Subgroups — #26147466

Order 12032.

2718 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688993	C₂	1	2	yes	maximal, complemented
2	C₂×#15688983	C₂	1	6	yes	maximal
3	#20241618	C₂	1	6	yes	maximal, complemented
4	C₂³×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	C₂²	1	12	yes	—
6	C₂²×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
7	#15688993	—	2	8	no	—
8	#15688983	—	2	24	no	—
9	C₂×C₄₇⋊#95	C₂²	1	12	yes	—
10	C₂⁴×C₄×C₄₇	C₂²	1	3	yes	—
11	C₂²×C₄²×C₄₇	—	2	8	no	—
12	C₂²×C₄²×C₄₇	C₄	1	8	yes	complemented
13	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
14	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₂³×Dic₄₇	C₂³	1	8	yes	—
16	C₂×C₄⋊Dic₄₇	—	2	24	no	—
17	C₂×C₄⋊Dic₄₇	—	2	24	no	—
18	C₂×C₄⋊Dic₄₇	—	2	24	no	—
19	C₂×C₂²⋊Dic₄₇	—	2	24	no	—
20	C₄₇⋊#95	—	2	24	no	—
21	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	—	2	12	no	—
23	C₂³×C₄×C₄₇	—	2	12	no	—
24	C₂³×C₄×C₄₇	C₂×C₄	1	12	yes	—
25	C₂³×C₄×C₄₇	D₄	1	12	yes	—
26	C₂³×C₄×C₄₇	Q₈	1	12	yes	—
27	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
28	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
29	C₂×C₄²×C₄₇	—	2	12	no	—
30	C₂×C₄²×C₄₇	—	2	32	no	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	12	yes	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₄₇	D₄	1	8	yes	—
34	C₂⁴×C₄₇	—	2	2	no	—
35	C₂⁴×C₄₇	—	2	6	no	—
36	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
37	C₂⁴×C₄₇	C₂×D₄	1	6	yes	—
38	C₂⁴×C₄₇	C₂×Q₈	1	6	yes	—
39	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	—	2	6	no	—
41	C₂²×C₄×C₄₇	—	2	6	no	—
42	C₂²×C₄×C₄₇	—	2	24	no	—
43	C₂²×C₄×C₄₇	—	2	24	no	—
44	C₂²×C₄×C₄₇	—	2	48	no	—
45	C₂²×C₄×C₄₇	—	2	48	no	—
46	C₂²×C₄×C₄₇	C₄⋊C₄	1	48	yes	complemented
47	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
50	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
52	C₂²×C₄×C₄₇	C₂×Q₈	1	6	yes	—
53	C₂²×C₄×C₄₇	C₂×Q₈	1	6	yes	—
54	C₂²×C₄×C₄₇	#41	1	12	yes	—
55	C₂²×Dic₄₇	—	2	24	no	—
56	C₂²×Dic₄₇	—	2	8	no	—
57	C₂²×Dic₄₇	—	2	24	no	—
58	C₄²×C₄₇	—	2	24	no	—
59	C₄²×C₄₇	C₂²⋊C₄	1	16	yes	complemented
60	C₂²⋊Dic₄₇	—	4	16	no	—
61	C₄⋊Dic₄₇	—	4	48	no	—
62	C₂³×C₄₇	—	2	2	no	—
63	C₂³×C₄₇	—	2	6	no	—
64	C₂³×C₄₇	—	2	12	no	—
65	C₂³×C₄₇	—	2	12	no	—
66	C₂³×C₄₇	—	2	12	no	—
67	C₂³×C₄₇	—	2	12	no	—
68	C₂³×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
69	C₂³×C₄₇	#127	1	4	yes	—
70	C₂³×C₄₇	C₄⋊Q₈	1	12	yes	—
71	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
72	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
73	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
74	C₂³×C₄₇	C₂²×Q₈	1	3	yes	—
75	C₂³×C₄₇	C₂×#41	1	6	yes	—
76	C₂×C₄×C₄₇	—	2	6	no	—
77	C₂×C₄×C₄₇	—	2	6	no	—
78	C₂×C₄×C₄₇	—	2	12	no	—
79	C₂×C₄×C₄₇	—	2	12	no	—
80	C₂×C₄×C₄₇	—	2	96	no	—
81	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
