Subgroups — #26115081

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243540	C₂	1	1	yes	maximal, characteristic, complemented
2	#20243134	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₄₇⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₄×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#15688878	—	2	2	no	—
9	D₄⋊(C₄₇⋊C₈)	—	2	8	no	—
10	C₂×#127×C₄₇	C₄	1	1	yes	characteristic
11	C₂×C₄×D₄×C₄₇	—	2	2	no	—
12	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#390	—	2	2	no	—
14	C₄₇×#390	C₄	1	2	yes	—
15	C₄₇×#384	C₄	1	1	yes	characteristic
16	C₄³×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
21	C₄×C₄₇⋊C₈	—	2	2	no	—
22	C₄×C₄₇⋊C₈	D₄	1	4	yes	complemented
23	C₂²×D₄×C₄₇	—	2	2	no	—
24	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₄₇	—	2	2	no	—
26	#127×C₄₇	—	2	4	no	—
27	C₄×D₄×C₄₇	—	2	4	no	—
28	C₄×D₄×C₄₇	—	2	4	no	—
29	C₄×D₄×C₄₇	—	4	4	no	—
30	C₄×D₄×C₄₇	—	4	4	no	—
31	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
35	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	D₄	1	2	yes	—
38	C₂×C₄²×C₄₇	D₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	C₂⁴×C₄₇	—	2	2	no	—
41	C₂×D₄×C₄₇	—	2	8	no	—
42	C₂×D₄×C₄₇	—	2	4	no	—
43	C₂×D₄×C₄₇	—	2	2	no	—
44	C₂×D₄×C₄₇	—	2	4	no	—
45	C₂×D₄×C₄₇	—	4	4	no	—
46	C₂×D₄×C₄₇	—	4	4	no	—
47	C₂×D₄×C₄₇	M₁₆	1	4	yes	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	—	4	2	no	—
50	C₂²×C₄×C₄₇	—	4	2	no	—
51	C₂²×C₄×C₄₇	—	4	4	no	—
52	C₂²×C₄×C₄₇	—	4	4	no	—
53	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₄₇	—	2	4	no	—
59	C₄⋊C₄×C₄₇	—	4	4	no	—
60	C₂²⋊C₄×C₄₇	—	4	8	no	—
61	C₂²⋊C₄×C₄₇	—	4	4	no	—
62	C₄²×C₄₇	—	2	2	no	—
63	C₄²×C₄₇	—	2	2	no	—
64	C₄²×C₄₇	—	4	2	no	—
65	C₄²×C₄₇	D₈	1	4	yes	—
66	C₄²×C₄₇	SD₁₆	1	4	yes	—
67	C₄²×C₄₇	C₂×D₄	1	2	yes	—
68	C₄²×C₄₇	#41	1	2	yes	—
69	C₂×C₄₇⋊C₈	—	2	4	no	—
70	C₂×C₄₇⋊C₈	—	4	4	no	—
71	C₂×C₄₇⋊C₈	—	4	2	no	—
72	C₂×C₄₇⋊C₈	—	4	4	no	—
73	C₂³×C₄₇	—	2	4	no	—
74	C₂³×C₄₇	—	4	2	no	—
75	C₂³×C₄₇	—	4	2	no	—
76	C₂³×C₄₇	—	4	4	no	—
77	C₂³×C₄₇	—	4	4	no	—
78	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	D₄×C₄₇	—	2	8	no	—
80	D₄×C₄₇	—	4	16	no	—
81	D₄×C₄₇	—	4	4	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	4	2	no	—
88	C₂×C₄×C₄₇	—	4	4	no	—
89	C₂×C₄×C₄₇	—	8	4	no	—
90	C₂×C₄×C₄₇	—	8	2	no	—
91	C₂×C₄×C₄₇	—	8	2	no	—
92	C₂×C₄×C₄₇	#102	1	4	yes	—
93	C₂×C₄×C₄₇	#102	1	4	yes	—
94	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₄₇	#121	1	2	yes	—
96	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₄₇	#131	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
102	C₄₇⋊C₈	—	4	4	no	—
103	C₄₇⋊C₈	—	8	2	no	—
104	#9644	—	47	1	no	maximal, Sylow
105	C₂²×C₄₇	—	4	4	no	—
106	C₂²×C₄₇	—	4	8	no	—
107	C₂²×C₄₇	—	8	2	no	—
108	C₂²×C₄₇	—	8	2	no	—
109	C₂²×C₄₇	—	8	4	no	—
110	C₂²×C₄₇	#390	1	1	yes	characteristic
111	C₂²×C₄₇	#407	1	1	yes	characteristic
112	C₂²×C₄₇	#408	1	1	yes	characteristic
113	C₂²×C₄₇	C₂×#102	1	2	yes	—
114	C₂²×C₄₇	#421	1	2	yes	—
115	C₄×C₄₇	—	2	2	no	—
116	C₄×C₄₇	—	2	4	no	—
117	C₄×C₄₇	—	2	4	no	—
118	C₄×C₄₇	—	8	2	no	—
119	C₄×C₄₇	#435	1	4	yes	—
120	C₄×C₄₇	#486	1	2	yes	—
121	C₄×C₄₇	#492	1	1	yes	characteristic
122	C₄×C₄₇	#493	1	1	yes	characteristic
123	C₄×#127	D₄₇	1	1	yes	characteristic
124	#1842	—	47	1	no	—
125	#1744	—	47	1	no	—
126	C₂×D₄⋊C₈	—	47	4	no	—
127	C₂×C₄₇	—	8	4	no	—
128	C₂×C₄₇	#1480	1	2	yes	—
129	C₂×C₄₇	#1486	1	2	yes	—
130	C₂×C₄₇	#1955	1	1	yes	characteristic
131	C₂×C₄₇	#1957	1	1	yes	characteristic
132	C₂×C₄₇	#1963	1	1	yes	characteristic
133	C₂×#127	Dic₄₇	1	1	yes	characteristic
134	C₂×C₄×D₄	—	2	2	no	—
135	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
