Subgroups — C₄×#15689236 #26214764

Order 12032.

1326 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689236	C₂	1	2	yes	maximal
2	C₂×C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic
3	C₄²×D₁₈₈	C₂	1	2	yes	maximal, complemented
4	#20243285	C₂	1	2	yes	maximal, complemented
5	C₄×#13093956	C₂	1	1	yes	maximal, characteristic, complemented
6	#20243214	C₂	1	4	yes	maximal, complemented
7	C₄×#13093946	C₂	1	2	yes	maximal, complemented
8	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₈₈	C₂²	1	2	yes	—
11	C₂×C₄×D₁₈₈	C₂²	1	4	yes	—
12	C₂×C₄²×D₄₇	—	2	2	no	—
13	#15689236	C₄	1	16	yes	direct product factor
14	C₄×D₃₇₆	C₄	1	8	yes	complemented
15	C₂×#13093958	C₂²	1	1	yes	characteristic
16	C₂×#13093956	C₂²	1	2	yes	—
17	C₂×#13093946	C₂²	1	4	yes	—
18	C₂×#13093946	C₂²	1	2	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
21	C₄×(C₂×D₄₇)⋊C₄	—	2	2	no	—
22	C₄×(C₂×D₄₇)⋊C₄	—	2	4	no	—
23	#15688886	C₂²	1	4	yes	—
24	#15688886	C₂²	1	2	yes	—
25	C₂×#97×C₄₇	C₂²	1	2	yes	—
26	C₂×#97×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₄³×C₄₇	C₂²	1	1	yes	characteristic
30	C₂²×D₁₈₈	C₂³	1	2	yes	—
31	C₂²×C₄×D₄₇	—	2	4	no	—
32	C₂²×C₄×D₄₇	—	2	2	no	—
33	C₂×D₃₇₆	C₂×C₄	1	8	yes	—
34	C₂×D₃₇₆	C₂×C₄	1	4	yes	—
35	C₄×D₁₈₈	—	2	4	no	—
36	C₄×D₁₈₈	—	2	8	no	—
37	C₄×D₁₈₈	C₂×C₄	1	16	yes	complemented
38	C₄×D₁₈₈	C₂×C₄	1	8	yes	complemented
39	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
40	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
41	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
42	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
43	C₂×C₄×Dic₄₇	—	2	4	no	—
44	C₂×C₄×Dic₄₇	—	2	2	no	—
45	C₄²×D₄₇	—	4	8	no	—
46	#13093958	C₂×C₄	1	4	yes	complemented
47	#13093956	C₂×C₄	1	8	yes	complemented
48	#13093946	C₂×C₄	1	16	yes	complemented
49	#13093946	C₂×C₄	1	8	yes	complemented
50	C₄₇⋊#95	—	2	2	no	—
51	C₄₇⋊#95	—	2	4	no	—
52	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	D₄	1	2	yes	—
56	C₂×C₄²×C₄₇	D₄	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
59	#97×C₄₇	—	2	4	no	—
60	#97×C₄₇	C₂×C₄	1	8	yes	complemented
61	C₄×C₈×C₄₇	—	2	2	no	—
62	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
63	C₂³×D₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	2	2	no	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂×D₁₈₈	—	2	2	no	—
71	C₂×D₁₈₈	—	2	4	no	—
72	C₂×D₁₈₈	C₂²×C₄	1	8	yes	—
73	C₂×D₁₈₈	C₂²×C₄	1	4	yes	—
74	C₂×C₄×D₄₇	—	2	16	no	—
75	C₂×C₄×D₄₇	—	2	8	no	—
76	C₂×C₄×D₄₇	—	4	8	no	—
77	C₂×C₄×D₄₇	—	4	8	no	—
78	C₂×C₄×D₄₇	—	4	4	no	—
79	C₂×C₄×D₄₇	—	4	4	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₄₇	—	2	4	no	—
88	C₄²×C₄₇	—	4	2	no	—
89	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
91	C₄²×C₄₇	#41	1	2	yes	—
92	C₄²×C₄₇	#41	1	4	yes	—
93	(C₂×D₄₇)⋊C₄	—	2	8	no	—
94	(C₂×D₄₇)⋊C₄	—	2	16	no	—
95	C₄⋊Dic₄₇	—	2	4	no	—
96	C₄⋊Dic₄₇	—	2	2	no	—
97	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
98	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
99	C₄×Dic₄₇	—	4	8	no	—
100	C₄×Dic₄₇	—	4	4	no	—
101	D₃₇₆	C₄²	1	16	yes	complemented
102	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
103	C₂²×D₄₇	—	2	4	no	—
104	C₂²×D₄₇	—	2	8	no	—
105	C₂²×D₄₇	—	4	2	no	—
106	C₂²×D₄₇	—	4	2	no	—
107	C₂²×D₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
113	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
115	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