82	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
83	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
84	C₂×C₄×C₄₇	#117	1	12	yes	—
85	C₂×C₄×C₄₇	#121	1	24	yes	—
86	C₂×C₄×C₄₇	C₂²⋊Q₈	1	24	yes	—
87	C₂×Dic₄₇	—	4	24	no	—
88	C₂×Dic₄₇	—	4	8	no	—
89	C₂×Dic₄₇	—	4	24	no	—
90	#14686	—	47	1	no	maximal, Sylow
91	C₂²×C₄₇	—	2	8	no	—
92	C₂²×C₄₇	—	2	24	no	—
93	C₂²×C₄₇	—	2	12	no	—
94	C₂²×C₄₇	—	2	12	no	—
95	C₂²×C₄₇	#384	1	8	yes	complemented
96	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₄₇	C₂²×C₄⋊C₄	1	3	yes	—
98	C₂²×C₄₇	C₂×#117	1	3	yes	—
99	C₂²×C₄₇	C₂×#121	1	6	yes	—
100	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	6	yes	—
101	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
102	C₂²×C₄₇	C₂×C₄⋊Q₈	1	3	yes	—
103	C₂²×C₄₇	#532	1	6	yes	—
104	C₂²×C₄₇	#533	1	6	yes	—
105	C₄×C₄₇	—	2	12	no	—
106	C₄×C₄₇	#380	1	24	yes	complemented
107	Dic₄₇	—	8	8	no	—
108	C₂³×C₄²	D₄₇	1	1	yes	characteristic
109	C₂×#384	—	47	2	no	—
110	C₂×#380	—	47	6	no	—
111	C₄⋊#95	—	47	6	no	—
112	C₂×C₄₇	—	2	8	no	—
113	C₂×C₄₇	C₂×#380	1	3	yes	—
114	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
115	C₂×C₄₇	#2285	1	3	yes	—
116	C₂×C₄₇	#2856	1	2	yes	—
117	C₂×C₄₇	#2857	1	6	yes	—
118	C₂⁴×C₄	C₂×D₄₇	1	3	yes	—
119	C₂²×C₄⋊C₄	—	47	12	no	—
120	C₂²×C₂²⋊C₄	—	47	4	no	—
121	C₂²×C₄²	—	2	8	no	—
122	C₂²×C₄²	Dic₄₇	1	8	yes	complemented
123	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
124	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
125	#384	—	94	8	no	—
126	#380	—	94	24	no	—
127	C₂×#95	—	47	12	no	—
128	C₄₇	#14686	1	1	yes	characteristic, complemented, Sylow
129	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
130	C₂³×C₄	—	2	12	no	—
131	C₂³×C₄	—	2	12	no	—
132	C₂³×C₄	—	47	8	no	—
133	C₂³×C₄	Dic₉₄	1	12	yes	—
134	C₂³×C₄	D₁₈₈	1	12	yes	—
135	C₂³×C₄	C₂×Dic₄₇	1	12	yes	—
136	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
137	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
138	C₂×C₄⋊C₄	—	94	24	no	—
139	C₂×C₄⋊C₄	—	94	24	no	—
140	C₂×C₄⋊C₄	—	94	24	no	—
141	C₂×C₂²⋊C₄	—	94	24	no	—
142	C₂×C₄²	—	2	12	no	—
143	C₂×C₄²	—	2	32	no	—
144	C₂×C₄²	C₂×Dic₄₇	1	12	yes	—
145	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
146	C₂×C₄²	C₂²⋊D₄₇	1	8	yes	—
147	#95	—	94	24	no	—
148	C₂⁴	—	2	2	no	—
149	C₂⁴	—	2	6	no	—
150	C₂⁴	C₂×Dic₉₄	1	6	yes	—
151	C₂⁴	C₂×D₁₈₈	1	6	yes	—
152	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
153	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
154	C₂²×C₄	—	2	6	no	—
155	C₂²×C₄	—	2	6	no	—
156	C₂²×C₄	—	2	24	no	—
157	C₂²×C₄	—	2	24	no	—
158	C₂²×C₄	—	2	48	no	—
159	C₂²×C₄	—	2	48	no	—
160	C₂²×C₄	—	94	8	no	—
161	C₂²×C₄	—	94	24	no	—
162	C₂²×C₄	—	94	24	no	—
163	C₂²×C₄	C₄⋊Dic₄₇	1	48	yes	complemented
164	C₂²×C₄	C₂×Dic₉₄	1	6	yes	—
