Subgroups — C₄×#15689185 #26214427

Order 12032.

970 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689185	C₂	1	2	yes	maximal
2	C₄×#13093981	C₂	1	1	yes	maximal, characteristic
3	#20243248	C₂	1	2	yes	maximal, complemented
4	C₄×Dic₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	#20243157	C₂	1	2	yes	maximal
6	C₄×(C₂×D₄₇)⋊C₈	C₂	1	2	yes	maximal
7	#20243141	C₂	1	2	yes	maximal, complemented
8	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13093981	C₂²	1	1	yes	characteristic
11	C₂×#13093981	C₂²	1	1	yes	characteristic
12	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
13	#15689185	C₄	1	16	yes	direct product factor
14	C₂×Dic₄₇⋊C₈	C₂²	1	4	yes	—
15	C₂×(C₂×D₄₇)⋊C₈	C₂²	1	4	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
19	C₄×Dic₄₇⋊C₄	C₄	1	4	yes	—
20	C₄×(C₂×D₄₇)⋊C₄	C₄	1	4	yes	—
21	#15688969	C₂²	1	1	yes	characteristic
22	#15688969	C₂²	1	1	yes	characteristic
23	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
24	#15688877	C₂²	1	4	yes	—
25	#15688877	C₂²	1	4	yes	—
26	C₂×#97×C₄₇	C₂²	1	2	yes	—
27	C₂×#97×C₄₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
29	C₄³×C₄₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄×D₄₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
33	#13093981	—	2	4	no	—
34	#13093981	C₂×C₄	1	8	yes	—
35	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
43	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
44	C₄²×D₄₇	—	2	4	no	—
45	Dic₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
46	(C₂×D₄₇)⋊C₈	C₂×C₄	1	16	yes	—
47	C₄×C₄₇⋊C₈	—	2	4	no	—
48	C₄×C₄₇⋊C₈	—	2	4	no	—
49	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
50	C₄₇⋊#95	C₂×C₄	1	4	yes	—
51	C₄₇⋊#95	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	#97×C₄₇	—	2	4	no	—
58	#97×C₄₇	C₂×C₄	1	8	yes	complemented
59	C₄×C₈×C₄₇	—	2	4	no	—
60	C₂³×D₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
66	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
67	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×D₄₇	—	2	2	no	—
70	C₂×C₄×D₄₇	—	2	2	no	—
71	C₂×C₄×D₄₇	—	2	4	no	—
72	C₂×C₄×D₄₇	—	2	8	no	—
73	C₂×C₄×D₄₇	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
77	C₄²×C₄₇	—	2	8	no	—
78	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
79	C₄²×C₄₇	#41	1	4	yes	—
80	C₄²×C₄₇	#41	1	4	yes	—
81	(C₂×D₄₇)⋊C₄	—	2	4	no	—
82	(C₂×D₄₇)⋊C₄	C₄²	1	8	yes	—
83	Dic₄₇⋊C₄	—	2	4	no	—
84	Dic₄₇⋊C₄	C₄²	1	8	yes	—
85	C₄×Dic₄₇	—	2	4	no	—
86	C₄×Dic₄₇	—	2	8	no	—
87	C₄×Dic₄₇	—	2	2	no	—
88	C₄×Dic₄₇	—	2	2	no	—
89	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
93	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
94	C₂²×D₄₇	—	2	1	no	—
95	C₂²×D₄₇	—	2	1	no	—
96	C₂²×D₄₇	—	2	4	no	—
97	C₂²×D₄₇	C₂×C₄²	1	2	yes	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
101	C₂×C₄×C₄₇	#117	1	4	yes	—
102	C₂×C₄×C₄₇	#117	1	4	yes	—
103	C₂×C₄×C₄₇	#131	1	8	yes	—
