Subgroups — #26216712

Order 12032.

1102 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689163	C₂	1	4	yes	maximal
2	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
3	#20245795	C₂	1	1	yes	maximal, characteristic
4	#20243557	C₂	1	4	yes	maximal, complemented
5	#20243550	C₂	1	4	yes	maximal, complemented
6	C₂×C₄₇×#382	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂×#13093983	C₂²	1	2	yes	—
9	#15689163	—	2	16	no	—
10	C₂×#13093957	C₂²	1	8	yes	—
11	C₂×#13093955	C₂²	1	8	yes	—
12	C₄×#10625329	C₄	1	8	yes	complemented
13	C₂×#13093935	C₂²	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	#15688993	C₂²	1	2	yes	—
16	#15688989	C₂²	1	2	yes	—
17	#15688983	—	2	2	no	—
18	C₄×C₄⋊Dic₄₇	—	2	2	no	—
19	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#382	—	2	2	no	—
22	C₄₇×#382	C₂²	1	2	yes	—
23	C₄₇×#382	C₂²	1	2	yes	—
24	C₂×#95×C₄₇	—	2	2	no	—
25	C₂×#10625329	C₂×C₄	1	8	yes	—
26	C₂×#10625329	D₄	1	8	yes	—
27	C₂×#10625329	Q₈	1	8	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
29	#13093983	—	2	8	no	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	2	no	—
34	#13093957	—	2	16	no	—
35	#13093955	—	2	16	no	—
36	#13093935	C₂×C₄	1	8	yes	complemented
37	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
38	C₄₇⋊#95	—	2	4	no	—
39	C₂³×C₄×C₄₇	—	2	2	no	—
40	C₂³×C₄×C₄₇	D₄	1	2	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	#95×C₄₇	—	2	4	no	—
51	#95×C₄₇	—	2	4	no	—
52	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	4	4	no	—
55	C₂²×C₄×C₄₇	—	4	8	no	—
56	C₂²×C₄×C₄₇	—	4	4	no	—
57	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₄₇	—	2	4	no	—
67	C₄⋊C₄×C₄₇	—	2	8	no	—
68	C₄⋊C₄×C₄₇	—	4	4	no	—
69	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
70	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
71	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
72	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
73	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
74	C₂²⋊Dic₄₇	—	4	4	no	—
75	C₄⋊Dic₄₇	—	2	4	no	—
76	C₄⋊Dic₄₇	—	2	4	no	—
77	C₄⋊Dic₄₇	—	4	8	no	—
78	C₄×Dic₄₇	—	4	4	no	—
79	#10625329	C₄⋊C₄	1	32	yes	complemented
80	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
81	C₂×C₄₇⋊C₈	C₂×D₄	1	8	yes	—
82	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
83	C₂³×C₄₇	—	2	4	no	—
84	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₄₇	#127	1	2	yes	—
86	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
87	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	—	2	4	no	—
89	C₂×C₄×C₄₇	—	2	4	no	—
90	C₂×C₄×C₄₇	—	4	8	no	—
91	C₂×C₄×C₄₇	—	4	4	no	—
92	C₂×C₄×C₄₇	—	4	8	no	—
93	C₂×C₄×C₄₇	—	8	8	no	—
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
95	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₇	#127	1	2	yes	—
99	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
100	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
101	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
104	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
105	C₂×Dic₄₇	—	4	4	no	—
106	C₂×Dic₄₇	—	4	8	no	—
107	C₄₇⋊C₈	C₂×C₄⋊C₄	1	16	yes	—
108	#20801	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	—	2	4	no	—
110	C₂²×C₄₇	—	2	4	no	—
111	C₂²×C₄₇	—	2	4	no	—
112	C₂²×C₄₇	#384	1	4	yes	—
113	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
115	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂²×C₄₇	#575	1	4	yes	—
117	C₄×C₄₇	—	2	4	no	—
118	C₄×C₄₇	—	2	4	no	—
119	C₄×C₄₇	—	8	4	no	—
120	C₄×C₄₇	#384	1	4	yes	—
121	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
122	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
123	Dic₄₇	—	8	4	no	—
124	C₂×#428	—	47	4	no	—
125	C₂×C₄×M₁₆	—	47	1	no	—
126	#2285	—	47	1	no	—
127	C₂×#382	D₄₇	1	1	yes	characteristic
128	#1847	—	47	4	no	—
129	#1846	—	47	4	no	—
130	C₂×C₄₇	—	2	4	no	—
131	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
132	C₂×C₄₇	#2911	1	2	yes	—
133	C₂×C₄₇	#3153	1	2	yes	—
134	C₂×C₄₇	#3166	1	2	yes	—
135	C₂²×M₁₆	—	47	2	no	—
136	C₂×#117	—	47	2	no	—
137	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
138	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
139	#428	—	94	16	no	—
140	C₂×#107	—	47	8	no	—
141	C₂×#106	—	47	8	no	—
