Subgroups — #26181921

Order 12032.

772 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	2	yes	maximal
2	#20241685	C₂	1	4	yes	maximal, complemented
3	#20241681	C₂	1	4	yes	maximal, complemented
4	#20241672	C₂	1	4	yes	maximal, complemented
5	C₄₇×#2920	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₄	1	4	yes	—
7	#15689176	C₄	1	4	yes	—
8	#15689176	C₂²	1	2	yes	—
9	#15689176	C₂²	1	2	yes	—
10	#15689176	C₂²	1	4	yes	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	4	yes	—
12	C₂×C₂²⋊(C₄₇⋊C₈)	C₄	1	4	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
14	#15688845	C₂²	1	8	yes	complemented
15	#15688844	C₂²	1	4	yes	complemented
16	C₈×C₄₇⋊C₈	C₂²	1	4	yes	complemented
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#432	C₄	1	4	yes	—
19	C₄₇×#431	C₂²	1	4	yes	—
20	C₄₇×#423	C₄	1	2	yes	—
21	C₄₇×#407	C₄	1	2	yes	—
22	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
23	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
24	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
25	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
26	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
27	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
28	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
29	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
30	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
31	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
32	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
44	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
45	#97×C₄₇	C₂×C₄	1	4	yes	—
46	C₄×C₈×C₄₇	C₂³	1	4	yes	—
47	C₂⁴×C₄₇	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	2	8	no	—
49	C₂²×C₄×C₄₇	—	4	2	no	—
50	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
51	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
52	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
53	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	M₁₆×C₄₇	—	4	4	no	—
61	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
62	C₄²×C₄₇	C₄²	1	1	yes	characteristic
63	C₄²×C₄₇	C₄²	1	2	yes	—
64	C₄²×C₄₇	C₄²	1	2	yes	—
65	C₄²×C₄₇	C₄²	1	1	yes	characteristic
66	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂×C₄₇⋊C₈	—	2	16	no	—
73	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
74	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
75	C₂³×C₄₇	—	2	4	no	—
76	C₂³×C₄₇	—	4	1	no	—
77	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
78	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	—	2	4	no	—
81	C₂×C₄×C₄₇	—	2	4	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	4	2	no	—
87	C₂×C₄×C₄₇	—	4	1	no	—
88	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
89	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₄₇	#131	1	8	yes	—
94	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
96	C₈×C₄₇	—	2	8	no	—
97	C₄₇⋊C₈	—	2	8	no	—
98	C₄₇⋊C₈	—	2	8	no	—
99	#17325	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	—	2	2	no	—
101	C₂²×C₄₇	—	2	2	no	—
102	C₂²×C₄₇	—	2	2	no	—
103	C₂²×C₄₇	—	2	1	no	—
104	C₂²×C₄₇	—	4	4	no	—
105	C₂²×C₄₇	#405	1	4	yes	—
106	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
107	C₄×C₄₇	—	2	2	no	—
108	C₄×C₄₇	—	2	2	no	—
109	C₄×C₄₇	—	2	1	no	—
110	C₄×C₄₇	—	2	2	no	—
111	C₄×C₄₇	#405	1	4	yes	—
112	C₄×C₄₇	C₂×#131	1	4	yes	—
113	C₄×C₄₇	#568	1	2	yes	—
114	#2920	D₄₇	1	1	yes	characteristic
115	C₂×#431	—	47	2	no	—
116	#1453	—	47	4	no	—
117	#1452	—	47	4	no	—
118	#1450	—	47	4	no	—
119	C₂×C₄₇	—	2	4	no	—
120	C₂×C₄₇	—	4	1	no	—
121	C₂×C₄₇	C₂×#405	1	2	yes	—
122	C₂×C₄₇	#2873	1	1	yes	characteristic
123	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
124	#432	Dic₄₇	1	4	yes	—
125	#431	—	47	4	no	—
126	#431	—	47	4	no	—
127	#431	—	47	2	no	—
128	#431	—	47	2	no	—
129	#431	—	47	4	no	—
130	#431	C₂×D₄₇	1	4	yes	—
131	#423	Dic₄₇	1	2	yes	—