104	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
107	C₂×Dic₄₇	—	2	1	no	—
108	C₂×Dic₄₇	—	2	1	no	—
109	C₂×Dic₄₇	—	2	4	no	—
110	C₂×Dic₄₇	—	2	2	no	—
111	C₂×Dic₄₇	—	2	2	no	—
112	C₂×Dic₄₇	—	2	4	no	—
113	C₂×Dic₄₇	—	2	4	no	—
114	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
115	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
116	C₄×D₄₇	—	2	8	no	—
117	C₄×D₄₇	—	4	8	no	—
118	C₈×C₄₇	—	2	8	no	—
119	C₄₇⋊C₈	—	2	8	no	—
120	C₄×#433	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
123	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×#131	1	4	yes	—
125	C₂×D₄₇	—	2	4	no	—
126	C₂×D₄₇	—	2	2	no	—
127	C₂×D₄₇	—	4	2	no	—
128	C₂×D₄₇	—	4	2	no	—
129	C₄×C₄₇	—	2	4	no	—
130	C₄×C₄₇	—	2	4	no	—
131	C₄×C₄₇	#433	1	8	yes	complemented
132	C₄×C₄₇	C₄×#41	1	4	yes	—
133	Dic₄₇	—	2	4	no	—
134	Dic₄₇	—	2	2	no	—
135	Dic₄₇	—	4	2	no	—
136	Dic₄₇	—	4	2	no	—
137	C₂×#433	—	47	2	no	—
138	C₄×#117	—	47	1	no	—
139	#1768	—	47	2	no	—
140	C₄×C₄⋊C₈	—	47	2	no	—
141	#1750	—	47	2	no	—
142	C₄×C₂²⋊C₈	—	47	2	no	—
143	#1746	—	47	2	no	—
144	C₄×#97	D₄₇	1	1	yes	characteristic, complemented
145	C₄²×C₈	—	47	1	no	—
146	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
147	C₂×C₄₇	C₄×#131	1	2	yes	—
148	C₂×C₄₇	C₂×#433	1	2	yes	—
149	C₂×C₄₇	#2922	1	2	yes	—
150	D₄₇	—	4	2	no	—
151	C₂×#117	—	47	1	no	—
152	C₂×#117	—	47	1	no	—
153	C₂²×C₄²	—	47	1	no	—
154	#433	—	47	16	no	—
155	C₂×C₄⋊C₈	—	47	4	no	—
156	C₂×C₂²⋊C₈	—	47	4	no	—
157	C₂×#97	C₂×D₄₇	1	2	yes	—
158	C₂×#97	C₂×D₄₇	1	2	yes	—
159	C₂×C₄×C₈	—	47	2	no	—
160	C₂×C₄×C₈	—	47	2	no	—
161	C₂×C₄×C₈	—	47	2	no	—
162	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
163	C₄×C₄⋊C₄	—	47	4	no	—
164	C₄×C₂²⋊C₄	—	47	4	no	—
165	#376	—	47	1	no	—
166	#376	—	47	1	no	—
167	C₄³	—	47	1	no	—
168	C₄³	C₂×D₄₇	1	1	yes	characteristic
169	#336	—	47	4	no	—
170	#336	—	47	4	no	—
171	C₄₇	C₄×#433	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	—	47	1	no	—
173	C₂³×C₄	—	47	2	no	—
174	C₂²×C₈	—	47	4	no	—
175	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
176	#117	—	47	8	no	—
177	#117	—	94	4	no	—
178	C₂×C₄⋊C₄	—	47	2	no	—
179	C₂×C₄⋊C₄	—	47	2	no	—
180	C₂×C₂²⋊C₄	—	47	2	no	—
181	C₂×C₂²⋊C₄	—	47	2	no	—
182	C₂×C₄²	—	47	1	no	—
183	C₂×C₄²	—	47	4	no	—
184	C₂×C₄²	—	47	2	no	—
185	C₂×C₄²	—	47	1	no	—
186	C₂×C₄²	—	47	1	no	—
187	C₂×C₄²	—	94	4	no	—
188	C₂×C₄²	C₄×D₄₇	1	4	yes	—
189	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	C₄⋊C₈	—	47	16	no	—
193	C₂²⋊C₈	—	47	16	no	—
194	#97	—	2	4	no	—
195	#97	C₄×D₄₇	1	8	yes	complemented
196	C₄×C₈	—	2	4	no	—
197	C₄×C₈	—	47	8	no	—
198	C₄×C₈	—	94	4	no	—
199	C₄×C₈	—	94	4	no	—
200	#95	—	47	4	no	—
201	#95	—	47	4	no	—
202	C₂⁴	—	47	1	no	—
203	C₂²×C₄	—	47	2	no	—
204	C₂²×C₄	—	47	1	no	—
205	C₂²×C₄	—	47	4	no	—
206	C₂²×C₄	—	47	2	no	—
207	C₂²×C₄	—	47	2	no	—
208	C₂²×C₄	—	94	2	no	—
209	C₂²×C₄	—	94	2	no	—
210	C₂²×C₄	—	94	4	no	—
211	C₂²×C₄	—	94	8	no	—
