Subgroups — C₄×#15689500 #26224708

Order 12032.

1422 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₃₇₆	C₂	1	4	yes	maximal, complemented
3	C₄×#13094021	C₂	1	2	yes	maximal, complemented
4	#20244254	C₂	1	4	yes	maximal, complemented
5	#20244194	C₂	1	2	yes	maximal, complemented
6	#20243548	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₇₆	C₂²	1	4	yes	—
9	C₂×#13094021	C₂²	1	2	yes	—
10	C₂×C₄×D₁₈₈	—	2	4	no	—
11	C₂×C₄×D₁₈₈	C₂²	1	4	yes	—
12	#15689500	C₄	1	8	yes	direct product factor
13	C₄×D₃₇₆	—	2	16	no	—
14	C₂×#13093958	C₂²	1	4	yes	—
15	C₂×#13093946	C₂²	1	8	yes	—
16	#15689025	—	2	4	no	—
17	#15689025	C₂²	1	4	yes	—
18	#15688993	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
23	C₄³×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×D₁₈₈	—	2	4	no	—
25	C₂²×D₁₈₈	C₂³	1	4	yes	—
26	C₂²×C₄×D₄₇	—	2	4	no	—
27	C₂×D₃₇₆	—	2	8	no	—
28	C₂×D₃₇₆	—	2	8	no	—
29	C₂×D₃₇₆	C₂×C₄	1	16	yes	complemented
30	#13094021	—	2	4	no	—
31	#13094021	C₂×C₄	1	8	yes	complemented
32	C₄×D₁₈₈	—	2	16	no	—
33	C₄×D₁₈₈	—	4	16	no	—
34	C₂×C₄⋊Dic₄₇	—	2	4	no	—
35	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
36	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
37	C₂×(C₂×D₄₇)⋊C₄	—	2	8	no	—
38	#13093958	—	2	8	no	—
39	#13093946	—	2	16	no	—
40	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
42	C₂×C₄²×C₄₇	D₄	1	2	yes	—
43	C₂×C₄²×C₄₇	D₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₄₇	—	2	4	no	—
48	C₄×C₈×C₄₇	—	2	2	no	—
49	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
50	C₄×C₈×C₄₇	D₄	1	8	yes	complemented
51	C₂³×D₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
57	C₂²×Dic₄₇	—	2	4	no	—
58	C₂×D₁₈₈	—	2	8	no	—
59	C₂×D₁₈₈	—	2	8	no	—
60	C₂×D₁₈₈	—	2	16	no	—
61	C₂×D₁₈₈	—	2	4	no	—
62	C₂×D₁₈₈	—	4	8	no	—
63	C₂×D₁₈₈	—	4	8	no	—
64	C₂×D₁₈₈	C₂²×C₄	1	8	yes	—
65	C₂×C₄×D₄₇	—	4	8	no	—
66	C₂×C₄×D₄₇	—	4	16	no	—
67	C₂×C₈×C₄₇	—	2	2	no	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
72	C₂×C₈×C₄₇	#41	1	4	yes	—
73	C₄²×C₄₇	—	2	4	no	—
74	C₄²×C₄₇	—	2	2	no	—
75	C₄²×C₄₇	D₈	1	8	yes	complemented
76	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₄₇	C₂×D₄	1	4	yes	—
78	C₄²×C₄₇	#41	1	2	yes	—
79	(C₂×D₄₇)⋊C₄	—	4	8	no	—
80	(C₂×D₄₇)⋊C₄	—	4	16	no	—
81	C₄⋊Dic₄₇	—	2	8	no	—
82	C₄⋊Dic₄₇	—	4	8	no	—
83	D₃₇₆	—	2	32	no	—
84	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
85	C₂²×D₄₇	—	2	8	no	—
86	C₂²×D₄₇	—	4	4	no	—
87	C₂²×D₄₇	—	4	4	no	—
88	C₂²×D₄₇	—	4	8	no	—
89	C₂²×D₄₇	—	4	8	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
94	C₂×C₄×C₄₇	#127	1	2	yes	—
95	C₂×C₄×C₄₇	#127	1	2	yes	—
96	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
97	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
98	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
99	C₂×C₄×C₄₇	#135	1	4	yes	—
100	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
104	C₂×Dic₄₇	—	4	4	no	—
105	C₂×Dic₄₇	—	4	8	no	—
106	D₁₈₈	—	2	16	no	—
107	D₁₈₈	—	4	8	no	—
108	D₁₈₈	—	4	32	no	—
109	C₄×D₄₇	—	8	16	no	—
