Subgroups — #26132487

Order 12032.

542 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	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#20241465	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₇⋊C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₄₇	C₄	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
8	C₂×C₈×Dic₄₇	—	2	2	no	—
9	C₂×C₈×Dic₄₇	—	2	2	no	—
10	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
11	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
12	C₂²⋊(C₄₇⋊C₁₆)	—	2	2	no	—
13	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
15	C₄₇×C₂²⋊C₁₆	—	2	2	no	—
16	C₂³×Dic₄₇	C₈	1	1	yes	characteristic
17	C₂³×Dic₄₇	C₈	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	C₂²×C₄₇⋊C₈	—	2	4	no	—
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₄₇	—	2	2	no	—
22	C₂×C₄×Dic₄₇	—	2	2	no	—
23	C₂×C₄×Dic₄₇	C₈	1	2	yes	—
24	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
25	C₂×C₄₇⋊C₁₆	—	2	2	no	—
26	C₂×C₄₇⋊C₁₆	—	2	4	no	—
27	C₈×Dic₄₇	—	2	4	no	—
28	C₈×Dic₄₇	—	2	2	no	—
29	C₈×Dic₄₇	—	2	2	no	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	—	2	4	no	—
32	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	D₄	1	2	yes	—
34	C₂²×C₈×C₄₇	D₄	1	2	yes	—
35	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×C₁₆×C₄₇	—	2	2	no	—
37	C₂×C₁₆×C₄₇	—	2	4	no	—
38	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	—	2	2	no	—
41	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₄₇	—	2	2	no	—
46	C₂²×Dic₄₇	—	2	2	no	—
47	C₂²×Dic₄₇	—	2	2	no	—
48	C₂²×Dic₄₇	—	2	4	no	—
49	C₂²×Dic₄₇	C₁₆	1	4	yes	complemented
50	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
51	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₄₇	—	2	2	no	—
53	C₂×C₈×C₄₇	—	2	4	no	—
54	C₂×C₈×C₄₇	—	2	4	no	—
55	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₄₇	#41	1	4	yes	—
59	C₄×Dic₄₇	—	2	4	no	—
60	C₄×Dic₄₇	—	2	2	no	—
61	C₄×Dic₄₇	—	2	2	no	—
62	C₂×C₄₇⋊C₈	—	2	8	no	—
63	C₂×C₄₇⋊C₈	—	2	2	no	—
64	C₂×C₄₇⋊C₈	—	2	2	no	—
65	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
66	C₁₆×C₄₇	—	4	2	no	—
67	C₄₇⋊C₁₆	—	4	2	no	—
68	C₂³×C₄₇	—	2	2	no	—
69	C₂³×C₄₇	—	2	2	no	—
70	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
71	C₂³×C₄₇	C₂×C₁₆	1	2	yes	—
72	C₂³×C₄₇	M₃₂	1	2	yes	—
73	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₄₇	—	2	2	no	—
75	C₂×C₄×C₄₇	—	2	2	no	—
76	C₂×C₄×C₄₇	—	2	2	no	—
77	C₂×C₄×C₄₇	—	2	4	no	—
78	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
79	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₄₇	#117	1	2	yes	—
81	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
82	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
83	C₂×Dic₄₇	—	2	8	no	—
84	C₂×Dic₄₇	—	2	8	no	—
85	C₂×Dic₄₇	—	2	2	no	—
86	C₂×Dic₄₇	—	2	2	no	—
87	C₂×Dic₄₇	—	2	2	no	—
88	C₂×Dic₄₇	—	2	2	no	—
89	C₂×Dic₄₇	C₂×C₁₆	1	4	yes	—
90	C₂×Dic₄₇	M₃₂	1	4	yes	—
91	C₈×C₄₇	—	2	2	no	—
92	C₈×C₄₇	#123	1	2	yes	—
93	C₈×C₄₇	#123	1	2	yes	—
94	C₄₇⋊C₈	—	2	4	no	—
95	#12844	—	47	1	no	maximal, Sylow
96	C₂²×C₄₇	—	2	2	no	—
97	C₂²×C₄₇	—	2	2	no	—
98	C₂²×C₄₇	—	2	4	no	—
99	C₂²×C₄₇	—	2	4	no	—
100	C₂²×C₄₇	C₂²⋊C₁₆	1	4	yes	—
101	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₂²×C₄₇	#431	1	2	yes	—
103	C₂²×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
104	C₂²×C₄₇	C₂×M₃₂	1	2	yes	—
105	C₂²×C₄₇	C₂×M₃₂	1	1	yes	characteristic
106	C₄×C₄₇	—	2	2	no	—
107	C₄×C₄₇	#381	1	1	yes	characteristic
108	C₄×C₄₇	#407	1	1	yes	characteristic
109	C₄×C₄₇	#432	1	2	yes	—
110	Dic₄₇	—	2	4	no	—
111	Dic₄₇	—	2	4	no	—
112	C₂²×C₄×C₈	—	47	1	no	—
113	C₂×C₂²⋊C₁₆	—	47	1	no	—
114	C₂×C₂²⋊C₁₆	D₄₇	1	1	yes	characteristic
115	#1374	—	47	4	no	—
116	C₂×C₄₇	—	2	4	no	—
117	C₂×C₄₇	#1817	1	1	yes	characteristic
118	C₂×C₄₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
119	C₂×C₄₇	#2112	1	1	yes	characteristic
