Subgroups — C₂²×#15689128 #27079801

Order 12032.

7214 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#10625329	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13093983	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15689128	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13093945	C₂	1	4	yes	maximal, complemented
5	C₂⁴×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	—	2	8	no	—
7	C₂²×#10625329	C₂²	1	12	yes	complemented
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	complemented
9	C₂×#13093983	—	2	6	no	—
10	C₂×#13093983	C₂²	1	12	yes	complemented
11	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	complemented
12	C₂²×C₂²⋊Dic₄₇	—	2	1	no	—
13	C₂²×C₄×Dic₄₇	—	2	1	no	—
14	#15689128	C₂²	1	64	yes	direct product factor
15	C₂×#13093945	C₂²	1	48	yes	complemented
16	C₂⁶×C₄₇	—	2	1	no	—
17	C₂³×D₄×C₄₇	—	2	1	no	—
18	C₂³×D₄×C₄₇	—	2	6	no	—
19	C₂³×D₄×C₄₇	—	2	8	no	—
20	C₂³×D₄×C₄₇	C₄	1	16	yes	complemented
21	C₂³×D₄×C₄₇	C₂²	1	2	yes	—
22	C₂³×D₄×C₄₇	C₂²	1	12	yes	—
23	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
24	C₂³×Dic₄₇	—	2	2	no	—
25	C₂×#10625329	—	2	48	no	—
26	C₂×#10625329	C₂³	1	16	yes	complemented
27	C₂²×C₄₇⋊C₈	—	2	8	no	—
28	C₂²×C₄₇⋊C₈	C₂³	1	12	yes	complemented
29	#13093983	—	2	24	no	—
30	#13093983	C₂³	1	16	yes	complemented
31	C₂×C₄⋊Dic₄₇	—	2	6	no	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	complemented
33	C₂×C₂²⋊Dic₄₇	—	2	12	no	—
34	C₂×C₄×Dic₄₇	—	2	12	no	—
35	#13093945	C₂³	1	64	yes	complemented
36	C₂⁵×C₄₇	—	2	16	no	—
37	C₂⁵×C₄₇	—	2	2	no	—
38	C₂⁵×C₄₇	—	2	12	no	—
39	C₂⁵×C₄₇	—	4	16	no	—
40	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
41	C₂²×D₄×C₄₇	—	2	96	no	—
42	C₂²×D₄×C₄₇	—	2	16	no	—
43	C₂²×D₄×C₄₇	—	2	12	no	—
44	C₂²×D₄×C₄₇	—	2	6	no	—
45	C₂²×D₄×C₄₇	—	2	24	no	—
46	C₂²×D₄×C₄₇	—	2	8	no	—
47	C₂²×D₄×C₄₇	—	2	48	no	—
48	C₂²×D₄×C₄₇	C₂×C₄	1	16	yes	—
49	C₂²×D₄×C₄₇	C₂×C₄	1	96	yes	complemented
50	C₂²×D₄×C₄₇	C₂³	1	12	yes	—
51	C₂²×D₄×C₄₇	C₂³	1	16	yes	—
52	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
53	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₄₇	D₄	1	6	yes	—
55	C₂³×C₄×C₄₇	D₄	1	8	yes	—
56	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
58	C₂⁴×C₄₇	—	2	8	no	—
59	C₂⁴×C₄₇	—	2	16	no	—
60	C₂⁴×C₄₇	—	2	96	no	—
61	C₂⁴×C₄₇	—	2	12	no	—
62	C₂⁴×C₄₇	—	2	16	no	—
63	C₂⁴×C₄₇	—	4	96	no	—
64	C₂⁴×C₄₇	—	4	128	no	—
65	C₂⁴×C₄₇	—	4	16	no	—
66	C₂⁴×C₄₇	C₂²⋊C₄	1	8	yes	—
67	C₂⁴×C₄₇	C₂×D₄	1	6	yes	—
68	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×D₄×C₄₇	—	2	48	no	—
70	C₂×D₄×C₄₇	—	2	192	no	—
71	C₂×D₄×C₄₇	—	2	96	no	—
72	C₂×D₄×C₄₇	—	2	16	no	—
73	C₂×D₄×C₄₇	—	2	12	no	—
74	C₂×D₄×C₄₇	—	2	24	no	—
75	C₂×D₄×C₄₇	—	2	32	no	—
76	C₂×D₄×C₄₇	C₂²×C₄	1	48	yes	—
77	C₂×D₄×C₄₇	C₂²×C₄	1	64	yes	complemented
78	C₂×D₄×C₄₇	C₂⁴	1	8	yes	—
79	C₂²×C₄×C₄₇	C₂²⋊C₄	1	48	yes	—
80	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₄₇	C₂²×C₄	1	24	yes	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
