Subgroups — #26235221

Order 12032.

734 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	#20246343	C₂	1	2	yes	maximal
3	#20244157	C₂	1	4	yes	maximal
4	#20244146	C₂	1	2	yes	maximal
5	#20243517	C₂	1	2	yes	maximal
6	#20243516	C₂	1	1	yes	maximal, characteristic
7	#20243280	C₂	1	2	yes	maximal
8	C₄×#106×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
10	C₈×Dic₉₄	C₄	1	8	yes	complemented
11	C₂×#13093957	C₂²	1	4	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
14	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
15	#15689029	—	2	2	no	—
16	#15689010	—	2	4	no	—
17	#15688993	C₂²	1	2	yes	—
18	#15688988	C₂²	1	4	yes	—
19	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
20	C₄×C₄⋊Dic₄₇	—	2	2	no	—
21	#15688886	C₂²	1	2	yes	—
22	#15688886	C₂²	1	2	yes	—
23	#15688885	C₂²	1	4	yes	—
24	C₂×#106×C₄₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
27	C₄₇×#340	C₂²	1	1	yes	characteristic
28	C₄₇×#340	C₂²	1	1	yes	characteristic
29	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
32	C₄×Dic₉₄	Q₈	1	4	yes	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
37	C₂×Dic₄₇⋊C₄	D₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	—	2	4	no	—
39	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
40	#13093957	—	2	8	no	—
41	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
42	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
43	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
44	C₄₇⋊#95	—	2	4	no	—
45	C₄₇⋊#95	—	2	4	no	—
46	C₄₇⋊#95	—	2	4	no	—
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₄₇	—	2	2	no	—
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	#106×C₄₇	—	2	4	no	—
54	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
55	#95×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₇	—	2	4	no	—
62	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
63	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
64	C₂×Dic₉₄	—	2	4	no	—
65	C₂×Dic₉₄	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₄₇	—	2	2	no	—
67	C₂×C₈×C₄₇	—	2	2	no	—
68	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
70	C₄⋊C₄×C₄₇	—	2	2	no	—
71	C₄⋊C₄×C₄₇	—	2	2	no	—
72	C₄⋊C₄×C₄₇	—	2	2	no	—
73	C₄⋊C₄×C₄₇	#41	1	4	yes	—
74	C₄²×C₄₇	—	4	4	no	—
75	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
77	C₄⋊Dic₄₇	—	2	4	no	—
78	C₄⋊Dic₄₇	—	2	4	no	—
79	C₄⋊Dic₄₇	—	4	4	no	—
80	C₄⋊Dic₄₇	—	8	2	no	—
81	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
82	Dic₄₇⋊C₄	C₄⋊C₄	1	8	yes	—
83	C₄×Dic₄₇	—	8	4	no	—
84	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
85	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
88	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	—	4	2	no	—
93	C₂×C₄×C₄₇	—	4	2	no	—
94	C₂×C₄×C₄₇	—	4	2	no	—
95	C₂×C₄×C₄₇	—	4	2	no	—
96	C₂×C₄×C₄₇	—	4	2	no	—
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₄₇	#135	1	4	yes	—
100	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
103	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
104	C₂×Dic₄₇	—	2	4	no	—
105	C₂×Dic₄₇	—	2	2	no	—
106	C₂×Dic₄₇	—	2	2	no	—
107	C₂×Dic₄₇	—	4	4	no	—
108	C₂×Dic₄₇	—	4	4	no	—
109	C₂×Dic₄₇	—	4	4	no	—
110	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
112	Dic₉₄	—	2	8	no	—
113	C₈×C₄₇	—	2	2	no	—
114	C₈×C₄₇	—	2	2	no	—
115	C₄₇⋊C₈	—	2	4	no	—
116	#22739	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₄₇	#548	1	2	yes	—
119	C₂²×C₄₇	C₂×#135	1	2	yes	—
120	C₂²×C₄₇	#576	1	1	yes	characteristic
121	C₂²×C₄₇	#578	1	1	yes	characteristic
122	C₄×C₄₇	—	2	2	no	—
123	C₄×C₄₇	—	4	4	no	—
124	C₄×C₄₇	—	8	2	no	—
125	C₄×C₄₇	#427	1	4	yes	—
126	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
127	C₄×C₄₇	#519	1	1	yes	characteristic
128	C₄×C₄₇	#557	1	2	yes	—
129	Dic₄₇	—	2	4	no	—
130	Dic₄₇	—	2	4	no	—
131	Dic₄₇	—	8	4	no	—
132	C₂×C₈×Q₈	—	47	1	no	—
133	#2355	—	47	2	no	—
134	#1949	—	47	4	no	—
135	#1947	—	47	2	no	—
136	#1835	—	47	2	no	—
137	#1834	—	47	1	no	—
138	C₄×#107	—	47	2	no	—
139	C₄×#106	D₄₇	1	1	yes	characteristic
140	C₂×C₄₇	#2351	1	1	yes	characteristic
