Subgroups — #26216892

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252019	C₂	1	2	yes	maximal
2	C₂×C₄×#10625324	C₂	1	1	yes	maximal, characteristic
3	#20243561	C₂	1	2	yes	maximal, complemented
4	#20243527	C₂	1	2	yes	maximal, complemented
5	#20243429	C₂	1	2	yes	maximal
6	#20243427	C₂	1	2	yes	maximal
7	C₄×#13093959	C₂	1	2	yes	maximal, complemented
8	C₄×#13093960	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#108×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10625324	C₂²	1	1	yes	characteristic
11	C₂²×#10625324	C₂²	1	1	yes	characteristic
12	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
13	#15689178	—	2	4	no	—
14	#15689174	—	2	4	no	—
15	C₂×#13093960	C₂²	1	2	yes	—
16	C₂×#13093959	C₂²	1	4	yes	—
17	#15689146	C₂²	1	4	yes	—
18	#15689086	C₂²	1	4	yes	—
19	C₄×#10625329	C₂²	1	2	yes	—
20	C₄×#10625324	C₄	1	8	yes	—
21	C₂×#13093934	C₂²	1	1	yes	characteristic
22	C₂×#13093934	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
24	#15688888	C₂²	1	2	yes	—
25	#15688887	C₂²	1	4	yes	—
26	C₂×#108×C₄₇	C₂²	1	2	yes	—
27	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#341	C₂²	1	2	yes	—
30	C₂²×C₄×D₄₇	Q₈	1	2	yes	—
31	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
32	C₂×#10625329	C₂³	1	4	yes	—
33	C₂×#10625324	—	2	4	no	—
34	C₂×#10625324	—	2	4	no	—
35	C₂×#10625324	C₂×C₄	1	4	yes	—
36	C₂×#10625324	C₂×C₄	1	4	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
39	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
41	C₄²×D₄₇	C₂×C₄	1	4	yes	—
42	C₄²×D₄₇	D₄	1	2	yes	—
43	C₄²×D₄₇	D₄	1	2	yes	—
44	#13093960	—	2	4	no	—
45	#13093959	—	2	8	no	—
46	Dic₄₇⋊C₈	—	2	8	no	—
47	(C₂×D₄₇)⋊C₈	—	2	8	no	—
48	#13093935	—	2	2	no	—
49	#13093934	C₂×C₄	1	4	yes	—
50	#13093934	C₂×C₄	1	4	yes	—
51	C₄×C₄₇⋊C₈	—	2	2	no	—
52	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
53	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	#108×C₄₇	—	2	4	no	—
58	#97×C₄₇	—	2	2	no	—
59	C₄×C₈×C₄₇	—	2	2	no	—
60	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
61	C₂³×D₄₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
65	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
66	C₂×C₄×D₄₇	—	2	2	no	—
67	C₂×C₄×D₄₇	—	2	2	no	—
68	C₂×C₄×D₄₇	—	4	4	no	—
69	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
70	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
71	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
72	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
73	M₁₆×C₄₇	—	2	4	no	—
74	M₁₆×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₇	#41	1	4	yes	—
80	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
81	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
84	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
85	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
86	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
87	#10625329	—	2	4	no	—
88	#10625329	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₄₇⋊C₈	#41	1	4	yes	—
94	#10625324	—	2	8	no	—
95	#10625324	—	2	8	no	—
96	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂²×D₄₇	—	2	2	no	—
98	C₂²×D₄₇	—	4	2	no	—
99	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
107	C₂×Dic₄₇	—	2	2	no	—
108	C₂×Dic₄₇	—	4	2	no	—
109	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
111	C₄×D₄₇	—	2	2	no	—
112	C₄×D₄₇	—	2	2	no	—
113	C₄×D₄₇	—	2	4	no	—
114	C₄×D₄₇	—	2	8	no	—
115	C₄×D₄₇	—	4	4	no	—
116	C₈×C₄₇	—	2	2	no	—
117	C₈×C₄₇	—	4	4	no	—
118	C₈×C₄₇	#117	1	4	yes	—
119	C₄₇⋊C₈	—	2	2	no	—
120	C₄₇⋊C₈	—	4	4	no	—
121	C₄₇⋊C₈	#117	1	4	yes	—
122	#20827	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₄₇	#533	1	2	yes	—
126	C₂×D₄₇	—	2	4	no	—
127	C₂×D₄₇	—	2	2	no	—
128	C₂×D₄₇	—	4	4	no	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	#430	1	4	yes	—
131	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
132	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
133	C₄×C₄₇	#532	1	2	yes	—
134	Dic₄₇	—	2	4	no	—
135	Dic₄₇	—	2	2	no	—
136	#2920	—	47	2	no	—
137	C₂×C₄×M₁₆	—	47	1	no	—
138	#1849	—	47	2	no	—
139	#1838	—	47	2	no	—
140	#1811	—	47	2	no	—
141	#1811	—	47	2	no	—
142	C₄×#108	—	47	2	no	—
143	C₄×#108	—	47	1	no	—
144	C₄×#108	D₄₇	1	1	yes	characteristic, complemented
145	C₂×C₄₇	—	2	2	no	—
146	C₂×C₄₇	#2285	1	1	yes	characteristic
