Subgroups — #26147469

Order 12032.

1710 conjugacy classes of subgroups.

The subgroup lattice has 246 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	1	yes	maximal, characteristic
2	C₂×#15688988	C₂	1	4	yes	maximal
3	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20241617	C₂	1	8	yes	maximal
5	C₂×C₄₇×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
9	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
10	#15688993	—	2	4	no	—
11	#15688988	—	2	16	no	—
12	C₂×C₄₇⋊#95	C₂²	1	16	yes	—
13	C₄²×Dic₄₇	C₄	1	8	yes	complemented
14	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#384	—	2	4	no	—
18	C₂³×Dic₄₇	C₂³	1	4	yes	—
19	C₂³×Dic₄₇	C₂³	1	4	yes	—
20	C₂×C₄⋊Dic₄₇	—	2	8	no	—
21	C₂×C₄⋊Dic₄₇	—	2	8	no	—
22	C₂×C₄⋊Dic₄₇	—	2	4	no	—
23	C₂×C₄⋊Dic₄₇	—	2	16	no	—
24	C₂×C₄×Dic₄₇	—	2	8	no	—
25	C₂×C₄×Dic₄₇	—	2	8	no	—
26	C₂×C₄×Dic₄₇	C₂×C₄	1	16	yes	—
27	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
28	C₂×C₄×Dic₄₇	D₄	1	8	yes	—
29	C₂×C₄×Dic₄₇	Q₈	1	8	yes	—
30	C₄₇⋊#95	—	2	32	no	—
31	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
34	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	16	no	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₄₇	D₄	1	4	yes	—
40	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
41	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	2	8	no	—
44	C₂²×C₄×C₄₇	—	2	16	no	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
48	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
53	C₂²×C₄×C₄₇	#41	1	8	yes	—
54	C₂²×C₄×C₄₇	#41	1	8	yes	—
55	C₂²×Dic₄₇	—	2	8	no	—
56	C₂²×Dic₄₇	—	2	8	no	—
57	C₂²×Dic₄₇	—	2	8	no	—
58	C₂²×Dic₄₇	—	2	4	no	—
59	C₂²×Dic₄₇	—	2	16	no	—
60	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
61	C₂²×Dic₄₇	C₂×D₄	1	8	yes	—
62	C₂²×Dic₄₇	C₂×Q₈	1	8	yes	—
63	C₄⋊C₄×C₄₇	—	4	16	no	—
64	C₄⋊C₄×C₄₇	—	4	8	no	—
65	C₄²×C₄₇	C₄⋊C₄	1	16	yes	—
66	C₄⋊Dic₄₇	—	4	16	no	—
67	C₄⋊Dic₄₇	—	4	8	no	—
68	C₄×Dic₄₇	—	2	32	no	—
69	C₄×Dic₄₇	C₄⋊C₄	1	32	yes	complemented
70	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
71	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
72	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
73	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
74	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
75	C₂³×C₄₇	C₂×#41	1	4	yes	—
76	C₂³×C₄₇	C₂×#41	1	4	yes	—
77	C₂×C₄×C₄₇	—	4	8	no	—
78	C₂×C₄×C₄₇	—	4	4	no	—
79	C₂×C₄×C₄₇	—	4	16	no	—
80	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
81	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
82	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
83	C₂×C₄×C₄₇	#117	1	16	yes	—
84	C₂×C₄×C₄₇	#124	1	8	yes	—
85	C₂×C₄×C₄₇	#124	1	8	yes	—
86	C₂×C₄×C₄₇	#125	1	16	yes	—
87	C₂×C₄×C₄₇	#127	1	4	yes	—
88	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
89	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
90	C₂×Dic₄₇	—	2	16	no	—
91	C₂×Dic₄₇	—	2	8	no	—
92	C₂×Dic₄₇	—	2	8	no	—
93	C₂×Dic₄₇	—	2	16	no	—
94	C₂×Dic₄₇	—	4	8	no	—
95	C₂×Dic₄₇	—	4	4	no	—
96	C₂×Dic₄₇	—	4	16	no	—
97	C₂×Dic₄₇	C₂×C₄⋊C₄	1	16	yes	—
98	#14687	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
100	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₄₇	C₂×#117	1	4	yes	—
102	C₂²×C₄₇	C₂×#124	1	2	yes	—
103	C₂²×C₄₇	C₂×#124	1	2	yes	—
104	C₂²×C₄₇	C₂×#125	1	4	yes	—
105	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
106	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
107	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
108	C₂²×C₄₇	#532	1	4	yes	—
109	C₂²×C₄₇	#532	1	4	yes	—
110	C₂²×C₄₇	#533	1	8	yes	—
111	C₄×C₄₇	—	8	4	no	—
112	C₄×C₄₇	#382	1	16	yes	complemented
113	C₄×C₄₇	#384	1	8	yes	—
114	Dic₄₇	—	2	16	no	—
115	Dic₄₇	—	8	4	no	—
116	C₂×#384	—	47	1	no	—
117	C₂×#384	D₄₇	1	1	yes	characteristic
118	C₂×#382	—	47	4	no	—
119	C₂×C₄³	—	47	1	no	—
120	C₄⋊#95	—	47	8	no	—
121	C₂×C₄₇	C₂×#382	1	2	yes	—
122	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
123	C₂×C₄₇	#2285	1	4	yes	—
124	C₂×C₄₇	#2858	1	1	yes	characteristic
125	C₂×C₄₇	#2858	1	1	yes	characteristic
126	C₂×C₄₇	#2859	1	4	yes	—
127	C₂×C₄₇	#2862	1	2	yes	—
128	C₂²×C₄⋊C₄	—	47	4	no	—
