Subgroups — C₂²×#15688998 #26374586

Order 12032.

8670 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₄₇	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688998	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄₇⋊#95	C₂	1	2	yes	maximal, complemented
6	C₂³×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₄₇	—	2	16	no	—
8	C₂⁴×Dic₄₇	—	2	8	no	—
9	C₂⁴×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₄₇	C₂²	1	12	yes	—
12	C₂⁴×Dic₄₇	C₂²	1	2	yes	—
13	C₂²×C₄⋊Dic₄₇	—	2	16	no	—
14	C₂²×C₄⋊Dic₄₇	C₂²	1	24	yes	complemented
15	C₂²×C₂²⋊Dic₄₇	—	2	8	no	—
16	C₂²×C₂²⋊Dic₄₇	C₂²	1	12	yes	—
17	#15688998	C₂²	1	64	yes	direct product factor
18	C₂×C₄₇⋊#95	C₂²	1	24	yes	complemented
19	C₂⁶×C₄₇	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
21	C₂²×C₂²⋊C₄×C₄₇	C₄	1	16	yes	complemented
22	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	12	yes	—
23	C₂³×Dic₄₇	—	2	128	no	—
24	C₂³×Dic₄₇	—	2	48	no	—
25	C₂³×Dic₄₇	—	2	8	no	—
26	C₂³×Dic₄₇	—	2	8	no	—
27	C₂³×Dic₄₇	—	2	96	no	—
28	C₂³×Dic₄₇	—	2	4	no	—
29	C₂³×Dic₄₇	—	2	4	no	—
30	C₂³×Dic₄₇	—	2	8	no	—
31	C₂³×Dic₄₇	—	2	8	no	—
32	C₂³×Dic₄₇	D₄	1	16	yes	—
33	C₂³×Dic₄₇	C₂³	1	6	yes	—
34	C₂³×Dic₄₇	C₂³	1	16	yes	—
35	C₂³×Dic₄₇	C₂³	1	6	yes	—
36	C₂³×Dic₄₇	C₂³	1	12	yes	—
37	C₂×C₄⋊Dic₄₇	—	2	96	no	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	32	yes	complemented
39	C₂×C₂²⋊Dic₄₇	—	2	48	no	—
40	C₂×C₂²⋊Dic₄₇	C₂³	1	16	yes	—
41	C₄₇⋊#95	C₂³	1	32	yes	complemented
42	C₂⁵×C₄₇	—	2	8	no	—
43	C₂⁵×C₄₇	—	2	8	no	—
44	C₂⁵×C₄₇	C₂×C₄	1	8	yes	—
45	C₂⁵×C₄₇	D₄	1	8	yes	—
46	C₂⁵×C₄₇	Q₈	1	8	yes	—
47	C₂⁵×C₄₇	C₂³	1	6	yes	—
48	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₄₇	—	2	8	no	—
50	C₂³×C₄×C₄₇	—	2	8	no	—
51	C₂³×C₄×C₄₇	C₂×C₄	1	16	yes	complemented
52	C₂³×C₄×C₄₇	C₂³	1	12	yes	—
53	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	96	yes	complemented
54	C₂×C₂²⋊C₄×C₄₇	C₂³	1	16	yes	—
55	C₂⁴×C₄₇	—	2	64	no	—
56	C₂⁴×C₄₇	—	2	32	no	—
57	C₂⁴×C₄₇	—	2	64	no	—
58	C₂⁴×C₄₇	—	2	32	no	—
59	C₂⁴×C₄₇	—	2	24	no	—
60	C₂⁴×C₄₇	—	2	24	no	—
61	C₂⁴×C₄₇	C₄⋊C₄	1	64	yes	complemented
62	C₂⁴×C₄₇	C₂²×C₄	1	24	yes	—
63	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
64	C₂⁴×C₄₇	C₂×D₄	1	24	yes	—
65	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
66	C₂⁴×C₄₇	C₂×D₄	1	8	yes	—
67	C₂⁴×C₄₇	C₂×Q₈	1	24	yes	—
68	C₂⁴×C₄₇	C₂×Q₈	1	4	yes	—