132	C₂×C₄⋊C₈	—	47	4	no	—
133	#407	Dic₄₇	1	2	yes	—
134	C₂×C₂²⋊C₈	—	47	4	no	—
135	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
136	C₂×C₄×C₈	—	47	4	no	—
137	#322	—	47	4	no	—
138	#322	—	47	8	no	—
139	C₈²	—	47	4	no	—
140	C₄₇	#17325	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₄	C₄×D₄₇	1	2	yes	—
142	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
143	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
144	C₂²×C₈	—	47	8	no	—
145	C₂×C₄²	C₄×D₄₇	1	2	yes	—
146	C₂×C₄²	C₄×D₄₇	1	2	yes	—
147	C₂×C₄²	C₄×D₄₇	1	2	yes	—
148	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
149	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
150	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
151	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
152	C₄⋊C₈	—	47	4	no	—
153	C₄⋊C₈	—	47	4	no	—
154	C₄⋊C₈	—	47	8	no	—
155	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
156	C₂²⋊C₈	—	47	4	no	—
157	C₂²⋊C₈	—	47	4	no	—
158	C₂²⋊C₈	—	47	8	no	—
159	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
160	#97	C₂×Dic₄₇	1	4	yes	—
161	C₄×C₈	—	47	8	no	—
162	C₄×C₈	—	47	4	no	—
163	C₄×C₈	—	47	4	no	—
164	C₄×C₈	C₂²×D₄₇	1	4	yes	—
165	C₂⁴	C₄×Dic₄₇	1	1	yes	characteristic
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
169	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
170	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
171	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
172	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
173	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
174	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
175	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
176	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
177	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
178	M₁₆	—	4	4	no	—
179	C₂×C₈	—	47	8	no	—
180	C₂×C₈	—	47	8	no	—
181	C₂×C₈	—	94	16	no	—
182	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
183	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
184	C₄²	C₄×Dic₄₇	1	2	yes	—
185	C₄²	C₄×Dic₄₇	1	2	yes	—
186	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
187	C₄²	C₂×C₄×D₄₇	1	2	yes	—
188	C₄²	C₂×C₄×D₄₇	1	2	yes	—
189	C₄²	C₂×C₄×D₄₇	1	2	yes	—
190	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
191	C₄²	C₂²×Dic₄₇	1	2	yes	—
192	C₄²	C₂³×D₄₇	1	1	yes	characteristic
193	C₂³	—	2	4	no	—
194	C₂³	—	4	1	no	—
195	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
196	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
207	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
208	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
209	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
210	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
211	C₂×C₄	#13094032	1	8	yes	—
212	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
213	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
214	C₈	—	2	8	no	—
215	C₈	—	94	8	no	—
216	C₈	—	94	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	4	4	no	—
222	C₂²	#15689081	1	4	yes	—
223	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	2	no	—
228	C₄	#15689081	1	4	yes	—
229	C₄	C₂×#13094032	1	4	yes	—
230	C₄	#15689890	1	2	yes	—
231	C₂	—	2	4	no	—
232	C₂	—	4	1	no	—
233	C₂	C₂×#15689081	1	2	yes	—
234	C₂	#20251807	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#2920	C₂
prime radical	C₂⁴×C₄₇	C₄²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#2920	D₄₇
hypocenter	C₄₇	#17325
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
2. #20241685	C₂	semidirect
3. #20241681	C₂	semidirect
4. #20241672	C₂	semidirect
14. #15688845	C₂²	semidirect
15. #15688844	C₂²	semidirect
16. C₈×C₄₇⋊C₈	C₂²	semidirect
140. C₄₇	#17325	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
#20241685	2
#20241681	2
#20241672	2
C₄₇×#2920	2
#17325	47