119	C₂×Dic₄₇	—	2	4	no	—
120	C₂×Dic₄₇	—	2	8	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	4	2	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	D₁₈₈	—	2	8	no	—
126	D₁₈₈	—	2	4	no	—
127	D₁₈₈	C₂×C₄²	1	8	yes	complemented
128	C₄×D₄₇	—	4	32	no	—
129	C₄×D₄₇	—	4	16	no	—
130	C₈×C₄₇	—	2	4	no	—
131	C₈×C₄₇	—	2	2	no	—
132	C₈×C₄₇	C₂×C₄²	1	4	yes	complemented
133	C₄×#442	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
136	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
137	C₂²×C₄₇	#575	1	2	yes	—
138	C₂×D₄₇	—	2	8	no	—
139	C₂×D₄₇	—	4	4	no	—
140	C₂×D₄₇	—	4	4	no	—
141	C₂×D₄₇	—	4	8	no	—
142	C₂×D₄₇	—	4	8	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	2	8	no	—
145	C₄×C₄₇	#442	1	4	yes	complemented
146	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
147	C₄×C₄₇	C₄×#41	1	2	yes	—
148	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
149	Dic₄₇	—	4	4	no	—
150	Dic₄₇	—	4	8	no	—
151	C₂×#442	—	47	2	no	—
152	C₂×C₄×D₈	—	47	1	no	—
153	C₄²×D₄	—	47	2	no	—
154	#1776	—	47	2	no	—
155	C₄×#106	—	47	1	no	—
156	#1762	—	47	4	no	—
157	C₄×#102	—	47	2	no	—
158	C₄×#97	D₄₇	1	1	yes	characteristic, complemented
159	C₂×C₄₇	C₄²×D₄	1	1	yes	characteristic
160	C₂×C₄₇	C₂×#442	1	1	yes	characteristic
161	C₂×C₄₇	#2943	1	1	yes	characteristic
162	C₂×C₄₇	C₄×AGL(1,ℤ/8)	1	4	yes	—
163	D₄₇	—	4	8	no	—
164	C₂²×D₈	—	47	1	no	—
165	C₂×C₄×D₄	—	47	2	no	—
166	C₂×C₄×D₄	—	47	4	no	—
167	C₂²×C₄²	—	94	2	no	—
168	#442	—	47	16	no	—
169	C₄×D₈	—	47	8	no	—
170	C₂×#107	—	47	1	no	—
171	C₂×#106	—	47	2	no	—
172	C₂×#102	—	47	2	no	—
173	C₂×#102	—	47	4	no	—
174	C₂×#97	C₂×D₄₇	1	2	yes	—
175	C₂×#97	C₂×D₄₇	1	2	yes	—
176	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
178	C₄×C₄⋊C₄	—	47	2	no	—
179	C₄×C₄⋊C₄	—	47	4	no	—
180	C₄×C₂²⋊C₄	—	94	4	no	—
181	C₄×C₂²⋊C₄	—	94	2	no	—
182	C₄³	C₂×D₄₇	1	1	yes	characteristic
183	#340	—	47	4	no	—
184	#340	—	47	2	no	—
185	C₄₇	C₄×#442	1	1	yes	characteristic, complemented, Sylow
186	C₂²×D₄	—	47	2	no	—
187	C₂³×C₄	—	94	4	no	—
188	C₂³×C₄	—	94	2	no	—
189	C₂×D₈	—	47	4	no	—
190	C₂×D₈	—	47	8	no	—
191	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
192	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
194	C₄×D₄	—	47	8	no	—
195	C₄×D₄	—	47	16	no	—
196	C₄×D₄	—	94	4	no	—
197	C₄×D₄	—	94	8	no	—
198	C₂×C₄⋊C₄	—	47	4	no	—
199	C₂×C₄⋊C₄	—	47	2	no	—
200	C₂×C₂²⋊C₄	—	94	2	no	—
201	C₂×C₂²⋊C₄	—	94	4	no	—
202	C₂×C₄²	—	94	4	no	—
203	C₂×C₄²	—	94	2	no	—
204	C₂×C₄²	—	188	8	no	—
205	C₂×C₄²	D₁₈₈	1	2	yes	—
206	C₂×C₄²	D₁₈₈	1	2	yes	—
207	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
208	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
209	#107	—	47	4	no	—
210	#106	—	47	8	no	—
211	#102	—	47	8	no	—
212	#102	—	47	16	no	—
213	#97	—	2	4	no	—
214	#97	C₄×D₄₇	1	8	yes	complemented
215	C₄×C₈	—	2	2	no	—
216	C₄×C₈	C₄×D₄₇	1	4	yes	complemented
217	#95	—	94	2	no	—
218	#95	—	94	4	no	—
219	C₂⁴	—	94	2	no	—
220	C₂×D₄	—	47	4	no	—
221	C₂×D₄	—	47	8	no	—
222	C₂×D₄	—	94	2	no	—
223	C₂×D₄	—	94	4	no	—
224	C₂²×C₄	—	94	2	no	—
225	C₂²×C₄	—	94	4	no	—
226	C₂²×C₄	—	94	16	no	—
227	C₂²×C₄	—	94	8	no	—
228	C₂²×C₄	—	188	8	no	—
229	C₂²×C₄	—	188	8	no	—
230	C₂²×C₄	—	188	4	no	—
231	C₂²×C₄	—	188	4	no	—
232	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
234	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
235	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
236	D₈	—	47	16	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
