C₄₇	—	2	4	no	—
49	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	4	2	no	—
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
56	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
57	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₄₇	—	4	2	no	—
60	C₂×C₈×C₄₇	C₄²	1	4	yes	—
61	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
64	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
65	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
67	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
73	C₄⋊C₄×C₄₇	—	2	4	no	—
74	C₄⋊C₄×C₄₇	—	4	2	no	—
75	C₂²⋊C₄×C₄₇	—	4	4	no	—
76	C₄²×C₄₇	—	4	4	no	—
77	C₂²⋊Dic₄₇	—	4	4	no	—
78	C₄⋊Dic₄₇	—	2	4	no	—
79	C₄⋊Dic₄₇	—	4	2	no	—
80	C₄×Dic₄₇	—	4	4	no	—
81	C₄₇⋊C₁₆	—	2	4	no	—
82	C₂³×C₄₇	—	2	4	no	—
83	C₂³×C₄₇	#102	1	2	yes	—
84	C₂³×C₄₇	#102	1	2	yes	—
85	C₂³×C₄₇	#107	1	2	yes	—
86	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₄₇	—	4	4	no	—
88	C₂×C₄×C₄₇	—	8	4	no	—
89	C₂×C₄×C₄₇	#95	1	4	yes	—
90	C₂×C₄×C₄₇	#102	1	2	yes	—
91	C₂×C₄×C₄₇	#102	1	2	yes	—
92	C₂×C₄×C₄₇	#103	1	4	yes	—
93	C₂×C₄×C₄₇	#103	1	4	yes	—
94	C₂×C₄×C₄₇	#106	1	4	yes	—
95	C₂×C₄×C₄₇	#107	1	2	yes	—
96	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×Dic₄₇	—	4	4	no	—
103	C₂×Dic₄₇	—	8	4	no	—
104	C₈×C₄₇	#95	1	4	yes	—
105	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
106	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	#12556	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	—	2	4	no	—
111	C₂²×C₄₇	—	2	4	no	—
112	C₂²×C₄₇	—	2	4	no	—
113	C₂²×C₄₇	#340	1	4	yes	—
114	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
115	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
116	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
117	C₂²×C₄₇	#508	1	4	yes	—
118	C₄×C₄₇	—	8	4	no	—
119	C₄×C₄₇	#340	1	4	yes	—
120	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
121	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
122	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
123	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
124	Dic₄₇	—	8	4	no	—
125	C₂³×C₁₆	—	47	1	no	—
126	C₂×#427	—	47	1	no	—
127	C₂×#427	D₄₇	1	1	yes	characteristic
128	#1380	—	47	4	no	—
129	C₂×C₄₇	—	2	4	no	—
130	C₂×C₄₇	C₂×#340	1	1	yes	characteristic
131	C₂×C₄₇	#2138	1	2	yes	—
132	C₂×C₄₇	#2138	1	2	yes	—
133	C₂×C₄₇	#2158	1	2	yes	—
134	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
135	C₂×#117	—	2	1	no	—
136	C₂×#117	—	94	1	no	—
137	C₂²×C₁₆	—	47	4	no	—
138	C₂²×C₁₆	—	47	2	no	—
139	C₂²×C₁₆	—	94	2	no	—
140	C₂²×C₁₆	—	94	2	no	—
141	#427	—	2	2	no	—
142	#427	—	47	4	no	—
143	#427	—	94	2	no	—
144	#427	Dic₄₇	1	4	yes	complemented
145	C₂×#107	—	47	2	no	—
146	C₂×#107	C₂×D₄₇	1	2	yes	—
147	C₂×#106	—	2	1	no	—
148	C₂×#106	—	94	1	no	—
149	C₄₇	#12556	1	1	yes	characteristic, complemented, Sylow
150	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
151	C₂²×C₈	Dic₉₄	1	2	yes	—
152	C₂²×C₈	C₄×D₄₇	1	2	yes	—
153	C₂²×C₈	C₄×D₄₇	1	2	yes	—
154	C₂²×C₈	D₁₈₈	1	2	yes	—
155	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
156	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
158	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
159	#117	—	2	4	no	—
160	#117	—	4	2	no	—
161	#117	—	94	4	no	—
162	#117	—	188	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	94	2	no	—
165	C₂×C₂²⋊C₄	—	4	1	no	—
166	C₂×C₂²⋊C₄	—	188	1	no	—
167	C₂×C₄²	—	4	1	no	—
168	C₂×C₄²	—	188	1	no	—
169	C₂×C₁₆	—	47	4	no	—
170	C₂×C₁₆	—	94	8	no	—
171	C₂×C₁₆	—	94	2	no	—
172	C₂×C₁₆	—	94	2	no	—
173	#107	—	2	2	no	—
174	#107	—	47	4	no	—
175	#107	—	94	2	no	—
176	#107	C₂×Dic₄₇	1	4	yes	—
177	#106	—	2	4	no	—
