Subgroups — #26125190

Order 12032.

1142 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094024	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688891	C₂	1	4	yes	maximal
3	#20243082	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1728	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₉₄	C₄	1	2	yes	—
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13094024	—	2	8	no	—
8	C₂×#13094024	—	2	2	no	—
9	C₂×#13094024	—	2	2	no	—
10	C₂²×C₄⋊Dic₄₇	C₄	1	2	yes	—
11	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	#15688891	—	2	8	no	—
13	#15688878	—	2	2	no	—
14	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#336	—	2	2	no	—
17	C₂²×Dic₉₄	—	2	4	no	—
18	C₂²×Dic₉₄	—	2	8	no	—
19	C₂²×Dic₉₄	—	4	8	no	—
20	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	4	no	—
23	C₂×#10625329	—	2	4	no	—
24	C₂×#10625329	Q₈	1	4	yes	complemented
25	C₂²×C₄₇⋊C₈	—	2	2	no	—
26	#13094024	—	2	16	no	—
27	#13094024	—	2	16	no	—
28	C₂×C₄⋊Dic₄₇	—	2	4	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	8	no	—
33	C₂×C₄⋊Dic₄₇	D₄	1	8	yes	—
34	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₄₇	—	2	2	no	—
38	C₂×M₁₆×C₄₇	—	2	4	no	—
39	C₂×M₁₆×C₄₇	—	2	4	no	—
40	C₂×M₁₆×C₄₇	Q₈	1	4	yes	complemented
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂×C₄²×C₄₇	—	2	4	no	—
43	C₂×C₄²×C₄₇	D₄	1	2	yes	—
44	C₂×C₄²×C₄₇	D₄	1	2	yes	—
45	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	2	4	no	—
49	C₂²×C₄×C₄₇	—	2	4	no	—
50	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₄₇	#41	1	2	yes	—
58	C₂²×C₄×C₄₇	#41	1	2	yes	—
59	C₂²×Dic₄₇	—	2	4	no	—
60	C₂²×Dic₄₇	—	2	4	no	—
61	C₂²×Dic₄₇	—	4	8	no	—
62	C₂²×Dic₄₇	C₂²⋊C₄	1	8	yes	—
63	C₂×Dic₉₄	—	2	16	no	—
64	C₂×Dic₉₄	—	4	32	no	—
65	C₂×Dic₉₄	—	4	8	no	—
66	C₂×Dic₉₄	—	4	8	no	—
67	M₁₆×C₄₇	—	2	8	no	—
68	M₁₆×C₄₇	—	4	4	no	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	4	2	no	—
71	C₂×C₈×C₄₇	—	4	2	no	—
72	C₄²×C₄₇	—	2	4	no	—
73	C₄²×C₄₇	—	2	4	no	—
74	C₄⋊Dic₄₇	—	2	16	no	—
75	C₄⋊Dic₄₇	—	2	8	no	—
76	C₄⋊Dic₄₇	—	2	8	no	—
77	#10625329	—	2	8	no	—
78	#10625329	—	4	4	no	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	4	2	no	—
81	C₂×C₄₇⋊C₈	—	4	2	no	—
82	C₂³×C₄₇	—	2	2	no	—
83	C₂³×C₄₇	—	2	2	no	—
84	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
85	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
87	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
88	C₂×C₄×C₄₇	—	2	8	no	—
89	C₂×C₄×C₄₇	—	2	8	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	#100	1	4	yes	—
95	C₂×C₄×C₄₇	#101	1	4	yes	—
96	C₂×C₄×C₄₇	#120	1	2	yes	—
97	C₂×C₄×C₄₇	#120	1	2	yes	—
98	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
99	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
100	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
101	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₄₇	#124	1	4	yes	—
103	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
107	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
108	C₂×Dic₄₇	—	2	8	no	—
109	C₂×Dic₄₇	—	4	8	no	—
110	C₂×Dic₄₇	—	4	8	no	—
111	C₂×Dic₄₇	—	4	8	no	—
112	Dic₉₄	—	4	32	no	—
113	C₈×C₄₇	—	4	4	no	—
114	C₄₇⋊C₈	—	4	4	no	—
115	#11382	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	—	2	4	no	—
117	C₂²×C₄₇	—	2	4	no	—
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	#379	1	1	yes	characteristic
121	C₂²×C₄₇	#391	1	4	yes	—
122	C₂²×C₄₇	#391	1	1	yes	characteristic
123	C₂²×C₄₇	#391	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×#100	1	2	yes	—
125	C₂²×C₄₇	C₂×#101	1	2	yes	—
126	C₄×C₄₇	—	2	4	no	—
127	C₄×C₄₇	—	2	4	no	—
128	C₄×C₄₇	#393	1	2	yes	—
129	C₄×C₄₇	#393	1	2	yes	—
130	C₄×C₄₇	#395	1	2	yes	—
131	C₄×C₄₇	#395	1	2	yes	—
132	Dic₄₇	—	4	8	no	—
133	C₂²×C₄⋊Q₈	—	47	1	no	—
134	C₂×#343	—	47	4	no	—
135	#1728	—	47	1	no	—
136	#1728	D₄₇	1	1	yes	characteristic
137	C₂×C₄₇	—	2	4	no	—
138	C₂×C₄₇	#1446	1	1	yes	characteristic
139	C₂×C₄₇	#1783	1	1	yes	characteristic
140	C₂×C₄₇	#1784	1	1	yes	characteristic
141	C₂×C₄₇	#1997	1	2	yes	—
142	C₂×C₄₇	#1997	1	2	yes	—
143	C₂³×Q₈	—	47	2	no	—
144	C₂²×M₁₆	—	47	2	no	—
145	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