69	C₂⁴×C₄₇	#41	1	8	yes	—
70	C₂⁴×C₄₇	C₂⁴	1	4	yes	—
71	C₂⁴×C₄₇	C₂⁴	1	3	yes	—
72	C₂²×C₄×C₄₇	—	2	64	no	—
73	C₂²×C₄×C₄₇	—	2	24	no	—
74	C₂²×C₄×C₄₇	—	2	24	no	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	48	yes	complemented
76	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
77	C₂²×Dic₄₇	—	2	384	no	—
78	C₂²×Dic₄₇	—	2	64	no	—
79	C₂²×Dic₄₇	—	2	32	no	—
80	C₂²×Dic₄₇	—	2	24	no	—
81	C₂²×Dic₄₇	—	2	24	no	—
82	C₂²×Dic₄₇	—	2	64	no	—
83	C₂²×Dic₄₇	—	2	12	no	—
84	C₂²×Dic₄₇	—	2	12	no	—
85	C₂²×Dic₄₇	—	2	24	no	—
86	C₂²×Dic₄₇	—	2	24	no	—
87	C₂²×Dic₄₇	—	4	16	no	—
88	C₂²×Dic₄₇	—	4	16	no	—
89	C₂²×Dic₄₇	C₂×D₄	1	48	yes	—
90	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
91	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
92	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
93	C₂²⋊C₄×C₄₇	C₂²×C₄	1	64	yes	complemented
94	C₂²⋊Dic₄₇	—	2	32	no	—
95	C₄⋊Dic₄₇	—	2	64	no	—
96	C₂³×C₄₇	—	2	96	no	—
97	C₂³×C₄₇	—	2	48	no	—
98	C₂³×C₄₇	—	2	96	no	—
99	C₂³×C₄₇	—	2	48	no	—
100	C₂³×C₄₇	—	2	8	no	—
101	C₂³×C₄₇	—	2	8	no	—
102	C₂³×C₄₇	—	2	256	no	—
103	C₂³×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
104	C₂³×C₄₇	C₂×C₄⋊C₄	1	96	yes	complemented
105	C₂³×C₄₇	C₄×D₄	1	32	yes	—
106	C₂³×C₄₇	#120	1	16	yes	—
107	C₂³×C₄₇	C₂²⋊Q₈	1	32	yes	—
108	C₂³×C₄₇	#123	1	16	yes	—
109	C₂³×C₄₇	C₂³×C₄	1	8	yes	—
110	C₂³×C₄₇	C₂³×C₄	1	6	yes	—
111	C₂³×C₄₇	C₂²×D₄	1	12	yes	—
112	C₂³×C₄₇	C₂²×D₄	1	8	yes	—
113	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
114	C₂³×C₄₇	C₂²×Q₈	1	8	yes	—
115	C₂³×C₄₇	C₂²×Q₈	1	6	yes	—
116	C₂³×C₄₇	C₂×#41	1	12	yes	—
117	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
118	C₂×C₄×C₄₇	—	2	96	no	—
119	C₂×C₄×C₄₇	—	2	8	no	—
120	C₂×C₄×C₄₇	—	2	8	no	—
121	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	complemented
122	C₂×Dic₄₇	—	2	128	no	—
123	C₂×Dic₄₇	—	2	96	no	—
124	C₂×Dic₄₇	—	2	8	no	—
125	C₂×Dic₄₇	—	2	8	no	—
126	C₂×Dic₄₇	—	2	8	no	—
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	2	4	no	—
129	C₂×Dic₄₇	—	2	8	no	—
130	C₂×Dic₄₇	—	4	24	no	—
131	C₂×Dic₄₇	—	4	24	no	—
132	C₂×Dic₄₇	C₂²×D₄	1	16	yes	—
133	C₂²×#385	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	16	no	—
135	C₂²×C₄₇	—	2	8	no	—
136	C₂²×C₄₇	—	2	16	no	—
137	C₂²×C₄₇	—	2	8	no	—
138	C₂²×C₄₇	—	2	192	no	—
139	C₂²×C₄₇	#385	1	64	yes	complemented
140	C₂²×C₄₇	C₂²×C₄⋊C₄	1	12	yes	—
