Subgroups — #26226704

Order 12032.

754 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	#20245997	C₂	1	2	yes	maximal
2	#20244158	C₂	1	4	yes	maximal, complemented
3	#20244147	C₂	1	4	yes	maximal, complemented
4	#20243421	C₂	1	2	yes	maximal
5	#20243417	C₂	1	2	yes	maximal
6	C₂×C₄₇×#424	C₂	1	1	yes	maximal, characteristic
7	C₂×#13093958	C₂²	1	2	yes	—
8	C₂×#13093958	C₂²	1	2	yes	—
9	C₂×#13093958	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	#13093958	—	4	2	no	—
31	#13093958	—	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	2	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	4	no	—
91	C₂×C₄×C₄₇	—	2	4	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	—	2	2	no	—
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
103	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
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	C₂×Dic₄₇	—	4	4	no	—
110	C₈×C₄₇	—	2	4	no	—
111	C₈×C₄₇	—	2	4	no	—
112	#21968	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	2	4	no	—
115	C₂²×C₄₇	—	4	1	no	—
116	C₂²×C₄₇	—	4	1	no	—
117	C₂²×C₄₇	—	4	2	no	—
118	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₄₇	#537	1	1	yes	characteristic
120	C₂²×C₄₇	#537	1	1	yes	characteristic
121	C₂²×C₄₇	#576	1	2	yes	—
122	C₂²×C₄₇	#578	1	2	yes	—
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	—	2	8	no	—
125	C₄×C₄₇	#518	1	1	yes	characteristic
126	C₄×C₄₇	#519	1	1	yes	characteristic
127	C₄×C₄₇	#544	1	1	yes	characteristic
128	C₄×C₄₇	#544	1	1	yes	characteristic
129	Dic₄₇	—	8	4	no	—
130	Dic₄₇	—	8	4	no	—
131	C₂×#424	D₄₇	1	1	yes	characteristic
132	#2314	—	47	2	no	—
133	#1949	—	47	4	no	—
134	#1947	—	47	4	no	—
135	#1809	—	47	2	no	—
136	#1808	—	47	2	no	—
137	C₂×C₄₇	—	4	2	no	—
138	C₂×C₄₇	#2316	1	1	yes	characteristic
139	C₂×C₄₇	#2913	1	2	yes	—
140	C₂×C₄₇	#3259	1	1	yes	characteristic
141	C₂×C₄₇	#3261	1	1	yes	characteristic
142	C₂×C₄₇	#3262	1	2	yes	—
143	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
144	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
145	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
146	C₂×#107	—	47	2	no	—
147	C₂×#107	—	47	2	no	—
148	C₂×#107	—	47	2	no	—
149	C₂×#106	—	47	2	no	—
150	#424	Dic₄₇	1	8	yes	complemented
151	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
152	#388	—	94	4	no	—
153	#387	—	94	4	no	—
154	#384	—	47	2	no	—
155	#382	—	47	2	no	—
156	#380	—	47	4	no	—
157	C₄×C₄⋊C₄	—	47	4	no	—
158	#340	—	47	4	no	—
159	#340	—	47	4	no	—
160	C₄₇	#21968	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
162	C₂×M₁₆	—	4	1	no	—
163	C₂×M₁₆	—	4	1	no	—
164	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
165	C₂²×C₈	—	4	1	no	—
166	C₂²×C₈	—	4	1	no	—
167	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
168	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
169	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
170	#117	Dic₉₄	1	4	yes	—
171	#117	C₂×Dic₄₇	1	4	yes	—
172	C₂×C₄⋊C₄	—	47	4	no	—
173	C₂×C₄⋊C₄	—	47	4	no	—
174	C₂×C₄⋊C₄	—	94	4	no	—
175	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
176	C₂×C₂²⋊C₄	—	94	4	no	—
177	C₂×C₂²⋊C₄	D₁₈₈	1	2	yes	—
178	C₂×C₄²	—	94	4	no	—
179	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
180	#107	—	188	2	no	—
181	#107	—	188	2	no	—
182	#107	—	188	2	no	—
183	#106	—	188	2	no	—
184	C₄⋊C₈	C₂×Dic₄₇	1	16	yes	—
185	#95	—	94	4	no	—
186	#95	—	94	4	no	—
187	#95	—	94	4	no	—
188	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	94	4	no	—
194	C₂²×C₄	—	94	4	no	—
195	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
196	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
198	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
199	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
200	M₁₆	—	4	4	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
207	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
208	C₄⋊C₄	—	94	4	no	—
209	C₄⋊C₄	—	94	4	no	—
210	C₄⋊C₄	—	94	4	no	—
211	C₄⋊C₄	—	94	4	no	—
212	C₄⋊C₄	—	376	4	no	—
213	C₄⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
214	C₂²⋊C₄	—	376	4	no	—
215	C₂²⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
216	C₄²	—	376	4	no	—
217	C₄²	C₂×Dic₉₄	1	4	yes	—
218	C₄²	C₂²×Dic₄₇	1	4	yes	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
224	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	188	4	no	—
234	C₂×C₄	—	188	4	no	—
235	C₂×C₄	—	188	4	no	—
236	C₂×C₄	—	188	4	no	—
237	C₂×C₄	—	188	4	no	—
238	C₂×C₄	—	188	4	no	—
239	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
240	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
241	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
242	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
243	C₂×C₄	#13094069	1	1	yes	characteristic
244	C₂×C₄	#13094069	1	1	yes	characteristic
245	C₂×C₄	#13094072	1	1	yes	characteristic
246	C₂×C₄	#13094072	1	1	yes	characteristic
247	C₈	—	2	4	no	—
248	C₈	—	2	4	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
255	C₂²	#15689704	1	1	yes	characteristic
256	C₂²	#15689704	1	1	yes	characteristic
257	C₂²	#15689928	1	2	yes	—
258	C₂²	#15689940	1	2	yes	—
259	C₄	—	2	8	no	—
260	C₄	—	2	2	no	—
261	C₄	—	376	4	no	—
262	C₄	—	376	4	no	—
263	C₄	#15689598	1	1	yes	characteristic
264	C₄	#15689601	1	1	yes	characteristic
265	C₄	#15689754	1	1	yes	characteristic
266	C₄	#15689754	1	1	yes	characteristic
267	C₂	—	4	2	no	—
268	C₂	#20246006	1	1	yes	characteristic
269	C₂	#20251988	1	2	yes	—
270	C₂	#20254827	1	1	yes	characteristic
271	C₂	#20254842	1	1	yes	characteristic
272	C₂	#20254849	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	#21968
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. #20244147	C₂	semidirect
22. C₄₇×#424	C₄	semidirect
150. #424	Dic₄₇	semidirect
160. C₄₇	#21968	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
#20244147	2
#20243421	2
#20243417	2
C₂×C₄₇×#424	2
#21968	47