110	C₈×C₄₇	—	2	4	no	—
111	C₈×C₄₇	C₄×D₄	1	8	yes	complemented
112	C₄×#493	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
115	C₂²×C₄₇	#532	1	1	yes	characteristic
116	C₂²×C₄₇	C₂²×D₈	1	2	yes	—
117	C₂²×C₄₇	C₂×#135	1	2	yes	—
118	C₂×D₄₇	—	4	8	no	—
119	C₂×D₄₇	—	4	16	no	—
120	C₂×D₄₇	—	8	4	no	—
121	C₂×D₄₇	—	8	4	no	—
122	C₂×D₄₇	—	8	8	no	—
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	—	2	4	no	—
125	C₄×C₄₇	C₄×D₈	1	8	yes	complemented
126	C₄×C₄₇	#493	1	4	yes	complemented
127	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
128	C₄×C₄₇	#532	1	2	yes	—
129	Dic₄₇	—	8	4	no	—
130	C₂×#493	—	47	1	no	—
131	C₂×C₄×D₈	—	47	4	no	—
132	C₄×#127	—	47	2	no	—
133	#1970	—	47	4	no	—
134	#1957	—	47	2	no	—
135	#1845	—	47	1	no	—
136	C₄²×C₈	D₄₇	1	1	yes	characteristic, complemented
137	C₂×C₄₇	C₄×#127	1	1	yes	characteristic
138	C₂×C₄₇	C₂×C₄×D₈	1	2	yes	—
139	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
140	C₂×C₄₇	#3147	1	1	yes	characteristic
141	C₂×C₄₇	#3169	1	2	yes	—
142	D₄₇	—	8	8	no	—
143	C₂²×D₈	—	47	4	no	—
144	C₂×#127	—	47	2	no	—
145	C₂×C₄×D₄	—	47	4	no	—
146	C₂×C₄×D₄	—	94	4	no	—
147	#493	—	47	8	no	—
148	C₄×D₈	—	94	16	no	—
149	C₂×#107	—	47	4	no	—
150	C₂×#102	—	47	8	no	—
151	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
153	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
155	#390	—	47	4	no	—
156	#390	—	94	4	no	—
157	#384	—	47	2	no	—
158	C₄³	C₂×D₄₇	1	1	yes	characteristic
159	C₄₇	C₄×#493	1	1	yes	characteristic, complemented, Sylow
160	C₂²×D₄	—	47	4	no	—
161	C₂²×D₄	—	94	4	no	—
162	C₂³×C₄	—	94	4	no	—
163	C₂×D₈	—	47	16	no	—
164	C₂×D₈	—	94	8	no	—
165	C₂×D₈	—	94	8	no	—
166	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
167	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
168	#127	—	47	8	no	—
169	#127	—	94	4	no	—
170	C₄×D₄	—	94	16	no	—
171	C₄×D₄	—	188	16	no	—
172	C₂×C₄⋊C₄	—	47	4	no	—
173	C₂×C₄⋊C₄	—	94	4	no	—
174	C₂×C₂²⋊C₄	—	94	4	no	—
175	C₂×C₂²⋊C₄	—	94	8	no	—
176	C₂×C₄²	D₁₈₈	1	2	yes	—
177	C₂×C₄²	D₁₈₈	1	2	yes	—
178	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
179	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
181	#107	—	94	8	no	—
182	#102	—	94	16	no	—
183	C₄×C₈	—	2	4	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	C₄×D₄₇	1	4	yes	complemented
186	C₄×C₈	D₁₈₈	1	8	yes	complemented
187	C₂⁴	—	94	4	no	—
188	C₂×D₄	—	47	8	no	—
189	C₂×D₄	—	94	16	no	—
190	C₂×D₄	—	94	8	no	—
191	C₂×D₄	—	94	8	no	—
192	C₂×D₄	—	94	4	no	—
193	C₂×D₄	—	188	8	no	—
194	C₂×D₄	—	188	8	no	—
195	C₂²×C₄	—	94	4	no	—
196	C₂²×C₄	—	188	8	no	—
197	C₂²×C₄	—	188	16	no	—
198	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
200	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
201	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
202	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
203	D₈	—	94	32	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
207	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
208	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
