Subgroups — #26147641

Order 12032.

1854 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689022	C₂	1	6	yes	maximal
2	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20241617	C₂	1	6	yes	maximal
5	C₂×C₄₇×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₄₇	C₂²	1	6	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	3	yes	—
9	C₂²×C₄×Dic₄₇	C₂²	1	3	yes	—
10	#15689022	—	2	24	no	—
11	#15688993	—	2	4	no	—
12	C₂×C₄₇⋊#95	C₂²	1	12	yes	—
13	C₄²×Dic₄₇	—	2	4	no	—
14	C₂²×C₄⋊C₄×C₄₇	C₂²	1	6	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#384	C₄	1	8	yes	complemented
17	C₂³×Dic₄₇	C₂³	1	4	yes	—
18	C₂³×Dic₄₇	C₂³	1	1	yes	characteristic
19	C₂³×Dic₄₇	C₂³	1	3	yes	—
20	C₂×C₄⋊Dic₄₇	—	2	12	no	—
21	C₂×C₄⋊Dic₄₇	—	2	12	no	—
22	C₂×C₄⋊Dic₄₇	—	2	12	no	—
23	C₂×C₄⋊Dic₄₇	—	2	24	no	—
24	C₂×C₄×Dic₄₇	—	2	6	no	—
25	C₂×C₄×Dic₄₇	—	2	6	no	—
26	C₂×C₄×Dic₄₇	—	2	6	no	—
27	C₂×C₄×Dic₄₇	—	2	6	no	—
28	C₂×C₄×Dic₄₇	D₄	1	12	yes	—
29	C₂×C₄×Dic₄₇	Q₈	1	12	yes	—
30	C₄₇⋊#95	—	2	24	no	—
31	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
33	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
35	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	24	yes	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₄₇	D₄	1	4	yes	—
38	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
39	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	—	2	12	no	—
41	C₂²×C₄×C₄₇	—	2	12	no	—
42	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
43	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
44	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
45	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
46	C₂²×C₄×C₄₇	C₂×Q₈	1	6	yes	—
47	C₂²×C₄×C₄₇	C₂×Q₈	1	6	yes	—
48	C₂²×C₄×C₄₇	#41	1	6	yes	—
49	C₂²×C₄×C₄₇	#41	1	6	yes	—
50	C₂²×Dic₄₇	—	2	1	no	—
51	C₂²×Dic₄₇	—	2	3	no	—
52	C₂²×Dic₄₇	—	2	3	no	—
53	C₂²×Dic₄₇	—	2	3	no	—
54	C₂²×Dic₄₇	—	2	6	no	—
55	C₂²×Dic₄₇	—	2	12	no	—
56	C₂²×Dic₄₇	—	2	4	no	—
57	C₂²×Dic₄₇	—	2	12	no	—
58	C₂²×Dic₄₇	C₂×D₄	1	6	yes	—
59	C₂²×Dic₄₇	C₂×D₄	1	6	yes	—
60	C₂²×Dic₄₇	C₂×Q₈	1	12	yes	—
61	C₄⋊C₄×C₄₇	—	2	48	no	—
62	C₄²×C₄₇	C₄⋊C₄	1	16	yes	complemented
63	C₄⋊Dic₄₇	—	4	24	no	—
64	C₄⋊Dic₄₇	—	4	16	no	—
65	C₄×Dic₄₇	—	2	24	no	—
66	C₄×Dic₄₇	—	2	24	no	—
67	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
68	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
69	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
70	C₂³×C₄₇	C₂²×Q₈	1	3	yes	—
71	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
72	C₂³×C₄₇	C₂×#41	1	3	yes	—
73	C₂³×C₄₇	C₂×#41	1	3	yes	—
74	C₂×C₄×C₄₇	—	2	24	no	—
75	C₂×C₄×C₄₇	—	4	4	no	—
76	C₂×C₄×C₄₇	—	4	12	no	—
77	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
78	C₂×C₄×C₄₇	C₄×D₄	1	12	yes	—
79	C₂×C₄×C₄₇	C₄×Q₈	1	12	yes	—