178	#106	—	94	4	no	—
179	C₂⁴	#10625334	1	1	yes	characteristic
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	—	188	2	no	—
182	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
183	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
184	C₂²×C₄	#10625334	1	1	yes	characteristic
185	C₂²×C₄	#10625335	1	2	yes	—
186	C₂²×C₄	#10625336	1	2	yes	—
187	C₂²×C₄	#10625337	1	2	yes	—
188	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
189	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂×C₈	C₄×Dic₄₇	1	4	yes	—
191	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
192	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
193	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
194	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
195	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
196	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
197	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
198	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
200	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
201	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	94	4	no	—
206	C₄⋊C₄	—	188	2	no	—
207	C₂²⋊C₄	—	4	4	no	—
208	C₂²⋊C₄	—	188	4	no	—
209	C₄²	—	4	4	no	—
210	C₄²	—	188	4	no	—
211	C₁₆	—	94	4	no	—
212	C₂³	—	2	4	no	—
213	C₂³	#13093944	1	2	yes	—
214	C₂³	#13093945	1	2	yes	—
215	C₂³	#13093957	1	2	yes	—
216	C₂³	C₂×#10625334	1	1	yes	characteristic
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	—	376	4	no	—
220	C₂×C₄	—	188	4	no	—
221	C₂×C₄	C₄₇⋊#95	1	4	yes	—
222	C₂×C₄	#13093944	1	2	yes	—
223	C₂×C₄	#13093945	1	2	yes	—
224	C₂×C₄	#13093947	1	4	yes	—
225	C₂×C₄	#13093948	1	4	yes	—
226	C₂×C₄	#13093955	1	4	yes	—
227	C₂×C₄	#13093957	1	2	yes	—
228	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
229	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
230	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
231	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
232	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
233	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
234	C₈	C₄₇⋊#95	1	4	yes	—
235	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
236	C₈	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
237	C₈	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
238	C₈	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
239	C₂²	—	2	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	#15688885	1	4	yes	—
243	C₂²	C₂×#13093944	1	1	yes	characteristic
244	C₂²	C₂×#13093945	1	1	yes	characteristic
245	C₂²	C₂×#13093957	1	1	yes	characteristic
246	C₂²	#15689553	1	4	yes	—
247	C₄	—	8	4	no	—
248	C₄	—	376	4	no	—
249	C₄	#15688885	1	4	yes	—
250	C₄	C₂×C₄₇⋊#95	1	1	yes	characteristic
251	C₄	C₂×#13093947	1	1	yes	characteristic
252	C₄	C₂×#13093948	1	1	yes	characteristic
253	C₄	C₂×#13093955	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	C₂×#15688885	1	1	yes	characteristic
256	C₂	#20245093	1	2	yes	—
257	C₂	#20245095	1	2	yes	—
258	C₂	#20245170	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#10625334
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#427	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂⁴×C₄₇	D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#427	D₄₇
hypocenter	C₄₇	#12556
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #20241479	C₂	semidirect
8. C₂²×C₄₇⋊C₁₆	C₄	semidirect
11. #15689161	C₄	semidirect
17. C₄₇×#427	C₄	semidirect
144. #427	Dic₄₇	semidirect
149. C₄₇	#12556	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄₇⋊C₁₆	2
C₂×#15689161	2
#20241479	2
C₂×C₄₇×#427	2
#12556	47