Subgroups — C₄×#15689492 #26224360

Order 12032.

802 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689492	C₂	1	1	yes	maximal, characteristic
2	C₄×#13094024	C₂	1	1	yes	maximal, characteristic
3	C₄×#13094016	C₂	1	1	yes	maximal, characteristic
4	#20244357	C₂	1	1	yes	maximal, characteristic
5	#20244208	C₂	1	1	yes	maximal, characteristic
6	#20244078	C₂	1	4	yes	maximal
7	#20243522	C₂	1	1	yes	maximal, characteristic
8	C₄×#13093949	C₂	1	4	yes	maximal
9	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094024	C₂²	1	1	yes	characteristic
11	C₂×#13094016	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
13	#15689492	C₄	1	8	yes	direct product factor
14	C₂×#13093949	C₂²	1	4	yes	—
15	C₂×#13093949	C₂²	1	4	yes	—
16	#15689029	C₂²	1	2	yes	—
17	#15689021	—	2	2	no	—
18	#15689021	C₂²	1	2	yes	—
19	#15689007	—	2	2	no	—
20	#15688993	C₂²	1	1	yes	characteristic
21	#15688989	C₂²	1	1	yes	characteristic
22	C₄×Dic₄₇⋊C₄	—	2	2	no	—
23	C₄×C₄⋊Dic₄₇	—	2	2	no	—
24	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
25	#15688886	C₂²	1	4	yes	—
26	#15688886	C₂²	1	4	yes	—
27	C₂×#97×C₄₇	C₂²	1	2	yes	—
28	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
31	C₄³×C₄₇	C₂²	1	1	yes	characteristic
32	C₂²×Dic₉₄	C₂³	1	2	yes	—
33	#13094024	—	2	2	no	—
34	#13094024	C₂×C₄	1	4	yes	—
35	#13094016	—	2	2	no	—
36	#13094016	C₂×C₄	1	4	yes	—
37	C₄×Dic₉₄	—	2	8	no	—
38	C₂×C₄⋊Dic₄₇	—	2	2	no	—
39	C₂×C₄⋊Dic₄₇	—	2	2	no	—
40	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
43	C₂×Dic₄₇⋊C₄	—	2	2	no	—
44	C₂×Dic₄₇⋊C₄	—	2	2	no	—
45	C₂×Dic₄₇⋊C₄	—	2	2	no	—
46	C₂×C₄×Dic₄₇	—	2	2	no	—
47	C₂×C₄×Dic₄₇	—	2	2	no	—
48	#13093949	—	2	8	no	—
49	#13093949	C₂×C₄	1	16	yes	—
50	C₄₇⋊#95	—	2	4	no	—
51	C₄₇⋊#95	—	2	2	no	—
52	C₄₇⋊#95	—	2	2	no	—
53	C₄₇⋊#95	—	2	2	no	—
54	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₄₇	D₄	1	2	yes	—
57	C₂×C₄²×C₄₇	D₄	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	#97×C₄₇	—	2	4	no	—
62	#97×C₄₇	—	2	2	no	—
63	#97×C₄₇	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₇	—	2	2	no	—
71	C₂²×Dic₄₇	—	2	2	no	—
72	C₂²×Dic₄₇	—	2	2	no	—
73	C₂²×Dic₄₇	—	2	2	no	—
74	C₂×Dic₉₄	—	2	4	no	—
75	C₂×Dic₉₄	—	2	4	no	—
76	C₂×Dic₉₄	—	2	2	no	—
77	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₄₇	#41	1	4	yes	—
84	C₄²×C₄₇	—	2	4	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	4	2	no	—
87	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₄₇	#41	1	2	yes	—
89	C₄²×C₄₇	#41	1	4	yes	—
90	C₄⋊Dic₄₇	—	2	8	no	—
91	C₄⋊Dic₄₇	—	2	2	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	2	4	no	—
94	C₄⋊Dic₄₇	—	4	4	no	—
95	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
96	Dic₄₇⋊C₄	—	2	8	no	—
97	Dic₄₇⋊C₄	—	4	4	no	—
98	Dic₄₇⋊C₄	—	4	4	no	—
99	C₄×Dic₄₇	—	4	4	no	—
100	C₄×Dic₄₇	—	4	4	no	—
101	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
