Subgroups — C₂²×#15689059 #26382508

Order 12032.

5838 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	C₂³×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×Dic₄₇⋊C₄	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊Dic₄₇	C₂	1	2	yes	maximal
4	C₂×#15689059	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄₇⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₄₇	C₂²	1	2	yes	—
8	C₂⁴×Dic₄₇	C₂²	1	2	yes	—
9	C₂²×C₄⋊Dic₄₇	—	2	8	no	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	12	yes	complemented
11	C₂²×Dic₄₇⋊C₄	—	2	16	no	—
12	C₂²×Dic₄₇⋊C₄	C₂²	1	24	yes	complemented
13	C₂²×C₂²⋊Dic₄₇	—	2	16	no	—
14	C₂²×C₂²⋊Dic₄₇	C₂²	1	24	yes	—
15	#15689059	C₂²	1	64	yes	direct product factor
16	C₂×C₄₇⋊#95	C₂²	1	12	yes	complemented
17	C₂⁶×C₄₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₂²⋊C₄×C₄₇	—	2	8	no	—
21	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	12	yes	—
22	C₂³×Dic₄₇	—	2	8	no	—
23	C₂³×Dic₄₇	—	2	8	no	—
24	C₂³×Dic₄₇	—	2	8	no	—
25	C₂³×Dic₄₇	—	2	8	no	—
26	C₂³×Dic₄₇	—	2	8	no	—
27	C₂³×Dic₄₇	D₄	1	16	yes	—
28	C₂³×Dic₄₇	C₂³	1	12	yes	—
29	C₂³×Dic₄₇	C₂³	1	12	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	48	no	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	complemented
32	C₂×Dic₄₇⋊C₄	—	2	96	no	—
33	C₂×Dic₄₇⋊C₄	C₂³	1	32	yes	complemented
34	C₂×C₂²⋊Dic₄₇	—	2	96	no	—
35	C₂×C₂²⋊Dic₄₇	C₂³	1	32	yes	—
36	C₄₇⋊#95	C₂³	1	16	yes	complemented
37	C₂⁵×C₄₇	—	2	4	no	—
38	C₂⁵×C₄₇	—	2	8	no	—
39	C₂⁵×C₄₇	—	2	4	no	—
40	C₂⁵×C₄₇	—	2	8	no	—
41	C₂⁵×C₄₇	Q₈	1	8	yes	—
42	C₂⁵×C₄₇	C₂³	1	6	yes	—
43	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	—	2	8	no	—
45	C₂³×C₄×C₄₇	—	2	4	no	—
46	C₂³×C₄×C₄₇	—	2	8	no	—
47	C₂³×C₄×C₄₇	D₄	1	8	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
49	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
50	C₂×C₂²⋊C₄×C₄₇	—	2	48	no	—
51	C₂×C₂²⋊C₄×C₄₇	C₂³	1	16	yes	—
52	C₂⁴×C₄₇	—	2	64	no	—
53	C₂⁴×C₄₇	—	2	32	no	—
54	C₂⁴×C₄₇	—	2	12	no	—
55	C₂⁴×C₄₇	—	2	24	no	—
56	C₂⁴×C₄₇	—	2	24	no	—
57	C₂⁴×C₄₇	—	2	12	no	—
58	C₂⁴×C₄₇	—	4	16	no	—
59	C₂⁴×C₄₇	—	4	32	no	—
60	C₂⁴×C₄₇	—	4	16	no	—
61	C₂⁴×C₄₇	C₂×D₄	1	8	yes	—
62	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
63	C₂⁴×C₄₇	C₂×Q₈	1	24	yes	—
64	C₂⁴×C₄₇	C₂×Q₈	1	4	yes	—
65	C₂⁴×C₄₇	#41	1	4	yes	—
66	C₂⁴×C₄₇	#41	1	8	yes	—
67	C₂⁴×C₄₇	C₂⁴	1	4	yes	—
68	C₂⁴×C₄₇	C₂⁴	1	3	yes	—
69	C₂²×C₄×C₄₇	—	2	32	no	—
70	C₂²×C₄×C₄₇	—	2	24	no	—
71	C₂²×C₄×C₄₇	—	2	12	no	—
72	C₂²×C₄×C₄₇	—	2	24	no	—
73	C₂²×C₄×C₄₇	—	4	16	no	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	24	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
77	C₂²×Dic₄₇	—	2	64	no	—
78	C₂²×Dic₄₇	—	2	24	no	—
79	C₂²×Dic₄₇	—	2	24	no	—