129	C₂²×C₄⋊C₄	—	47	2	no	—
130	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
131	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
132	C₂²×C₄²	—	47	2	no	—
133	C₂²×C₄²	—	47	4	no	—
134	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
135	#384	—	2	4	no	—
136	#384	—	94	4	no	—
137	#382	—	94	16	no	—
138	C₂×#95	—	47	16	no	—
139	C₄³	—	47	8	no	—
140	C₄₇	#14687	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₄	—	47	4	no	—
142	C₂³×C₄	—	47	4	no	—
143	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
144	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
145	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
146	C₂×C₄⋊C₄	—	2	8	no	—
147	C₂×C₄⋊C₄	—	2	16	no	—
148	C₂×C₄⋊C₄	—	2	8	no	—
149	C₂×C₄⋊C₄	—	2	4	no	—
150	C₂×C₄⋊C₄	—	94	8	no	—
151	C₂×C₄⋊C₄	—	94	16	no	—
152	C₂×C₄⋊C₄	—	94	8	no	—
153	C₂×C₄⋊C₄	—	94	4	no	—
154	C₂×C₄²	—	47	8	no	—
155	C₂×C₄²	—	47	8	no	—
156	C₂×C₄²	—	47	8	no	—
157	C₂×C₄²	—	47	16	no	—
158	C₂×C₄²	—	94	8	no	—
159	C₂×C₄²	—	94	8	no	—
160	C₂×C₄²	Dic₉₄	1	4	yes	—
161	C₂×C₄²	C₄×D₄₇	1	4	yes	—
162	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
163	#95	—	94	32	no	—
164	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	2	16	no	—
168	C₂²×C₄	—	47	8	no	—
169	C₂²×C₄	—	47	8	no	—
170	C₂²×C₄	—	47	8	no	—
171	C₂²×C₄	—	94	8	no	—
172	C₂²×C₄	—	94	8	no	—
173	C₂²×C₄	—	94	8	no	—
174	C₂²×C₄	—	94	4	no	—
175	C₂²×C₄	—	94	16	no	—
176	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
177	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
179	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
180	C₂²×C₄	D₄×D₄₇	1	4	yes	—
181	C₂²×C₄	#10625352	1	8	yes	—
182	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
183	C₂²×C₄	#10625354	1	8	yes	—
184	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
185	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
186	C₄⋊C₄	—	4	16	no	—
187	C₄⋊C₄	—	4	8	no	—
188	C₄⋊C₄	—	188	16	no	—
189	C₄⋊C₄	—	188	8	no	—
190	C₄²	—	47	32	no	—
191	C₄²	—	94	32	no	—
192	C₄²	Dic₄₇⋊C₄	1	16	yes	—
193	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
194	C₂³	C₂×D₄×D₄₇	1	2	yes	—
195	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
196	C₂³	C₂×Q₈×D₄₇	1	2	yes	—
197	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
198	C₂³	C₂×#10625352	1	4	yes	—
199	C₂³	C₂×#10625354	1	4	yes	—
200	C₂×C₄	—	4	8	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	16	no	—
203	C₂×C₄	—	47	16	no	—
204	C₂×C₄	—	94	16	no	—
205	C₂×C₄	—	94	8	no	—
206	C₂×C₄	—	94	8	no	—
207	C₂×C₄	—	94	16	no	—
208	C₂×C₄	—	188	8	no	—
209	C₂×C₄	—	188	4	no	—
210	C₂×C₄	—	188	16	no	—
211	C₂×C₄	C₄⋊C₄×D₄₇	1	8	yes	—
212	C₂×C₄	C₂×Dic₄₇⋊C₄	1	8	yes	—
213	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
214	C₂×C₄	#13093982	1	16	yes	—
215	C₂×C₄	#13094011	1	8	yes	—
216	C₂×C₄	#13094013	1	8	yes	—
217	C₂×C₄	#13094015	1	16	yes	—
218	C₂×C₄	#13094020	1	4	yes	—
219	C₂×C₄	#13094022	1	4	yes	—
220	C₂×C₄	#13094023	1	8	yes	—
221	C₂²	C₂×C₄⋊C₄×D₄₇	1	2	yes	—
222	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
223	C₂²	C₂×#13093982	1	4	yes	—
224	C₂²	C₂×#13094011	1	2	yes	—
225	C₂²	C₂×#13094013	1	2	yes	—
226	C₂²	C₂×#13094015	1	4	yes	—
227	C₂²	C₂×#13094020	1	1	yes	characteristic
228	C₂²	C₂×#13094022	1	1	yes	characteristic
229	C₂²	C₂×#13094023	1	2	yes	—
230	C₂²	#15689671	1	4	yes	—
231	C₂²	#15689674	1	4	yes	—
232	C₂²	#15689678	1	8	yes	—
233	C₄	—	8	4	no	—
234	C₄	—	94	16	no	—
235	C₄	—	376	4	no	—
236	C₄	#15688988	1	16	yes	complemented
237	C₄	#15688992	1	8	yes	—
238	C₂	C₂×#15688988	1	2	yes	—
239	C₂	C₂×#15688992	1	1	yes	characteristic
240	C₂	#20245793	1	4	yes	—
241	C₂	#20251744	1	1	yes	characteristic
242	C₂	#20251746	1	1	yes	characteristic
243	C₂	#20251749	1	4	yes	—
244	C₂	#20251765	1	2	yes	—

	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	#14687
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
13. C₄²×Dic₄₇	C₄	semidirect
69. C₄×Dic₄₇	C₄⋊C₄	semidirect
112. C₄×C₄₇	#382	semidirect
140. C₄₇	#14687	semidirect
236. C₄	#15688988	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×#15688988	2
C₂×C₄²×Dic₄₇	2
#20241617	2
C₂×C₄₇×#384	2
#14687	47