Subgroups — C₂²×#15689436 #27084002

Order 12032.

3006 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093998	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689436	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13093957	C₂	1	2	yes	maximal, complemented
4	C₂²×#13093944	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₈₈	C₂²	1	1	yes	characteristic
8	C₂³×C₂²⋊D₄₇	—	2	1	no	—
9	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
10	C₂×#13093998	—	2	6	no	—
11	C₂×#13093998	C₂²	1	12	yes	—
12	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂²×(C₂×D₄₇)⋊C₄	—	2	1	no	—
14	#15689436	C₂²	1	64	yes	direct product factor
15	C₂×#13093957	C₂²	1	24	yes	complemented
16	C₂×#13093944	C₂²	1	24	yes	complemented
17	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	12	yes	complemented
18	C₂⁴×C₄×C₄₇	—	2	1	no	—
19	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₄₇	—	2	6	no	—
21	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
22	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₄₇	C₂²	1	12	yes	complemented
24	C₂⁴×D₄₇	—	2	1	no	—
25	C₂²×D₁₈₈	—	2	8	no	—
26	C₂²×D₁₈₈	—	2	3	no	—
27	C₂²×D₁₈₈	C₂³	1	6	yes	—
28	C₂²×C₂²⋊D₄₇	—	2	12	no	—
29	C₂²×C₂²⋊D₄₇	—	4	8	no	—
30	C₂³×Dic₄₇	—	2	1	no	—
31	C₂²×C₄₇⋊C₈	—	2	8	no	—
32	C₂²×C₄₇⋊C₈	C₂³	1	12	yes	—
33	#13093998	—	2	24	no	—
34	#13093998	C₂³	1	16	yes	—
35	C₂×C₄⋊Dic₄₇	—	2	3	no	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	6	yes	—
37	C₂×(C₂×D₄₇)⋊C₄	—	2	12	no	—
38	#13093957	C₂³	1	32	yes	complemented
39	#13093944	C₂³	1	32	yes	complemented
40	C₂²⋊(C₄₇⋊C₈)	C₂³	1	16	yes	complemented
41	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	—	2	8	no	—
43	C₂³×C₄×C₄₇	—	2	2	no	—
44	C₂³×C₄×C₄₇	—	2	12	no	—
45	C₂³×C₄×C₄₇	—	4	8	no	—
46	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₇	D₄	1	6	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
49	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₄₇	—	2	6	no	—
51	C₂×C₄⋊C₄×C₄₇	—	2	24	no	—
52	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
53	C₂×C₄⋊C₄×C₄₇	C₂³	1	12	yes	—
54	C₂×C₄⋊C₄×C₄₇	C₂³	1	16	yes	complemented
55	C₂⁴×C₄₇	—	2	4	no	—
56	C₂⁴×C₄₇	—	2	4	no	—
57	C₂⁴×C₄₇	D₈	1	8	yes	—
58	C₂⁴×C₄₇	C₂×D₄	1	6	yes	—
59	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂³×D₄₇	—	2	6	no	—
61	C₂³×D₄₇	—	4	4	no	—
62	C₂³×D₄₇	—	4	4	no	—
63	C₂³×D₄₇	—	4	4	no	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	2	24	no	—
66	C₂²×C₄×C₄₇	—	2	24	no	—
67	C₂²×C₄×C₄₇	—	2	12	no	—
68	C₂²×C₄×C₄₇	—	2	16	no	—
69	C₂²×C₄×C₄₇	—	4	48	no	—
70	C₂²×C₄×C₄₇	—	4	8	no	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