80	C₂²×Dic₄₇	—	2	24	no	—
81	C₂²×Dic₄₇	—	2	24	no	—
82	C₂²×Dic₄₇	—	2	24	no	—
83	C₂²×Dic₄₇	—	4	32	no	—
84	C₂²×Dic₄₇	C₂×D₄	1	48	yes	—
85	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
86	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
87	C₂²⋊C₄×C₄₇	—	2	32	no	—
88	C₂²⋊Dic₄₇	—	2	64	no	—
89	C₄⋊Dic₄₇	—	2	32	no	—
90	Dic₄₇⋊C₄	—	2	64	no	—
91	C₂³×C₄₇	—	2	96	no	—
92	C₂³×C₄₇	—	2	48	no	—
93	C₂³×C₄₇	—	2	8	no	—
94	C₂³×C₄₇	—	2	4	no	—
95	C₂³×C₄₇	—	2	8	no	—
96	C₂³×C₄₇	—	2	4	no	—
97	C₂³×C₄₇	—	4	24	no	—
98	C₂³×C₄₇	—	4	48	no	—
99	C₂³×C₄₇	—	4	24	no	—
100	C₂³×C₄₇	—	4	128	no	—
101	C₂³×C₄₇	C₂²⋊Q₈	1	32	yes	—
102	C₂³×C₄₇	C₂²⋊Q₈	1	16	yes	—
103	C₂³×C₄₇	#123	1	16	yes	—
104	C₂³×C₄₇	#123	1	32	yes	—
105	C₂³×C₄₇	#127	1	16	yes	—
106	C₂³×C₄₇	C₂²×D₄	1	12	yes	—
107	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
108	C₂³×C₄₇	C₂²×Q₈	1	8	yes	—
109	C₂³×C₄₇	C₂²×Q₈	1	6	yes	—
110	C₂³×C₄₇	C₂×#41	1	12	yes	—
111	C₂³×C₄₇	C₂×#41	1	6	yes	—
112	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
113	C₂×C₄×C₄₇	—	2	48	no	—
114	C₂×C₄×C₄₇	—	2	8	no	—
115	C₂×C₄×C₄₇	—	2	8	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	4	24	no	—
118	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
119	C₂×Dic₄₇	—	2	96	no	—
120	C₂×Dic₄₇	—	2	8	no	—
121	C₂×Dic₄₇	—	2	8	no	—
122	C₂×Dic₄₇	—	2	8	no	—
123	C₂×Dic₄₇	—	2	8	no	—
124	C₂×Dic₄₇	—	2	8	no	—
125	C₂×Dic₄₇	—	4	48	no	—
126	C₂×Dic₄₇	C₂²×D₄	1	16	yes	—
127	C₂²×#399	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	16	no	—
129	C₂²×C₄₇	—	2	8	no	—
130	C₂²×C₄₇	—	4	96	no	—
131	C₂²×C₄₇	—	4	4	no	—
132	C₂²×C₄₇	—	4	8	no	—
133	C₂²×C₄₇	—	4	4	no	—
134	C₂²×C₄₇	#399	1	64	yes	complemented
135	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	24	yes	—
136	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	12	yes	—
137	C₂²×C₄₇	C₂×#123	1	12	yes	—
138	C₂²×C₄₇	C₂×#123	1	24	yes	—
139	C₂²×C₄₇	C₂×#127	1	12	yes	—
140	C₂²×C₄₇	C₂³×D₄	1	2	yes	—
141	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
142	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
143	C₂²×C₄₇	C₂²×#41	1	2	yes	—
144	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
145	C₄×C₄₇	—	2	8	no	—
146	C₄×C₄₇	—	4	4	no	—
147	Dic₄₇	—	2	16	no	—
148	Dic₄₇	—	4	8	no	—
149	C₂³×C₄⋊C₄	—	47	2	no	—
150	C₂³×C₄⋊C₄	—	47	1	no	—
151	C₂³×C₂²⋊C₄	—	47	2	no	—
152	C₂³×C₂²⋊C₄	D₄₇	1	1	yes	characteristic
153	C₂×#399	—	47	24	no	—
154	C₂²×#95	—	47	1	no	—
155	C₂×C₄₇	—	4	8	no	—
156	C₂×C₄₇	C₂×#399	1	24	yes	complemented
157	C₂×C₄₇	C₂²×C₂²⋊Q₈	1	2	yes	—
158	C₂×C₄₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×C₄₇	C₂²×#123	1	1	yes	characteristic
160	C₂×C₄₇	C₂²×#123	1	2	yes	—
161	C₂×C₄₇	C₂²×#127	1	1	yes	characteristic
