Subgroups — C₄×#15689246 #26214854

Order 12032.

1022 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689246	C₂	1	2	yes	maximal
2	C₄×#13094046	C₂	1	1	yes	maximal, characteristic
3	C₄×#13094030	C₂	1	2	yes	maximal, complemented
4	C₄×#13093960	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243272	C₂	1	2	yes	maximal, complemented
6	#20243232	C₂	1	4	yes	maximal, complemented
7	C₄×#13093952	C₂	1	2	yes	maximal, complemented
8	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	C₂²	1	1	yes	characteristic
10	C₂×#13094030	C₂²	1	4	yes	—
11	C₄×#10625350	C₂²	1	2	yes	—
12	#15689246	C₄	1	16	yes	direct product factor
13	C₄×Dic₁₈₈	C₄	1	2	yes	complemented
14	C₄×#10625325	C₄	1	4	yes	complemented
15	C₄×D₃₇₆	C₄	1	2	yes	complemented
16	C₂×#13093960	C₂²	1	2	yes	—
17	#15689161	C₂²	1	1	yes	characteristic
18	C₂×#13093952	C₂²	1	4	yes	—
19	#15689120	C₂²	1	2	yes	—
20	#15689078	C₂²	1	4	yes	—
21	#15689077	—	2	4	no	—
22	C₄×#10625329	C₂²	1	2	yes	—
23	C₄×#10625324	—	2	2	no	—
24	C₄×C₈×D₄₇	—	2	2	no	—
25	#15688888	C₂²	1	2	yes	—
26	#15688883	C₂²	1	4	yes	—
27	C₂×#97×C₄₇	C₂²	1	2	yes	—
28	C₂×#97×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄³×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×#10625350	C₂³	1	2	yes	—
33	#13094046	C₂×C₄	1	8	yes	—
34	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
35	C₂×#10625325	C₂×C₄	1	2	yes	—
36	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
37	#13094030	—	2	8	no	—
38	#13094030	C₂×C₄	1	16	yes	complemented
39	C₂×#10625329	C₂³	1	4	yes	—
40	C₂×#10625324	—	2	4	no	—
41	C₂×C₈×D₄₇	—	2	4	no	—
42	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
43	C₄×D₁₈₈	C₂×C₄	1	4	yes	—
44	C₄×C₂²⋊D₄₇	—	2	4	no	—
45	#13093983	C₂³	1	2	yes	—
46	#13093981	—	2	2	no	—
47	C₄²×D₄₇	—	2	2	no	—
48	#13093960	C₂×C₄	1	8	yes	complemented
49	#13093958	C₂×C₄	1	2	yes	—
50	#13093956	C₂×C₄	1	2	yes	—
51	#13093952	C₂×C₄	1	16	yes	complemented
52	#13093949	C₂×C₄	1	4	yes	—
53	#13093946	C₂×C₄	1	4	yes	—
54	#13093935	—	2	2	no	—
55	#13093934	—	2	4	no	—
56	C₄×C₄₇⋊C₈	—	2	2	no	—
57	C₈×Dic₄₇	—	2	4	no	—
58	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
61	C₂×C₄²×C₄₇	—	2	2	no	—
62	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	#97×C₄₇	—	2	4	no	—
65	#97×C₄₇	C₂×C₄	1	8	yes	complemented
66	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₄₇	D₄	1	2	yes	—
69	C₄×C₈×C₄₇	D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	#41	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂×C₂²⋊D₄₇	—	2	2	no	—
74	#10625350	—	2	4	no	—
75	#10625350	C₂²×C₄	1	8	yes	—
76	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
77	C₂×C₄×D₄₇	—	2	2	no	—
78	C₂×C₄×D₄₇	—	2	4	no	—
79	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₄₇	#41	1	4	yes	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	—	4	4	no	—
90	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
91	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂²⋊Dic₄₇	—	2	2	no	—
95	(C₂×D₄₇)⋊C₄	—	2	4	no	—
96	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
97	Dic₄₇⋊C₄	—	2	4	no	—
98	C₄×Dic₄₇	—	2	4	no	—
99	C₄×Dic₄₇	—	2	2	no	—
100	#10625329	—	2	4	no	—
101	#10625329	C₂²×C₄	1	8	yes	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	Dic₁₈₈	C₄²	1	4	yes	complemented
104	D₃₇₆	C₄²	1	4	yes	complemented
