Subgroups — #26226698

Order 12032.

754 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245997	C₂	1	2	yes	maximal
2	#20244158	C₂	1	4	yes	maximal, complemented
3	#20244152	C₂	1	4	yes	maximal, complemented
4	#20243421	C₂	1	2	yes	maximal
5	#20243413	C₂	1	2	yes	maximal
6	C₂×C₄₇×#424	C₂	1	1	yes	maximal, characteristic
7	C₂×#13093958	C₂²	1	2	yes	—
8	C₂×#13093956	C₂²	1	2	yes	—
9	C₂×#13093956	C₂²	1	2	yes	—
10	C₂×#13093956	C₂²	1	2	yes	—
11	#15689016	—	2	4	no	—
12	#15689009	—	2	4	no	—
13	#15688993	C₂²	1	2	yes	—
14	#15688989	C₂²	1	2	yes	—
15	#15688983	C₂²	1	4	yes	—
16	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
17	#15688886	C₂²	1	4	yes	—
18	#15688886	C₂²	1	4	yes	—
19	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#424	C₄	1	8	yes	complemented
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
24	C₂×C₄⋊Dic₄₇	—	2	4	no	—
25	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
26	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
27	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
28	C₂×C₄×Dic₄₇	—	2	4	no	—
29	#13093958	—	4	2	no	—
30	#13093956	—	4	2	no	—
31	#13093956	—	4	2	no	—
32	#13093956	—	4	2	no	—
33	C₄₇⋊#95	—	2	4	no	—
34	C₄₇⋊#95	—	2	4	no	—
35	C₄₇⋊#95	—	2	4	no	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₄₇	—	4	1	no	—
38	C₂×M₁₆×C₄₇	—	4	1	no	—
39	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₄₇	—	4	1	no	—
41	C₂²×C₈×C₄₇	—	4	1	no	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
45	#117×C₄₇	C₂×C₄	1	4	yes	—
46	#117×C₄₇	Q₈	1	4	yes	—
47	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
50	C₄⋊C₈×C₄₇	C₂×C₄	1	16	yes	—
51	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	1	no	—
53	C₂²×C₄×C₄₇	—	2	1	no	—
54	C₂²×C₄×C₄₇	—	2	1	no	—
55	C₂²×C₄×C₄₇	—	2	1	no	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₇	—	2	4	no	—
62	C₂²×Dic₄₇	—	2	4	no	—
63	M₁₆×C₄₇	—	4	4	no	—
64	C₂×C₈×C₄₇	—	2	2	no	—
65	C₂×C₈×C₄₇	—	2	2	no	—
66	C₂×C₈×C₄₇	—	2	2	no	—
67	C₂×C₈×C₄₇	—	2	2	no	—
68	C₂×C₈×C₄₇	—	4	4	no	—
69	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
71	C₄⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
72	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
73	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
74	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
75	C₂²⋊Dic₄₇	—	8	4	no	—
76	C₄⋊Dic₄₇	—	2	4	no	—
77	C₄⋊Dic₄₇	—	2	4	no	—
78	C₄⋊Dic₄₇	—	2	4	no	—
79	C₄⋊Dic₄₇	—	2	4	no	—
80	C₄⋊Dic₄₇	—	8	4	no	—
81	C₄×Dic₄₇	—	8	4	no	—
82	C₂³×C₄₇	—	2	1	no	—
83	C₂³×C₄₇	—	2	1	no	—
84	C₂³×C₄₇	—	2	4	no	—
85	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	4	no	—
90	C₂×C₄×C₄₇	—	2	4	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	—	2	2	no	—
95	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
103	C₂×Dic₄₇	—	4	4	no	—
104	C₂×Dic₄₇	—	4	4	no	—
105	C₂×Dic₄₇	—	4	4	no	—
106	C₂×Dic₄₇	—	4	4	no	—
107	C₂×Dic₄₇	—	4	4	no	—
108	C₂×Dic₄₇	—	4	4	no	—
109	C₈×C₄₇	—	2	4	no	—
110	C₈×C₄₇	—	2	4	no	—
111	#21967	—	47	1	no	maximal, Sylow
112	C₂²×C₄₇	—	2	2	no	—
113	C₂²×C₄₇	—	2	4	no	—
114	C₂²×C₄₇	—	4	2	no	—
115	C₂²×C₄₇	—	4	2	no	—
116	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₄₇	#537	1	1	yes	characteristic
118	C₂²×C₄₇	#537	1	1	yes	characteristic
119	C₂²×C₄₇	#577	1	4	yes	—
120	C₄×C₄₇	—	2	2	no	—
121	C₄×C₄₇	—	2	8	no	—
122	C₄×C₄₇	#518	1	1	yes	characteristic
123	C₄×C₄₇	#519	1	1	yes	characteristic
124	C₄×C₄₇	#544	1	1	yes	characteristic
125	C₄×C₄₇	#544	1	1	yes	characteristic
126	Dic₄₇	—	8	4	no	—
127	Dic₄₇	—	8	4	no	—
128	C₂×#424	D₄₇	1	1	yes	characteristic
129	#2314	—	47	2	no	—
130	#1949	—	47	4	no	—
131	#1948	—	47	4	no	—
132	#1809	—	47	2	no	—
133	#1807	—	47	2	no	—
134	C₂×C₄₇	—	4	2	no	—
135	C₂×C₄₇	#2316	1	1	yes	characteristic
136	C₂×C₄₇	#2912	1	2	yes	—
137	C₂×C₄₇	#3260	1	2	yes	—
138	C₂×C₄₇	#3263	1	2	yes	—
139	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
140	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