141	C₂²×C₄₇	C₂²×C₄⋊C₄	1	16	yes	complemented
142	C₂²×C₄₇	C₂×C₄×D₄	1	24	yes	—
143	C₂²×C₄₇	C₂×#120	1	12	yes	—
144	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	24	yes	—
145	C₂²×C₄₇	C₂×#123	1	12	yes	—
146	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
147	C₂²×C₄₇	C₂³×D₄	1	2	yes	—
148	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
149	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
150	C₂²×C₄₇	C₂²×#41	1	2	yes	—
151	C₄×C₄₇	—	2	16	no	—
152	Dic₄₇	—	2	16	no	—
153	Dic₄₇	—	4	4	no	—
154	Dic₄₇	—	4	4	no	—
155	C₂⁵×C₄	—	47	1	no	—
156	C₂³×C₄⋊C₄	—	47	2	no	—
157	C₂³×C₂²⋊C₄	—	47	1	no	—
158	C₂³×C₂²⋊C₄	D₄₇	1	1	yes	characteristic
159	C₂×#385	—	47	24	no	—
160	C₂²×#95	—	47	2	no	—
161	C₂×C₄₇	—	2	16	no	—
162	C₂×C₄₇	C₂×#385	1	24	yes	complemented
163	C₂×C₄₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄₇	C₂²×C₄×D₄	1	2	yes	—
165	C₂×C₄₇	C₂²×#120	1	1	yes	characteristic
166	C₂×C₄₇	C₂²×C₂²⋊Q₈	1	2	yes	—
167	C₂×C₄₇	C₂²×#123	1	1	yes	characteristic
168	C₂⁶	C₂×D₄₇	1	1	yes	characteristic
169	C₂⁴×C₄	—	47	2	no	—
170	C₂⁴×C₄	—	47	12	no	—
171	C₂⁴×C₄	—	47	1	no	—
172	C₂⁴×C₄	—	47	1	no	—
173	C₂⁴×C₄	—	94	16	no	—
174	C₂⁴×C₄	—	94	8	no	—
175	C₂⁴×C₄	C₂×D₄₇	1	2	yes	—
176	C₂²×C₄⋊C₄	—	47	24	no	—
177	C₂²×C₄⋊C₄	—	94	16	no	—
178	C₂²×C₂²⋊C₄	—	47	12	no	—
179	C₂²×C₂²⋊C₄	—	94	8	no	—
180	C₂²×C₂²⋊C₄	Dic₄₇	1	16	yes	complemented
181	C₂²×C₂²⋊C₄	C₂×D₄₇	1	12	yes	—
182	#385	—	47	64	no	—
183	C₂×#95	—	47	24	no	—
184	C₄₇	C₂²×#385	1	1	yes	characteristic, complemented, Sylow
185	C₂⁵	—	2	8	no	—
186	C₂⁵	—	2	8	no	—
187	C₂⁵	Dic₉₄	1	8	yes	—
188	C₂⁵	D₁₈₈	1	8	yes	—
189	C₂⁵	C₂×Dic₄₇	1	8	yes	—
190	C₂⁵	C₂²×D₄₇	1	6	yes	—
191	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
192	C₂³×C₄	—	2	8	no	—
193	C₂³×C₄	—	2	8	no	—
194	C₂³×C₄	—	47	16	no	—
195	C₂³×C₄	—	47	12	no	—
196	C₂³×C₄	—	47	6	no	—
197	C₂³×C₄	—	47	6	no	—
198	C₂³×C₄	—	47	16	no	—
199	C₂³×C₄	—	94	96	no	—
200	C₂³×C₄	—	94	128	no	—
201	C₂³×C₄	—	94	48	no	—
202	C₂³×C₄	—	94	8	no	—
203	C₂³×C₄	—	94	8	no	—
204	C₂³×C₄	—	94	4	no	—
205	C₂³×C₄	—	94	4	no	—
206	C₂³×C₄	—	94	8	no	—
207	C₂³×C₄	—	94	8	no	—
208	C₂³×C₄	C₂×Dic₄₇	1	16	yes	complemented
209	C₂³×C₄	C₂²×D₄₇	1	12	yes	—
210	C₂×C₄⋊C₄	—	47	32	no	—
211	C₂×C₄⋊C₄	—	94	96	no	—
212	C₂×C₂²⋊C₄	—	47	16	no	—
213	C₂×C₂²⋊C₄	—	94	48	no	—
214	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	96	yes	complemented