212	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
213	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
214	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
215	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	47	8	no	—
219	C₂×C₈	—	94	4	no	—
220	C₂×C₈	—	94	4	no	—
221	C₂×C₈	C₂×C₄×D₄₇	1	8	yes	—
222	C₄⋊C₄	—	47	8	no	—
223	C₄⋊C₄	—	94	4	no	—
224	C₂²⋊C₄	—	47	8	no	—
225	C₂²⋊C₄	—	94	4	no	—
226	C₄²	—	2	8	no	—
227	C₄²	—	47	4	no	—
228	C₄²	—	94	4	no	—
229	C₄²	—	94	8	no	—
230	C₄²	—	94	2	no	—
231	C₄²	—	94	2	no	—
232	C₄²	C₂×C₄×D₄₇	1	4	yes	—
233	C₄²	#10625350	1	4	yes	—
234	C₄²	#10625350	1	4	yes	—
235	C₂³	—	47	2	no	—
236	C₂³	—	94	1	no	—
237	C₂³	—	94	1	no	—
238	C₂³	—	94	4	no	—
239	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	47	4	no	—
243	C₂×C₄	—	47	2	no	—
244	C₂×C₄	—	94	1	no	—
245	C₂×C₄	—	94	1	no	—
246	C₂×C₄	—	94	4	no	—
247	C₂×C₄	—	94	8	no	—
248	C₂×C₄	—	94	2	no	—
249	C₂×C₄	—	94	2	no	—
250	C₂×C₄	—	94	4	no	—
251	C₂×C₄	—	94	4	no	—
252	C₂×C₄	—	188	8	no	—
253	C₂×C₄	C₄²×D₄₇	1	4	yes	—
254	C₂×C₄	#13093981	1	4	yes	—
255	C₂×C₄	#13093981	1	4	yes	—
256	C₂×C₄	#13094032	1	8	yes	—
257	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
258	C₂×C₄	C₂×#10625350	1	2	yes	—
259	C₂×C₄	C₂×#10625350	1	2	yes	—
260	C₈	—	2	8	no	—
261	C₈	—	94	8	no	—
262	C₂²	—	94	4	no	—
263	C₂²	—	94	2	no	—
264	C₂²	—	188	2	no	—
265	C₂²	—	188	2	no	—
266	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
267	C₂²	C₂×#13093981	1	1	yes	characteristic
268	C₂²	C₂×#13093981	1	1	yes	characteristic
269	C₂²	C₂×#13094032	1	4	yes	—
270	C₄	—	2	4	no	—
271	C₄	—	2	4	no	—
272	C₄	—	94	4	no	—
273	C₄	—	94	2	no	—
274	C₄	—	188	2	no	—
275	C₄	—	188	2	no	—
276	C₄	#15689185	1	8	yes	direct product factor
277	C₄	C₄×#10625350	1	4	yes	—
278	C₂	—	188	2	no	—
279	C₂	C₄×#13093981	1	1	yes	characteristic
280	C₂	C₄×#13094032	1	2	yes	—
281	C₂	C₂×#15689185	1	2	yes	—
282	C₂	#20252033	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#97×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂³×D₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	C₄×#433
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
3. #20243248	C₂	semidirect
4. C₄×Dic₄₇⋊C₈	C₂	semidirect
7. #20243141	C₂	semidirect
8. C₄²×C₄₇⋊C₈	C₂	semidirect
9. C₄×#97×C₄₇	C₂	semidirect
13. #15689185	C₄	direct (N × H)
45. Dic₄₇⋊C₈	C₂×C₄	semidirect
49. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
58. #97×C₄₇	C₂×C₄	semidirect
131. C₄×C₄₇	#433	semidirect
144. C₄×#97	D₄₇	semidirect
171. C₄₇	C₄×#433	semidirect
195. #97	C₄×D₄₇	semidirect
276. C₄	#15689185	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×#15689185	2
C₄×#13093981	2
#20243248	2
C₄×Dic₄₇⋊C₈	2
#20243157	2
C₄×(C₂×D₄₇)⋊C₈	2
#20243141	2
C₄²×C₄₇⋊C₈	2
C₄×#97×C₄₇	2
C₄×#433	47

Subgroups — C4×#15689185 #26214427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689185 #26214427