Subgroups — #26116340

Order 12032.

600 conjugacy classes of subgroups.

The subgroup lattice has 328 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	#20243150	C₂	1	1	yes	maximal, characteristic
3	#20241231	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×#13093935	—	2	1	no	—
9	#15688878	—	2	1	no	—
10	#15688878	—	2	1	no	—
11	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
14	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#389	—	2	1	no	—
16	C₄₇×#389	—	2	1	no	—
17	C₄₇×#389	—	2	1	no	—
18	C₄₇×#389	—	2	1	no	—
19	C₂×#95×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
25	C₂²⋊(C₄₇⋊C₈)	Q₈	1	4	yes	—
26	#13093935	—	4	2	no	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	4	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	2	no	—
39	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
40	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
41	C₂×C₄²×C₄₇	—	4	2	no	—
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	#95×C₄₇	—	2	2	no	—
47	#95×C₄₇	—	2	2	no	—
48	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
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	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	4	1	no	—
59	C₂²×C₄×C₄₇	—	4	1	no	—
60	C₂²×C₄×C₄₇	—	4	1	no	—
61	C₂²×C₄×C₄₇	—	4	2	no	—
62	C₂²×C₄×C₄₇	—	4	2	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₄₇	#41	1	2	yes	—
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	—	4	2	no	—
77	C₄⋊C₄×C₄₇	—	4	4	no	—
78	C₄⋊C₄×C₄₇	—	4	2	no	—
79	C₄⋊C₄×C₄₇	—	4	4	no	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	2	2	no	—
82	C₄²×C₄₇	—	4	1	no	—
83	C₄²×C₄₇	—	4	1	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	4	2	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂×C₄₇⋊C₈	—	4	2	no	—
90	C₂³×C₄₇	—	2	1	no	—
91	C₂³×C₄₇	—	2	1	no	—
92	C₂³×C₄₇	—	2	1	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
96	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
97	C₂³×C₄₇	#131	1	2	yes	—
98	C₂×C₄×C₄₇	—	2	8	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
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	4	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	—	4	4	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	4	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	8	4	no	—
124	C₂×C₄×C₄₇	#99	1	2	yes	—
125	C₂×C₄×C₄₇	#100	1	2	yes	—
126	C₂×C₄×C₄₇	#101	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
130	C₄₇⋊C₈	—	4	4	no	—
131	C₄₇⋊C₈	—	8	2	no	—
132	#9883	—	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	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	#391	1	1	yes	characteristic
140	C₂²×C₄₇	#407	1	1	yes	characteristic
141	C₂²×C₄₇	#408	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
143	C₂²×C₄₇	#410	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
146	C₂²×C₄₇	#446	1	4	yes	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	—	2	1	no	—
153	C₄×C₄₇	—	4	4	no	—
154	C₄×C₄₇	—	4	1	no	—
155	C₄×C₄₇	—	8	2	no	—
156	C₂×#389	D₄₇	1	1	yes	characteristic
157	#1797	—	47	1	no	—
158	#1749	—	47	1	no	—
159	#1280	—	47	4	no	—
160	C₂×C₄₇	—	2	4	no	—
161	C₂×C₄₇	#1463	1	1	yes	characteristic
162	C₂×C₄₇	#1466	1	1	yes	characteristic
163	C₂×C₄₇	#1470	1	1	yes	characteristic
164	C₂×C₄₇	#1471	1	1	yes	characteristic
165	C₂×C₄₇	#1985	1	1	yes	characteristic
166	C₂×C₄₇	#1996	1	1	yes	characteristic
167	C₂×C₄₇	#1997	1	1	yes	characteristic
168	C₂²×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
169	C₂²×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
170	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	—
171	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
172	C₂×C₄⋊C₈	—	94	1	no	—
173	C₂×C₂²⋊C₈	—	47	2	no	—
174	C₂×C₂²⋊C₈	—	47	2	no	—
175	C₂×#97	—	94	1	no	—
176	#389	—	2	1	no	—
177	#389	—	2	1	no	—
178	#389	—	2	1	no	—
179	#389	—	2	1	no	—
180	C₂×#95	C₂×D₄₇	1	2	yes	—
181	#336	—	94	1	no	—
182	#336	—	94	1	no	—
183	C₄₇	#9883	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
185	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
186	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
187	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
188	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
189	C₂²×C₈	—	94	2	no	—
190	C₂²×C₈	—	94	2	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	4	2	no	—
199	C₂×C₄⋊C₄	—	4	4	no	—
200	C₂×C₄²	—	4	2	no	—
201	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
205	C₄⋊C₈	—	188	2	no	—
206	C₂²⋊C₈	—	47	4	no	—
207	C₂²⋊C₈	—	94	2	no	—
208	C₂²⋊C₈	—	94	4	no	—
209	#97	—	188	2	no	—
210	#95	—	2	2	no	—
211	#95	—	2	2	no	—
212	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	#10625329	1	4	yes	—
230	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
234	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
235	C₂²×C₄	#10625354	1	2	yes	—
236	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
238	C₂×C₈	—	94	4	no	—
239	C₂×C₈	—	188	2	no	—
240	C₂×C₈	—	188	2	no	—
241	C₂×C₈	—	188	2	no	—
242	C₂×C₈	—	188	2	no	—
243	C₂×C₈	—	188	2	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	4	1	no	—
253	C₄²	—	4	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	—	2	1	no	—
258	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
259	C₂³	Q₈×Dic₄₇	1	2	yes	—
260	C₂³	C₂×#10625329	1	2	yes	—
261	C₂³	#13094031	1	2	yes	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	8	4	no	—
288	C₂×C₄	#13093939	1	2	yes	—
289	C₂×C₄	#13093941	1	2	yes	—
290	C₂×C₄	#13093943	1	2	yes	—
291	C₂×C₄	#13094000	1	2	yes	—
292	C₂×C₄	#13094013	1	1	yes	characteristic
293	C₂×C₄	#13094022	1	1	yes	characteristic
294	C₈	—	376	2	no	—
295	C₈	—	188	4	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	#15689030	1	1	yes	characteristic
303	C₂²	#15689087	1	1	yes	characteristic
304	C₂²	#15689090	1	1	yes	characteristic
305	C₂²	C₂×#13093939	1	1	yes	characteristic
306	C₂²	#15689097	1	1	yes	characteristic
307	C₂²	C₂×#13093941	1	1	yes	characteristic
308	C₂²	C₂×#13093943	1	1	yes	characteristic
309	C₂²	#15689254	1	4	yes	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	1	no	—
318	C₄	—	8	2	no	—
319	C₂	—	2	4	no	—
320	C₂	#20241718	1	1	yes	characteristic
321	C₂	#20241729	1	1	yes	characteristic
322	C₂	#20241744	1	1	yes	characteristic
323	C₂	#20241748	1	1	yes	characteristic
324	C₂	#20244329	1	1	yes	characteristic
325	C₂	#20244374	1	1	yes	characteristic
326	C₂	#20244378	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#389	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#389	D₄₇
hypocenter	C₄₇	#9883
LCS	G ▷ C₂⁴×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
183. C₄₇	#9883	semidirect

Maximal subgroups

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

subgroup	index
#20243368	2
#20243150	2
#20241231	2
C₂×C₄₇×#389	2
#9883	47