105	#10625325	C₄²	1	8	yes	complemented
106	#10625324	—	2	16	no	—
107	C₈×D₄₇	—	2	16	no	—
108	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
109	C₂²×D₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
119	C₂²⋊D₄₇	—	2	8	no	—
120	C₂×Dic₄₇	—	2	2	no	—
121	C₂×Dic₄₇	—	2	4	no	—
122	D₁₈₈	—	2	2	no	—
123	D₁₈₈	C₂×C₄²	1	4	yes	—
124	C₄×D₄₇	—	2	8	no	—
125	C₄×D₄₇	—	4	8	no	—
126	Dic₉₄	—	2	2	no	—
127	Dic₉₄	C₂×C₄²	1	4	yes	—
128	C₈×C₄₇	—	2	2	no	—
129	C₈×C₄₇	C₂×C₄²	1	2	yes	—
130	C₈×C₄₇	C₂×C₄²	1	2	yes	—
131	C₈×C₄₇	C₄×D₄	1	8	yes	—
132	C₄₇⋊C₈	—	2	8	no	—
133	C₄×#444	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₄₇	C₄×#41	1	2	yes	—
137	C₂×D₄₇	—	2	4	no	—
138	C₂×D₄₇	—	4	4	no	—
139	C₄×C₄₇	—	2	2	no	—
140	C₄×C₄₇	—	4	4	no	—
141	C₄×C₄₇	#444	1	4	yes	complemented
142	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
143	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
144	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
145	Dic₄₇	—	2	4	no	—
146	Dic₄₇	—	4	4	no	—
147	C₂×#444	—	47	2	no	—
148	C₄×#135	—	47	1	no	—
149	C₄×#131	—	47	2	no	—
150	C₄×#108	—	47	1	no	—
151	#1773	—	47	2	no	—
152	#1765	—	47	4	no	—
153	C₄×#104	—	47	2	no	—
154	C₄×#97	D₄₇	1	1	yes	characteristic, complemented
155	C₂×C₄₇	—	2	2	no	—
156	C₂×C₄₇	C₄²×D₄	1	1	yes	characteristic
157	C₂×C₄₇	C₂×#444	1	2	yes	—
158	D₄₇	—	4	4	no	—
159	C₂×#135	—	47	1	no	—
160	C₂×#131	—	47	4	no	—
161	C₄×#41	—	47	2	no	—
162	#444	—	47	16	no	—
163	C₄×Q₁₆	—	47	2	no	—
164	C₄×SD₁₆	—	47	4	no	—
165	C₄×D₈	—	47	2	no	—
166	C₂×#108	—	47	2	no	—
167	#427	—	47	1	no	—
168	C₂×#104	—	47	4	no	—
169	#416	—	47	2	no	—
170	#405	—	47	4	no	—
171	#405	—	94	4	no	—
172	C₄×M₁₆	—	47	2	no	—
173	C₄×M₁₆	—	94	2	no	—
174	C₂×#97	C₂×D₄₇	1	2	yes	—
175	C₂×#97	C₂×D₄₇	1	2	yes	—
176	C₂×C₄×C₈	—	94	2	no	—
177	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
179	C₄³	C₂×D₄₇	1	1	yes	characteristic
180	#341	—	47	2	no	—
181	#339	—	47	4	no	—
182	C₄₇	C₄×#444	1	1	yes	characteristic, complemented, Sylow
183	C₂×#41	—	47	2	no	—
184	#135	—	47	8	no	—
185	C₂×Q₁₆	—	47	1	no	—
186	C₂×SD₁₆	—	47	2	no	—
187	C₂×D₈	—	47	1	no	—
188	#131	—	47	16	no	—
189	#131	—	94	8	no	—
190	C₂×M₁₆	—	47	4	no	—
191	C₂×M₁₆	—	94	4	no	—
192	C₂²×C₈	—	94	4	no	—
193	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
196	C₄×Q₈	—	47	4	no	—
197	C₄×D₄	—	47	4	no	—
198	C₄×D₄	—	94	4	no	—
199	#117	—	47	2	no	—
200	#117	—	94	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	94	2	no	—
203	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
204	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
205	#108	—	47	8	no	—
206	#107	—	47	2	no	—
207	#106	—	47	2	no	—
208	#104	—	47	16	no	—
209	#103	—	47	4	no	—
210	#102	—	47	4	no	—
211	#97	—	2	4	no	—
212	#97	—	94	4	no	—
213	#97	—	94	2	no	—
214	#97	C₄×D₄₇	1	8	yes	complemented
215	C₄×C₈	—	94	4	no	—
216	C₄×C₈	—	94	2	no	—
217	C₄×C₈	C₄×D₄₇	1	2	yes	—
218	C₄×C₈	C₄×D₄₇	1	2	yes	—
219	C₄×C₈	D₁₈₈	1	2	yes	—
220	C₄×C₈	D₁₈₈	1	2	yes	—
221	#41	—	47	8	no	—
222	#41	—	94	4	no	—
223	C₂×Q₈	—	47	2	no	—
224	C₂×D₄	—	47	2	no	—
225	C₂×D₄	—	94	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	—	94	4	no	—
229	C₂²×C₄	#10625350	1	2	yes	—
230	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
231	Q₁₆	—	47	4	no	—
232	SD₁₆	—	47	8	no	—
