Subgroups — #26216896

Order 12032.

718 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252023	C₂	1	2	yes	maximal
2	C₂×#15689078	C₂	1	1	yes	maximal, characteristic
3	#20243558	C₂	1	2	yes	maximal, complemented
4	#20243527	C₂	1	2	yes	maximal, complemented
5	#20243428	C₂	1	2	yes	maximal
6	#20243426	C₂	1	1	yes	maximal, characteristic
7	#20243293	C₂	1	2	yes	maximal, complemented
8	#108×Dic₄₇	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1810	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10625329	C₂²	1	1	yes	characteristic
11	C₂×#13093983	C₂²	1	1	yes	characteristic
12	#15689181	—	2	8	no	—
13	C₂×#13093960	C₂²	1	2	yes	—
14	C₂×#13093959	C₂²	1	4	yes	—
15	#15689146	C₂²	1	4	yes	—
16	#15689087	C₂²	1	2	yes	—
17	#15689078	C₄	1	8	yes	—
18	M₁₆×Dic₄₇	C₂²	1	4	yes	—
19	C₂×#13093934	C₂²	1	1	yes	characteristic
20	C₂×#13093934	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
23	#15688888	C₂²	1	2	yes	—
24	#15688887	C₂²	1	4	yes	—
25	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#108×C₄₇	C₂²	1	2	yes	—
27	C₄₇×#407	C₂²	1	2	yes	—
28	C₄₇×#341	C₂²	1	2	yes	—
29	C₂×#10625329	—	2	4	no	—
30	C₂×#10625329	C₂×C₄	1	4	yes	—
31	C₂×#10625329	C₂³	1	4	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
34	#13093983	C₂×C₄	1	4	yes	—
35	#13093983	D₄	1	4	yes	—
36	C₂×C₄⋊Dic₄₇	Q₈	1	2	yes	—
37	C₂×C₂²⋊Dic₄₇	Q₈	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
40	#13093960	—	2	4	no	—
41	#13093959	—	2	8	no	—
42	Dic₄₇⋊C₈	—	2	8	no	—
43	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
44	#13093934	—	2	4	no	—
45	#13093934	C₂×C₄	1	4	yes	—
46	#13093934	C₂×C₄	1	4	yes	—
47	C₈×Dic₄₇	—	2	4	no	—
48	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
49	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
52	C₂²×C₈×C₄₇	—	2	4	no	—
53	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	#108×C₄₇	—	2	4	no	—
57	C₂²⋊C₈×C₄₇	—	2	4	no	—
58	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
66	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
67	M₁₆×C₄₇	—	2	4	no	—
68	M₁₆×C₄₇	—	2	4	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	8	no	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₄₇	#41	1	4	yes	—
75	C₂²⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
76	C₄⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
77	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
78	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
79	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
80	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
81	#10625329	—	2	8	no	—
82	#10625329	—	2	4	no	—
83	#10625329	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	2	2	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₄₇⋊C₈	#41	1	4	yes	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	—	4	2	no	—
91	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
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	—	2	4	no	—
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×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
102	C₂×Dic₄₇	—	4	2	no	—
103	C₂×Dic₄₇	—	4	2	no	—
104	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
106	C₈×C₄₇	—	2	2	no	—
107	C₈×C₄₇	—	4	4	no	—
108	C₈×C₄₇	#117	1	4	yes	—
109	C₄₇⋊C₈	—	2	2	no	—
110	C₄₇⋊C₈	—	4	4	no	—
111	C₄₇⋊C₈	#117	1	4	yes	—
112	#20828	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	2	4	no	—
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	4	4	no	—
117	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₄₇	#533	1	2	yes	—
119	C₄×C₄₇	—	2	2	no	—
120	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
121	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
122	C₄×C₄₇	#532	1	2	yes	—
123	Dic₄₇	—	2	4	no	—
124	Dic₄₇	—	2	4	no	—
125	#2921	—	47	2	no	—
126	C₂×#405	—	47	1	no	—
127	#1848	—	47	2	no	—
128	#1838	—	47	2	no	—
129	#1811	—	47	2	no	—
130	#1810	—	47	1	no	—
131	#1810	D₄₇	1	1	yes	characteristic
132	#1778	—	47	2	no	—
133	C₄×#108	—	47	2	no	—