142	C₄×M₁₆	—	47	8	no	—
143	C₂×#97	—	47	2	no	—
144	C₂×C₄×C₈	—	47	2	no	—
145	#384	—	47	2	no	—
146	#382	—	2	2	no	—
147	#382	—	47	2	no	—
148	#382	C₂×D₄₇	1	2	yes	—
149	#382	C₂×D₄₇	1	2	yes	—
150	#380	—	94	2	no	—
151	C₄×C₄⋊C₄	—	94	2	no	—
152	C₂×#95	—	2	2	no	—
153	C₄₇	#20801	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₄	—	2	2	no	—
155	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
156	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
157	C₂×M₁₆	—	47	8	no	—
158	C₂×M₁₆	—	47	8	no	—
159	C₂×M₁₆	—	47	8	no	—
160	C₂²×C₈	—	47	4	no	—
161	#117	—	94	8	no	—
162	C₂×C₄⋊C₄	—	2	8	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	47	4	no	—
165	C₂×C₄⋊C₄	—	94	4	no	—
166	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
167	C₂×C₂²⋊C₄	—	94	2	no	—
168	C₂×C₄²	—	94	2	no	—
169	C₂×C₄²	Dic₉₄	1	4	yes	—
170	C₂×C₄²	C₄×D₄₇	1	4	yes	—
171	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
172	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
174	#107	—	94	16	no	—
175	#106	—	94	16	no	—
176	#97	—	47	8	no	—
177	C₄×C₈	—	47	8	no	—
178	#95	—	2	4	no	—
179	#95	—	2	4	no	—
180	#95	—	94	4	no	—
181	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	4	4	no	—
184	C₂²×C₄	—	4	8	no	—
185	C₂²×C₄	—	4	4	no	—
186	C₂²×C₄	—	94	4	no	—
187	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
188	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
189	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
190	C₂²×C₄	D₄×D₄₇	1	4	yes	—
191	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
192	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
193	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
195	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
196	M₁₆	—	47	32	no	—
197	C₂×C₈	—	47	8	no	—
198	C₂×C₈	—	47	8	no	—
199	C₂×C₈	—	47	8	no	—
200	C₄⋊C₄	—	2	8	no	—
201	C₄⋊C₄	—	4	4	no	—
202	C₄⋊C₄	—	94	4	no	—
203	C₄⋊C₄	—	94	4	no	—
204	C₄⋊C₄	—	188	8	no	—
205	C₂²⋊C₄	—	188	4	no	—
206	C₄²	—	188	4	no	—
207	C₄²	Dic₄₇⋊C₄	1	8	yes	—
208	C₄²	C₂×Dic₉₄	1	2	yes	—
209	C₄²	C₂×Dic₉₄	1	2	yes	—
210	C₄²	C₂×C₄×D₄₇	1	2	yes	—
211	C₄²	C₂×C₄×D₄₇	1	2	yes	—
212	C₂³	—	2	4	no	—
213	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
214	C₂³	#13094020	1	2	yes	—
215	C₂³	#13094022	1	2	yes	—
216	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	8	no	—
222	C₂×C₄	—	8	8	no	—
223	C₂×C₄	—	188	4	no	—
224	C₂×C₄	—	188	8	no	—
225	C₂×C₄	C₄⋊C₄×D₄₇	1	8	yes	—
226	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
227	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
228	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
229	C₂×C₄	#13094020	1	2	yes	—
230	C₂×C₄	#13094022	1	2	yes	—
231	C₂×C₄	#13094023	1	8	yes	—
232	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
233	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
234	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
235	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
236	C₈	—	47	16	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	4	no	—
240	C₂²	#15688992	1	4	yes	—
241	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
242	C₂²	C₂×#13094020	1	1	yes	characteristic
243	C₂²	C₂×#13094022	1	1	yes	characteristic
244	C₂²	#15689921	1	4	yes	—
245	C₄	—	2	4	no	—
246	C₄	—	2	4	no	—
247	C₄	—	8	4	no	—
248	C₄	—	376	4	no	—
249	C₄	#15688992	1	4	yes	—
250	C₄	C₂×C₄⋊C₄×D₄₇	1	2	yes	—
251	C₄	C₂×#13094023	1	2	yes	—
252	C₂	—	2	4	no	—
253	C₂	C₂×#15688992	1	1	yes	characteristic
254	C₂	#20251977	1	2	yes	—
255	C₂	#20253852	1	2	yes	—
256	C₂	#20253937	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#382	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#382	D₄₇
hypocenter	C₄₇	#20801
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#382

Complements

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

N	H ≅ G/N	type
4. #20243557	C₂	semidirect
5. #20243550	C₂	semidirect
12. C₄×#10625329	C₄	semidirect
36. #13093935	C₂×C₄	semidirect
37. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
79. #10625329	C₄⋊C₄	semidirect
153. C₄₇	#20801	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689163	2
C₂×C₄×#10625329	2
#20245795	2
#20243557	2
#20243550	2
C₂×C₄₇×#382	2
#20801	47