86	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₄₇	C₂×D₄	1	24	yes	—
88	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
89	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
90	C₂²×Dic₄₇	—	2	12	no	—
91	C₂²×Dic₄₇	—	4	8	no	—
92	C₂²⋊Dic₄₇	—	2	16	no	—
93	C₄⋊Dic₄₇	—	2	12	no	—
94	C₄⋊Dic₄₇	C₂⁴	1	8	yes	complemented
95	C₄×Dic₄₇	—	2	16	no	—
96	#10625329	—	2	32	no	—
97	C₂×C₄₇⋊C₈	—	2	24	no	—
98	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	complemented
99	C₂³×C₄₇	—	2	24	no	—
100	C₂³×C₄₇	—	2	32	no	—
101	C₂³×C₄₇	—	2	48	no	—
102	C₂³×C₄₇	—	2	64	no	—
103	C₂³×C₄₇	—	2	8	no	—
104	C₂³×C₄₇	—	4	384	no	—
105	C₂³×C₄₇	—	4	64	no	—
106	C₂³×C₄₇	—	4	48	no	—
107	C₂³×C₄₇	—	4	64	no	—
108	C₂³×C₄₇	C₂×C₂²⋊C₄	1	24	yes	—
109	C₂³×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂³×C₄₇	AGL(1,ℤ/8)	1	8	yes	—
111	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
112	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
113	D₄×C₄₇	—	2	48	no	—
114	D₄×C₄₇	—	2	64	no	—
115	D₄×C₄₇	—	2	8	no	—
116	D₄×C₄₇	C₂³×C₄	1	16	yes	—
117	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	48	yes	—
118	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
119	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	24	yes	—
120	C₂×C₄×C₄₇	C₂³×C₄	1	6	yes	—
121	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
122	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
123	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
124	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
125	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
126	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
127	C₂×Dic₄₇	—	2	8	no	—
128	C₂×Dic₄₇	—	4	24	no	—
129	C₄₇⋊C₈	—	2	8	no	—
130	C₂²×#418	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	48	no	—
132	C₂²×C₄₇	—	2	8	no	—
133	C₂²×C₄₇	—	2	16	no	—
134	C₂²×C₄₇	—	4	128	no	—
135	C₂²×C₄₇	—	4	96	no	—
136	C₂²×C₄₇	—	4	16	no	—
137	C₂²×C₄₇	#418	1	16	yes	complemented
138	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	8	yes	—
139	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	6	yes	—
140	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	12	yes	—
141	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
142	C₄×C₄₇	C₂²×C₂²⋊C₄	1	6	yes	—
143	C₄×C₄₇	C₂²×C₂²⋊C₄	1	8	yes	—
144	C₄×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
145	C₄×C₄₇	C₂³×D₄	1	1	yes	characteristic
146	Dic₄₇	—	4	8	no	—
147	C₂⁴×D₄	D₄₇	1	1	yes	characteristic
148	C₂³×M₁₆	—	47	1	no	—
149	C₂²×#117	—	47	1	no	—
150	C₂×#418	—	47	24	no	—
151	C₂²×#102	—	47	4	no	—
152	C₂×C₄₇	—	2	8	no	—
153	C₂×C₄₇	—	4	16	no	—
154	C₂×C₄₇	C₂×#418	1	12	yes	complemented
155	C₂×C₄₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₄₇	C₂²×AGL(1,ℤ/8)	1	2	yes	—
157	C₂⁶	—	2	1	no	—
158	C₂³×D₄	—	2	6	no	—
159	C₂³×D₄	—	2	1	no	—
160	C₂³×D₄	—	2	8	no	—
161	C₂³×D₄	Dic₄₇	1	16	yes	complemented
162	C₂³×D₄	C₂×D₄₇	1	2	yes	—
163	C₂³×D₄	C₂×D₄₇	1	12	yes	—
