Subgroups — C₂²×#15689049 #26382484

Order 12032.

5230 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×(C₂×D₄₇)⋊C₄	C₂	1	2	yes	maximal
2	C₂³×(C₂×D₄₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689049	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄₇⋊#95	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄₇⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₄₇	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₄₇	C₂²	1	2	yes	—
9	C₂⁴×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂²×(C₂×D₄₇)⋊C₄	—	2	16	no	—
11	C₂²×(C₂×D₄₇)⋊C₄	—	2	8	no	—
12	C₂²×(C₂×D₄₇)⋊C₄	C₂²	1	24	yes	—
13	C₂²×(C₂×D₄₇)⋊C₄	C₂²	1	12	yes	—
14	#15689049	C₂²	1	64	yes	direct product factor
15	C₂×C₄₇⋊#95	C₂²	1	24	yes	complemented
16	C₂×C₄₇⋊#95	C₂²	1	12	yes	complemented
17	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
18	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₄₇	—	2	8	no	—
20	C₂²×C₄⋊C₄×C₄₇	C₂²	1	12	yes	complemented
21	C₂⁴×D₄₇	—	2	4	no	—
22	C₂⁴×D₄₇	—	2	4	no	—
23	C₂⁴×D₄₇	—	2	8	no	—
24	C₂⁴×D₄₇	—	2	8	no	—
25	C₂⁴×D₄₇	D₄	1	8	yes	—
26	C₂⁴×D₄₇	C₂³	1	6	yes	—
27	C₂³×Dic₄₇	—	2	8	no	—
28	C₂³×Dic₄₇	—	2	4	no	—
29	C₂³×Dic₄₇	—	2	8	no	—
30	C₂³×Dic₄₇	—	2	8	no	—
31	C₂³×Dic₄₇	—	2	8	no	—
32	C₂³×Dic₄₇	C₂³	1	12	yes	—
33	C₂³×Dic₄₇	C₂³	1	6	yes	—
34	C₂×(C₂×D₄₇)⋊C₄	—	2	96	no	—
35	C₂×(C₂×D₄₇)⋊C₄	—	2	48	no	—
36	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	32	yes	—
37	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	16	yes	—
38	C₄₇⋊#95	C₂³	1	32	yes	complemented
39	C₄₇⋊#95	C₂³	1	16	yes	complemented
40	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	—	2	8	no	—
42	C₂³×C₄×C₄₇	—	2	8	no	—
43	C₂³×C₄×C₄₇	—	2	8	no	—
44	C₂³×C₄×C₄₇	—	2	8	no	—
45	C₂³×C₄×C₄₇	D₄	1	8	yes	—
46	C₂³×C₄×C₄₇	C₂³	1	12	yes	—
47	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
48	C₂×C₄⋊C₄×C₄₇	—	2	48	no	—
49	C₂×C₄⋊C₄×C₄₇	C₂³	1	16	yes	complemented
50	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
51	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
52	C₂⁴×C₄₇	#41	1	4	yes	—
53	C₂⁴×C₄₇	#41	1	8	yes	—
54	C₂⁴×C₄₇	#41	1	8	yes	—
55	C₂⁴×C₄₇	C₂⁴	1	3	yes	—
56	C₂³×D₄₇	—	2	32	no	—
57	C₂³×D₄₇	—	2	64	no	—
58	C₂³×D₄₇	—	2	12	no	—
59	C₂³×D₄₇	—	2	12	no	—
60	C₂³×D₄₇	—	2	24	no	—
61	C₂³×D₄₇	—	2	24	no	—
62	C₂³×D₄₇	—	4	16	no	—
63	C₂³×D₄₇	—	4	16	no	—
64	C₂³×D₄₇	—	4	32	no	—
65	C₂³×D₄₇	C₂×D₄	1	24	yes	—
66	C₂³×D₄₇	C₂⁴	1	4	yes	—
67	C₂²×C₄×C₄₇	—	2	32	no	—
68	C₂²×C₄×C₄₇	—	2	24	no	—
69	C₂²×C₄×C₄₇	—	2	24	no	—
70	C₂²×C₄×C₄₇	—	2	24	no	—
71	C₂²×C₄×C₄₇	—	2	24	no	—
72	C₂²×C₄×C₄₇	—	4	32	no	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	24	yes	—