141	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
142	C₂×C₄₇	#2913	1	1	yes	characteristic
143	C₂×C₄₇	#3361	1	2	yes	—
144	C₂×C₄₇	#3364	1	2	yes	—
145	C₂×C₄×Q₈	—	47	1	no	—
146	C₈×Q₈	—	47	8	no	—
147	C₂×#107	—	47	4	no	—
148	C₂×#106	C₂×D₄₇	1	2	yes	—
149	C₂×C₄⋊C₈	—	47	2	no	—
150	C₂×C₄⋊C₈	—	47	1	no	—
151	C₂×C₄×C₈	—	47	2	no	—
152	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
153	#391	—	94	2	no	—
154	#387	—	94	4	no	—
155	#384	—	47	2	no	—
156	#382	—	47	4	no	—
157	C₄×C₄⋊C₄	—	47	4	no	—
158	C₄×C₄⋊C₄	—	94	2	no	—
159	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
160	#340	—	47	2	no	—
161	#340	—	47	2	no	—
162	#340	—	47	4	no	—
163	#340	C₂×D₄₇	1	1	yes	characteristic
164	#340	C₂×D₄₇	1	1	yes	characteristic
165	C₄₇	#22739	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₈	—	47	1	no	—
167	C₂²×C₈	—	47	2	no	—
168	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
170	C₄×Q₈	—	47	4	no	—
171	C₄×Q₈	—	47	4	no	—
172	C₂×C₄⋊C₄	—	47	2	no	—
173	C₂×C₄⋊C₄	—	47	1	no	—
174	C₂×C₄⋊C₄	—	47	4	no	—
175	C₂×C₄⋊C₄	—	94	2	no	—
176	C₂×C₄⋊C₄	—	94	2	no	—
177	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
178	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	—	47	2	no	—
181	C₂×C₄²	—	94	4	no	—
182	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
183	#107	—	94	8	no	—
184	#106	—	2	4	no	—
185	C₄⋊C₈	—	47	8	no	—
186	C₄⋊C₈	—	47	4	no	—
187	C₄×C₈	—	47	8	no	—
188	C₄×C₈	C₄×D₄₇	1	4	yes	—
189	#95	—	2	2	no	—
190	#95	—	94	4	no	—
191	#95	—	94	4	no	—
192	#95	—	94	4	no	—
193	C₂×Q₈	—	47	4	no	—
194	C₂×Q₈	—	94	4	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	47	2	no	—
198	C₂²×C₄	—	47	2	no	—
199	C₂²×C₄	—	94	4	no	—
200	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
201	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
202	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	47	4	no	—
206	C₂×C₈	—	94	4	no	—
207	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
208	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	47	8	no	—
213	C₄⋊C₄	—	47	4	no	—
214	C₄⋊C₄	—	94	4	no	—
215	C₄⋊C₄	—	94	4	no	—
216	C₄⋊C₄	—	376	2	no	—
217	C₄⋊C₄	—	188	4	no	—
218	C₄⋊C₄	#10625350	1	4	yes	—
219	C₄²	—	4	4	no	—
220	C₄²	—	47	4	no	—
221	C₄²	—	47	4	no	—
222	C₄²	—	376	4	no	—
223	C₄²	C₂×C₄×D₄₇	1	2	yes	—
224	C₄²	Q₈×D₄₇	1	2	yes	—
225	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
226	Q₈	—	94	8	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	47	4	no	—
236	C₂×C₄	—	47	4	no	—
237	C₂×C₄	—	94	4	no	—
238	C₂×C₄	—	94	2	no	—
239	C₂×C₄	—	94	2	no	—
240	C₂×C₄	—	188	4	no	—
241	C₂×C₄	—	188	4	no	—
242	C₂×C₄	—	188	4	no	—
243	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
244	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
245	C₂×C₄	#13094045	1	4	yes	—
246	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
247	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
248	C₂×C₄	C₂×#10625350	1	2	yes	—
249	C₂×C₄	#13094071	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	2	2	no	—
252	C₈	—	94	4	no	—
253	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
254	C₂²	#15689782	1	2	yes	—
255	C₂²	C₂×#13094045	1	2	yes	—
256	C₂²	#15689925	1	1	yes	characteristic
257	C₂²	#15689937	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	4	4	no	—
260	C₄	—	8	2	no	—
261	C₄	—	94	4	no	—
262	C₄	—	94	4	no	—
263	C₄	—	376	4	no	—
264	C₄	#15689158	1	4	yes	—
265	C₄	C₄×#10625350	1	1	yes	characteristic
266	C₄	#15689600	1	1	yes	characteristic
267	C₄	#15689851	1	2	yes	—
268	C₂	#20246308	1	1	yes	characteristic
269	C₂	C₂×#15689158	1	1	yes	characteristic
270	C₂	#20251985	1	1	yes	characteristic
271	C₂	#20256046	1	2	yes	—
272	C₂	#20256071	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#106×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#106	D₄₇
hypocenter	C₄₇	#22739
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
10. C₈×Dic₉₄	C₄	semidirect
165. C₄₇	#22739	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₉₄	2
#20246343	2
#20244157	2
#20244146	2
#20243517	2
#20243516	2
#20243280	2
C₄×#106×C₄₇	2
#22739	47