Subgroups — #26115107

Order 12032.

656 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243368	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688849	C₂	1	4	yes	maximal
4	C₂×C₄₇×#389	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
9	#15688878	—	2	1	no	—
10	#15688878	—	2	1	no	—
11	#15688849	—	2	8	no	—
12	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#389	—	2	1	no	—
17	C₄₇×#389	—	2	1	no	—
18	C₄₇×#389	—	2	1	no	—
19	C₄₇×#389	—	2	1	no	—
20	C₂×#95×C₄₇	C₄	1	2	yes	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
26	C₂²⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
27	C₄×C₄₇⋊C₈	—	4	2	no	—
28	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
43	C₂×C₄⋊C₄×C₄₇	C₈	1	4	yes	complemented
44	C₂×C₄²×C₄₇	—	4	2	no	—
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	#95×C₄₇	—	2	2	no	—
50	#95×C₄₇	—	2	2	no	—
51	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	4	1	no	—
63	C₂²×C₄×C₄₇	—	4	1	no	—
64	C₂²×C₄×C₄₇	—	4	1	no	—
65	C₂²×C₄×C₄₇	—	4	2	no	—
66	C₂²×C₄×C₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×C₈	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₇	#41	1	2	yes	—
78	C₄⋊C₄×C₄₇	—	2	4	no	—
79	C₄⋊C₄×C₄₇	—	2	4	no	—
80	C₄⋊C₄×C₄₇	—	4	2	no	—
81	C₄⋊C₄×C₄₇	—	4	2	no	—
82	C₄⋊C₄×C₄₇	—	4	4	no	—
83	C₄⋊C₄×C₄₇	—	4	2	no	—
84	C₄⋊C₄×C₄₇	—	4	2	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	2	2	no	—
87	C₄²×C₄₇	—	4	1	no	—
88	C₄²×C₄₇	—	4	1	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	4	2	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂³×C₄₇	—	2	1	no	—
96	C₂³×C₄₇	—	2	1	no	—
97	C₂³×C₄₇	—	2	1	no	—
98	C₂³×C₄₇	—	2	1	no	—
99	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₄₇	C₄×D₄	1	2	yes	—
101	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
102	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂³×C₄₇	#131	1	2	yes	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	8	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	4	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	4	4	no	—
125	C₂×C₄×C₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	8	4	no	—
129	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
130	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
131	C₂×C₄×C₄₇	#99	1	2	yes	—
132	C₂×C₄×C₄₇	#101	1	2	yes	—
133	C₂×C₄×C₄₇	#101	1	2	yes	—
134	C₂×C₄×C₄₇	#121	1	2	yes	—
135	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
136	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
137	C₄₇⋊C₈	—	4	4	no	—
138	C₄₇⋊C₈	—	8	2	no	—
139	#9650	—	47	1	no	maximal, Sylow
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	1	no	—
143	C₂²×C₄₇	—	2	1	no	—
144	C₂²×C₄₇	—	2	1	no	—
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	#390	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₂²×C₄₇	#408	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
152	C₂²×C₄₇	#410	1	1	yes	characteristic
153	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
155	C₂²×C₄₇	C₈×D₄	1	2	yes	—
156	C₂²×C₄₇	#436	1	2	yes	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	1	no	—
159	C₄×C₄₇	—	2	1	no	—
160	C₄×C₄₇	—	2	2	no	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	4	4	no	—
163	C₄×C₄₇	—	4	1	no	—
164	C₄×C₄₇	—	8	2	no	—
165	C₄×C₄₇	#324	1	4	yes	complemented
166	C₂×#389	D₄₇	1	1	yes	characteristic
167	#1797	—	47	1	no	—
168	C₄×C₂²⋊C₈	—	47	1	no	—
169	C₂×#324	—	47	4	no	—
170	C₂×C₄₇	—	2	2	no	—
171	C₂×C₄₇	—	2	2	no	—
172	C₂×C₄₇	C₂×#324	1	1	yes	characteristic
173	C₂×C₄₇	#1461	1	1	yes	characteristic
174	C₂×C₄₇	#1470	1	1	yes	characteristic
175	C₂×C₄₇	#1472	1	1	yes	characteristic
176	C₂×C₄₇	#1954	1	1	yes	characteristic
177	C₂×C₄₇	#1964	1	1	yes	characteristic
178	C₂×C₄₇	#1966	1	1	yes	characteristic
179	C₂²×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
180	C₂²×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
181	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
