Subgroups — #26245344

Order 12032.

874 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	C₂×#15689149	C₂	1	1	yes	maximal, characteristic
2	#20246012	C₂	1	2	yes	maximal
3	#20244882	C₂	1	4	yes	maximal, complemented
4	#20244877	C₂	1	4	yes	maximal, complemented
5	#20243107	C₂	1	2	yes	maximal
6	#20243105	C₂	1	1	yes	maximal, characteristic
7	C₂²×#107×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	1	yes	characteristic
9	C₂×#13093983	C₂²	1	2	yes	—
10	C₂×#13093983	C₂²	1	1	yes	characteristic
11	#15689149	—	2	4	no	—
12	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
13	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
14	#15689019	C₂²	1	8	yes	—
15	#15688996	—	2	4	no	—
16	#15688992	—	2	2	no	—
17	#15688988	—	2	2	no	—
18	#15688886	C₂²	1	2	yes	—
19	#15688886	C₂²	1	2	yes	—
20	#15688885	C₂²	1	8	yes	—
21	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂×#107×C₄₇	—	2	4	no	—
24	C₂×#107×C₄₇	C₂²	1	2	yes	—
25	C₂×#107×C₄₇	C₂²	1	2	yes	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	4	no	—
28	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
29	#13093983	—	2	2	no	—
30	#13093983	—	2	4	no	—
31	#13093983	—	2	2	no	—
32	#13093983	Q₈	1	8	yes	—
33	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
36	C₂×Dic₄₇⋊C₄	—	2	8	no	—
37	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
38	C₂×C₂²⋊Dic₄₇	D₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	—	2	2	no	—
40	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
41	Dic₄₇⋊C₈	—	2	4	no	—
42	Dic₄₇⋊C₈	—	2	4	no	—
43	C₄₇⋊#95	—	2	8	no	—
44	C₂³×C₄×C₄₇	—	2	2	no	—
45	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	—	2	4	no	—
47	C₂²×C₈×C₄₇	D₄	1	4	yes	—
48	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
51	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
52	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
54	#107×C₄₇	—	2	4	no	—
55	#107×C₄₇	—	2	4	no	—
56	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	4	4	no	—
59	C₂²×C₄×C₄₇	—	8	4	no	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₄₇	—	2	4	no	—
67	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
68	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₄₇	—	2	8	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	2	2	no	—
72	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	—	4	4	no	—
77	C₂²⋊Dic₄₇	—	2	4	no	—
78	C₂²⋊Dic₄₇	—	2	8	no	—
79	C₄⋊Dic₄₇	—	2	2	no	—
80	C₄⋊Dic₄₇	—	2	2	no	—
81	C₄⋊Dic₄₇	—	2	4	no	—
82	C₄⋊Dic₄₇	C₂×Q₈	1	8	yes	—
83	Dic₄₇⋊C₄	—	8	8	no	—
84	C₄×Dic₄₇	—	2	8	no	—
85	C₄×Dic₄₇	—	2	2	no	—
86	C₄×Dic₄₇	—	2	2	no	—
87	#10625329	—	4	4	no	—
88	C₂×C₄₇⋊C₈	—	2	2	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	#121	1	2	yes	—
93	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
94	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	—	4	8	no	—
97	C₂×C₄×C₄₇	—	8	8	no	—
98	C₂×C₄×C₄₇	#121	1	2	yes	—
99	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
100	C₂×C₄×C₄₇	#123	1	4	yes	—
101	C₂×C₄×C₄₇	#125	1	8	yes	—
102	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
103	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
107	C₂×Dic₄₇	—	2	2	no	—
108	C₂×Dic₄₇	—	2	2	no	—
109	C₂×Dic₄₇	—	2	8	no	—
110	C₂×Dic₄₇	—	2	4	no	—
111	C₂×Dic₄₇	—	2	4	no	—
112	C₂×Dic₄₇	—	4	4	no	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₈×C₄₇	—	2	2	no	—
115	C₈×C₄₇	—	2	2	no	—
116	C₄₇⋊C₈	—	4	2	no	—
117	#23667	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	—	2	8	no	—
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	#398	1	4	yes	—
122	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
123	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
124	C₂²×C₄₇	#576	1	2	yes	—
125	C₂²×C₄₇	#578	1	2	yes	—
126	C₄×C₄₇	—	8	4	no	—
127	C₄×C₄₇	#398	1	4	yes	—
128	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
129	C₄×C₄₇	C₂×#125	1	2	yes	—
130	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
131	Dic₄₇	—	4	8	no	—
132	Dic₄₇	—	4	2	no	—
133	Dic₄₇	—	4	2	no	—
134	C₂²×#107	D₄₇	1	1	yes	characteristic
135	C₂×#424	—	47	1	no	—
136	#2317	—	47	2	no	—
137	#2087	—	47	4	no	—
138	#2086	—	47	4	no	—
139	#1736	—	47	2	no	—
140	#1735	—	47	1	no	—
141	C₂×C₄₇	—	2	4	no	—
