Subgroups — #26103422

Order 12032.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#20243417	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×#340	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
5	C₂×#13093958	C₄	1	2	yes	complemented
6	#15688983	—	2	1	no	—
7	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	C₂²⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	complemented
9	#15688886	—	2	1	no	—
10	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
12	C₄₇×#340	—	2	1	no	—
13	C₄₇×#340	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₄₇	—	2	2	no	—
15	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
16	C₂×C₄₇⋊C₁₆	—	2	2	no	—
17	C₂×C₄₇⋊C₁₆	—	4	2	no	—
18	#13093958	—	4	2	no	—
19	C₄₇⋊#95	—	4	1	no	—
20	C₂³×C₄×C₄₇	—	4	1	no	—
21	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₄₇	—	4	2	no	—
23	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
32	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
33	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	—	2	2	no	—
35	C₂²×C₄×C₄₇	—	2	2	no	—
36	C₂²×C₄×C₄₇	—	4	4	no	—
37	C₂²×C₄×C₄₇	—	4	2	no	—
38	C₂²×C₄×C₄₇	—	8	2	no	—
39	C₂²×C₄×C₄₇	—	8	1	no	—
40	C₂²×C₄×C₄₇	—	8	1	no	—
41	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
47	C₂²×Dic₄₇	—	4	2	no	—
48	C₂×C₈×C₄₇	—	2	2	no	—
49	C₂×C₈×C₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	—	4	1	no	—
51	C₂×C₈×C₄₇	—	4	1	no	—
52	C₂×C₈×C₄₇	—	4	1	no	—
53	C₂×C₈×C₄₇	—	4	1	no	—
54	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₄₇	—	2	4	no	—
59	C₄⋊C₄×C₄₇	—	4	4	no	—
60	C₄⋊C₄×C₄₇	—	4	2	no	—
61	C₂²⋊Dic₄₇	—	8	1	no	—
62	C₂²⋊Dic₄₇	—	8	1	no	—
63	C₄⋊Dic₄₇	—	4	2	no	—
64	C₄⋊Dic₄₇	—	4	2	no	—
65	C₄₇⋊C₁₆	—	4	2	no	—
66	C₂³×C₄₇	—	2	2	no	—
67	C₂³×C₄₇	—	2	2	no	—
68	C₂³×C₄₇	#102	1	1	yes	characteristic
69	C₂³×C₄₇	#102	1	1	yes	characteristic
70	C₂³×C₄₇	#107	1	1	yes	characteristic
71	C₂³×C₄₇	D₁₆	1	1	yes	characteristic
72	C₂³×C₄₇	SD₃₂	1	1	yes	characteristic
73	C₂³×C₄₇	SD₃₂	1	1	yes	characteristic
74	C₂³×C₄₇	Q₃₂	1	1	yes	characteristic
75	C₂×C₄×C₄₇	—	2	2	no	—
76	C₂×C₄×C₄₇	—	2	2	no	—
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	2	1	no	—
79	C₂×C₄×C₄₇	—	2	1	no	—
80	C₂×C₄×C₄₇	—	2	1	no	—
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	4	4	no	—
83	C₂×C₄×C₄₇	—	8	4	no	—
84	C₂×C₄×C₄₇	—	8	4	no	—
85	C₂×C₄×C₄₇	—	8	2	no	—
86	C₂×C₄×C₄₇	—	8	2	no	—
87	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
88	C₂×C₄×C₄₇	#99	1	2	yes	—
89	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
90	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#104	1	2	yes	—
92	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
93	C₂×Dic₄₇	—	8	2	no	—
94	C₂×Dic₄₇	—	8	2	no	—
95	C₂×Dic₄₇	—	8	2	no	—
96	C₈×C₄₇	—	2	1	no	—
97	C₈×C₄₇	—	2	1	no	—
98	C₈×C₄₇	—	2	1	no	—
99	C₈×C₄₇	—	2	1	no	—
100	#7258	—	47	1	no	maximal, Sylow
101	C₂²×C₄₇	—	2	1	no	—
102	C₂²×C₄₇	—	2	1	no	—
103	C₂²×C₄₇	—	2	2	no	—
104	C₂²×C₄₇	—	2	2	no	—
105	C₂²×C₄₇	—	2	2	no	—
106	C₂²×C₄₇	—	2	2	no	—
107	C₂²×C₄₇	—	2	1	no	—
108	C₂²×C₄₇	—	2	1	no	—
109	C₂²×C₄₇	#327	1	2	yes	—
110	C₂²×C₄₇	#340	1	1	yes	characteristic
111	C₂²×C₄₇	#357	1	1	yes	characteristic
112	C₂²×C₄₇	#357	1	1	yes	characteristic
113	C₂²×C₄₇	#358	1	1	yes	characteristic
114	C₂²×C₄₇	#358	1	1	yes	characteristic
115	C₂²×C₄₇	#360	1	2	yes	—
116	C₂²×C₄₇	#366	1	1	yes	characteristic
117	C₂²×C₄₇	#367	1	1	yes	characteristic
118	C₄×C₄₇	—	2	1	no	—
119	C₄×C₄₇	—	2	1	no	—
120	C₄×C₄₇	—	8	4	no	—
121	C₄×C₄₇	#328	1	2	yes	—
122	C₄×C₄₇	#339	1	1	yes	characteristic
123	C₄×C₄₇	#342	1	1	yes	characteristic
124	Dic₄₇	—	16	2	no	—
125	C₂×C₂²⋊C₁₆	—	47	1	no	—
126	#1808	—	47	1	no	—
127	C₂×#340	D₄₇	1	1	yes	characteristic
128	C₂×C₄₇	—	2	1	no	—
129	C₂×C₄₇	—	2	1	no	—
130	C₂×C₄₇	—	2	1	no	—
131	C₂×C₄₇	—	2	1	no	—
132	C₂×C₄₇	#1294	1	1	yes	characteristic
133	C₂×C₄₇	#1347	1	2	yes	—
134	C₂×C₄₇	#1348	1	2	yes	—
135	C₂×C₄₇	#1380	1	1	yes	characteristic