162	C₂⁶	C₂×D₄₇	1	1	yes	characteristic
163	C₂⁴×C₄	—	47	2	no	—
164	C₂⁴×C₄	—	47	2	no	—
165	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
166	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
167	C₂²×C₄⋊C₄	—	47	24	no	—
168	C₂²×C₄⋊C₄	—	47	12	no	—
169	C₂²×C₄⋊C₄	—	94	16	no	—
170	C₂²×C₄⋊C₄	—	94	8	no	—
171	C₂²×C₂²⋊C₄	—	2	8	no	—
172	C₂²×C₂²⋊C₄	—	47	24	no	—
173	C₂²×C₂²⋊C₄	—	94	16	no	—
174	C₂²×C₂²⋊C₄	C₂×D₄₇	1	12	yes	—
175	#399	—	47	64	no	—
176	C₂×#95	—	47	12	no	—
177	C₄₇	C₂²×#399	1	1	yes	characteristic, complemented, Sylow
178	C₂⁵	—	2	4	no	—
179	C₂⁵	—	2	8	no	—
180	C₂⁵	—	2	8	no	—
181	C₂⁵	—	2	4	no	—
182	C₂⁵	Dic₉₄	1	8	yes	—
183	C₂⁵	C₂²×D₄₇	1	6	yes	—
184	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
185	C₂³×C₄	—	2	8	no	—
186	C₂³×C₄	—	2	8	no	—
187	C₂³×C₄	—	2	4	no	—
188	C₂³×C₄	—	47	16	no	—
189	C₂³×C₄	—	47	12	no	—
190	C₂³×C₄	—	47	12	no	—
191	C₂³×C₄	—	94	8	no	—
192	C₂³×C₄	—	94	8	no	—
193	C₂³×C₄	—	94	8	no	—
194	C₂³×C₄	—	94	8	no	—
195	C₂³×C₄	—	94	8	no	—
196	C₂³×C₄	C₂²⋊D₄₇	1	8	yes	—
197	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
198	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
199	C₂×C₄⋊C₄	—	47	32	no	—
200	C₂×C₄⋊C₄	—	47	16	no	—
201	C₂×C₄⋊C₄	—	94	96	no	—
202	C₂×C₄⋊C₄	—	94	48	no	—
203	C₂×C₂²⋊C₄	—	2	48	no	—
204	C₂×C₂²⋊C₄	—	47	32	no	—
205	C₂×C₂²⋊C₄	—	94	96	no	—
206	C₂×C₂²⋊C₄	C₂²×D₄₇	1	16	yes	—
207	#95	—	47	16	no	—
208	C₂⁴	—	2	64	no	—
209	C₂⁴	—	2	32	no	—
210	C₂⁴	—	2	24	no	—
211	C₂⁴	—	2	12	no	—
212	C₂⁴	—	2	24	no	—
213	C₂⁴	—	2	12	no	—
214	C₂⁴	—	4	16	no	—
215	C₂⁴	—	4	32	no	—
216	C₂⁴	—	4	16	no	—
217	C₂⁴	C₂×Dic₉₄	1	24	yes	—
218	C₂⁴	C₂×Dic₉₄	1	4	yes	—
219	C₂⁴	#10625350	1	4	yes	—
220	C₂⁴	D₄×D₄₇	1	8	yes	—
221	C₂⁴	#10625352	1	8	yes	—
222	C₂⁴	C₂×C₂²⋊D₄₇	1	4	yes	—
223	C₂⁴	C₂³×D₄₇	1	4	yes	—
224	C₂⁴	C₂³×D₄₇	1	3	yes	—
225	C₂²×C₄	—	2	32	no	—
226	C₂²×C₄	—	2	24	no	—
227	C₂²×C₄	—	2	24	no	—
228	C₂²×C₄	—	2	12	no	—
229	C₂²×C₄	—	4	16	no	—
230	C₂²×C₄	—	47	48	no	—
231	C₂²×C₄	—	47	8	no	—
232	C₂²×C₄	—	47	8	no	—
233	C₂²×C₄	—	94	64	no	—
234	C₂²×C₄	—	94	24	no	—
235	C₂²×C₄	—	94	24	no	—
236	C₂²×C₄	—	94	24	no	—
237	C₂²×C₄	—	94	24	no	—
238	C₂²×C₄	—	94	24	no	—
239	C₂²×C₄	—	188	32	no	—
240	C₂²×C₄	C₂×C₂²⋊D₄₇	1	24	yes	—
241	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
242	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
243	C₄⋊C₄	—	94	64	no	—
244	C₄⋊C₄	—	94	32	no	—
245	C₂²⋊C₄	—	2	32	no	—
246	C₂²⋊C₄	—	94	64	no	—
247	C₂³	—	2	96	no	—
248	C₂³	—	2	48	no	—
249	C₂³	—	2	8	no	—
250	C₂³	—	2	4	no	—
251	C₂³	—	2	8	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	4	128	no	—
254	C₂³	—	4	24	no	—
255	C₂³	—	4	48	no	—
256	C₂³	—	4	24	no	—