241	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
242	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
243	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
244	C₄⋊C₄	—	47	8	no	—
245	C₄⋊C₄	—	47	4	no	—
246	C₄⋊C₄	—	94	4	no	—
247	C₄⋊C₄	—	94	2	no	—
248	C₂²⋊C₄	—	94	8	no	—
249	C₂²⋊C₄	—	94	16	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	4	2	no	—
252	C₄²	—	188	8	no	—
253	C₄²	—	188	4	no	—
254	C₄²	C₂×C₄×D₄₇	1	2	yes	—
255	C₄²	C₂×C₄×D₄₇	1	4	yes	—
256	C₄²	#10625350	1	2	yes	—
257	C₄²	#10625350	1	4	yes	—
258	C₂³	—	94	4	no	—
259	C₂³	—	94	8	no	—
260	C₂³	—	188	2	no	—
261	C₂³	—	188	2	no	—
262	C₂³	—	188	4	no	—
263	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
264	D₄	—	47	8	no	—
265	D₄	—	94	4	no	—
266	D₄	—	94	8	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	94	4	no	—
271	C₂×C₄	—	94	8	no	—
272	C₂×C₄	—	188	32	no	—
273	C₂×C₄	—	188	2	no	—
274	C₂×C₄	—	188	4	no	—
275	C₂×C₄	—	188	2	no	—
276	C₂×C₄	—	188	4	no	—
277	C₂×C₄	—	188	16	no	—
278	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
279	C₂×C₄	C₄×D₁₈₈	1	8	yes	—
280	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
281	C₂×C₄	#13094052	1	4	yes	—
282	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
284	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
285	C₂×C₄	C₂×#10625350	1	2	yes	—
286	C₈	—	2	4	no	—
287	C₈	—	2	2	no	—
288	C₈	C₄²×D₄₇	1	4	yes	complemented
289	C₂²	—	94	8	no	—
290	C₂²	—	188	4	no	—
291	C₂²	—	188	4	no	—
292	C₂²	—	188	8	no	—
293	C₂²	—	188	8	no	—
294	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
295	C₂²	C₂×C₄×D₁₈₈	1	2	yes	—
296	C₂²	C₂×#13094052	1	2	yes	—
297	C₂²	#15689923	1	2	yes	—
298	C₄	—	2	2	no	—
299	C₄	—	2	8	no	—
300	C₄	—	188	4	no	—
301	C₄	—	188	8	no	—
302	C₄	#15689236	1	4	yes	direct product factor
303	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
304	C₄	C₄×#10625350	1	2	yes	—
305	C₄	C₄×#10625350	1	1	yes	characteristic
306	C₂	—	188	8	no	—
307	C₂	C₄²×D₁₈₈	1	1	yes	characteristic
308	C₂	C₂×#15689236	1	1	yes	characteristic
309	C₂	#20252190	1	1	yes	characteristic
310	C₂	C₄×#13094052	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#97×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₃₇₆	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	C₄×#442
LCS	G ▷ 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
3. C₄²×D₁₈₈	C₂	semidirect
4. #20243285	C₂	semidirect
5. C₄×#13093956	C₂	semidirect
6. #20243214	C₂	semidirect
7. C₄×#13093946	C₂	semidirect
8. C₄×#97×C₄₇	C₂	semidirect
13. #15689236	C₄	direct (N × H)
14. C₄×D₃₇₆	C₄	semidirect
37. C₄×D₁₈₈	C₂×C₄	semidirect
38. C₄×D₁₈₈	C₂×C₄	semidirect
46. #13093958	C₂×C₄	semidirect
47. #13093956	C₂×C₄	semidirect
48. #13093946	C₂×C₄	semidirect
49. #13093946	C₂×C₄	semidirect
60. #97×C₄₇	C₂×C₄	semidirect
62. C₄×C₈×C₄₇	C₂×C₄	semidirect
101. D₃₇₆	C₄²	semidirect
127. D₁₈₈	C₂×C₄²	semidirect
132. C₈×C₄₇	C₂×C₄²	semidirect
145. C₄×C₄₇	#442	semidirect
158. C₄×#97	D₄₇	semidirect
185. C₄₇	C₄×#442	semidirect
214. #97	C₄×D₄₇	semidirect
216. C₄×C₈	C₄×D₄₇	semidirect
288. C₈	C₄²×D₄₇	semidirect
302. C₄	#15689236	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×#15689236	2
C₂×C₄×D₃₇₆	2
C₄²×D₁₈₈	2
#20243285	2
C₄×#13093956	2
#20243214	2
C₄×#13093946	2
C₄×#97×C₄₇	2
C₄×#442	47

Subgroups — C4×#15689236 #26214764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689236 #26214764