Subgroups — #26115353

Order 12032.

486 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251766	C₂	1	1	yes	maximal, characteristic
2	#20243241	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688871	C₂	1	2	yes	maximal
4	C₂×#15688870	C₂	1	2	yes	maximal
5	C₄₇×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	1	yes	characteristic
7	C₂×#13094024	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×#13093935	—	2	1	no	—
10	#15689029	—	2	2	no	—
11	#15689029	C₄	1	4	yes	—
12	#15688989	—	2	1	no	—
13	#15688989	—	2	1	no	—
14	#15688878	—	2	1	no	—
15	#15688871	—	2	4	no	—
16	#15688870	—	2	4	no	—
17	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
18	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	#13094024	—	2	2	no	—
24	#13094024	—	2	2	no	—
25	#13094024	—	2	2	no	—
26	#13094024	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	1	no	—
28	C₂×C₄⋊Dic₄₇	—	2	1	no	—
29	C₂×C₄⋊Dic₄₇	—	2	1	no	—
30	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
31	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
33	#13093935	—	4	2	no	—
34	C₄₇⋊#95	—	2	4	no	—
35	C₄₇⋊#95	—	2	4	no	—
36	C₂²×C₈×C₄₇	—	2	2	no	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	D₄	1	2	yes	—
40	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₄₇	—	2	2	no	—
43	C₄⋊C₈×C₄₇	—	2	2	no	—
44	C₄⋊C₈×C₄₇	—	4	4	no	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₄₇	—	2	2	no	—
51	C₂²×Dic₄₇	—	2	2	no	—
52	C₂×Dic₉₄	—	2	4	no	—
53	C₂×Dic₉₄	—	4	1	no	—
54	C₂×Dic₉₄	—	4	1	no	—
55	C₂×Dic₉₄	—	4	2	no	—
56	C₂×Dic₉₄	—	4	2	no	—
57	C₂×Dic₉₄	—	4	4	no	—
58	C₂×C₈×C₄₇	—	4	2	no	—
59	C₂×C₈×C₄₇	—	4	4	no	—
60	C₄²×C₄₇	—	2	2	no	—
61	C₄²×C₄₇	—	2	2	no	—
62	C₄²×C₄₇	—	2	2	no	—
63	C₄²×C₄₇	—	4	2	no	—
64	C₄²×C₄₇	SD₁₆	1	2	yes	—
65	C₄²×C₄₇	Q₁₆	1	2	yes	—
66	C₄²×C₄₇	C₂×D₄	1	2	yes	—
67	C₄²×C₄₇	#41	1	2	yes	—
68	C₄⋊Dic₄₇	—	2	4	no	—
69	C₄⋊Dic₄₇	—	4	2	no	—
70	C₄⋊Dic₄₇	—	4	2	no	—
71	C₄⋊Dic₄₇	—	4	1	no	—
72	C₄⋊Dic₄₇	—	4	1	no	—
73	C₂×C₄₇⋊C₈	—	2	4	no	—
74	C₂×C₄₇⋊C₈	—	4	2	no	—
75	C₂×C₄₇⋊C₈	—	4	2	no	—
76	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	#103	1	4	yes	—
84	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₄₇	#121	1	2	yes	—
86	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
87	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
90	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
91	C₂×C₄×C₄₇	#137	1	2	yes	—
92	C₂×C₄×C₄₇	#137	1	2	yes	—
93	C₂×Dic₄₇	—	4	2	no	—
94	C₂×Dic₄₇	—	4	2	no	—
95	C₂×Dic₄₇	—	4	4	no	—
96	C₂×Dic₄₇	—	4	4	no	—
97	Dic₉₄	—	8	2	no	—
98	Dic₉₄	—	8	1	no	—
99	Dic₉₄	—	8	1	no	—
100	Dic₉₄	—	8	4	no	—
101	C₈×C₄₇	—	8	2	no	—
102	C₄₇⋊C₈	—	4	4	no	—
103	#9701	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	#390	1	1	yes	characteristic
105	C₂²×C₄₇	#413	1	2	yes	—
106	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
107	C₂²×C₄₇	#417	1	2	yes	—
108	C₂²×C₄₇	#419	1	1	yes	characteristic
109	C₄×C₄₇	—	2	4	no	—
110	C₄×C₄₇	—	2	2	no	—
111	C₄×C₄₇	—	4	2	no	—
112	C₄×C₄₇	#461	1	2	yes	—
113	C₄×C₄₇	#462	1	2	yes	—
114	C₄×C₄₇	#489	1	2	yes	—
115	C₄×C₄₇	#496	1	1	yes	characteristic
116	C₄×C₄₇	#497	1	1	yes	characteristic
117	Dic₄₇	—	8	2	no	—
118	Dic₄₇	—	8	2	no	—
119	#2862	—	47	1	no	—
120	#1842	D₄₇	1	1	yes	characteristic
121	#1767	—	47	1	no	—
122	C₂×#333	—	47	2	no	—
123	C₂×#332	—	47	2	no	—
124	C₂×C₄₇	#1550	1	2	yes	—
125	C₂×C₄₇	#1559	1	2	yes	—
126	C₂×C₄₇	#1959	1	1	yes	characteristic
127	C₂×C₄₇	#1967	1	1	yes	characteristic
128	C₂×C₄₇	#1981	1	1	yes	characteristic
129	C₂×C₄⋊Q₈	—	47	1	no	—