74	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
76	C₂²×Dic₄₇	—	2	24	no	—
77	C₂²×Dic₄₇	—	2	12	no	—
78	C₂²×Dic₄₇	—	2	24	no	—
79	C₂²×Dic₄₇	—	2	24	no	—
80	C₂²×Dic₄₇	—	2	24	no	—
81	C₂²×Dic₄₇	—	4	32	no	—
82	C₂²×Dic₄₇	—	4	16	no	—
83	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
84	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
85	C₄⋊C₄×C₄₇	—	2	32	no	—
86	(C₂×D₄₇)⋊C₄	—	2	64	no	—
87	(C₂×D₄₇)⋊C₄	—	2	32	no	—
88	C₂³×C₄₇	#121	1	16	yes	—
89	C₂³×C₄₇	#123	1	16	yes	—
90	C₂³×C₄₇	#123	1	32	yes	—
91	C₂³×C₄₇	#124	1	16	yes	—
92	C₂³×C₄₇	#126	1	32	yes	—
93	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
94	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
95	C₂³×C₄₇	C₂×#41	1	6	yes	—
96	C₂³×C₄₇	C₂×#41	1	12	yes	—
97	C₂³×C₄₇	C₂×#41	1	12	yes	—
98	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
99	C₂²×D₄₇	—	2	48	no	—
100	C₂²×D₄₇	—	2	96	no	—
101	C₂²×D₄₇	—	2	4	no	—
102	C₂²×D₄₇	—	2	4	no	—
103	C₂²×D₄₇	—	2	8	no	—
104	C₂²×D₄₇	—	2	8	no	—
105	C₂²×D₄₇	—	4	24	no	—
106	C₂²×D₄₇	—	4	24	no	—
107	C₂²×D₄₇	—	4	48	no	—
108	C₂²×D₄₇	—	4	128	no	—
109	C₂²×D₄₇	C₂²×D₄	1	8	yes	—
110	C₂×C₄×C₄₇	—	2	48	no	—
111	C₂×C₄×C₄₇	—	2	8	no	—
112	C₂×C₄×C₄₇	—	2	8	no	—
113	C₂×C₄×C₄₇	—	2	8	no	—
114	C₂×C₄×C₄₇	—	2	8	no	—
115	C₂×C₄×C₄₇	—	4	48	no	—
116	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
117	C₂×Dic₄₇	—	2	8	no	—
118	C₂×Dic₄₇	—	2	4	no	—
119	C₂×Dic₄₇	—	2	8	no	—
120	C₂×Dic₄₇	—	2	8	no	—
121	C₂×Dic₄₇	—	2	8	no	—
122	C₂×Dic₄₇	—	4	48	no	—
123	C₂×Dic₄₇	—	4	24	no	—
124	C₂²×#397	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	#397	1	64	yes	complemented
126	C₂²×C₄₇	C₂×#121	1	12	yes	—
127	C₂²×C₄₇	C₂×#123	1	12	yes	—
128	C₂²×C₄₇	C₂×#123	1	24	yes	—
129	C₂²×C₄₇	C₂×#124	1	12	yes	—
130	C₂²×C₄₇	C₂×#126	1	24	yes	—
131	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
132	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
133	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
134	C₂²×C₄₇	C₂²×#41	1	2	yes	—
135	C₂²×C₄₇	C₂²×#41	1	2	yes	—
136	C₂×D₄₇	—	2	8	no	—
137	C₂×D₄₇	—	2	16	no	—
138	C₂×D₄₇	—	4	96	no	—
139	C₂×D₄₇	—	4	4	no	—
140	C₂×D₄₇	—	4	4	no	—
141	C₂×D₄₇	—	4	8	no	—
142	C₄×C₄₇	—	2	8	no	—
143	C₄×C₄₇	—	4	8	no	—
144	Dic₄₇	—	4	8	no	—
145	Dic₄₇	—	4	4	no	—
146	C₂³×C₄⋊C₄	D₄₇	1	1	yes	characteristic, complemented
147	C₂³×C₂²⋊C₄	—	47	2	no	—
148	C₂³×C₂²⋊C₄	—	47	1	no	—
149	C₂×#397	—	47	24	no	—
150	C₂²×#95	—	47	2	no	—
151	C₂²×#95	—	47	1	no	—
152	C₂×C₄₇	C₂×#397	1	24	yes	complemented
153	C₂×C₄₇	C₂²×#121	1	1	yes	characteristic
154	C₂×C₄₇	C₂²×#123	1	1	yes	characteristic
155	C₂×C₄₇	C₂²×#123	1	2	yes	—
156	C₂×C₄₇	C₂²×#124	1	1	yes	characteristic
