Subgroups — #26377217

Order 12032.

2858 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689601	C₂	1	1	yes	maximal, characteristic
2	C₂²×Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20246309	C₂	1	8	yes	maximal, complemented
4	Q₈×C₄⋊Dic₄₇	C₂	1	8	yes	maximal, complemented
5	#20246011	C₂	1	6	yes	maximal
6	#20245872	C₂	1	6	yes	maximal
7	C₂×C₄₇×#519	C₂	1	1	yes	maximal, characteristic
8	#15689601	—	2	8	no	—
9	C₂×Q₈×Dic₄₇	—	2	8	no	—
10	C₂×Q₈×Dic₄₇	C₂²	1	4	yes	—
11	C₂×Q₈×Dic₄₇	C₂²	1	8	yes	—
12	C₂×Q₈×Dic₄₇	C₂²	1	4	yes	—
13	C₂×#13093983	C₂²	1	6	yes	—
14	C₂²×C₄⋊Dic₄₇	C₂²	1	3	yes	—
15	C₂²×C₄×Dic₄₇	C₂²	1	3	yes	—
16	#15689022	C₂²	1	12	yes	—
17	#15689022	C₂²	1	24	yes	—
18	#15689022	C₂²	1	12	yes	—
19	#15688998	C₂²	1	4	yes	—
20	#15688998	C₂²	1	12	yes	—
21	#15688993	C₂²	1	12	yes	—
22	#15688989	C₂²	1	12	yes	—
23	C₄×C₄⋊Dic₄₇	C₂²	1	24	yes	—
24	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#519	C₄	1	16	yes	complemented
26	C₂×C₄×Q₈×C₄₇	C₂²	1	8	yes	—
27	C₂×#117×C₄₇	C₂²	1	6	yes	—
28	C₂³×Dic₄₇	C₂³	1	1	yes	characteristic
29	C₂³×Dic₄₇	C₂³	1	3	yes	—
30	Q₈×Dic₄₇	—	2	32	no	—
31	Q₈×Dic₄₇	—	2	16	no	—
32	Q₈×Dic₄₇	—	2	16	no	—
33	#13093983	—	2	24	no	—
34	C₂×C₄⋊Dic₄₇	—	2	12	no	—
35	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	6	yes	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	6	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	24	yes	—
40	C₂×C₂²⋊Dic₄₇	C₂³	1	4	yes	—
41	C₂×C₄×Dic₄₇	—	2	12	no	—
42	C₂×C₄×Dic₄₇	C₂³	1	12	yes	—
43	C₂×C₄×Dic₄₇	C₂³	1	12	yes	—
44	C₄₇⋊#95	C₂³	1	8	yes	—
45	C₄₇⋊#95	C₂³	1	24	yes	—
46	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
47	C₂²×Q₈×C₄₇	D₄	1	8	yes	—
48	C₂²×Q₈×C₄₇	Q₈	1	8	yes	—
49	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
50	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
51	C₄×Q₈×C₄₇	C₂×C₄	1	64	yes	complemented
52	#117×C₄₇	C₂×C₄	1	48	yes	—
53	C₂×C₄⋊C₄×C₄₇	C₂³	1	12	yes	—
54	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	12	yes	—
56	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
57	C₂×Q₈×C₄₇	C₄⋊C₄	1	64	yes	complemented
58	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
59	C₂×Q₈×C₄₇	C₂×D₄	1	16	yes	—
60	C₂×Q₈×C₄₇	C₂×Q₈	1	16	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
63	C₂²×C₄×C₄₇	C₂×Q₈	1	12	yes	—
64	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
65	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
66	C₂²×Dic₄₇	—	2	2	no	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	—	2	6	no	—
69	C₂²×Dic₄₇	—	4	6	no	—
70	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
71	C₂²×Dic₄₇	C₂⁴	1	6	yes	—
72	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
73	C₂²×Dic₄₇	C₂⁴	1	2	yes	—
74	C₄⋊C₄×C₄₇	C₂²×C₄	1	48	yes	—
75	C₂²⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
76	C₄²×C₄₇	C₂²×C₄	1	48	yes	—
77	C₂²⋊Dic₄₇	—	2	8	no	—
78	C₄⋊Dic₄₇	—	2	24	no	—
79	C₄⋊Dic₄₇	—	2	12	no	—
80	C₄⋊Dic₄₇	—	2	12	no	—
81	C₄⋊Dic₄₇	—	2	16	no	—
82	C₄⋊Dic₄₇	—	4	24	no	—
83	C₄×Dic₄₇	—	2	24	no	—
84	C₄×Dic₄₇	—	2	24	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
88	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