80	C₂×C₄×C₄₇	#121	1	12	yes	—
81	C₂×C₄×C₄₇	C₂²⋊Q₈	1	12	yes	—
82	C₂×C₄×C₄₇	#125	1	12	yes	—
83	C₂×C₄×C₄₇	C₄⋊Q₈	1	12	yes	—
84	C₂×Dic₄₇	—	2	6	no	—
85	C₂×Dic₄₇	—	2	6	no	—
86	C₂×Dic₄₇	—	2	12	no	—
87	C₂×Dic₄₇	—	2	6	no	—
88	C₂×Dic₄₇	—	2	6	no	—
89	C₂×Dic₄₇	—	2	12	no	—
90	C₂×Dic₄₇	—	4	12	no	—
91	C₂×Dic₄₇	—	4	4	no	—
92	C₂×Dic₄₇	—	4	12	no	—
93	C₂×Dic₄₇	#127	1	4	yes	—
94	C₂×Dic₄₇	C₄⋊Q₈	1	12	yes	—
95	#14712	—	47	1	no	maximal, Sylow
96	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₄₇	C₂×C₄×D₄	1	3	yes	—
98	C₂²×C₄₇	C₂×C₄×Q₈	1	3	yes	—
99	C₂²×C₄₇	C₂×#121	1	3	yes	—
100	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	3	yes	—
101	C₂²×C₄₇	C₂×#125	1	3	yes	—
102	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
103	C₂²×C₄₇	C₂×C₄⋊Q₈	1	3	yes	—
104	C₂²×C₄₇	C₂×C₄⋊Q₈	1	3	yes	—
105	C₂²×C₄₇	#532	1	3	yes	—
106	C₂²×C₄₇	#532	1	3	yes	—
107	C₂²×C₄₇	#533	1	6	yes	—
108	C₄×C₄₇	—	4	8	no	—
109	C₄×C₄₇	#389	1	24	yes	complemented
110	Dic₄₇	—	2	4	no	—
111	Dic₄₇	—	2	12	no	—
112	Dic₄₇	—	8	4	no	—
113	C₂×#389	—	47	6	no	—
114	C₂×#384	—	47	1	no	—
115	C₂×#384	D₄₇	1	1	yes	characteristic
116	C₂×C₄³	—	47	1	no	—
117	C₄⋊#95	—	47	6	no	—
118	C₂×C₄₇	C₂×#389	1	3	yes	—
119	C₂×C₄₇	C₄×#127	1	1	yes	characteristic
120	C₂×C₄₇	C₄×C₄⋊Q₈	1	3	yes	—
121	C₂×C₄₇	#2856	1	1	yes	characteristic
122	C₂×C₄₇	#2857	1	3	yes	—
123	C₂×C₄₇	#2859	1	3	yes	—
124	C₂×C₄₇	#2868	1	1	yes	characteristic
125	C₂²×C₄⋊C₄	—	47	6	no	—
126	C₂²×C₄⋊C₄	—	47	4	no	—
127	C₂²×C₄⋊C₄	C₂×D₄₇	1	6	yes	—
128	C₂²×C₄²	—	47	3	no	—
129	C₂²×C₄²	—	47	3	no	—
130	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
131	#389	—	94	24	no	—
132	#384	—	94	4	no	—
133	#384	Dic₄₇	1	8	yes	complemented
134	C₂×#95	—	47	12	no	—
135	C₄³	—	94	4	no	—
136	C₄₇	#14712	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	—	47	1	no	—
138	C₂³×C₄	—	47	3	no	—
139	C₂³×C₄	—	47	4	no	—
140	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
141	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
142	C₂×C₄⋊C₄	—	2	12	no	—
143	C₂×C₄⋊C₄	—	2	12	no	—
144	C₂×C₄⋊C₄	—	94	24	no	—
145	C₂×C₄⋊C₄	—	94	12	no	—
146	C₂×C₄⋊C₄	—	94	12	no	—
147	C₂×C₄⋊C₄	—	94	12	no	—
148	C₂×C₄⋊C₄	C₂×Dic₄₇	1	24	yes	—
149	C₂×C₄²	—	47	12	no	—
150	C₂×C₄²	—	47	12	no	—
151	C₂×C₄²	—	94	6	no	—
152	C₂×C₄²	—	94	6	no	—
153	C₂×C₄²	—	94	6	no	—
154	C₂×C₄²	—	94	6	no	—
155	C₂×C₄²	Dic₉₄	1	4	yes	—
156	C₂×C₄²	D₁₈₈	1	4	yes	—
157	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
158	#95	—	94	24	no	—
159	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
160	C₂²×C₄	—	2	12	no	—
161	C₂²×C₄	—	2	12	no	—
162	C₂²×C₄	—	47	6	no	—
163	C₂²×C₄	—	47	6	no	—
164	C₂²×C₄	—	47	12	no	—
165	C₂²×C₄	—	94	1	no	—
166	C₂²×C₄	—	94	3	no	—
167	C₂²×C₄	—	94	6	no	—
168	C₂²×C₄	—	94	3	no	—