157	C₂×C₄₇	C₂²×#126	1	2	yes	—
158	D₄₇	—	4	8	no	—
159	C₂⁶	—	47	1	no	—
160	C₂⁴×C₄	—	47	2	no	—
161	C₂⁴×C₄	—	47	1	no	—
162	C₂⁴×C₄	C₂×D₄₇	1	2	yes	—
163	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	—	2	8	no	—
165	C₂²×C₄⋊C₄	C₂×D₄₇	1	12	yes	complemented
166	C₂²×C₂²⋊C₄	—	47	24	no	—
167	C₂²×C₂²⋊C₄	—	47	12	no	—
168	C₂²×C₂²⋊C₄	—	94	16	no	—
169	C₂²×C₂²⋊C₄	—	94	8	no	—
170	#397	—	47	64	no	—
171	C₂×#95	—	47	24	no	—
172	C₂×#95	—	47	12	no	—
173	C₄₇	C₂²×#397	1	1	yes	characteristic, complemented, Sylow
174	C₂⁵	—	47	8	no	—
175	C₂⁵	—	47	6	no	—
176	C₂⁵	—	94	8	no	—
177	C₂⁵	—	94	4	no	—
178	C₂⁵	—	94	8	no	—
179	C₂⁵	—	94	4	no	—
180	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
181	C₂³×C₄	—	2	8	no	—
182	C₂³×C₄	—	2	8	no	—
183	C₂³×C₄	—	2	8	no	—
184	C₂³×C₄	—	2	8	no	—
185	C₂³×C₄	—	47	12	no	—
186	C₂³×C₄	—	47	6	no	—
187	C₂³×C₄	—	94	8	no	—
188	C₂³×C₄	—	94	8	no	—
189	C₂³×C₄	—	94	8	no	—
190	C₂³×C₄	—	94	8	no	—
191	C₂³×C₄	—	94	4	no	—
192	C₂³×C₄	C₂²⋊D₄₇	1	8	yes	—
193	C₂³×C₄	C₂²×D₄₇	1	12	yes	—
194	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
195	C₂×C₄⋊C₄	—	2	48	no	—
196	C₂×C₄⋊C₄	C₂²×D₄₇	1	16	yes	complemented
197	C₂×C₂²⋊C₄	—	47	32	no	—
198	C₂×C₂²⋊C₄	—	47	16	no	—
199	C₂×C₂²⋊C₄	—	94	96	no	—
200	C₂×C₂²⋊C₄	—	94	48	no	—
201	#95	—	47	32	no	—
202	#95	—	47	16	no	—
203	C₂⁴	—	47	24	no	—
204	C₂⁴	—	47	4	no	—
205	C₂⁴	—	94	32	no	—
206	C₂⁴	—	94	64	no	—
207	C₂⁴	—	94	12	no	—
208	C₂⁴	—	94	12	no	—
209	C₂⁴	—	94	24	no	—
210	C₂⁴	—	94	24	no	—
211	C₂⁴	—	188	16	no	—
212	C₂⁴	—	188	16	no	—
213	C₂⁴	—	188	32	no	—
214	C₂⁴	#10625350	1	8	yes	—
215	C₂⁴	D₄×D₄₇	1	4	yes	—
216	C₂⁴	#10625352	1	4	yes	—
217	C₂⁴	#10625354	1	8	yes	—
218	C₂⁴	C₂×C₂²⋊D₄₇	1	4	yes	—
219	C₂⁴	C₂³×D₄₇	1	3	yes	—
220	C₂²×C₄	—	2	32	no	—
221	C₂²×C₄	—	2	24	no	—
222	C₂²×C₄	—	2	24	no	—
223	C₂²×C₄	—	2	24	no	—
224	C₂²×C₄	—	2	24	no	—
225	C₂²×C₄	—	4	32	no	—
226	C₂²×C₄	—	47	8	no	—
227	C₂²×C₄	—	47	4	no	—
228	C₂²×C₄	—	94	24	no	—
229	C₂²×C₄	—	94	12	no	—
230	C₂²×C₄	—	94	24	no	—
231	C₂²×C₄	—	94	24	no	—
232	C₂²×C₄	—	94	24	no	—
233	C₂²×C₄	—	188	32	no	—
234	C₂²×C₄	—	188	16	no	—
235	C₂²×C₄	C₂×C₂²⋊D₄₇	1	24	yes	—
236	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
237	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
238	C₄⋊C₄	—	2	32	no	—
239	C₂²⋊C₄	—	94	64	no	—
240	C₂²⋊C₄	—	94	32	no	—
241	C₂³	—	47	8	no	—
242	C₂³	—	94	48	no	—
243	C₂³	—	94	96	no	—
244	C₂³	—	94	4	no	—
245	C₂³	—	94	4	no	—
246	C₂³	—	94	8	no	—
247	C₂³	—	94	8	no	—
248	C₂³	—	188	128	no	—
249	C₂³	—	188	24	no	—
250	C₂³	—	188	24	no	—
251	C₂³	—	188	48	no	—
252	C₂³	#13093995	1	16	yes	—
253	C₂³	#13094007	1	16	yes	—