89	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
90	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
91	Q₈×C₄₇	C₂×C₄⋊C₄	1	64	yes	—
92	C₂×C₄×C₄₇	—	2	8	no	—
93	C₂×C₄×C₄₇	—	2	8	no	—
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	48	yes	—
95	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
96	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
97	C₂×C₄×C₄₇	C₂²×D₄	1	12	yes	—
98	C₂×C₄×C₄₇	C₂²×Q₈	1	12	yes	—
99	C₂×Dic₄₇	—	2	8	no	—
100	C₂×Dic₄₇	—	2	2	no	—
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	4	no	—
103	C₂×Dic₄₇	—	2	4	no	—
104	C₂×Dic₄₇	—	2	6	no	—
105	C₂×Dic₄₇	—	2	8	no	—
106	C₂×Dic₄₇	—	2	8	no	—
107	C₂×Dic₄₇	—	2	12	no	—
108	C₂×Dic₄₇	—	4	12	no	—
109	C₂×Dic₄₇	2₋¹⁺⁴	1	4	yes	—
110	#34249	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	—	2	4	no	—
112	C₂²×C₄₇	—	2	2	no	—
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	2	4	no	—
115	C₂²×C₄₇	C₂²×C₄⋊C₄	1	4	yes	—
116	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
117	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
118	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
119	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
120	C₂²×C₄₇	#585	1	2	yes	—
121	C₄×C₄₇	—	2	16	no	—
122	C₄×C₄₇	C₂²×C₄⋊C₄	1	24	yes	—
123	Dic₄₇	—	2	4	no	—
124	Dic₄₇	—	4	4	no	—
125	Dic₄₇	—	4	4	no	—
126	Dic₄₇	—	4	6	no	—
127	C₂×#519	—	47	1	no	—
128	C₂×#519	D₄₇	1	1	yes	characteristic
129	C₂²×C₄×Q₈	—	47	1	no	—
130	#2351	—	47	8	no	—
131	Q₈×C₄⋊C₄	—	47	8	no	—
132	#2317	—	47	6	no	—
133	#2296	—	47	6	no	—
134	C₂×C₄₇	—	2	4	no	—
135	C₂×C₄₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄₇	C₄×2₋¹⁺⁴	1	2	yes	—
137	C₂×C₄₇	#3486	1	1	yes	characteristic
138	C₂×C₄₇	#3487	1	1	yes	characteristic
139	C₂×C₄₇	#3502	1	2	yes	—
140	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
141	#519	—	94	8	no	—
142	#519	Dic₄₇	1	16	yes	complemented
143	C₂×C₄×Q₈	—	47	4	no	—
144	C₂×C₄×Q₈	—	47	8	no	—
145	C₂×C₄×Q₈	—	47	4	no	—
146	C₂×C₄×Q₈	—	94	8	no	—
147	C₂×C₄×Q₈	C₂×D₄₇	1	8	yes	—
148	C₂×#117	—	47	6	no	—
149	C₂×#117	C₂×D₄₇	1	6	yes	—
150	C₂²×C₄⋊C₄	—	47	3	no	—
151	C₂²×C₄²	—	47	3	no	—
152	#389	—	47	24	no	—
153	#389	—	47	12	no	—
154	#389	—	47	12	no	—
155	#385	—	47	12	no	—
156	#385	—	47	4	no	—
157	#384	—	47	12	no	—
158	#382	—	47	12	no	—
159	C₄×C₄⋊C₄	—	47	24	no	—
160	C₄₇	#34249	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₈	Dic₉₄	1	8	yes	—
162	C₂²×Q₈	D₁₈₈	1	8	yes	—
163	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	—
164	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
165	C₂³×C₄	—	47	1	no	—
166	C₂³×C₄	—	47	3	no	—
167	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
168	C₄×Q₈	—	94	32	no	—
169	C₄×Q₈	—	94	16	no	—
170	C₄×Q₈	—	94	16	no	—
171	C₄×Q₈	C₂×Dic₄₇	1	64	yes	complemented
172	#117	—	94	24	no	—
173	#117	C₂×Dic₄₇	1	48	yes	—
174	C₂×C₄⋊C₄	—	47	12	no	—
175	C₂×C₄⋊C₄	—	47	6	no	—
176	C₂×C₄⋊C₄	—	47	6	no	—
177	C₂×C₄⋊C₄	—	47	8	no	—
178	C₂×C₄⋊C₄	—	47	24	no	—
179	C₂×C₄⋊C₄	—	94	12	no	—
180	C₂×C₄⋊C₄	C₂²×D₄₇	1	12	yes	—
181	C₂×C₂²⋊C₄	—	47	4	no	—
182	C₂×C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