136	C₂×C₄⋊C₈	—	47	2	no	—
137	C₂×C₄⋊C₈	—	94	2	no	—
138	C₂×C₄×C₈	—	47	2	no	—
139	#390	—	2	2	no	—
140	#390	Dic₄₇	1	2	yes	—
141	#384	Dic₄₇	1	1	yes	characteristic
142	C₄³	C₂×D₄₇	1	1	yes	characteristic
143	#336	—	94	2	no	—
144	D₄⋊C₈	—	94	8	no	—
145	C₄₇	#9644	1	1	yes	characteristic, complemented, Sylow
146	C₂²×D₄	—	2	2	no	—
147	C₂²×D₄	C₂×Dic₄₇	1	2	yes	—
148	C₂³×C₄	—	2	2	no	—
149	C₂²×C₈	—	94	2	no	—
150	C₂²×C₈	—	94	2	no	—
151	#127	—	2	4	no	—
152	C₄×D₄	—	2	4	no	—
153	C₄×D₄	—	2	4	no	—
154	C₄×D₄	—	4	4	no	—
155	C₄×D₄	—	4	4	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
158	C₂×C₂²⋊C₄	—	2	4	no	—
159	C₂×C₂²⋊C₄	—	2	2	no	—
160	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
161	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
162	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
163	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
164	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
165	C₄⋊C₈	—	94	4	no	—
166	C₄⋊C₈	—	188	4	no	—
167	C₄×C₈	—	47	4	no	—
168	C₄×C₈	—	94	2	no	—
169	C₂⁴	—	2	2	no	—
170	C₂×D₄	—	2	8	no	—
171	C₂×D₄	—	2	4	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	—	4	4	no	—
176	C₂×D₄	#10625329	1	4	yes	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
183	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
184	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
187	C₂×C₈	—	94	4	no	—
188	C₂×C₈	—	188	2	no	—
189	C₂×C₈	—	188	4	no	—
190	C₂×C₈	—	188	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₂²⋊C₄	—	4	8	no	—
194	C₂²⋊C₄	—	4	4	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	#10625334	1	4	yes	—
199	C₄²	#10625335	1	4	yes	—
200	C₄²	D₄×D₄₇	1	2	yes	—
201	C₄²	#10625352	1	2	yes	—
202	C₂³	—	2	4	no	—
203	C₂³	—	4	2	no	—
204	C₂³	—	4	2	no	—
205	C₂³	—	4	4	no	—
206	C₂³	—	4	4	no	—
207	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
208	D₄	—	2	8	no	—
209	D₄	—	4	16	no	—
210	D₄	—	4	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	8	4	no	—
221	C₂×C₄	#13093945	1	4	yes	—
222	C₂×C₄	#13093945	1	4	yes	—
223	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
224	C₂×C₄	#13093995	1	2	yes	—
225	C₂×C₄	#13094011	1	1	yes	characteristic
226	C₂×C₄	#13094020	1	1	yes	characteristic
227	C₂×C₄	C₂×#10625329	1	2	yes	—
228	C₂×C₄	#13094031	1	2	yes	—
229	C₂×C₄	C₂×#10625334	1	2	yes	—
230	C₂×C₄	C₂×#10625335	1	2	yes	—
231	C₈	—	376	2	no	—
232	C₈	—	188	4	no	—
233	C₂²	—	4	8	no	—
234	C₂²	—	4	4	no	—
235	C₂²	—	8	4	no	—
236	C₂²	—	8	2	no	—
237	C₂²	—	8	2	no	—
238	C₂²	#15689026	1	1	yes	characteristic
239	C₂²	#15689087	1	1	yes	characteristic
240	C₂²	#15689090	1	1	yes	characteristic
241	C₂²	C₂×#13093945	1	2	yes	—
242	C₂²	#15689140	1	2	yes	—
243	C₄	—	2	4	no	—
244	C₄	—	2	4	no	—
245	C₄	—	2	2	no	—
246	C₄	—	8	2	no	—
247	C₄	#15689195	1	4	yes	—
248	C₄	#15689469	1	2	yes	—
249	C₄	#15689495	1	1	yes	characteristic
250	C₄	#15689499	1	1	yes	characteristic
251	C₂	—	8	4	no	—
252	C₂	#20241787	1	2	yes	—
253	C₂	#20241813	1	2	yes	—
254	C₂	#20244187	1	1	yes	characteristic
255	C₂	#20244195	1	1	yes	characteristic
256	C₂	#20244221	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₄₇	C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#127	D₄₇
hypocenter	C₄₇	#9644
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#127

Complements

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

N	H ≅ G/N	type
1. #20243540	C₂	semidirect
2. #20243134	C₂	semidirect
3. C₂×D₄⋊(C₄₇⋊C₈)	C₂	semidirect
22. C₄×C₄₇⋊C₈	D₄	semidirect
145. C₄₇	#9644	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
#20243134	2
C₂×D₄⋊(C₄₇⋊C₈)	2
C₄×#127×C₄₇	2
#9644	47