146	C₂×C₄⋊Q₈	—	94	8	no	—
147	C₂×C₄⋊Q₈	—	94	2	no	—
148	C₂×C₄⋊Q₈	—	94	2	no	—
149	C₂²×C₄⋊C₄	—	47	2	no	—
150	C₂²×C₄⋊C₄	—	47	2	no	—
151	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
152	#343	—	94	8	no	—
153	#336	—	2	2	no	—
154	#336	—	94	2	no	—
155	C₄₇	#11382	1	1	yes	characteristic, complemented, Sylow
156	C₂²×Q₈	—	94	8	no	—
157	C₂²×Q₈	—	94	4	no	—
158	C₂²×Q₈	—	188	8	no	—
159	C₂³×C₄	—	47	4	no	—
160	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
161	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
162	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	47	4	no	—
167	C₂×M₁₆	—	94	2	no	—
168	C₂×M₁₆	—	94	4	no	—
169	C₂×M₁₆	—	94	4	no	—
170	C₂×M₁₆	Dic₉₄	1	4	yes	complemented
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	94	2	no	—
173	C₄⋊Q₈	—	94	16	no	—
174	C₄⋊Q₈	—	94	16	no	—
175	C₂×C₄⋊C₄	—	47	8	no	—
176	C₂×C₄⋊C₄	—	94	8	no	—
177	C₂×C₄⋊C₄	—	94	4	no	—
178	C₂×C₄⋊C₄	—	94	2	no	—
179	C₂×C₄⋊C₄	—	94	4	no	—
180	C₂×C₄⋊C₄	—	94	2	no	—
181	C₂×C₄²	—	2	4	no	—
182	C₂×C₄²	D₁₈₈	1	2	yes	—
183	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
184	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
185	C₂×Q₈	—	94	16	no	—
186	C₂×Q₈	—	188	32	no	—
187	C₂×Q₈	—	188	8	no	—
188	C₂×Q₈	—	188	8	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	47	8	no	—
194	C₂²×C₄	—	94	4	no	—
195	C₂²×C₄	—	94	4	no	—
196	C₂²×C₄	—	188	8	no	—
197	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
198	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
199	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
200	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
201	C₂²×C₄	#10625350	1	2	yes	—
202	C₂²×C₄	D₄×D₄₇	1	4	yes	—
203	C₂²×C₄	#10625352	1	2	yes	—
204	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
205	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
206	M₁₆	—	2	8	no	—
207	M₁₆	—	4	4	no	—
208	M₁₆	—	94	8	no	—
209	M₁₆	—	188	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	94	4	no	—
214	C₂×C₈	—	188	2	no	—
215	C₂×C₈	—	188	2	no	—
216	C₄⋊C₄	—	94	16	no	—
217	C₄⋊C₄	—	94	8	no	—
218	C₄⋊C₄	—	94	8	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	4	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	C₂²⋊C₄×D₄₇	1	2	yes	—
224	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
225	C₂³	#13093989	1	2	yes	—
226	C₂³	C₄×Dic₉₄	1	2	yes	—
227	Q₈	—	188	32	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	94	8	no	—
235	C₂×C₄	—	188	8	no	—
236	C₂×C₄	—	188	8	no	—
237	C₂×C₄	—	188	8	no	—
238	C₂×C₄	#13093940	1	4	yes	—
239	C₂×C₄	#13093942	1	4	yes	—
240	C₂×C₄	#13093992	1	2	yes	—
241	C₂×C₄	#13093994	1	2	yes	—
242	C₂×C₄	#13094001	1	4	yes	—
243	C₂×C₄	#13094002	1	4	yes	—
244	C₂×C₄	#13094003	1	2	yes	—
245	C₂×C₄	#13094004	1	2	yes	—
246	C₂×C₄	#13094010	1	4	yes	—
247	C₂×C₄	#13094011	1	1	yes	characteristic
248	C₂×C₄	#13094012	1	1	yes	characteristic
249	C₂×C₄	#13094022	1	1	yes	characteristic
250	C₂×C₄	#13094023	1	4	yes	—
251	C₂×C₄	#13094024	1	1	yes	characteristic
252	C₈	—	4	4	no	—
253	C₈	—	188	4	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	#15688978	1	1	yes	characteristic
259	C₂²	#15689027	1	4	yes	—
260	C₂²	#15689028	1	1	yes	characteristic
261	C₂²	#15689029	1	1	yes	characteristic
262	C₂²	C₂×#13093940	1	2	yes	—
263	C₂²	C₂×#13093942	1	2	yes	—
264	C₄	—	2	4	no	—
265	C₄	—	2	4	no	—
266	C₄	—	188	8	no	—
267	C₄	#15689035	1	2	yes	—
268	C₄	#15689036	1	2	yes	—
269	C₄	#15689043	1	2	yes	—
270	C₄	#15689044	1	2	yes	—
271	C₂	—	2	4	no	—
272	C₂	#20241658	1	1	yes	characteristic
273	C₂	#20243308	1	1	yes	characteristic
274	C₂	#20243311	1	1	yes	characteristic
275	C₂	#20244376	1	2	yes	—
276	C₂	#20244377	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1728	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1728	D₄₇
hypocenter	C₄₇	#11382
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
24. C₂×#10625329	Q₈	semidirect
40. C₂×M₁₆×C₄₇	Q₈	semidirect
155. C₄₇	#11382	semidirect
170. C₂×M₁₆	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13094024	2
C₂×#15688891	2
#20243082	2
C₄₇×#1728	2
#11382	47