183	C₂×C₄²	—	47	12	no	—
184	C₂×C₄²	—	47	12	no	—
185	C₂×C₄²	—	94	12	no	—
186	C₂×C₄²	C₂²×D₄₇	1	12	yes	—
187	#95	—	47	8	no	—
188	#95	—	47	24	no	—
189	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
190	C₂×Q₈	C₄⋊Dic₄₇	1	64	yes	complemented
191	C₂×Q₈	C₂×Dic₉₄	1	16	yes	—
192	C₂×Q₈	C₂×D₁₈₈	1	16	yes	—
193	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
194	C₂²×C₄	—	47	2	no	—
195	C₂²×C₄	—	47	4	no	—
196	C₂²×C₄	—	47	6	no	—
197	C₂²×C₄	—	47	4	no	—
198	C₂²×C₄	—	94	4	no	—
199	C₂²×C₄	—	94	2	no	—
200	C₂²×C₄	—	94	6	no	—
201	C₂²×C₄	—	188	6	no	—
202	C₂²×C₄	C₂×Dic₉₄	1	12	yes	—
203	C₂²×C₄	C₂×D₁₈₈	1	12	yes	—
204	C₂²×C₄	C₂²×Dic₄₇	1	12	yes	—
205	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
206	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
207	C₄⋊C₄	—	94	16	no	—
208	C₄⋊C₄	—	94	24	no	—
209	C₄⋊C₄	—	94	12	no	—
210	C₄⋊C₄	—	94	12	no	—
211	C₄⋊C₄	—	188	24	no	—
212	C₄⋊C₄	C₂²×Dic₄₇	1	48	yes	—
213	C₂²⋊C₄	—	94	8	no	—
214	C₂²⋊C₄	C₂²×Dic₄₇	1	16	yes	—
215	C₄²	—	94	24	no	—
216	C₄²	—	94	24	no	—
217	C₄²	C₂²×Dic₄₇	1	48	yes	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	C₂³×Dic₄₇	1	2	yes	—
221	C₂³	C₂²×D₁₈₈	1	2	yes	—
222	C₂³	C₂²×Dic₉₄	1	2	yes	—
223	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
224	Q₈	C₂×C₄⋊Dic₄₇	1	64	yes	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	47	4	no	—
228	C₂×C₄	—	94	8	no	—
229	C₂×C₄	—	94	2	no	—
230	C₂×C₄	—	94	2	no	—
231	C₂×C₄	—	94	4	no	—
232	C₂×C₄	—	94	4	no	—
233	C₂×C₄	—	94	6	no	—
234	C₂×C₄	—	94	8	no	—
235	C₂×C₄	—	94	8	no	—
236	C₂×C₄	—	94	12	no	—
237	C₂×C₄	—	188	12	no	—
238	C₂×C₄	C₂×C₄⋊Dic₄₇	1	48	yes	—
239	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
240	C₂×C₄	C₂³×Dic₄₇	1	12	yes	—
241	C₂×C₄	C₂²×D₁₈₈	1	12	yes	—
242	C₂×C₄	C₂²×Dic₉₄	1	12	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	4	no	—
247	C₂²	C₂²×C₄⋊Dic₄₇	1	4	yes	—
248	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
249	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
250	C₂²	C₂³×Dic₉₄	1	1	yes	characteristic
251	C₂²	2₋¹⁺⁴×D₄₇	1	2	yes	—
252	C₂²	#15689960	1	2	yes	—
253	C₄	—	2	16	no	—
254	C₄	—	94	4	no	—
255	C₄	—	188	4	no	—
256	C₄	—	188	6	no	—
257	C₄	—	188	4	no	—
258	C₄	C₂²×C₄⋊Dic₄₇	1	24	yes	—
259	C₂	—	2	4	no	—
260	C₂	C₂³×C₄⋊Dic₄₇	1	1	yes	characteristic
261	C₂	2₋¹⁺⁴×Dic₄₇	1	2	yes	—
262	C₂	#20257040	1	1	yes	characteristic
263	C₂	#20257047	1	1	yes	characteristic
264	C₂	#20257194	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Fitting	C₂×C₄₇×#519	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#519	D₄₇
hypocenter	C₄₇	#34249
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#519

Complements

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

N	H ≅ G/N	type
3. #20246309	C₂	semidirect
4. Q₈×C₄⋊Dic₄₇	C₂	semidirect
25. C₄₇×#519	C₄	semidirect
51. C₄×Q₈×C₄₇	C₂×C₄	semidirect
57. C₂×Q₈×C₄₇	C₄⋊C₄	semidirect
142. #519	Dic₄₇	semidirect
160. C₄₇	#34249	semidirect
171. C₄×Q₈	C₂×Dic₄₇	semidirect
190. C₂×Q₈	C₄⋊Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689601	2
C₂²×Q₈×Dic₄₇	2
#20246309	2
Q₈×C₄⋊Dic₄₇	2
#20246011	2
#20245872	2
C₂×C₄₇×#519	2
#34249	47