120	C₂×C₄₇	#2162	1	2	yes	—
121	C₂×C₄₇	#2163	1	2	yes	—
122	C₂³×C₈	—	47	1	no	—
123	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
124	C₂²×C₄²	—	47	1	no	—
125	C₂²×C₁₆	—	47	2	no	—
126	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
127	C₂×C₄×C₈	—	47	2	no	—
128	C₂×C₄×C₈	—	47	2	no	—
129	C₂×C₄×C₈	—	94	2	no	—
130	C₂×C₄×C₈	—	94	2	no	—
131	C₂²⋊C₁₆	—	2	2	no	—
132	C₂²⋊C₁₆	—	94	2	no	—
133	C₄₇	#12844	1	1	yes	characteristic, complemented, Sylow
134	C₂³×C₄	—	47	1	no	—
135	C₂³×C₄	—	47	1	no	—
136	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
137	C₂²×C₈	—	2	4	no	—
138	C₂²×C₈	—	47	2	no	—
139	C₂²×C₈	—	94	4	no	—
140	C₂²×C₈	—	94	2	no	—
141	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
142	C₂²×C₈	D₁₈₈	1	2	yes	—
143	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
144	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
145	C₂×C₄²	—	47	2	no	—
146	C₂×C₄²	—	47	2	no	—
147	C₂×C₄²	—	94	2	no	—
148	C₂×C₄²	—	94	2	no	—
149	C₂×C₁₆	—	2	4	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	94	2	no	—
152	C₂×C₁₆	—	94	4	no	—
153	C₄×C₈	—	94	4	no	—
154	C₄×C₈	—	94	2	no	—
155	C₄×C₈	—	94	2	no	—
156	C₂⁴	C₈×D₄₇	1	1	yes	characteristic
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	47	4	no	—
160	C₂²×C₄	—	47	2	no	—
161	C₂²×C₄	—	47	2	no	—
162	C₂²×C₄	—	94	4	no	—
163	C₂²×C₄	—	94	2	no	—
164	C₂²×C₄	—	94	2	no	—
165	C₂²×C₄	—	94	2	no	—
166	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
167	C₂²×C₄	#10625324	1	2	yes	—
168	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
169	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	47	4	no	—
174	C₂×C₈	—	94	8	no	—
175	C₂×C₈	—	94	2	no	—
176	C₂×C₈	—	94	2	no	—
177	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
178	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
179	C₂×C₈	#10625352	1	4	yes	—
180	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
181	C₄²	—	94	4	no	—
182	C₄²	—	94	2	no	—
183	C₄²	—	94	2	no	—
184	C₁₆	—	4	2	no	—
185	C₁₆	—	188	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	(C₂×D₄₇)⋊C₈	1	2	yes	—
189	C₂³	C₁₆×D₄₇	1	2	yes	—
190	C₂³	#13093963	1	2	yes	—
191	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	47	4	no	—
197	C₂×C₄	—	47	4	no	—
198	C₂×C₄	—	94	8	no	—
199	C₂×C₄	—	94	2	no	—
200	C₂×C₄	—	94	2	no	—
201	C₂×C₄	—	94	2	no	—
202	C₂×C₄	—	94	2	no	—
203	C₂×C₄	—	94	8	no	—
204	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
205	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
206	C₂×C₄	#13093980	1	2	yes	—
207	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
208	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	94	4	no	—
211	C₈	#13094005	1	2	yes	—
212	C₈	#13094008	1	2	yes	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	(C₂×D₄₇)⋊C₁₆	1	4	yes	—
218	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
219	C₂²	#15689173	1	2	yes	—
220	C₂²	C₂×C₁₆×D₄₇	1	1	yes	characteristic
221	C₂²	M₃₂×D₄₇	1	2	yes	—
222	C₂²	C₂×#13093963	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	94	4	no	—
225	C₄	—	94	4	no	—
226	C₄	#15688984	1	1	yes	characteristic
227	C₄	#15689086	1	1	yes	characteristic
228	C₄	#15689177	1	2	yes	—
229	C₂	—	2	4	no	—
230	C₂	#20243454	1	1	yes	characteristic
231	C₂	C₂×(C₂×D₄₇)⋊C₁₆	1	1	yes	characteristic
232	C₂	#20244998	1	1	yes	characteristic
233	C₂	#20245184	1	2	yes	—
234	C₂	#20245188	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₁₆
Fitting	C₂×C₄₇×C₂²⋊C₁₆	C₂
socle	C₂³×C₄₇	C₂²×C₈
prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₄₇
hypocenter	C₄₇	#12844
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #20241465	C₂	semidirect
49. C₂²×Dic₄₇	C₁₆	semidirect
133. C₄₇	#12844	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₄₇	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
#20241465	2
C₂×C₄₇×C₂²⋊C₁₆	2
#12844	47