257	C₂³	#13094001	1	32	yes	—
258	C₂³	#13094003	1	16	yes	—
259	C₂³	#13094005	1	16	yes	—
260	C₂³	#13094006	1	32	yes	—
261	C₂³	#13094020	1	16	yes	—
262	C₂³	C₂×D₄×D₄₇	1	12	yes	—
263	C₂³	C₂²×C₂²⋊D₄₇	1	6	yes	—
264	C₂³	C₂²×Dic₉₄	1	8	yes	—
265	C₂³	C₂²×Dic₉₄	1	6	yes	—
266	C₂³	C₂×#10625352	1	12	yes	—
267	C₂³	C₂×#10625350	1	6	yes	—
268	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
269	C₂×C₄	—	2	48	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	24	no	—
274	C₂×C₄	—	47	16	no	—
275	C₂×C₄	—	94	96	no	—
276	C₂×C₄	—	94	8	no	—
277	C₂×C₄	—	94	8	no	—
278	C₂×C₄	—	94	8	no	—
279	C₂×C₄	—	94	8	no	—
280	C₂×C₄	—	94	8	no	—
281	C₂×C₄	—	188	48	no	—
282	C₂×C₄	C₂²×C₂²⋊D₄₇	1	8	yes	—
283	C₂²	—	2	16	no	—
284	C₂²	—	2	8	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	8	no	—
287	C₂²	—	4	96	no	—
288	C₂²	—	4	4	no	—
289	C₂²	#15689059	1	64	yes	direct product factor
290	C₂²	C₂×#13094001	1	24	yes	—
291	C₂²	C₂×#13094003	1	12	yes	—
292	C₂²	C₂×#13094005	1	12	yes	—
293	C₂²	C₂×#13094006	1	24	yes	—
294	C₂²	C₂×#13094020	1	12	yes	—
295	C₂²	C₂²×D₄×D₄₇	1	2	yes	—
296	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
297	C₂²	C₂³×Dic₉₄	1	1	yes	characteristic
298	C₂²	C₂²×#10625352	1	2	yes	—
299	C₂²	C₂²×#10625350	1	1	yes	characteristic
300	C₄	—	2	8	no	—
301	C₄	—	4	4	no	—
302	C₄	—	94	16	no	—
303	C₄	—	188	8	no	—
304	C₂	—	4	8	no	—
305	C₂	C₂×#15689059	1	24	yes	direct product factor
306	C₂	C₂²×#13094001	1	2	yes	—
307	C₂	C₂²×#13094003	1	1	yes	characteristic
308	C₂	C₂²×#13094005	1	1	yes	characteristic
309	C₂	C₂²×#13094006	1	2	yes	—
310	C₂	C₂²×#13094020	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₄₇
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂³×C₂²⋊C₄×C₄₇	C₂
socle	C₂⁵×C₄₇	C₂³
prime radical	C₂⁶×C₄₇	C₂²
hypercenter	C₂³×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	C₂²×#399
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄⋊Dic₄₇	C₂	semidirect
2. C₂³×Dic₄₇⋊C₄	C₂	semidirect
4. C₂×#15689059	C₂	direct (N × H)
5. C₂²×C₄₇⋊#95	C₂	semidirect
10. C₂²×C₄⋊Dic₄₇	C₂²	semidirect
12. C₂²×Dic₄₇⋊C₄	C₂²	semidirect
15. #15689059	C₂²	direct (N × H)
16. C₂×C₄₇⋊#95	C₂²	semidirect
31. C₂×C₄⋊Dic₄₇	C₂³	semidirect
33. C₂×Dic₄₇⋊C₄	C₂³	semidirect
36. C₄₇⋊#95	C₂³	semidirect
134. C₂²×C₄₇	#399	semidirect
156. C₂×C₄₇	C₂×#399	semidirect
177. C₄₇	C₂²×#399	semidirect
289. C₂²	#15689059	direct (N × H)
305. C₂	C₂×#15689059	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×C₄⋊Dic₄₇	2
C₂³×Dic₄₇⋊C₄	2
C₂³×C₂²⋊Dic₄₇	2
C₂×#15689059	2
C₂²×C₄₇⋊#95	2
C₂³×C₂²⋊C₄×C₄₇	2
C₂²×#399	47

Subgroups — C22×#15689059 #26382508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689059 #26382508