215	C₂×C₂²⋊C₄	C₂²×D₄₇	1	16	yes	—
216	#95	—	47	32	no	—
217	C₂⁴	—	2	64	no	—
218	C₂⁴	—	2	32	no	—
219	C₂⁴	—	2	64	no	—
220	C₂⁴	—	2	32	no	—
221	C₂⁴	—	2	24	no	—
222	C₂⁴	—	2	24	no	—
223	C₂⁴	C₄⋊Dic₄₇	1	64	yes	complemented
224	C₂⁴	C₂×Dic₉₄	1	24	yes	—
225	C₂⁴	C₂×Dic₉₄	1	4	yes	—
226	C₂⁴	C₂×D₁₈₈	1	24	yes	—
227	C₂⁴	C₂×D₁₈₈	1	4	yes	—
228	C₂⁴	D₄×D₄₇	1	8	yes	—
229	C₂⁴	#10625352	1	8	yes	—
230	C₂⁴	C₂²×Dic₄₇	1	24	yes	—
231	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
232	C₂⁴	C₂³×D₄₇	1	4	yes	—
233	C₂⁴	C₂³×D₄₇	1	3	yes	—
234	C₂²×C₄	—	2	64	no	—
235	C₂²×C₄	—	2	24	no	—
236	C₂²×C₄	—	2	24	no	—
237	C₂²×C₄	—	47	48	no	—
238	C₂²×C₄	—	47	8	no	—
239	C₂²×C₄	—	47	4	no	—
240	C₂²×C₄	—	47	4	no	—
241	C₂²×C₄	—	94	64	no	—
242	C₂²×C₄	—	94	384	no	—
243	C₂²×C₄	—	94	64	no	—
244	C₂²×C₄	—	94	24	no	—
245	C₂²×C₄	—	94	32	no	—
246	C₂²×C₄	—	94	24	no	—
247	C₂²×C₄	—	94	24	no	—
248	C₂²×C₄	—	94	12	no	—
249	C₂²×C₄	—	94	12	no	—
250	C₂²×C₄	—	94	24	no	—
251	C₂²×C₄	—	188	16	no	—
252	C₂²×C₄	—	188	16	no	—
253	C₂²×C₄	C₂²×Dic₄₇	1	48	yes	complemented
254	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
255	C₄⋊C₄	—	94	64	no	—
256	C₂²⋊C₄	—	94	32	no	—
257	C₂²⋊C₄	C₂²×Dic₄₇	1	64	yes	complemented
258	C₂³	—	2	256	no	—
259	C₂³	—	2	96	no	—
260	C₂³	—	2	48	no	—
261	C₂³	—	2	96	no	—
262	C₂³	—	2	48	no	—
263	C₂³	—	2	8	no	—
264	C₂³	—	2	8	no	—
265	C₂³	C₂×C₄⋊Dic₄₇	1	16	yes	—
266	C₂³	C₂×C₄⋊Dic₄₇	1	96	yes	complemented
267	C₂³	D₄×Dic₄₇	1	32	yes	—
268	C₂³	#13093994	1	16	yes	—
269	C₂³	#13094001	1	32	yes	—
270	C₂³	#13094008	1	16	yes	—
271	C₂³	C₂³×Dic₄₇	1	8	yes	—
272	C₂³	C₂³×Dic₄₇	1	6	yes	—
273	C₂³	C₂×D₄×D₄₇	1	12	yes	—
274	C₂³	C₂²×D₁₈₈	1	8	yes	—
275	C₂³	C₂²×D₁₈₈	1	6	yes	—
276	C₂³	C₂²×Dic₉₄	1	8	yes	—
277	C₂³	C₂²×Dic₉₄	1	6	yes	—
278	C₂³	C₂×#10625352	1	12	yes	—
279	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
280	C₂×C₄	—	2	96	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	47	16	no	—
284	C₂×C₄	—	94	128	no	—
285	C₂×C₄	—	94	96	no	—
286	C₂×C₄	—	94	8	no	—
287	C₂×C₄	—	94	8	no	—
288	C₂×C₄	—	94	8	no	—
289	C₂×C₄	—	94	4	no	—
290	C₂×C₄	—	94	4	no	—
291	C₂×C₄	—	94	8	no	—
292	C₂×C₄	—	188	24	no	—
293	C₂×C₄	—	188	24	no	—
294	C₂×C₄	C₂³×Dic₄₇	1	16	yes	complemented
295	C₂²	—	2	192	no	—
296	C₂²	—	2	16	no	—
297	C₂²	—	2	8	no	—
298	C₂²	—	2	16	no	—
299	C₂²	—	2	8	no	—
300	C₂²	#15688998	1	64	yes	direct product factor
