Subgroups — #26215112

Order 12032.

966 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×#15689180	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
3	#20243263	C₂	1	4	yes	maximal, complemented
4	#20243241	C₂	1	2	yes	maximal, complemented
5	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal, complemented
6	#20243172	C₂	1	4	yes	maximal
7	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	#15689180	C₄	1	8	yes	complemented
10	#15689147	C₄	1	8	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×#13093935	C₂²	1	4	yes	—
16	C₂×#13093935	C₂²	1	2	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
18	#15688878	C₂²	1	4	yes	—
19	#15688878	C₂²	1	4	yes	—
20	C₂×#117×C₄₇	C₄	1	2	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
24	C₄×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
25	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
26	C₄₇×#376	C₄	1	2	yes	—
27	C₄³×C₄₇	C₂²	1	1	yes	characteristic
28	C₄³×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#10625329	—	4	4	no	—
30	C₂×#10625329	C₂×C₄	1	8	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
33	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
34	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
35	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
36	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
37	#13093935	—	4	4	no	—
38	#13093935	C₂×C₄	1	8	yes	—
39	C₄×C₄₇⋊C₈	—	4	4	no	—
40	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	#117×C₄₇	—	2	8	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	#95×C₄₇	C₂×C₄	1	4	yes	—
59	#95×C₄₇	C₂×C₄	1	4	yes	—
60	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
66	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
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	4	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
77	C₄⋊C₄×C₄₇	—	2	8	no	—
78	C₂²⋊C₄×C₄₇	—	2	8	no	—
79	C₄²×C₄₇	—	2	4	no	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	2	2	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	4	2	no	—
85	C₄²×C₄₇	—	4	2	no	—
86	C₄²×C₄₇	C₄²	1	4	yes	—
87	C₄²×C₄₇	C₄²	1	4	yes	—
88	C₄²×C₄₇	M₁₆	1	8	yes	complemented
89	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
91	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
93	#10625329	—	4	8	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	C₂×C₄₇⋊C₈	—	2	4	no	—
98	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
99	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	—	2	2	no	—
102	C₂³×C₄₇	—	2	2	no	—
103	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
104	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	—	2	4	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	4	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	8	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	4	4	no	—
119	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
125	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
126	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
129	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
130	C₄₇⋊C₈	—	2	8	no	—
131	C₄₇⋊C₈	—	2	8	no	—
132	#20549	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	4	4	no	—
137	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
138	C₂²×C₄₇	#520	1	2	yes	—
139	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
140	C₂²×C₄₇	#568	1	2	yes	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	2	8	no	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	4	4	no	—
147	C₄×C₄₇	C₄×M₁₆	1	4	yes	—
148	C₄×C₄₇	#518	1	1	yes	characteristic
149	C₄×C₄₇	#519	1	1	yes	characteristic
150	C₄×C₄₇	#520	1	2	yes	—
151	C₂×#432	—	47	1	no	—
152	C₂×#423	—	47	1	no	—
153	C₄×#117	D₄₇	1	1	yes	characteristic
154	#1771	—	47	4	no	—
155	#1767	—	47	2	no	—
156	C₄×C₄⋊C₈	—	47	2	no	—
157	#1753	—	47	4	no	—
158	C₂×C₄₇	—	4	2	no	—
159	C₂×C₄₇	#2274	1	1	yes	characteristic
160	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
161	C₂×C₄₇	#2873	1	1	yes	characteristic
162	C₂×C₄₇	#2989	1	4	yes	—
163	C₂²×M₁₆	—	47	2	no	—
164	C₂×#117	Dic₄₇	1	2	yes	—
165	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
166	#432	—	47	8	no	—
167	#423	—	47	8	no	—
168	C₂×C₄⋊C₈	—	47	2	no	—
169	C₂×C₄⋊C₈	—	47	2	no	—
170	C₂×C₄⋊C₈	—	47	2	no	—
171	C₂×C₂²⋊C₈	—	47	2	no	—
172	C₂×#97	—	47	2	no	—