134	C₂×C₄₇	—	2	2	no	—
135	C₂×C₄₇	—	4	2	no	—
136	C₂×C₄₇	#2285	1	1	yes	characteristic
137	C₂×C₄₇	#2916	1	2	yes	—
138	C₂²×M₁₆	—	47	1	no	—
139	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
140	C₂×#117	—	47	1	no	—
141	#433	—	94	8	no	—
142	C₂×#108	—	47	4	no	—
143	C₂×#108	—	47	2	no	—
144	C₂×#108	C₂×D₄₇	1	2	yes	—
145	#423	—	47	4	no	—
146	#407	—	47	2	no	—
147	#407	C₂×D₄₇	1	2	yes	—
148	#405	—	47	8	no	—
149	C₄×M₁₆	—	47	4	no	—
150	C₂×#97	—	47	1	no	—
151	C₂×#97	—	47	1	no	—
152	C₂×C₄×C₈	—	47	1	no	—
153	C₂×C₄×C₈	—	47	1	no	—
154	#341	—	47	4	no	—
155	#341	—	47	2	no	—
156	#341	C₂×D₄₇	1	2	yes	—
157	C₄₇	#20828	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
159	C₂×M₁₆	—	47	4	no	—
160	C₂×M₁₆	—	47	4	no	—
161	C₂×M₁₆	—	94	4	no	—
162	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
163	C₂²×C₈	—	2	4	no	—
164	C₂²×C₈	—	47	1	no	—
165	C₂²×C₈	—	47	1	no	—
166	C₂²×C₈	C₄×D₄₇	1	4	yes	—
167	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
169	#117	—	47	4	no	—
170	#117	—	47	4	no	—
171	C₂×C₄⋊C₄	—	47	2	no	—
172	C₂×C₂²⋊C₄	—	47	2	no	—
173	C₂×C₄²	—	47	1	no	—
174	C₂×C₄²	—	47	1	no	—
175	#108	—	2	4	no	—
176	#108	—	94	8	no	—
177	#108	—	94	4	no	—
178	C₄⋊C₈	—	94	8	no	—
179	C₂²⋊C₈	—	2	4	no	—
180	C₂²⋊C₈	—	94	4	no	—
181	#97	—	47	4	no	—
182	#97	—	47	4	no	—
183	#97	—	94	4	no	—
184	C₄×C₈	—	47	4	no	—
185	C₄×C₈	—	47	4	no	—
186	C₄×C₈	—	94	4	no	—
187	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	47	2	no	—
191	C₂²×C₄	—	47	2	no	—
192	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
193	C₂²×C₄	D₄×D₄₇	1	2	yes	—
194	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
195	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
196	M₁₆	—	2	4	no	—
197	M₁₆	—	2	4	no	—
198	M₁₆	—	94	8	no	—
199	M₁₆	—	94	4	no	—
200	M₁₆	—	94	4	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	8	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	47	2	no	—
205	C₂×C₈	—	47	4	no	—
206	C₂×C₈	—	47	2	no	—
207	C₂×C₈	—	94	2	no	—
208	C₂×C₈	—	94	4	no	—
209	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
210	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
211	C₂×C₈	#10625352	1	4	yes	—
212	C₄⋊C₄	—	47	8	no	—
213	C₂²⋊C₄	—	47	8	no	—
214	C₄²	—	47	2	no	—
215	C₄²	—	47	2	no	—
216	C₄²	—	47	2	no	—
217	C₄²	—	47	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	4	2	no	—
220	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
221	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	47	4	no	—
227	C₂×C₄	—	47	4	no	—
228	C₂×C₄	—	188	2	no	—
229	C₂×C₄	—	188	2	no	—
230	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
231	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
232	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
233	C₂×C₄	#41×D₄₇	1	2	yes	—
234	C₂×C₄	C₂×#10625352	1	2	yes	—
235	C₈	—	2	2	no	—
236	C₈	—	4	4	no	—
237	C₈	—	47	4	no	—
238	C₈	—	94	2	no	—
239	C₈	—	188	4	no	—
240	C₈	#13093980	1	4	yes	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	4	4	no	—
245	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
246	C₂²	#15689675	1	2	yes	—
247	C₄	—	2	2	no	—
248	C₄	—	94	4	no	—
249	C₄	—	94	4	no	—
250	C₄	#117×D₄₇	1	1	yes	characteristic
251	C₄	C₂×#13093980	1	1	yes	characteristic
252	C₄	#15689668	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	4	2	no	—
255	C₂	#20245790	1	1	yes	characteristic
256	C₂	#20251996	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1810	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1810	D₄₇
hypocenter	C₄₇	#20828
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1810

Complements

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

N	H ≅ G/N	type
3. #20243558	C₂	semidirect
4. #20243527	C₂	semidirect
7. #20243293	C₂	semidirect
8. #108×Dic₄₇	C₂	semidirect
157. C₄₇	#20828	semidirect

Maximal subgroups

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

subgroup	index
#20252023	2
C₂×#15689078	2
#20243558	2
#20243527	2
#20243428	2
#20243426	2
#20243293	2
#108×Dic₄₇	2
C₄₇×#1810	2
#20828	47