74	C₂²×C₄×C₄₇	#41	1	8	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
77	C₂×C₂²⋊D₄₇	—	2	16	no	—
78	C₂×C₂²⋊D₄₇	—	4	48	no	—
79	C₂²×Dic₄₇	—	2	6	no	—
80	C₂²×Dic₄₇	—	4	4	no	—
81	C₂×D₁₈₈	—	2	24	no	—
82	C₂×D₁₈₈	—	2	6	no	—
83	C₂×D₁₈₈	—	4	12	no	—
84	C₂×D₁₈₈	C₂⁴	1	4	yes	—
85	C₄⋊C₄×C₄₇	—	2	12	no	—
86	C₄⋊C₄×C₄₇	—	2	16	no	—
87	C₄⋊C₄×C₄₇	C₂⁴	1	8	yes	—
88	(C₂×D₄₇)⋊C₄	—	2	16	no	—
89	C₄⋊Dic₄₇	—	2	6	no	—
90	C₄⋊Dic₄₇	C₂⁴	1	4	yes	—
91	C₂×C₄₇⋊C₈	—	2	24	no	—
92	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
93	C₂³×C₄₇	—	2	12	no	—
94	C₂³×C₄₇	—	2	12	no	—
95	C₂³×C₄₇	—	2	32	no	—
96	C₂³×C₄₇	#123	1	4	yes	—
97	C₂³×C₄₇	C₂×D₈	1	24	yes	—
98	C₂³×C₄₇	C₂×D₈	1	4	yes	—
99	C₂³×C₄₇	#137	1	4	yes	—
100	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
101	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
102	C₂²×D₄₇	—	2	4	no	—
103	C₂²×D₄₇	—	4	12	no	—
104	C₂²×D₄₇	—	4	12	no	—
105	C₂²×D₄₇	—	4	12	no	—
106	C₂²×D₄₇	—	8	16	no	—
107	C₂×C₄×C₄₇	—	2	12	no	—
108	C₂×C₄×C₄₇	—	2	24	no	—
109	C₂×C₄×C₄₇	—	2	8	no	—
110	C₂×C₄×C₄₇	—	2	8	no	—
111	C₂×C₄×C₄₇	—	4	32	no	—
112	C₂×C₄×C₄₇	—	4	24	no	—
113	C₂×C₄×C₄₇	#123	1	12	yes	—
114	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
115	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
116	C₂×C₄×C₄₇	C₂×#41	1	12	yes	—
117	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
118	C₂²⋊D₄₇	—	4	32	no	—
119	C₂×Dic₄₇	—	2	4	no	—
120	C₂×Dic₄₇	—	4	12	no	—
121	D₁₈₈	—	2	8	no	—
122	D₁₈₈	—	4	12	no	—
123	C₄₇⋊C₈	—	2	8	no	—
124	C₂²×#480	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	48	no	—
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	#480	1	16	yes	complemented
129	C₂²×C₄₇	C₂×#123	1	6	yes	—
130	C₂²×C₄₇	C₂²×D₈	1	8	yes	—
131	C₂²×C₄₇	C₂²×D₈	1	6	yes	—
132	C₂²×C₄₇	C₂×#137	1	6	yes	—
133	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
134	C₂×D₄₇	—	4	4	no	—
135	C₂×D₄₇	—	4	4	no	—
136	C₂×D₄₇	—	4	4	no	—
137	C₂×D₄₇	—	8	24	no	—
138	C₄×C₄₇	—	2	4	no	—
139	C₄×C₄₇	—	4	8	no	—
140	C₄×C₄₇	C₂×#123	1	6	yes	—
141	C₄×C₄₇	C₂²×#41	1	2	yes	—
142	Dic₄₇	—	4	4	no	—
143	C₂²×#121	—	47	1	no	—
144	C₂³×C₄⋊C₄	D₄₇	1	1	yes	characteristic, complemented
145	C₂×#480	—	47	24	no	—
146	C₂²×#107	—	47	2	no	—
147	C₂²×#102	—	47	2	no	—
148	C₂²×C₂²⋊C₈	—	47	1	no	—
149	C₂×C₄₇	—	2	8	no	—
150	C₂×C₄₇	C₂×#480	1	12	yes	complemented
151	C₂×C₄₇	C₂²×#123	1	1	yes	characteristic
152	C₂×C₄₇	C₂³×D₈	1	1	yes	characteristic
153	C₂×C₄₇	C₂²×#137	1	1	yes	characteristic
154	D₄₇	—	8	4	no	—
155	C₂³×D₄	—	47	1	no	—
156	C₂³×D₄	—	94	1	no	—
157	C₂⁴×C₄	—	2	1	no	—
158	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
159	C₂³×C₈	—	47	2	no	—