109	C₂×C₄×C₄₇	#124	1	2	yes	—
110	C₂×C₄×C₄₇	#124	1	2	yes	—
111	C₂×C₄×C₄₇	#137	1	4	yes	—
112	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
116	C₂×Dic₄₇	—	2	4	no	—
117	C₂×Dic₄₇	—	2	4	no	—
118	C₂×Dic₄₇	—	4	2	no	—
119	C₂×Dic₄₇	—	4	2	no	—
120	C₂×Dic₄₇	—	4	2	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	C₂×Dic₄₇	—	4	2	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	Dic₉₄	—	2	8	no	—
128	Dic₉₄	—	4	4	no	—
129	C₈×C₄₇	—	2	4	no	—
130	C₈×C₄₇	—	2	4	no	—
131	C₄×#491	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
134	C₂²×C₄₇	#532	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×#137	1	2	yes	—
136	C₂²×C₄₇	#575	1	2	yes	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	2	4	no	—
139	C₄×C₄₇	—	4	2	no	—
140	C₄×C₄₇	#491	1	4	yes	complemented
141	C₄×C₄₇	C₄×#41	1	2	yes	—
142	C₄×C₄₇	#529	1	2	yes	—
143	Dic₄₇	—	4	4	no	—
144	Dic₄₇	—	4	4	no	—
145	Dic₄₇	—	8	2	no	—
146	Dic₄₇	—	8	2	no	—
147	C₂×#491	—	47	1	no	—
148	C₄×C₄⋊Q₈	—	47	1	no	—
149	C₄×#125	—	47	1	no	—
150	#1992	—	47	1	no	—
151	#1960	—	47	1	no	—
152	#1935	—	47	4	no	—
153	#1836	—	47	1	no	—
154	C₄×#103	—	47	4	no	—
155	C₄×#97	D₄₇	1	1	yes	characteristic
156	C₂×C₄₇	C₄×#124	1	1	yes	characteristic
157	C₂×C₄₇	C₄×#137	1	2	yes	—
158	C₂×C₄₇	C₂×#491	1	1	yes	characteristic
159	C₂×C₄₇	#3135	1	1	yes	characteristic
160	C₂×C₄₇	#3172	1	2	yes	—
161	C₂×C₄⋊Q₈	—	47	1	no	—
162	C₂×#125	—	47	1	no	—
163	C₂×C₄×Q₈	—	47	2	no	—
164	#491	—	47	8	no	—
165	C₂×#103	—	47	4	no	—
166	C₂×#103	—	47	4	no	—
167	C₂×#97	C₂×D₄₇	1	2	yes	—
168	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
169	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
171	#391	—	47	2	no	—
172	#389	—	47	2	no	—
173	#389	—	94	2	no	—
174	#387	—	94	2	no	—
175	#384	—	47	1	no	—
176	#382	—	47	1	no	—
177	C₄×C₄⋊C₄	—	47	2	no	—
178	C₄×C₄⋊C₄	—	94	2	no	—
179	C₄×C₄⋊C₄	—	94	2	no	—
180	C₄³	C₂×D₄₇	1	1	yes	characteristic
181	#340	—	47	4	no	—
182	#340	—	47	4	no	—
183	C₄₇	C₄×#491	1	1	yes	characteristic, complemented, Sylow
184	C₂²×Q₈	—	47	2	no	—
185	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
186	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
187	C₄⋊Q₈	—	47	4	no	—
188	C₄⋊Q₈	—	94	2	no	—
189	#125	—	47	4	no	—
190	#125	—	94	2	no	—
191	C₄×Q₈	—	94	8	no	—
192	C₂×C₄⋊C₄	—	47	2	no	—
193	C₂×C₄⋊C₄	—	47	2	no	—
194	C₂×C₄⋊C₄	—	47	2	no	—
195	C₂×C₄⋊C₄	—	94	2	no	—
196	C₂×C₄⋊C₄	—	94	2	no	—
197	C₂×C₄⋊C₄	—	94	2	no	—
198	C₂×C₄⋊C₄	—	94	2	no	—
199	C₂×C₄⋊C₄	—	94	2	no	—
200	C₂×C₄²	—	94	2	no	—
201	C₂×C₄²	—	94	2	no	—
202	C₂×C₄²	D₁₈₈	1	2	yes	—
203	C₂×C₄²	D₁₈₈	1	2	yes	—
204	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
205	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
207	#103	—	47	16	no	—
208	#103	—	94	8	no	—
209	#97	—	2	4	no	—
210	#97	—	2	2	no	—
211	#97	C₄×D₄₇	1	4	yes	—
212	C₄×C₈	—	2	4	no	—
