Subgroups — #26215313

Order 12032.

1086 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	C₂×#15689201	C₂	1	2	yes	maximal
2	#20243257	C₂	1	2	yes	maximal, complemented
3	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243172	C₂	1	4	yes	maximal, complemented
5	#20243150	C₂	1	2	yes	maximal, complemented
6	#20243134	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243082	C₂	1	2	yes	maximal, complemented
8	C₄²×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625329	C₂²	1	4	yes	—
10	#15689201	C₄	1	16	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
13	C₂×#13093935	C₂²	1	4	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
16	#15688878	C₂²	1	2	yes	—
17	#15688878	C₂²	1	4	yes	—
18	#15688878	C₂²	1	1	yes	characteristic
19	#15688878	C₂²	1	1	yes	characteristic
20	#15688878	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₄₇	C₄	1	2	yes	—
22	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
26	C₄×C₂²⋊C₄×C₄₇	C₄	1	4	yes	—
27	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₄³×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#10625329	—	4	4	no	—
30	C₂×#10625329	—	4	4	no	—
31	C₂×#10625329	C₂×C₄	1	16	yes	complemented
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	complemented
36	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	complemented
37	#13093935	—	4	4	no	—
38	C₄×C₄₇⋊C₈	—	4	2	no	—
39	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
40	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
43	C₄×D₄×C₄₇	—	2	8	no	—
44	C₄×D₄×C₄₇	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₇	—	4	4	no	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	#95×C₄₇	C₂×C₄	1	2	yes	—
58	#95×C₄₇	C₂×C₄	1	4	yes	—
59	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂×D₄×C₄₇	—	2	4	no	—
61	C₂×D₄×C₄₇	C₄²	1	4	yes	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	8	no	—
66	C₂²×C₄×C₄₇	—	4	4	no	—
67	C₂²×C₄×C₄₇	—	4	4	no	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₄⋊C₄×C₄₇	C₄²	1	4	yes	—
78	C₂²⋊C₄×C₄₇	—	2	8	no	—
79	C₂²⋊C₄×C₄₇	C₄²	1	8	yes	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	2	4	no	—
82	C₄²×C₄₇	—	2	1	no	—
83	C₄²×C₄₇	—	2	1	no	—
84	C₄²×C₄₇	—	4	4	no	—
85	C₄²×C₄₇	—	4	2	no	—
86	C₄²×C₄₇	M₁₆	1	8	yes	complemented
87	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
88	#10625329	—	4	16	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
96	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
97	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
98	C₂³×C₄₇	—	2	2	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	4	no	—
101	C₂³×C₄₇	—	2	4	no	—
102	C₂³×C₄₇	C₂×C₄²	1	4	yes	—
103	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	D₄×C₄₇	—	2	8	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	8	no	—
113	C₂×C₄×C₄₇	—	2	8	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	4	16	no	—
118	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
120	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
123	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
124	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
126	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
128	C₄₇⋊C₈	—	2	8	no	—
129	C₄₇⋊C₈	—	2	4	no	—
130	C₄₇⋊C₈	—	2	4	no	—
131	#20568	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	—	2	4	no	—
133	C₂²×C₄₇	—	2	4	no	—
134	C₂²×C₄₇	—	2	4	no	—
135	C₂²×C₄₇	—	4	4	no	—
136	C₂²×C₄₇	—	4	4	no	—
137	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
138	C₂²×C₄₇	#518	1	1	yes	characteristic
139	C₂²×C₄₇	#520	1	1	yes	characteristic
140	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
141	C₂²×C₄₇	#568	1	2	yes	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	2	4	no	—
144	C₄×C₄₇	—	2	4	no	—
145	C₄×C₄₇	—	2	4	no	—
146	C₄×C₄₇	—	4	4	no	—
147	C₄×C₄₇	C₄×M₁₆	1	4	yes	—
148	C₄×C₄₇	#518	1	2	yes	—
149	C₄×C₄₇	#520	1	2	yes	—
150	C₂×#436	—	47	2	no	—
151	C₄²×D₄	D₄₇	1	1	yes	characteristic
152	#1770	—	47	2	no	—
153	C₄×C₄⋊C₈	—	47	1	no	—
154	#1753	—	47	4	no	—
155	#1749	—	47	2	no	—
156	#1744	—	47	1	no	—
157	#1728	—	47	2	no	—
158	C₂×C₄₇	—	4	4	no	—
159	C₂×C₄₇	#2275	1	1	yes	characteristic
160	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
161	C₂×C₄₇	#2875	1	1	yes	characteristic
162	C₂×C₄₇	#2975	1	2	yes	—
163	C₂×C₄₇	#2989	1	2	yes	—
164	C₂²×M₁₆	—	47	4	no	—
165	C₂×C₄×D₄	Dic₄₇	1	2	yes	—
166	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
167	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
168	#436	—	47	16	no	—
169	C₂×C₄⋊C₈	—	47	2	no	—
170	C₂×C₂²⋊C₈	—	47	4	no	—
171	C₂×#97	—	47	4	no	—
172	C₂×C₄×C₈	—	47	2	no	—
173	C₂×C₄×C₈	—	47	2	no	—
174	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
175	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
176	C₄×C₂²⋊C₄	Dic₄₇	1	4	yes	—