160	C₂×#121	—	47	12	no	—
161	C₂×#121	—	94	6	no	—
162	C₂²×C₄⋊C₄	—	2	6	no	—
163	C₂²×C₄⋊C₄	—	2	1	no	—
164	C₂²×C₄⋊C₄	—	47	1	no	—
165	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
166	C₂²×C₄⋊C₄	C₂×D₄₇	1	12	yes	complemented
167	C₂²×C₂²⋊C₄	—	94	1	no	—
168	#480	—	47	64	no	—
169	C₂×#107	—	47	24	no	—
170	C₂×#102	—	47	24	no	—
171	C₂×C₂²⋊C₈	—	47	12	no	—
172	C₄₇	C₂²×#480	1	1	yes	characteristic, complemented, Sylow
173	C₂⁵	—	94	1	no	—
174	C₂⁵	C₂²⋊D₄₇	1	1	yes	characteristic
175	C₂²×D₄	—	47	6	no	—
176	C₂²×D₄	—	94	12	no	—
177	C₂²×D₄	—	94	8	no	—
178	C₂²×D₄	—	94	3	no	—
179	C₂²×D₄	—	188	8	no	—
180	C₂³×C₄	—	2	8	no	—
181	C₂³×C₄	—	2	12	no	—
182	C₂³×C₄	—	2	2	no	—
183	C₂³×C₄	—	4	8	no	—
184	C₂³×C₄	—	94	1	no	—
185	C₂³×C₄	C₂²⋊D₄₇	1	6	yes	—
186	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
187	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
188	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
189	C₂²×C₈	—	47	12	no	—
190	C₂²×C₈	—	94	8	no	—
191	#121	—	47	16	no	—
192	#121	—	94	24	no	—
193	C₂×C₄⋊C₄	—	2	6	no	—
194	C₂×C₄⋊C₄	—	2	24	no	—
195	C₂×C₄⋊C₄	—	2	12	no	—
196	C₂×C₄⋊C₄	—	47	6	no	—
197	C₂×C₄⋊C₄	—	94	3	no	—
198	C₂×C₄⋊C₄	C₂²×D₄₇	1	12	yes	—
199	C₂×C₄⋊C₄	C₂²×D₄₇	1	16	yes	complemented
200	C₂×C₂²⋊C₄	—	94	12	no	—
201	#107	—	47	32	no	—
202	#102	—	47	32	no	—
203	C₂²⋊C₈	—	47	16	no	—
204	C₂⁴	—	2	4	no	—
205	C₂⁴	—	2	4	no	—
206	C₂⁴	—	94	6	no	—
207	C₂⁴	—	188	4	no	—
208	C₂⁴	—	188	4	no	—
209	C₂⁴	—	188	4	no	—
210	C₂⁴	#10625334	1	8	yes	—
211	C₂⁴	C₂×C₂²⋊D₄₇	1	6	yes	—
212	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂×D₄	—	47	4	no	—
214	C₂×D₄	—	94	16	no	—
215	C₂×D₄	—	94	6	no	—
216	C₂×D₄	—	94	24	no	—
217	C₂×D₄	—	188	48	no	—
218	C₂×D₄	—	188	12	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	24	no	—
221	C₂²×C₄	—	2	24	no	—
222	C₂²×C₄	—	2	12	no	—
223	C₂²×C₄	—	2	16	no	—
224	C₂²×C₄	—	4	48	no	—
225	C₂²×C₄	—	4	8	no	—
226	C₂²×C₄	—	94	6	no	—
227	C₂²×C₄	—	188	4	no	—
228	C₂²×C₄	#10625350	1	8	yes	—
229	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
230	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
232	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
233	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
234	C₂×C₈	—	47	8	no	—
235	C₂×C₈	—	94	24	no	—
236	C₄⋊C₄	—	2	12	no	—
237	C₄⋊C₄	—	2	16	no	—
238	C₄⋊C₄	—	47	4	no	—
239	C₄⋊C₄	—	94	6	no	—
240	C₄⋊C₄	C₂³×D₄₇	1	8	yes	—
241	C₂²⋊C₄	—	94	16	no	—
242	C₂³	—	2	12	no	—
243	C₂³	—	2	12	no	—
244	C₂³	—	2	32	no	—
245	C₂³	—	94	4	no	—
246	C₂³	—	376	16	no	—
247	C₂³	—	188	12	no	—
248	C₂³	—	188	12	no	—
249	C₂³	—	188	12	no	—
250	C₂³	#13094007	1	4	yes	—