209	C₂×C₈	#10625350	1	4	yes	—
210	C₄⋊C₄	—	94	8	no	—
211	C₄⋊C₄	—	188	8	no	—
212	C₂²⋊C₄	—	188	8	no	—
213	C₂²⋊C₄	—	188	16	no	—
214	C₄²	—	2	4	no	—
215	C₄²	—	2	2	no	—
216	C₄²	D₃₇₆	1	8	yes	complemented
217	C₄²	C₂×C₄×D₄₇	1	2	yes	—
218	C₄²	C₂×D₁₈₈	1	4	yes	—
219	C₄²	#10625350	1	2	yes	—
220	C₂³	—	94	8	no	—
221	C₂³	—	188	8	no	—
222	C₂³	—	188	8	no	—
223	C₂³	—	188	4	no	—
224	C₂³	—	188	4	no	—
225	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
226	D₄	—	94	16	no	—
227	D₄	—	188	8	no	—
228	D₄	—	188	32	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	376	16	no	—
233	C₂×C₄	—	188	8	no	—
234	C₂×C₄	—	188	4	no	—
235	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
236	C₂×C₄	#13094021	1	2	yes	—
237	C₂×C₄	#13094021	1	2	yes	—
238	C₂×C₄	C₂×D₃₇₆	1	4	yes	—
239	C₂×C₄	C₂×D₃₇₆	1	4	yes	—
240	C₂×C₄	C₂×D₃₇₆	1	4	yes	—
241	C₂×C₄	#13094046	1	4	yes	—
242	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
244	C₂×C₄	C₂×#10625350	1	2	yes	—
245	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
246	C₈	—	2	4	no	—
247	C₈	C₄×D₁₈₈	1	8	yes	complemented
248	C₂²	—	376	8	no	—
249	C₂²	—	376	4	no	—
250	C₂²	—	376	4	no	—
251	C₂²	—	188	8	no	—
252	C₂²	—	188	16	no	—
253	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
254	C₂²	C₂×#13094021	1	1	yes	characteristic
255	C₂²	#15689673	1	1	yes	characteristic
256	C₂²	C₂²×D₃₇₆	1	2	yes	—
257	C₂²	C₂×#13094046	1	2	yes	—
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	376	4	no	—
261	C₄	C₄×D₃₇₆	1	8	yes	complemented
262	C₄	#15689500	1	4	yes	direct product factor
263	C₄	C₂×C₄×D₁₈₈	1	2	yes	—
264	C₄	#15689673	1	2	yes	—
265	C₂	—	376	8	no	—
266	C₂	C₄×#13094021	1	1	yes	characteristic
267	C₂	C₂×C₄×D₃₇₆	1	2	yes	—
268	C₂	C₂×#15689500	1	1	yes	characteristic
269	C₂	#20253816	1	1	yes	characteristic
270	C₂	#20253967	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄²×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂×#15689500	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	C₄×#493
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₃₇₆	C₂	semidirect
3. C₄×#13094021	C₂	semidirect
4. #20244254	C₂	semidirect
5. #20244194	C₂	semidirect
6. #20243548	C₂	semidirect
7. C₄²×C₈×C₄₇	C₂	semidirect
12. #15689500	C₄	direct (N × H)
29. C₂×D₃₇₆	C₂×C₄	semidirect
31. #13094021	C₂×C₄	semidirect
49. C₄×C₈×C₄₇	C₂×C₄	semidirect
50. C₄×C₈×C₄₇	D₄	semidirect
75. C₄²×C₄₇	D₈	semidirect
111. C₈×C₄₇	C₄×D₄	semidirect
125. C₄×C₄₇	C₄×D₈	semidirect
126. C₄×C₄₇	#493	semidirect
136. C₄²×C₈	D₄₇	semidirect
159. C₄₇	C₄×#493	semidirect
185. C₄×C₈	C₄×D₄₇	semidirect
186. C₄×C₈	D₁₈₈	semidirect
216. C₄²	D₃₇₆	semidirect
247. C₈	C₄×D₁₈₈	semidirect
261. C₄	C₄×D₃₇₆	semidirect
262. C₄	#15689500	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#15689500	2
C₂×C₄×D₃₇₆	2
C₄×#13094021	2
#20244254	2
#20244194	2
#20243548	2
C₄²×C₈×C₄₇	2
C₄×#493	47

Subgroups — C4×#15689500 #26224708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689500 #26224708