142	C₂×C₄₇	C₂×#398	1	1	yes	characteristic
143	C₂×C₄₇	#3058	1	1	yes	characteristic
144	C₂×C₄₇	#3059	1	1	yes	characteristic
145	C₂×C₄₇	#3083	1	4	yes	—
146	C₂²×M₁₆	—	47	1	no	—
147	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
148	C₂×#117	—	47	1	no	—
149	C₂×#117	—	47	2	no	—
150	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
151	C₂×#107	—	2	4	no	—
152	C₂×#107	C₂×D₄₇	1	2	yes	—
153	C₂×#107	C₂×D₄₇	1	2	yes	—
154	#424	—	94	4	no	—
155	C₂×C₄⋊C₈	—	47	2	no	—
156	C₂×C₄⋊C₈	—	47	2	no	—
157	#389	—	47	8	no	—
158	#385	—	94	4	no	—
159	#384	—	94	2	no	—
160	#382	—	94	2	no	—
161	#340	—	47	2	no	—
162	#340	—	47	2	no	—
163	#340	—	47	8	no	—
164	C₄₇	#23667	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	—	2	2	no	—
166	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
167	C₂×M₁₆	—	94	4	no	—
168	C₂×M₁₆	—	94	2	no	—
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	—	47	2	no	—
171	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
172	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
174	#117	—	47	8	no	—
175	#117	—	94	4	no	—
176	#117	—	94	2	no	—
177	#117	—	94	2	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	47	4	no	—
182	C₂×C₄⋊C₄	—	47	1	no	—
183	C₂×C₄⋊C₄	—	47	1	no	—
184	C₂×C₄⋊C₄	—	94	8	no	—
185	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
186	C₂×C₂²⋊C₄	—	47	2	no	—
187	C₂×C₂²⋊C₄	—	94	2	no	—
188	C₂×C₄²	—	47	2	no	—
189	C₂×C₄²	—	94	2	no	—
190	#107	—	2	4	no	—
191	#107	—	2	4	no	—
192	C₄⋊C₈	—	94	4	no	—
193	C₄⋊C₈	—	94	4	no	—
194	#95	—	94	8	no	—
195	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	8	4	no	—
199	C₂²×C₄	—	47	2	no	—
200	C₂²×C₄	—	47	2	no	—
201	C₂²×C₄	—	94	4	no	—
202	C₂²×C₄	#10625350	1	4	yes	—
203	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
204	C₂²×C₄	#10625352	1	2	yes	—
205	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
206	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
207	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
208	M₁₆	—	188	4	no	—
209	C₂×C₈	—	2	8	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	94	2	no	—
213	C₂×C₈	—	94	4	no	—
214	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
215	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	47	8	no	—
220	C₄⋊C₄	—	94	2	no	—
221	C₄⋊C₄	—	94	2	no	—
222	C₄⋊C₄	—	94	4	no	—
223	C₄⋊C₄	—	376	8	no	—
224	C₂²⋊C₄	—	94	4	no	—
225	C₂²⋊C₄	—	94	8	no	—
226	C₄²	—	94	8	no	—
227	C₄²	—	94	2	no	—
228	C₄²	—	94	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	#13093995	1	2	yes	—
232	C₂³	#13094002	1	4	yes	—
233	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	—	8	8	no	—
237	C₂×C₄	—	94	2	no	—
238	C₂×C₄	—	94	2	no	—
239	C₂×C₄	—	94	8	no	—
240	C₂×C₄	—	94	4	no	—
241	C₂×C₄	—	94	4	no	—
242	C₂×C₄	—	188	4	no	—
243	C₂×C₄	—	188	4	no	—
244	C₂×C₄	#13093995	1	2	yes	—
245	C₂×C₄	#13094002	1	4	yes	—
246	C₂×C₄	#13094007	1	4	yes	—
247	C₂×C₄	#13094014	1	8	yes	—
248	C₂×C₄	#13094022	1	4	yes	—
249	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
250	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
251	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
252	C₂×C₄	C₂×#10625350	1	2	yes	—
253	C₈	—	2	2	no	—
254	C₈	—	2	2	no	—
255	C₈	—	188	2	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	#15689054	1	4	yes	—
260	C₂²	C₂×#13093995	1	1	yes	characteristic
261	C₂²	C₂×#13094002	1	2	yes	—
262	C₂²	#15689924	1	2	yes	—
263	C₂²	#15689936	1	2	yes	—
264	C₄	—	8	4	no	—
265	C₄	—	188	8	no	—
266	C₄	—	188	2	no	—
267	C₄	—	188	2	no	—
268	C₄	#15689054	1	4	yes	—
269	C₄	C₂×#13094007	1	1	yes	characteristic
270	C₄	C₂×#13094014	1	2	yes	—
271	C₄	C₂×#13094022	1	1	yes	characteristic
272	C₂	—	2	4	no	—
273	C₂	C₂×#15689054	1	1	yes	characteristic
274	C₂	#20253032	1	1	yes	characteristic
275	C₂	#20253039	1	1	yes	characteristic
276	C₂	#20253244	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#107×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂²×#107	D₄₇
hypocenter	C₄₇	#23667
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. #20244882	C₂	semidirect
4. #20244877	C₂	semidirect
164. C₄₇	#23667	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689149	2
#20246012	2
#20244882	2
#20244877	2
#20243107	2
#20243105	2
C₂²×#107×C₄₇	2
#23667	47