213	#95	—	94	2	no	—
214	#95	—	94	2	no	—
215	#95	—	94	2	no	—
216	#95	—	94	4	no	—
217	C₂×Q₈	—	47	4	no	—
218	C₂×Q₈	—	94	2	no	—
219	C₂×Q₈	—	94	4	no	—
220	C₂×Q₈	—	94	4	no	—
221	C₂²×C₄	—	94	2	no	—
222	C₂²×C₄	—	94	2	no	—
223	C₂²×C₄	—	94	2	no	—
224	C₂²×C₄	—	94	2	no	—
225	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
226	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
227	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
229	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
235	C₂×C₈	#10625350	1	4	yes	—
236	C₄⋊C₄	—	47	4	no	—
237	C₄⋊C₄	—	94	8	no	—
238	C₄⋊C₄	—	94	8	no	—
239	C₄⋊C₄	—	94	2	no	—
240	C₄⋊C₄	—	94	4	no	—
241	C₄⋊C₄	—	94	4	no	—
242	C₄⋊C₄	—	188	4	no	—
243	C₄⋊C₄	—	188	4	no	—
244	C₄⋊C₄	—	188	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	—	188	4	no	—
249	C₄²	—	188	4	no	—
250	C₄²	C₂×C₄×D₄₇	1	2	yes	—
251	C₄²	#10625350	1	2	yes	—
252	C₄²	#10625350	1	4	yes	—
253	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
254	Q₈	—	94	8	no	—
255	Q₈	—	188	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	94	4	no	—
263	C₂×C₄	—	94	4	no	—
264	C₂×C₄	—	188	2	no	—
265	C₂×C₄	—	188	2	no	—
266	C₂×C₄	—	188	4	no	—
267	C₂×C₄	—	188	2	no	—
268	C₂×C₄	—	188	2	no	—
269	C₂×C₄	—	188	2	no	—
270	C₂×C₄	—	188	2	no	—
271	C₂×C₄	—	188	2	no	—
272	C₂×C₄	—	188	2	no	—
273	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
274	C₂×C₄	#13094012	1	2	yes	—
275	C₂×C₄	#13094012	1	2	yes	—
276	C₂×C₄	#13094057	1	4	yes	—
277	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
278	C₂×C₄	C₂×#10625350	1	2	yes	—
279	C₂×C₄	C₂×#10625350	1	2	yes	—
280	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
284	C₂²	C₂×#13094012	1	1	yes	characteristic
285	C₂²	#15689673	1	1	yes	characteristic
286	C₂²	C₂×#13094057	1	2	yes	—
287	C₂²	#15689923	1	2	yes	—
288	C₄	—	2	2	no	—
289	C₄	—	2	4	no	—
290	C₄	—	4	2	no	—
291	C₄	—	376	2	no	—
292	C₄	—	376	2	no	—
293	C₄	—	188	4	no	—
294	C₄	—	188	4	no	—
295	C₄	#15689492	1	4	yes	direct product factor
296	C₄	C₄×#10625350	1	2	yes	—
297	C₄	#15689657	1	2	yes	—
298	C₂	C₄×#13094012	1	1	yes	characteristic
299	C₂	C₄×#13094057	1	2	yes	—
300	C₂	C₂×#15689492	1	1	yes	characteristic
301	C₂	#20253734	1	1	yes	characteristic
302	C₂	#20253994	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#97×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	C₄×#491
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
13. #15689492	C₄	direct (N × H)
140. C₄×C₄₇	#491	semidirect
183. C₄₇	C₄×#491	semidirect
295. C₄	#15689492	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×#15689492	2
C₄×#13094024	2
C₄×#13094016	2
#20244357	2
#20244208	2
#20244078	2
#20243522	2
C₄×#13093949	2
C₄×#97×C₄₇	2
C₄×#491	47

Subgroups — C4×#15689492 #26224360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689492 #26224360