147	C₂×C₄₇	C₂×#430	1	2	yes	—
148	D₄₇	—	4	2	no	—
149	C₂²×M₁₆	—	47	1	no	—
150	C₂²×M₁₆	—	47	1	no	—
151	C₂²×C₄²	—	47	1	no	—
152	#432	—	94	4	no	—
153	#431	—	94	4	no	—
154	C₂×#108	—	47	4	no	—
155	C₂×#108	—	47	2	no	—
156	C₂×#108	C₂×D₄₇	1	2	yes	—
157	#423	—	47	4	no	—
158	#407	—	47	4	no	—
159	C₄×M₁₆	—	47	8	no	—
160	C₄×M₁₆	—	47	2	no	—
161	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
162	C₂×#97	—	47	1	no	—
163	C₂×#97	—	47	1	no	—
164	C₂×C₄×C₈	—	47	1	no	—
165	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
166	#341	—	47	4	no	—
167	#341	—	47	2	no	—
168	#341	C₂×D₄₇	1	2	yes	—
169	C₄₇	#20827	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	—	47	1	no	—
171	C₂³×C₄	—	47	2	no	—
172	C₂×M₁₆	—	47	4	no	—
173	C₂×M₁₆	—	47	4	no	—
174	C₂×M₁₆	—	47	4	no	—
175	C₂×M₁₆	—	94	4	no	—
176	C₂×M₁₆	—	94	4	no	—
177	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
178	C₂²×C₈	—	47	1	no	—
179	C₂²×C₈	—	47	1	no	—
180	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
182	C₂×C₄²	—	47	2	no	—
183	C₂×C₄²	—	47	1	no	—
184	C₂×C₄²	—	47	4	no	—
185	C₂×C₄²	—	47	2	no	—
186	C₂×C₄²	—	47	2	no	—
187	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
188	#108	—	2	4	no	—
189	#108	—	94	8	no	—
190	#108	—	94	4	no	—
191	C₄⋊C₈	—	94	8	no	—
192	C₂²⋊C₈	—	94	8	no	—
193	#97	—	2	2	no	—
194	#97	—	47	4	no	—
195	#97	—	47	4	no	—
196	#97	—	94	2	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	—	47	4	no	—
199	C₄×C₈	—	94	2	no	—
200	C₄×C₈	C₄×D₄₇	1	4	yes	—
201	C₂⁴	—	47	1	no	—
202	C₂²×C₄	—	47	1	no	—
203	C₂²×C₄	—	47	2	no	—
204	C₂²×C₄	—	47	4	no	—
205	C₂²×C₄	—	47	2	no	—
206	C₂²×C₄	—	47	2	no	—
207	C₂²×C₄	—	47	4	no	—
208	C₂²×C₄	—	94	2	no	—
209	C₂²×C₄	—	94	2	no	—
210	C₂²×C₄	—	188	4	no	—
211	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
212	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	94	8	no	—
216	M₁₆	—	94	8	no	—
217	M₁₆	—	94	4	no	—
218	M₁₆	—	94	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	47	2	no	—
222	C₂×C₈	—	47	4	no	—
223	C₂×C₈	—	47	2	no	—
224	C₂×C₈	—	94	2	no	—
225	C₂×C₈	—	94	4	no	—
226	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
227	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
228	C₂×C₈	#10625350	1	4	yes	—
229	C₄²	—	47	4	no	—
230	C₄²	—	47	4	no	—
231	C₄²	—	47	2	no	—
232	C₄²	—	47	2	no	—
233	C₄²	C₂×C₄×D₄₇	1	2	yes	—
234	C₄²	D₄×D₄₇	1	1	yes	characteristic
235	C₄²	D₄×D₄₇	1	1	yes	characteristic
236	C₂³	—	47	2	no	—
237	C₂³	—	94	2	no	—
238	C₂³	—	188	2	no	—
239	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	47	2	no	—
242	C₂×C₄	—	47	4	no	—
243	C₂×C₄	—	94	2	no	—
244	C₂×C₄	—	94	2	no	—
245	C₂×C₄	—	94	2	no	—
246	C₂×C₄	—	94	4	no	—
247	C₂×C₄	—	94	8	no	—
248	C₂×C₄	—	188	4	no	—
249	C₂×C₄	—	188	2	no	—
250	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
251	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
252	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
253	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
254	C₂×C₄	#41×D₄₇	1	2	yes	—
255	C₂×C₄	C₂×#10625350	1	2	yes	—
256	C₈	—	2	2	no	—
257	C₈	—	4	4	no	—
258	C₈	—	47	4	no	—
259	C₈	—	94	2	no	—
260	C₈	—	188	4	no	—
261	C₈	#13093981	1	4	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	—	94	2	no	—
264	C₂²	—	94	4	no	—
265	C₂²	—	188	4	no	—
266	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
267	C₂²	#15689677	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	94	2	no	—
270	C₄	—	94	4	no	—
271	C₄	#15689170	1	4	yes	—
272	C₄	#117×D₄₇	1	1	yes	characteristic
273	C₄	C₂×#13093981	1	1	yes	characteristic
274	C₄	#15689670	1	2	yes	—
275	C₂	—	2	2	no	—
276	C₂	—	188	2	no	—
277	C₂	#20245792	1	1	yes	characteristic
278	C₂	C₂×#15689170	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#108×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#108	D₄₇
hypocenter	C₄₇	#20827
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
3. #20243561	C₂	semidirect
4. #20243527	C₂	semidirect
7. C₄×#13093959	C₂	semidirect
8. C₄×#13093960	C₂	semidirect
9. C₄×#108×C₄₇	C₂	semidirect
144. C₄×#108	D₄₇	semidirect
169. C₄₇	#20827	semidirect

Maximal subgroups

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

subgroup	index
#20252019	2
C₂×C₄×#10625324	2
#20243561	2
#20243527	2
#20243429	2
#20243427	2
C₄×#13093959	2
C₄×#13093960	2
C₄×#108×C₄₇	2
#20827	47