136	C₂×C₄₇	#1386	1	1	yes	characteristic
137	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
138	C₂²×C₄⋊C₄	—	2	1	no	—
139	C₂²×C₁₆	—	94	1	no	—
140	C₂×#107	—	47	2	no	—
141	#380	—	94	1	no	—
142	C₂²⋊C₁₆	—	47	2	no	—
143	C₂²⋊C₁₆	—	94	1	no	—
144	#340	—	2	1	no	—
145	#340	—	94	1	no	—
146	#340	Dic₄₇	1	2	yes	complemented
147	C₄₇	#7258	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₄	—	4	1	no	—
149	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
150	C₂²×C₈	—	4	2	no	—
151	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
152	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
153	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
154	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
155	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
156	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	2	4	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	4	2	no	—
160	C₂×C₄⋊C₄	—	4	1	no	—
161	C₂×C₄⋊C₄	—	94	2	no	—
162	C₂×C₂²⋊C₄	—	188	1	no	—
163	C₂×C₁₆	—	94	2	no	—
164	C₂×C₁₆	—	188	2	no	—
165	#107	—	188	2	no	—
166	#95	—	188	1	no	—
167	C₂⁴	#10625334	1	1	yes	characteristic
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	4	4	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	8	2	no	—
173	C₂²×C₄	—	8	1	no	—
174	C₂²×C₄	—	8	1	no	—
175	C₂²×C₄	—	188	2	no	—
176	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
178	C₂²×C₄	#10625335	1	1	yes	characteristic
179	C₂²×C₄	#10625336	1	1	yes	characteristic
180	C₂²×C₄	#10625337	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
189	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
190	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
191	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	188	2	no	—
196	C₄⋊C₄	—	188	2	no	—
197	C₂²⋊C₄	—	376	1	no	—
198	C₂²⋊C₄	—	376	1	no	—
199	C₁₆	—	188	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	#13093944	1	1	yes	characteristic
203	C₂³	#13093945	1	1	yes	characteristic
204	C₂³	#13093957	1	1	yes	characteristic
205	C₂³	#13093965	1	1	yes	characteristic
206	C₂³	#13093967	1	1	yes	characteristic
207	C₂³	#13093968	1	1	yes	characteristic
208	C₂³	#13093970	1	1	yes	characteristic
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	—	8	4	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	—	376	2	no	—
222	C₂×C₄	—	376	2	no	—
223	C₂×C₄	—	376	2	no	—
224	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
225	C₂×C₄	#13093938	1	2	yes	—
226	C₂×C₄	#13093947	1	1	yes	characteristic
227	C₂×C₄	#13093948	1	1	yes	characteristic
228	C₂×C₄	#13093952	1	2	yes	—
229	C₂×C₄	#13093955	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	#15688856	1	2	yes	—
243	C₂²	#15688885	1	1	yes	characteristic
244	C₂²	#15688923	1	1	yes	characteristic
245	C₂²	#15688924	1	1	yes	characteristic
246	C₂²	#15688926	1	1	yes	characteristic
247	C₂²	#15688927	1	1	yes	characteristic
248	C₂²	#15688932	1	2	yes	—
249	C₂²	#15688948	1	1	yes	characteristic
250	C₂²	#15688950	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	4	no	—
254	C₄	—	752	2	no	—
255	C₄	#15688859	1	2	yes	—
256	C₄	#15688883	1	1	yes	characteristic
257	C₄	#15688889	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#20241267	1	1	yes	characteristic
263	C₂	#20241396	1	2	yes	—
264	C₂	#20241399	1	2	yes	—
265	C₂	#20241479	1	1	yes	characteristic
266	C₂	#20241495	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093944
commutator	C₂×C₈×C₄₇	C₄²
Frattini	C₂³×C₈	C₂×D₄₇
Fitting	C₂×C₄₇×#340	C₂
socle	C₂³×C₄₇	#102
prime radical	C₂⁴×C₄₇	D₈
cosocle	C₂³×C₈×C₄₇	C₂²
hypercenter	C₂×#340	D₄₇
hypocenter	C₄₇	#7258
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
5. C₂×#13093958	C₄	semidirect
8. C₂²⋊(C₄₇⋊C₁₆)	C₄	semidirect
13. C₄₇×#340	C₄	semidirect
146. #340	Dic₄₇	semidirect
147. C₄₇	#7258	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
#20243417	2
C₂×C₄₇×#340	2
#7258	47