173	C₂×#97	—	47	4	no	—
174	C₂×C₄×C₈	—	47	4	no	—
175	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
176	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
177	C₄×C₂²⋊C₄	Dic₄₇	1	2	yes	—
178	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
179	#376	Dic₄₇	1	2	yes	—
180	C₄³	C₂×D₄₇	1	1	yes	characteristic
181	C₄³	C₂×D₄₇	1	1	yes	characteristic
182	#336	—	47	4	no	—
183	#336	—	47	4	no	—
184	C₄₇	#20549	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
186	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
187	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
188	C₂×M₁₆	—	47	8	no	—
189	C₂×M₁₆	—	188	4	no	—
190	C₂²×C₈	—	47	4	no	—
191	C₂²×C₈	—	47	4	no	—
192	#117	—	2	8	no	—
193	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
194	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
195	C₂×C₄²	C₄×D₄₇	1	4	yes	—
196	C₂×C₄²	C₄×D₄₇	1	4	yes	—
197	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
198	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
199	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
200	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
201	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
202	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
203	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
204	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
205	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
207	C₄⋊C₈	—	47	8	no	—
208	C₄⋊C₈	—	47	8	no	—
209	C₄⋊C₈	—	47	8	no	—
210	C₂²⋊C₈	—	47	8	no	—
211	#97	—	47	8	no	—
212	#97	—	188	4	no	—
213	C₄×C₈	—	188	4	no	—
214	#95	C₂×Dic₄₇	1	4	yes	—
215	#95	C₂×Dic₄₇	1	4	yes	—
216	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
222	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
223	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
224	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
225	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
226	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
231	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
232	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
233	M₁₆	—	188	8	no	—
234	C₂×C₈	—	47	8	no	—
235	C₂×C₈	—	47	8	no	—
236	C₂×C₈	—	94	4	no	—
237	C₂×C₈	—	94	4	no	—
238	C₂×C₈	—	94	4	no	—
239	C₂×C₈	—	94	4	no	—
240	C₄⋊C₄	—	2	8	no	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	—	4	2	no	—
249	C₄²	#10625329	1	8	yes	complemented
250	C₄²	C₄×Dic₄₇	1	4	yes	—
251	C₄²	C₄×Dic₄₇	1	4	yes	—
252	C₄²	C₂×C₄×D₄₇	1	2	yes	—
253	C₄²	C₂×C₄×D₄₇	1	2	yes	—
254	C₄²	C₂×C₄×D₄₇	1	2	yes	—
255	C₄²	C₂×C₄×D₄₇	1	2	yes	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
260	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
276	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
277	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
278	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
279	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
280	C₂×C₄	C₂×#10625329	1	8	yes	—
281	C₂×C₄	C₂×#10625329	1	4	yes	—
282	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
284	C₂×C₄	#13094070	1	2	yes	—
285	C₂×C₄	#13094071	1	2	yes	—
286	C₈	—	94	8	no	—
287	C₈	—	94	8	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	4	no	—
292	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
293	C₂²	#15689606	1	2	yes	—
294	C₂²	C₂²×#10625329	1	2	yes	—
295	C₂²	#15689890	1	2	yes	—
296	C₄	—	2	8	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	4	4	no	—
302	C₄	C₄×#10625329	1	4	yes	—
303	C₄	#15689597	1	1	yes	characteristic
304	C₄	#15689600	1	1	yes	characteristic
305	C₄	#15689603	1	2	yes	—
306	C₂	—	4	2	no	—
307	C₂	#20245744	1	1	yes	characteristic
308	C₂	C₂×C₄×#10625329	1	1	yes	characteristic
309	C₂	#20251807	1	1	yes	characteristic
310	C₂	#20252505	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#20549
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #20243263	C₂	semidirect
4. #20243241	C₂	semidirect
5. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
9. #15689180	C₄	semidirect
10. #15689147	C₄	semidirect
88. C₄²×C₄₇	M₁₆	semidirect
184. C₄₇	#20549	semidirect
249. 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×#15689180	2
C₂×#15689147	2
#20243263	2
#20243241	2
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243172	2
C₄×#117×C₄₇	2
#20549	47