130	C₂×C₄⋊Q₈	—	47	1	no	—
131	C₂×C₄⋊C₈	—	2	2	no	—
132	C₂×C₄⋊C₈	—	47	2	no	—
133	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
134	C₂×#97	—	94	1	no	—
135	#391	—	47	4	no	—
136	#391	—	94	2	no	—
137	#382	—	94	1	no	—
138	#382	—	94	1	no	—
139	C₄³	C₂×D₄₇	1	1	yes	characteristic
140	#336	—	94	1	no	—
141	#333	—	94	4	no	—
142	#332	—	94	4	no	—
143	C₄₇	#9701	1	1	yes	characteristic, complemented, Sylow
144	C₂²×Q₈	—	47	2	no	—
145	C₂²×Q₈	—	94	1	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	94	2	no	—
148	C₄⋊Q₈	—	94	2	no	—
149	C₄⋊Q₈	—	94	2	no	—
150	C₄⋊Q₈	—	94	2	no	—
151	C₄⋊Q₈	—	94	2	no	—
152	C₂×C₄⋊C₄	—	47	2	no	—
153	C₂×C₄⋊C₄	—	94	1	no	—
154	C₂×C₄⋊C₄	—	94	1	no	—
155	C₂×C₄⋊C₄	—	94	1	no	—
156	C₂×C₄²	C₄×D₄₇	1	2	yes	—
157	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
158	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
160	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	4	4	no	—
164	C₄⋊C₈	—	47	4	no	—
165	C₄⋊C₈	—	94	2	no	—
166	#97	—	188	2	no	—
167	#95	—	94	4	no	—
168	#95	—	94	4	no	—
169	C₂×Q₈	—	94	4	no	—
170	C₂×Q₈	—	188	4	no	—
171	C₂×Q₈	—	188	1	no	—
172	C₂×Q₈	—	188	1	no	—
173	C₂×Q₈	—	188	2	no	—
174	C₂×Q₈	—	188	2	no	—
175	C₂²×C₄	—	94	2	no	—
176	C₂²×C₄	—	94	2	no	—
177	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
178	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
179	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	4	no	—
184	C₂×C₈	—	94	4	no	—
185	C₂×C₈	—	188	2	no	—
186	C₂×C₈	—	188	2	no	—
187	C₄⋊C₄	—	94	4	no	—
188	C₄⋊C₄	—	188	2	no	—
189	C₄⋊C₄	—	188	2	no	—
190	C₄⋊C₄	—	188	1	no	—
191	C₄⋊C₄	—	188	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	#10625325	1	2	yes	—
197	C₄²	Dic₁₈₈	1	2	yes	—
198	C₄²	D₄×D₄₇	1	2	yes	—
199	C₄²	#10625354	1	2	yes	—
200	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
201	Q₈	—	376	4	no	—
202	Q₈	—	376	2	no	—
203	Q₈	—	376	1	no	—
204	Q₈	—	376	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	188	2	no	—
212	C₂×C₄	—	188	2	no	—
213	C₂×C₄	—	188	4	no	—
214	C₂×C₄	—	188	4	no	—
215	C₂×C₄	#13093949	1	4	yes	—
216	C₂×C₄	#13093986	1	2	yes	—
217	C₂×C₄	#13093999	1	2	yes	—
218	C₂×C₄	#13094013	1	1	yes	characteristic
219	C₂×C₄	#13094020	1	1	yes	characteristic
220	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
221	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
222	C₂×C₄	#13094050	1	2	yes	—
223	C₂×C₄	#13094055	1	2	yes	—
224	C₂×C₄	#13094057	1	2	yes	—
225	C₈	—	8	2	no	—
226	C₈	—	188	4	no	—
227	C₂²	#15689024	1	1	yes	characteristic
228	C₂²	#15689105	1	2	yes	—
229	C₂²	C₂×#13093949	1	1	yes	characteristic
230	C₂²	#15689122	1	2	yes	—
231	C₂²	#15689133	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	4	no	—
234	C₄	—	4	2	no	—
235	C₄	—	376	2	no	—
236	C₄	—	376	2	no	—
237	C₄	#15689329	1	2	yes	—
238	C₄	#15689334	1	2	yes	—
239	C₄	#15689485	1	2	yes	—
240	C₄	#15689510	1	1	yes	characteristic
241	C₄	#15689515	1	1	yes	characteristic
242	C₂	#20242076	1	2	yes	—
243	C₂	#20242116	1	2	yes	—
244	C₂	#20244202	1	1	yes	characteristic
245	C₂	#20244235	1	1	yes	characteristic
246	C₂	#20244310	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1842	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1842	D₄₇
hypocenter	C₄₇	#9701
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
143. C₄₇	#9701	semidirect

Maximal subgroups

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

subgroup	index
#20251766	2
#20243241	2
C₂×#15688871	2
C₂×#15688870	2
C₄₇×#1842	2
#9701	47