164	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
165	C₂²×M₁₆	—	47	12	no	—
166	C₂²×M₁₆	—	94	8	no	—
167	C₂³×C₈	—	47	2	no	—
168	C₂×#117	—	47	12	no	—
169	C₂×#117	—	94	6	no	—
170	C₂²×C₄⋊C₄	—	47	2	no	—
171	C₂²×C₂²⋊C₄	—	94	1	no	—
172	C₂²×C₄²	—	94	1	no	—
173	#418	—	47	64	no	—
174	C₂×#102	—	47	48	no	—
175	C₄₇	C₂²×#418	1	1	yes	characteristic, complemented, Sylow
176	C₂⁵	—	2	16	no	—
177	C₂⁵	—	2	12	no	—
178	C₂⁵	—	2	2	no	—
179	C₂⁵	—	4	16	no	—
180	C₂⁵	C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂²×D₄	—	2	96	no	—
182	C₂²×D₄	—	2	16	no	—
183	C₂²×D₄	—	2	12	no	—
184	C₂²×D₄	—	2	6	no	—
185	C₂²×D₄	—	2	24	no	—
186	C₂²×D₄	—	2	8	no	—
187	C₂²×D₄	—	2	48	no	—
188	C₂²×D₄	C₂×Dic₄₇	1	16	yes	—
189	C₂²×D₄	C₂×Dic₄₇	1	96	yes	complemented
190	C₂²×D₄	C₂²×D₄₇	1	12	yes	—
191	C₂²×D₄	C₂²×D₄₇	1	16	yes	—
192	C₂³×C₄	—	94	2	no	—
193	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
194	C₂³×C₄	C₂²⋊D₄₇	1	6	yes	—
195	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
196	C₂³×C₄	C₂²⋊D₄₇	1	8	yes	—
197	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
198	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
199	C₂×M₁₆	—	47	16	no	—
200	C₂×M₁₆	—	94	48	no	—
201	C₂²×C₈	—	47	12	no	—
202	C₂²×C₈	—	94	8	no	—
203	#117	—	47	16	no	—
204	#117	—	94	24	no	—
205	C₂×C₄⋊C₄	—	47	12	no	—
206	C₂×C₄⋊C₄	—	94	6	no	—
207	C₂×C₂²⋊C₄	—	94	12	no	—
208	C₂×C₄²	—	94	12	no	—
209	#102	—	47	64	no	—
210	C₂⁴	—	2	8	no	—
211	C₂⁴	—	2	16	no	—
212	C₂⁴	—	2	96	no	—
213	C₂⁴	—	2	12	no	—
214	C₂⁴	—	2	16	no	—
215	C₂⁴	—	4	128	no	—
216	C₂⁴	—	4	96	no	—
217	C₂⁴	—	4	16	no	—
218	C₂⁴	C₂²⋊Dic₄₇	1	8	yes	—
219	C₂⁴	C₂×C₂²⋊D₄₇	1	6	yes	—
220	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
221	C₂×D₄	—	2	48	no	—
222	C₂×D₄	—	2	192	no	—
223	C₂×D₄	—	2	96	no	—
224	C₂×D₄	—	2	16	no	—
225	C₂×D₄	—	2	12	no	—
226	C₂×D₄	—	2	24	no	—
227	C₂×D₄	—	2	32	no	—
228	C₂×D₄	C₂²×Dic₄₇	1	48	yes	—
229	C₂×D₄	C₂²×Dic₄₇	1	64	yes	complemented
230	C₂×D₄	C₂³×D₄₇	1	8	yes	—
231	C₂²×C₄	—	94	12	no	—
232	C₂²×C₄	—	188	8	no	—
233	C₂²×C₄	C₂²⋊Dic₄₇	1	48	yes	—
234	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
235	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
236	C₂²×C₄	C₂²×Dic₄₇	1	24	yes	—
237	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
238	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
239	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
240	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
241	C₂²×C₄	C₂×C₂²⋊D₄₇	1	24	yes	—
242	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
243	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
244	M₁₆	—	94	32	no	—
245	C₂×C₈	—	47	8	no	—
246	C₂×C₈	—	94	24	no	—
247	C₄⋊C₄	—	47	8	no	—
248	C₄⋊C₄	—	94	12	no	—
249	C₂²⋊C₄	—	94	16	no	—
250	C₄²	—	94	16	no	—
251	C₂³	—	2	32	no	—
252	C₂³	—	2	24	no	—
253	C₂³	—	2	48	no	—
254	C₂³	—	2	64	no	—
255	C₂³	—	2	8	no	—
256	C₂³	—	4	64	no	—
257	C₂³	—	4	384	no	—
258	C₂³	—	4	64	no	—