233	D₈	—	47	4	no	—
234	M₁₆	—	47	8	no	—
235	M₁₆	—	94	4	no	—
236	M₁₆	—	94	16	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	94	4	no	—
239	C₂×C₈	—	94	16	no	—
240	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
241	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
244	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
245	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
246	C₂×C₈	#10625350	1	4	yes	—
247	C₄⋊C₄	—	47	4	no	—
248	C₄⋊C₄	—	94	4	no	—
249	C₂²⋊C₄	—	94	2	no	—
250	C₂²⋊C₄	—	94	4	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	4	4	no	—
253	C₄²	—	94	4	no	—
254	C₄²	—	94	2	no	—
255	C₄²	C₂×C₄×D₄₇	1	4	yes	—
256	C₄²	C₂×C₄×D₄₇	1	2	yes	—
257	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
258	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
259	C₂³	—	94	2	no	—
260	C₂³	C₂×#10625350	1	1	yes	characteristic
261	Q₈	—	47	4	no	—
262	Q₈	—	94	2	no	—
263	D₄	—	47	4	no	—
264	D₄	—	94	8	no	—
265	D₄	—	94	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	94	2	no	—
270	C₂×C₄	—	94	8	no	—
271	C₂×C₄	—	94	4	no	—
272	C₂×C₄	—	188	8	no	—
273	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
274	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
275	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
276	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
278	C₂×C₄	C₂×#10625350	1	2	yes	—
279	C₈	—	2	2	no	—
280	C₈	—	94	8	no	—
281	C₈	C₄²×D₄₇	1	2	yes	—
282	C₈	C₄²×D₄₇	1	2	yes	—
283	C₈	C₄×D₁₈₈	1	8	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	94	4	no	—
286	C₂²	—	188	4	no	—
287	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
288	C₂²	C₄×#10625350	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	4	4	no	—
291	C₄	—	94	4	no	—
292	C₄	—	188	4	no	—
293	C₄	#15689246	1	4	yes	direct product factor
294	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
295	C₄	C₂×C₄×D₁₈₈	1	2	yes	—
296	C₄	C₄×#10625350	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	188	4	no	—
299	C₂	C₄²×D₁₈₈	1	1	yes	characteristic
300	C₂	C₂×#15689246	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₁₈₈
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#97×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#13094046	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	C₄×#444
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
3. C₄×#13094030	C₂	semidirect
4. C₄×#13093960	C₂	semidirect
5. #20243272	C₂	semidirect
6. #20243232	C₂	semidirect
7. C₄×#13093952	C₂	semidirect
8. C₄×#97×C₄₇	C₂	semidirect
12. #15689246	C₄	direct (N × H)
13. C₄×Dic₁₈₈	C₄	semidirect
14. C₄×#10625325	C₄	semidirect
15. C₄×D₃₇₆	C₄	semidirect
38. #13094030	C₂×C₄	semidirect
48. #13093960	C₂×C₄	semidirect
51. #13093952	C₂×C₄	semidirect
65. #97×C₄₇	C₂×C₄	semidirect
103. Dic₁₈₈	C₄²	semidirect
104. D₃₇₆	C₄²	semidirect
105. #10625325	C₄²	semidirect
141. C₄×C₄₇	#444	semidirect
154. C₄×#97	D₄₇	semidirect
182. C₄₇	C₄×#444	semidirect
214. #97	C₄×D₄₇	semidirect
293. C₄	#15689246	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×#15689246	2
C₄×#13094046	2
C₄×#13094030	2
C₄×#13093960	2
#20243272	2
#20243232	2
C₄×#13093952	2
C₄×#97×C₄₇	2
C₄×#444	47

Subgroups — C4×#15689246 #26214854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689246 #26214854