141	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
142	C₂×#107	—	47	2	no	—
143	C₂×#106	—	47	2	no	—
144	C₂×#106	—	47	2	no	—
145	C₂×#106	—	47	2	no	—
146	#424	Dic₄₇	1	8	yes	complemented
147	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
148	#388	—	94	4	no	—
149	#387	—	94	4	no	—
150	#384	—	47	2	no	—
151	#382	—	47	2	no	—
152	#380	—	47	4	no	—
153	C₄×C₄⋊C₄	—	47	4	no	—
154	#340	—	47	4	no	—
155	#340	—	47	4	no	—
156	C₄₇	#21967	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
158	C₂×M₁₆	—	4	1	no	—
159	C₂×M₁₆	—	4	1	no	—
160	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
161	C₂²×C₈	—	4	1	no	—
162	C₂²×C₈	—	4	1	no	—
163	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
164	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
165	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
166	#117	Dic₉₄	1	4	yes	—
167	#117	C₂×Dic₄₇	1	4	yes	—
168	C₂×C₄⋊C₄	—	47	4	no	—
169	C₂×C₄⋊C₄	—	47	4	no	—
170	C₂×C₄⋊C₄	—	94	4	no	—
171	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
172	C₂×C₂²⋊C₄	—	94	4	no	—
173	C₂×C₂²⋊C₄	D₁₈₈	1	2	yes	—
174	C₂×C₄²	—	94	4	no	—
175	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
176	#107	—	188	2	no	—
177	#106	—	188	2	no	—
178	#106	—	188	2	no	—
179	#106	—	188	2	no	—
180	C₄⋊C₈	C₂×Dic₄₇	1	16	yes	—
181	#95	—	94	4	no	—
182	#95	—	94	4	no	—
183	#95	—	94	4	no	—
184	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	94	4	no	—
190	C₂²×C₄	—	94	4	no	—
191	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
192	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
194	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
195	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
196	M₁₆	—	4	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
203	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
204	C₄⋊C₄	—	94	4	no	—
205	C₄⋊C₄	—	94	4	no	—
206	C₄⋊C₄	—	94	4	no	—
207	C₄⋊C₄	—	94	4	no	—
208	C₄⋊C₄	—	376	4	no	—
209	C₄⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
210	C₂²⋊C₄	—	376	4	no	—
211	C₂²⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
212	C₄²	—	376	4	no	—
213	C₄²	C₂×Dic₉₄	1	4	yes	—
214	C₄²	C₂²×Dic₄₇	1	4	yes	—
215	C₂³	—	2	4	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
219	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	188	4	no	—
229	C₂×C₄	—	188	4	no	—
230	C₂×C₄	—	188	4	no	—
231	C₂×C₄	—	188	4	no	—
232	C₂×C₄	—	188	4	no	—
233	C₂×C₄	—	188	4	no	—
234	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
235	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
236	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
237	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
238	C₂×C₄	#13094069	1	1	yes	characteristic
239	C₂×C₄	#13094069	1	1	yes	characteristic
240	C₂×C₄	#13094072	1	1	yes	characteristic
241	C₂×C₄	#13094072	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	—	2	4	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
249	C₂²	#15689704	1	1	yes	characteristic
250	C₂²	#15689704	1	1	yes	characteristic
251	C₂²	#15689935	1	4	yes	—
252	C₄	—	2	8	no	—
253	C₄	—	2	2	no	—
254	C₄	—	376	4	no	—
255	C₄	—	376	4	no	—
256	C₄	#15689598	1	1	yes	characteristic
257	C₄	#15689601	1	1	yes	characteristic
258	C₄	#15689754	1	1	yes	characteristic
259	C₄	#15689754	1	1	yes	characteristic
260	C₂	—	4	2	no	—
261	C₂	#20246006	1	1	yes	characteristic
262	C₂	#20251983	1	2	yes	—
263	C₂	#20254835	1	2	yes	—
264	C₂	#20254859	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#424	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#424	D₄₇
hypocenter	C₄₇	#21967
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
2. #20244158	C₂	semidirect
3. #20244152	C₂	semidirect
22. C₄₇×#424	C₄	semidirect
146. #424	Dic₄₇	semidirect
156. C₄₇	#21967	semidirect

Maximal subgroups

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

subgroup	index
#20245997	2
#20244158	2
#20244152	2
#20243421	2
#20243413	2
C₂×C₄₇×#424	2
#21967	47