165	C₂²×C₄	C₂×Dic₉₄	1	6	yes	—
166	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
167	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
168	C₂²×C₄	#10625350	1	12	yes	—
169	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
170	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
171	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
172	C₄⋊C₄	—	188	48	no	—
173	C₂²⋊C₄	—	188	16	no	—
174	C₄²	—	2	24	no	—
175	C₄²	C₂²⋊Dic₄₇	1	16	yes	complemented
176	C₂³	—	2	2	no	—
177	C₂³	—	2	6	no	—
178	C₂³	—	2	12	no	—
179	C₂³	—	2	12	no	—
180	C₂³	—	2	12	no	—
181	C₂³	—	2	12	no	—
182	C₂³	C₂×C₄⋊Dic₄₇	1	12	yes	—
183	C₂³	#13094021	1	4	yes	—
184	C₂³	#13094024	1	12	yes	—
185	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
186	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
187	C₂³	C₂²×D₁₈₈	1	3	yes	—
188	C₂³	C₂²×Dic₉₄	1	3	yes	—
189	C₂³	C₂×#10625350	1	6	yes	—
190	C₂×C₄	—	2	6	no	—
191	C₂×C₄	—	2	6	no	—
192	C₂×C₄	—	2	12	no	—
193	C₂×C₄	—	2	12	no	—
194	C₂×C₄	—	2	96	no	—
195	C₂×C₄	—	188	24	no	—
196	C₂×C₄	—	188	8	no	—
197	C₂×C₄	—	188	24	no	—
198	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	12	yes	—
199	C₂×C₄	C₂×C₄⋊Dic₄₇	1	12	yes	—
200	C₂×C₄	C₂×C₄⋊Dic₄₇	1	12	yes	—
201	C₂×C₄	#13093983	1	12	yes	—
202	C₂×C₄	#13093998	1	24	yes	—
203	C₂×C₄	#13094003	1	24	yes	—
204	C₂²	—	2	12	no	—
205	C₂²	—	2	12	no	—
206	C₂²	—	2	8	no	—
207	C₂²	—	2	24	no	—
208	C₂²	#15688993	1	8	yes	complemented
209	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
210	C₂²	C₂²×C₄⋊Dic₄₇	1	3	yes	—
211	C₂²	C₂×#13093983	1	3	yes	—
212	C₂²	C₂×#13093998	1	6	yes	—
213	C₂²	C₂×#13094003	1	6	yes	—
214	C₂²	C₂×#13094021	1	1	yes	characteristic
215	C₂²	C₂×#13094024	1	3	yes	—
216	C₂²	#15689673	1	6	yes	—
217	C₂²	#15689680	1	6	yes	—
218	C₄	—	2	12	no	—
219	C₄	—	376	8	no	—
220	C₄	#15688983	1	24	yes	complemented
221	C₂	—	2	8	no	—
222	C₂	C₂×#15688983	1	3	yes	—
223	C₂	C₂×#15688993	1	1	yes	characteristic
224	C₂	#20245795	1	3	yes	—
225	C₂	#20251733	1	2	yes	—
226	C₂	#20251740	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Fitting	C₂³×C₄²×C₄₇	C₂
socle, cosocle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂⁵×C₄₇	C₂³
hypercenter	C₂³×C₄²	D₄₇
hypocenter	C₄₇	#14686
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×#15688993	C₂	semidirect
3. #20241618	C₂	semidirect
12. C₂²×C₄²×C₄₇	C₄	semidirect
46. C₂²×C₄×C₄₇	C₄⋊C₄	semidirect
59. C₄²×C₄₇	C₂²⋊C₄	semidirect
95. C₂²×C₄₇	#384	semidirect
106. C₄×C₄₇	#380	semidirect
122. C₂²×C₄²	Dic₄₇	semidirect
128. C₄₇	#14686	semidirect
163. C₂²×C₄	C₄⋊Dic₄₇	semidirect
175. C₄²	C₂²⋊Dic₄₇	semidirect
208. C₂²	#15688993	semidirect
220. C₄	#15688983	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688993	2
C₂×#15688983	2
#20241618	2
C₂³×C₄²×C₄₇	2
#14686	47