254	C₂³	#13094009	1	32	yes	—
255	C₂³	#13094013	1	16	yes	—
256	C₂³	#13094019	1	32	yes	—
257	C₂³	C₂×D₄×D₄₇	1	6	yes	—
258	C₂³	C₂²×C₂²⋊D₄₇	1	6	yes	—
259	C₂³	C₂×#10625352	1	6	yes	—
260	C₂³	C₂×#10625350	1	12	yes	—
261	C₂³	C₂×#10625354	1	12	yes	—
262	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
263	C₂×C₄	—	2	48	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	4	48	no	—
269	C₂×C₄	—	94	8	no	—
270	C₂×C₄	—	94	4	no	—
271	C₂×C₄	—	94	8	no	—
272	C₂×C₄	—	94	8	no	—
273	C₂×C₄	—	94	8	no	—
274	C₂×C₄	—	188	48	no	—
275	C₂×C₄	—	188	24	no	—
276	C₂×C₄	C₂²×C₂²⋊D₄₇	1	8	yes	—
277	C₂²	—	94	8	no	—
278	C₂²	—	94	16	no	—
279	C₂²	—	188	96	no	—
280	C₂²	—	188	4	no	—
281	C₂²	—	188	4	no	—
282	C₂²	—	188	8	no	—
283	C₂²	#15689049	1	64	yes	direct product factor
284	C₂²	C₂×#13093995	1	12	yes	—
285	C₂²	C₂×#13094007	1	12	yes	—
286	C₂²	C₂×#13094009	1	24	yes	—
287	C₂²	C₂×#13094013	1	12	yes	—
288	C₂²	C₂×#13094019	1	24	yes	—
289	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
290	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
291	C₂²	C₂²×#10625352	1	1	yes	characteristic
292	C₂²	C₂²×#10625350	1	2	yes	—
293	C₂²	C₂²×#10625354	1	2	yes	—
294	C₄	—	2	8	no	—
295	C₄	—	4	8	no	—
296	C₄	—	188	8	no	—
297	C₄	—	188	4	no	—
298	C₂	—	188	8	no	—
299	C₂	C₂×#15689049	1	24	yes	direct product factor
300	C₂	C₂²×#13093995	1	1	yes	characteristic
301	C₂	C₂²×#13094007	1	1	yes	characteristic
302	C₂	C₂²×#13094009	1	2	yes	—
303	C₂	C₂²×#13094013	1	1	yes	characteristic
304	C₂	C₂²×#13094019	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₄₇
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂³×C₄⋊C₄×C₄₇	C₂
socle	C₂⁵×C₄₇	C₂³
prime radical	C₂⁵×D₄₇	C₂²
hypercenter	C₂³×C₄⋊C₄	D₄₇
hypocenter	C₄₇	C₂²×#397
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂×#15689049	C₂	direct (N × H)
4. C₂²×C₄₇⋊#95	C₂	semidirect
5. C₂²×C₄₇⋊#95	C₂	semidirect
6. C₂³×C₄⋊C₄×C₄₇	C₂	semidirect
14. #15689049	C₂²	direct (N × H)
15. C₂×C₄₇⋊#95	C₂²	semidirect
16. C₂×C₄₇⋊#95	C₂²	semidirect
20. C₂²×C₄⋊C₄×C₄₇	C₂²	semidirect
38. C₄₇⋊#95	C₂³	semidirect
39. C₄₇⋊#95	C₂³	semidirect
49. C₂×C₄⋊C₄×C₄₇	C₂³	semidirect
125. C₂²×C₄₇	#397	semidirect
146. C₂³×C₄⋊C₄	D₄₇	semidirect
152. C₂×C₄₇	C₂×#397	semidirect
165. C₂²×C₄⋊C₄	C₂×D₄₇	semidirect
173. C₄₇	C₂²×#397	semidirect
196. C₂×C₄⋊C₄	C₂²×D₄₇	semidirect
283. C₂²	#15689049	direct (N × H)
299. C₂	C₂×#15689049	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×(C₂×D₄₇)⋊C₄	2
C₂×#15689049	2
C₂²×C₄₇⋊#95	2
C₂³×C₄⋊C₄×C₄₇	2
C₂²×#397	47

Subgroups — C22×#15689049 #26382484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689049 #26382484