177	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
178	C₄³	C₂×D₄₇	1	1	yes	characteristic
179	#336	—	47	1	no	—
180	#336	—	47	2	no	—
181	#336	—	47	4	no	—
182	#336	—	47	1	no	—
183	#336	—	47	2	no	—
184	C₄₇	#20568	1	1	yes	characteristic, complemented, Sylow
185	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
186	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
187	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
188	C₂×M₁₆	—	47	16	no	—
189	C₂×M₁₆	—	188	4	no	—
190	C₂×M₁₆	—	188	4	no	—
191	C₂²×C₈	—	47	2	no	—
192	C₂²×C₈	—	47	4	no	—
193	C₂²×C₈	—	47	2	no	—
194	C₄×D₄	—	2	8	no	—
195	C₄×D₄	C₄×D₄₇	1	8	yes	—
196	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
197	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
198	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
199	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
200	C₂×C₄²	—	4	4	no	—
201	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
202	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
203	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
204	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
205	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
207	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
208	C₄⋊C₈	—	47	8	no	—
209	C₂²⋊C₈	—	47	16	no	—
210	#97	—	188	4	no	—
211	C₄×C₈	—	47	8	no	—
212	C₄×C₈	—	188	2	no	—
213	#95	C₂×Dic₄₇	1	2	yes	—
214	#95	C₂×Dic₄₇	1	4	yes	—
215	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	C₄×Dic₄₇	1	4	yes	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
225	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
226	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
228	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
229	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
230	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
232	M₁₆	—	188	16	no	—
233	C₂×C₈	—	47	4	no	—
234	C₂×C₈	—	47	4	no	—
235	C₂×C₈	—	47	8	no	—
236	C₂×C₈	—	94	2	no	—
237	C₂×C₈	—	94	4	no	—
238	C₂×C₈	—	94	2	no	—
239	C₂×C₈	—	94	4	no	—
240	C₂×C₈	—	94	2	no	—
241	C₂×C₈	—	94	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	C₄×Dic₄₇	1	4	yes	—
244	C₂²⋊C₄	—	2	8	no	—
245	C₂²⋊C₄	C₄×Dic₄₇	1	8	yes	—
246	C₄²	—	2	4	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	4	4	no	—
251	C₄²	—	4	2	no	—
252	C₄²	#10625329	1	8	yes	complemented
253	C₄²	C₂×C₄×D₄₇	1	4	yes	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	4	no	—
257	C₂³	—	2	4	no	—
258	C₂³	C₂×C₄×Dic₄₇	1	4	yes	—
259	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
260	D₄	—	2	8	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	16	no	—
274	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
275	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
276	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
277	C₂×C₄	C₂×#10625329	1	4	yes	—
278	C₂×C₄	C₂×#10625329	1	4	yes	—
279	C₂×C₄	C₂×#10625329	1	4	yes	—
280	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
281	C₂×C₄	#13094069	1	2	yes	—
282	C₂×C₄	#13094070	1	1	yes	characteristic
283	C₂×C₄	#13094071	1	1	yes	characteristic
284	C₈	—	94	8	no	—
285	C₈	—	94	4	no	—
286	C₈	—	94	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
293	C₂²	#15689598	1	1	yes	characteristic
294	C₂²	#15689606	1	1	yes	characteristic
295	C₂²	C₂²×#10625329	1	2	yes	—
296	C₂²	#15689890	1	2	yes	—
297	C₄	—	2	4	no	—
298	C₄	—	2	4	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	4	no	—
301	C₄	—	4	4	no	—
302	C₄	C₄×#10625329	1	4	yes	—
303	C₄	#15689595	1	2	yes	—
304	C₄	#15689602	1	2	yes	—
305	C₂	—	4	4	no	—
306	C₂	#20245749	1	1	yes	characteristic
307	C₂	C₂×C₄×#10625329	1	1	yes	characteristic
308	C₂	#20251816	1	1	yes	characteristic
309	C₂	#20252429	1	2	yes	—
310	C₂	#20252505	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄²×D₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄²×D₄	D₄₇
hypocenter	C₄₇	#20568
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄

Complements

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

N	H ≅ G/N	type
2. #20243257	C₂	semidirect
3. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
4. #20243172	C₂	semidirect
5. #20243150	C₂	semidirect
6. #20243134	C₂	semidirect
7. #20243082	C₂	semidirect
10. #15689201	C₄	semidirect
31. C₂×#10625329	C₂×C₄	semidirect
35. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
36. C₂²⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
39. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
86. C₄²×C₄₇	M₁₆	semidirect
184. C₄₇	#20568	semidirect
252. C₄²	#10625329	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689201	2
#20243257	2
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243172	2
#20243150	2
#20243134	2
#20243082	2
C₄²×D₄×C₄₇	2
#20568	47