251	C₂³	C₂×#10625334	1	24	yes	—
252	C₂³	C₂×#10625334	1	4	yes	—
253	C₂³	#13094055	1	4	yes	—
254	C₂³	C₂²×C₂²⋊D₄₇	1	4	yes	—
255	C₂³	C₂²×C₂²⋊D₄₇	1	3	yes	—
256	D₄	—	94	8	no	—
257	D₄	—	188	32	no	—
258	D₄	—	188	12	no	—
259	C₂×C₄	—	2	12	no	—
260	C₂×C₄	—	2	24	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	32	no	—
264	C₂×C₄	—	4	24	no	—
265	C₂×C₄	—	94	4	no	—
266	C₂×C₄	—	188	12	no	—
267	C₂×C₄	#13094007	1	12	yes	—
268	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
269	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
270	C₂×C₄	C₂×#10625350	1	12	yes	—
271	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
272	C₈	—	94	8	no	—
273	C₂²	—	2	48	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	376	24	no	—
277	C₂²	—	188	4	no	—
278	C₂²	—	188	4	no	—
279	C₂²	—	188	4	no	—
280	C₂²	#15689436	1	16	yes	direct product factor
281	C₂²	C₂×#13094007	1	6	yes	—
282	C₂²	C₂²×#10625334	1	8	yes	—
283	C₂²	C₂²×#10625334	1	6	yes	—
284	C₂²	C₂×#13094055	1	6	yes	—
285	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	4	8	no	—
288	C₄	—	188	4	no	—
289	C₄	C₂×#13094007	1	6	yes	—
290	C₄	C₂²×#10625350	1	2	yes	—
291	C₂	—	2	8	no	—
292	C₂	—	376	4	no	—
293	C₂	C₂×#15689436	1	12	yes	direct product factor
294	C₂	C₂²×#13094007	1	1	yes	characteristic
295	C₂	C₂³×#10625334	1	1	yes	characteristic
296	C₂	C₂²×#13094055	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂³×C₄⋊C₄×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
prime radical	C₂²×#13093998	C₂
hypercenter	C₂³×C₄⋊C₄	D₄₇
hypocenter	C₄₇	C₂²×#480
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#15689436	C₂	direct (N × H)
3. C₂²×#13093957	C₂	semidirect
4. C₂²×#13093944	C₂	semidirect
5. C₂²×C₂²⋊(C₄₇⋊C₈)	C₂	semidirect
6. C₂³×C₄⋊C₄×C₄₇	C₂	semidirect
14. #15689436	C₂²	direct (N × H)
15. C₂×#13093957	C₂²	semidirect
16. C₂×#13093944	C₂²	semidirect
17. C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	semidirect
23. C₂²×C₄⋊C₄×C₄₇	C₂²	semidirect
38. #13093957	C₂³	semidirect
39. #13093944	C₂³	semidirect
40. C₂²⋊(C₄₇⋊C₈)	C₂³	semidirect
54. C₂×C₄⋊C₄×C₄₇	C₂³	semidirect
128. C₂²×C₄₇	#480	semidirect
144. C₂³×C₄⋊C₄	D₄₇	semidirect
150. C₂×C₄₇	C₂×#480	semidirect
166. C₂²×C₄⋊C₄	C₂×D₄₇	semidirect
172. C₄₇	C₂²×#480	semidirect
199. C₂×C₄⋊C₄	C₂²×D₄₇	semidirect
280. C₂²	#15689436	direct (N × H)
293. C₂	C₂×#15689436	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13093998	2
C₂×#15689436	2
C₂²×#13093957	2
C₂²×#13093944	2
C₂²×C₂²⋊(C₄₇⋊C₈)	2
C₂³×C₄⋊C₄×C₄₇	2
C₂²×#480	47

Subgroups — C22×#15689436 #27084002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689436 #27084002