169	C₂²×C₄	—	94	3	no	—
170	C₂²×C₄	—	94	12	no	—
171	C₂²×C₄	—	94	4	no	—
172	C₂²×C₄	—	94	12	no	—
173	C₂²×C₄	C₂×Dic₉₄	1	6	yes	—
174	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
175	C₂²×C₄	D₄×D₄₇	1	6	yes	—
176	C₂²×C₄	#10625352	1	6	yes	—
177	C₂²×C₄	Q₈×D₄₇	1	6	yes	—
178	C₂²×C₄	#10625354	1	6	yes	—
179	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
180	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
181	C₄⋊C₄	—	2	48	no	—
182	C₄⋊C₄	—	188	16	no	—
183	C₄⋊C₄	—	188	24	no	—
184	C₄²	—	94	24	no	—
185	C₄²	—	94	24	no	—
186	C₄²	C₄⋊Dic₄₇	1	16	yes	complemented
187	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
188	C₂³	C₂×D₄×D₄₇	1	3	yes	—
189	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
190	C₂³	C₂×Q₈×D₄₇	1	3	yes	—
191	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
192	C₂³	C₂×#10625352	1	3	yes	—
193	C₂³	C₂×#10625354	1	3	yes	—
194	C₂×C₄	—	2	24	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	12	no	—
197	C₂×C₄	—	47	4	no	—
198	C₂×C₄	—	47	12	no	—
199	C₂×C₄	—	94	6	no	—
200	C₂×C₄	—	94	6	no	—
201	C₂×C₄	—	94	12	no	—
202	C₂×C₄	—	94	6	no	—
203	C₂×C₄	—	94	6	no	—
204	C₂×C₄	—	94	12	no	—
205	C₂×C₄	—	188	12	no	—
206	C₂×C₄	—	188	4	no	—
207	C₂×C₄	—	188	12	no	—
208	C₂×C₄	C₂×C₄⋊Dic₄₇	1	12	yes	—
209	C₂×C₄	D₄×Dic₄₇	1	12	yes	—
210	C₂×C₄	Q₈×Dic₄₇	1	12	yes	—
211	C₂×C₄	#13093999	1	12	yes	—
212	C₂×C₄	#13094004	1	12	yes	—
213	C₂×C₄	#13094015	1	12	yes	—
214	C₂×C₄	#13094023	1	12	yes	—
215	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
216	C₂²	C₂×D₄×Dic₄₇	1	3	yes	—
217	C₂²	C₂×Q₈×Dic₄₇	1	3	yes	—
218	C₂²	C₂×#13093999	1	3	yes	—
219	C₂²	C₂×#13094004	1	3	yes	—
220	C₂²	C₂×#13094015	1	3	yes	—
221	C₂²	#127×D₄₇	1	1	yes	characteristic
222	C₂²	C₄⋊Q₈×D₄₇	1	3	yes	—
223	C₂²	C₂×#13094023	1	3	yes	—
224	C₂²	#15689671	1	3	yes	—
225	C₂²	#15689674	1	3	yes	—
226	C₂²	#15689678	1	6	yes	—
227	C₄	—	4	8	no	—
228	C₄	—	94	4	no	—
229	C₄	—	94	12	no	—
230	C₄	—	376	4	no	—
231	C₄	#15689022	1	24	yes	complemented
232	C₂	C₂×#15689022	1	3	yes	—
233	C₂	#127×Dic₄₇	1	1	yes	characteristic
234	C₂	C₄⋊Q₈×Dic₄₇	1	3	yes	—
235	C₂	#20251734	1	1	yes	characteristic
236	C₂	#20251741	1	3	yes	—
237	C₂	#20251749	1	3	yes	—
238	C₂	#20251785	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#384	C₂
socle, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#384	D₄₇
hypocenter	C₄₇	#14712
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
16. C₄₇×#384	C₄	semidirect
62. C₄²×C₄₇	C₄⋊C₄	semidirect
109. C₄×C₄₇	#389	semidirect
133. #384	Dic₄₇	semidirect
136. C₄₇	#14712	semidirect
186. C₄²	C₄⋊Dic₄₇	semidirect
231. C₄	#15689022	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689022	2
C₂×#15688993	2
C₂×C₄²×Dic₄₇	2
#20241617	2
C₂×C₄₇×#384	2
#14712	47