182	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
183	C₂×C₄⋊C₈	—	94	1	no	—
184	C₂×C₂²⋊C₈	—	47	2	no	—
185	C₂×C₂²⋊C₈	—	47	2	no	—
186	C₂×C₄×C₈	—	94	1	no	—
187	#389	—	2	1	no	—
188	#389	—	2	1	no	—
189	#389	—	2	1	no	—
190	#389	—	2	1	no	—
191	C₂×#95	Dic₄₇	1	2	yes	—
192	#336	—	94	1	no	—
193	#336	—	94	1	no	—
194	#324	—	94	8	no	—
195	C₄₇	#9650	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
197	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
198	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
199	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
200	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
201	C₂²×C₈	—	94	2	no	—
202	C₂²×C₈	—	94	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₄⋊C₄	—	2	4	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	4	2	no	—
213	C₂×C₄⋊C₄	C₄₇⋊C₈	1	4	yes	complemented
214	C₂×C₄²	—	4	2	no	—
215	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
219	C₄⋊C₈	—	188	2	no	—
220	C₂²⋊C₈	—	47	4	no	—
221	C₂²⋊C₈	—	94	2	no	—
222	C₂²⋊C₈	—	94	4	no	—
223	C₄×C₈	—	188	2	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
243	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
244	C₂²×C₄	#10625329	1	2	yes	—
245	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
246	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
249	C₂²×C₄	D₄×D₄₇	1	2	yes	—
250	C₂²×C₄	#10625352	1	2	yes	—
251	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
253	C₂×C₈	—	94	4	no	—
254	C₂×C₈	—	188	2	no	—
255	C₂×C₈	—	188	2	no	—
256	C₂×C₈	—	188	2	no	—
257	C₂×C₈	—	188	2	no	—
258	C₂×C₈	—	188	2	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	4	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	4	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
275	C₂³	D₄×Dic₄₇	1	2	yes	—
276	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
277	C₂³	C₂×#10625329	1	1	yes	characteristic
278	C₂³	#13094031	1	2	yes	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	4	no	—
304	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
305	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
306	C₂×C₄	#13093939	1	2	yes	—
307	C₂×C₄	#13093943	1	2	yes	—
308	C₂×C₄	#13093943	1	2	yes	—
309	C₂×C₄	#13093995	1	2	yes	—
310	C₂×C₄	#13094011	1	1	yes	characteristic
311	C₂×C₄	#13094020	1	1	yes	characteristic
312	C₈	—	376	2	no	—
313	C₈	—	188	4	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	#15689026	1	1	yes	characteristic
323	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
324	C₂²	#15689090	1	1	yes	characteristic
325	C₂²	C₂×#13093939	1	1	yes	characteristic
326	C₂²	#15689097	1	1	yes	characteristic
327	C₂²	C₂×#13093943	1	1	yes	characteristic
328	C₂²	C₂×#13093943	1	1	yes	characteristic
329	C₂²	D₄×C₄₇⋊C₈	1	2	yes	—
330	C₂²	#15689201	1	2	yes	—
331	C₄	—	2	1	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	4	no	—
338	C₄	—	8	2	no	—
339	C₄	#15688849	1	4	yes	complemented
340	C₂	—	2	2	no	—
341	C₂	—	2	2	no	—
342	C₂	C₂×#15688849	1	1	yes	characteristic
343	C₂	#20241712	1	1	yes	characteristic
344	C₂	#20241744	1	1	yes	characteristic
345	C₂	#20241752	1	1	yes	characteristic
346	C₂	#20244183	1	1	yes	characteristic
347	C₂	#20244225	1	1	yes	characteristic
348	C₂	#20244233	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂²×C₈
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#389	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#389	D₄₇
hypocenter	C₄₇	#9650
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#389

Complements

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

N	H ≅ G/N	type
43. C₂×C₄⋊C₄×C₄₇	C₈	semidirect
165. C₄×C₄₇	#324	semidirect
195. C₄₇	#9650	semidirect
213. C₂×C₄⋊C₄	C₄₇⋊C₈	semidirect
339. C₄	#15688849	semidirect

Maximal subgroups

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

subgroup	index
#20243368	2
C₄×C₂²⋊(C₄₇⋊C₈)	2
C₂×#15688849	2
C₂×C₄₇×#389	2
#9650	47