259	C₂³	—	4	48	no	—
260	C₂³	C₂×C₂²⋊Dic₄₇	1	24	yes	—
261	C₂³	C₂×C₂²⋊Dic₄₇	1	4	yes	—
262	C₂³	#13094050	1	8	yes	—
263	C₂³	C₂²×C₂²⋊D₄₇	1	4	yes	—
264	C₂³	C₂²×C₂²⋊D₄₇	1	3	yes	—
265	D₄	—	2	48	no	—
266	D₄	—	2	64	no	—
267	D₄	—	2	8	no	—
268	D₄	C₂³×Dic₄₇	1	16	yes	—
269	C₂×C₄	—	94	8	no	—
270	C₂×C₄	—	188	24	no	—
271	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	48	yes	—
272	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	12	yes	—
273	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	24	yes	—
274	C₂×C₄	C₂³×Dic₄₇	1	6	yes	—
275	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
276	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
277	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
278	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
279	C₂×C₄	C₂²×C₂²⋊D₄₇	1	8	yes	—
280	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
281	C₈	—	94	8	no	—
282	C₂²	—	2	8	no	—
283	C₂²	—	2	48	no	—
284	C₂²	—	2	16	no	—
285	C₂²	—	4	128	no	—
286	C₂²	—	4	16	no	—
287	C₂²	—	4	96	no	—
288	C₂²	#15689128	1	16	yes	direct product factor
289	C₂²	C₂²×C₂²⋊Dic₄₇	1	8	yes	—
290	C₂²	C₂²×C₂²⋊Dic₄₇	1	6	yes	—
291	C₂²	C₂×#13094050	1	12	yes	—
292	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
293	C₄	—	188	8	no	—
294	C₄	C₂²×C₂²⋊Dic₄₇	1	6	yes	—
295	C₄	C₂²×C₂²⋊Dic₄₇	1	8	yes	—
296	C₄	C₂⁴×Dic₄₇	1	1	yes	characteristic
297	C₄	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
298	C₂	—	2	8	no	—
299	C₂	—	4	16	no	—
300	C₂	C₂×#15689128	1	12	yes	direct product factor
301	C₂	C₂³×C₂²⋊Dic₄₇	1	1	yes	characteristic
302	C₂	C₂²×#13094050	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting, prime radical	C₂⁴×D₄×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂⁴×D₄	D₄₇
hypocenter	C₄₇	C₂²×#418
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#10625329	C₂	semidirect
2. C₂²×#13093983	C₂	semidirect
3. C₂×#15689128	C₂	direct (N × H)
4. C₂²×#13093945	C₂	semidirect
7. C₂²×#10625329	C₂²	semidirect
8. C₂³×C₄₇⋊C₈	C₂²	semidirect
10. C₂×#13093983	C₂²	semidirect
11. C₂²×C₄⋊Dic₄₇	C₂²	semidirect
14. #15689128	C₂²	direct (N × H)
15. C₂×#13093945	C₂²	semidirect
20. C₂³×D₄×C₄₇	C₄	semidirect
26. C₂×#10625329	C₂³	semidirect
28. C₂²×C₄₇⋊C₈	C₂³	semidirect
30. #13093983	C₂³	semidirect
32. C₂×C₄⋊Dic₄₇	C₂³	semidirect
35. #13093945	C₂³	semidirect
49. C₂²×D₄×C₄₇	C₂×C₄	semidirect
77. C₂×D₄×C₄₇	C₂²×C₄	semidirect
94. C₄⋊Dic₄₇	C₂⁴	semidirect
98. C₂×C₄₇⋊C₈	C₂⁴	semidirect
137. C₂²×C₄₇	#418	semidirect
154. C₂×C₄₇	C₂×#418	semidirect
161. C₂³×D₄	Dic₄₇	semidirect
175. C₄₇	C₂²×#418	semidirect
189. C₂²×D₄	C₂×Dic₄₇	semidirect
229. C₂×D₄	C₂²×Dic₄₇	semidirect
288. C₂²	#15689128	direct (N × H)
300. C₂	C₂×#15689128	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×#10625329	2
C₂²×#13093983	2
C₂×#15689128	2
C₂²×#13093945	2
C₂⁴×D₄×C₄₇	2
C₂²×#418	47

Subgroups — C22×#15689128 #27079801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689128 #27079801