301	C₂²	C₂²×C₄⋊Dic₄₇	1	12	yes	—
302	C₂²	C₂²×C₄⋊Dic₄₇	1	16	yes	complemented
303	C₂²	C₂×D₄×Dic₄₇	1	24	yes	—
304	C₂²	C₂×#13093994	1	12	yes	—
305	C₂²	C₂×#13094001	1	24	yes	—
306	C₂²	C₂×#13094008	1	12	yes	—
307	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
308	C₂²	C₂²×D₄×D₄₇	1	2	yes	—
309	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
310	C₂²	C₂³×Dic₉₄	1	1	yes	characteristic
311	C₂²	C₂²×#10625352	1	2	yes	—
312	C₄	—	2	16	no	—
313	C₄	—	94	16	no	—
314	C₄	—	188	4	no	—
315	C₄	—	188	4	no	—
316	C₂	—	2	16	no	—
317	C₂	C₂×#15688998	1	24	yes	direct product factor
318	C₂	C₂³×C₄⋊Dic₄₇	1	1	yes	characteristic
319	C₂	C₂²×D₄×Dic₄₇	1	2	yes	—
320	C₂	C₂²×#13093994	1	1	yes	characteristic
321	C₂	C₂²×#13094001	1	2	yes	—
322	C₂	C₂²×#13094008	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂³×C₂²⋊C₄×C₄₇	C₂
socle	C₂⁵×C₄₇	C₂³
prime radical	C₂⁶×C₄₇	C₂²
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂³×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	C₂²×#385
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄⋊Dic₄₇	C₂	semidirect
4. C₂×#15688998	C₂	direct (N × H)
5. C₂²×C₄₇⋊#95	C₂	semidirect
14. C₂²×C₄⋊Dic₄₇	C₂²	semidirect
17. #15688998	C₂²	direct (N × H)
18. C₂×C₄₇⋊#95	C₂²	semidirect
21. C₂²×C₂²⋊C₄×C₄₇	C₄	semidirect
38. C₂×C₄⋊Dic₄₇	C₂³	semidirect
41. C₄₇⋊#95	C₂³	semidirect
51. C₂³×C₄×C₄₇	C₂×C₄	semidirect
53. C₂×C₂²⋊C₄×C₄₇	C₂×C₄	semidirect
61. C₂⁴×C₄₇	C₄⋊C₄	semidirect
75. C₂²×C₄×C₄₇	C₂²×C₄	semidirect
93. C₂²⋊C₄×C₄₇	C₂²×C₄	semidirect
104. C₂³×C₄₇	C₂×C₄⋊C₄	semidirect
121. C₂×C₄×C₄₇	C₂³×C₄	semidirect
139. C₂²×C₄₇	#385	semidirect
141. C₂²×C₄₇	C₂²×C₄⋊C₄	semidirect
162. C₂×C₄₇	C₂×#385	semidirect
180. C₂²×C₂²⋊C₄	Dic₄₇	semidirect
184. C₄₇	C₂²×#385	semidirect
208. C₂³×C₄	C₂×Dic₄₇	semidirect
214. C₂×C₂²⋊C₄	C₂×Dic₄₇	semidirect
223. C₂⁴	C₄⋊Dic₄₇	semidirect
253. C₂²×C₄	C₂²×Dic₄₇	semidirect
257. C₂²⋊C₄	C₂²×Dic₄₇	semidirect
266. C₂³	C₂×C₄⋊Dic₄₇	semidirect
294. C₂×C₄	C₂³×Dic₄₇	semidirect
300. C₂²	#15688998	direct (N × H)
302. C₂²	C₂²×C₄⋊Dic₄₇	semidirect
317. C₂	C₂×#15688998	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×Dic₄₇	2
C₂³×C₄⋊Dic₄₇	2
C₂³×C₂²⋊Dic₄₇	2
C₂×#15688998	2
C₂²×C₄₇⋊#95	2
C₂³×C₂²⋊C₄×C₄₇	2
C₂²×#385	47

Subgroups — C22×#15688998 #26374586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15688998 #26374586