Subgroups — #26316628

Order 12032.

590 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20249368	C₂	1	1	yes	maximal, characteristic
2	#20246130	C₂	1	1	yes	maximal, characteristic
3	#20244885	C₂	1	2	yes	maximal
4	#20244880	C₂	1	2	yes	maximal
5	#20244354	C₂	1	1	yes	maximal, characteristic
6	#20243983	C₂	1	2	yes	maximal
7	#20243962	C₂	1	2	yes	maximal
8	#20243547	C₂	1	1	yes	maximal, characteristic
9	#20243522	C₂	1	1	yes	maximal, characteristic
10	#20243248	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1992	C₂	1	1	yes	maximal, characteristic
12	C₂×#13093957	C₂²	1	2	yes	—
13	C₂×#13093957	C₂²	1	2	yes	—
14	C₂×Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	#15689062	—	2	2	no	—
18	#15689056	—	2	2	no	—
19	#15689022	—	2	2	no	—
20	#15689022	C₂²	1	2	yes	—
21	#15689022	C₂²	1	2	yes	—
22	#15689019	C₂²	1	2	yes	—
23	#15689009	—	2	2	no	—
24	#15688993	C₂²	1	1	yes	characteristic
25	#15688989	C₂²	1	1	yes	characteristic
26	C₄⋊C₄×Dic₄₇	C₂²	1	2	yes	—
27	#15688969	C₂²	1	1	yes	characteristic
28	#15688886	C₂²	1	2	yes	—
29	#15688886	C₂²	1	2	yes	—
30	#15688885	C₂²	1	2	yes	—
31	#15688885	C₂²	1	2	yes	—
32	#15688877	C₂²	1	1	yes	characteristic
33	#15688877	C₂²	1	1	yes	characteristic
34	C₂×#125×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
36	C₄₇×#384	C₂²	1	1	yes	characteristic
37	C₄₇×#340	C₂²	1	2	yes	—
38	C₄₇×#340	C₂²	1	2	yes	—
39	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊Dic₄₇	—	2	2	no	—
42	C₂×C₄⋊Dic₄₇	—	2	2	no	—
43	C₂×C₄⋊Dic₄₇	—	2	2	no	—
44	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
45	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
46	C₂×Dic₄₇⋊C₄	—	2	2	no	—
47	C₂×C₄×Dic₄₇	—	2	2	no	—
48	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
50	#13093957	—	2	4	no	—
51	#13093957	—	2	4	no	—
52	Dic₄₇⋊C₈	—	2	2	no	—
53	Dic₄₇⋊C₈	—	2	2	no	—
54	C₄×C₄₇⋊C₈	—	2	2	no	—
55	C₄₇⋊#95	—	2	2	no	—
56	C₄₇⋊#95	—	2	2	no	—
57	C₄₇⋊#95	—	2	2	no	—
58	C₄₇⋊#95	—	2	2	no	—
59	C₄₇⋊#95	—	2	2	no	—
60	C₄₇⋊#95	D₄	1	2	yes	—
61	C₄₇⋊#95	D₄	1	2	yes	—
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	#125×C₄₇	—	2	2	no	—
65	#125×C₄₇	Q₈	1	4	yes	—
66	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
67	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
68	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
70	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
71	#97×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	—	2	2	no	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	2	2	no	—
77	C₂²×Dic₄₇	—	2	2	no	—
78	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
80	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	—	2	1	no	—
82	C₂×C₈×C₄₇	—	2	1	no	—
83	C₂×C₈×C₄₇	—	2	1	no	—
84	C₂×C₈×C₄₇	—	2	1	no	—
85	C₂×C₈×C₄₇	—	2	1	no	—
86	C₂×C₈×C₄₇	—	2	1	no	—
87	C₄⋊C₄×C₄₇	—	2	8	no	—
88	C₄⋊C₄×C₄₇	—	2	2	no	—
89	C₄⋊C₄×C₄₇	—	2	2	no	—
90	C₄⋊C₄×C₄₇	—	2	2	no	—
91	C₄⋊C₄×C₄₇	—	8	2	no	—
92	C₄⋊C₄×C₄₇	C₂×Q₈	1	4	yes	—
93	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
94	C₄²×C₄₇	#41	1	2	yes	—
95	C₄⋊Dic₄₇	—	2	2	no	—
96	C₄⋊Dic₄₇	—	2	2	no	—
97	C₄⋊Dic₄₇	—	2	2	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	8	2	no	—
100	C₄⋊Dic₄₇	—	8	2	no	—
101	C₄⋊Dic₄₇	—	8	2	no	—
102	Dic₄₇⋊C₄	—	8	2	no	—
103	C₄×Dic₄₇	—	2	1	no	—
104	C₄×Dic₄₇	—	2	1	no	—
105	C₄×Dic₄₇	—	2	1	no	—
106	C₄×Dic₄₇	—	2	1	no	—
107	C₄×Dic₄₇	—	8	2	no	—
108	C₂×C₄₇⋊C₈	—	2	1	no	—
109	C₂×C₄₇⋊C₈	—	2	1	no	—
110	C₂×C₄₇⋊C₈	—	2	1	no	—
111	C₂×C₄₇⋊C₈	—	2	1	no	—
112	C₂×C₄₇⋊C₈	#41	1	2	yes	—
113	C₂×C₄₇⋊C₈	#41	1	2	yes	—
114	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	2	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	C₂²⋊Q₈	1	4	yes	—
124	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
129	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
130	C₂×Dic₄₇	—	2	1	no	—
131	C₂×Dic₄₇	—	2	1	no	—
132	C₂×Dic₄₇	—	2	2	no	—
133	C₂×Dic₄₇	—	2	2	no	—
134	C₂×Dic₄₇	—	2	2	no	—
135	C₂×Dic₄₇	—	2	2	no	—
136	C₂×Dic₄₇	—	2	2	no	—
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₈×C₄₇	—	4	1	no	—
144	C₈×C₄₇	—	4	1	no	—
145	C₄₇⋊C₈	—	2	2	no	—
146	C₄₇⋊C₈	—	2	2	no	—
147	#29186	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
149	C₂²×C₄₇	#532	1	1	yes	characteristic
150	C₂²×C₄₇	#536	1	1	yes	characteristic
151	C₂²×C₄₇	#576	1	2	yes	—
152	C₂²×C₄₇	#578	1	2	yes	—
153	C₄×C₄₇	—	2	4	no	—
154	C₄×C₄₇	—	4	4	no	—
155	C₄×C₄₇	—	8	2	no	—
156	C₄×C₄₇	#552	1	1	yes	characteristic
157	C₄×C₄₇	#552	1	1	yes	characteristic
158	C₄×C₄₇	#557	1	1	yes	characteristic
159	C₄×C₄₇	#557	1	1	yes	characteristic
160	Dic₄₇	—	4	1	no	—
161	Dic₄₇	—	4	1	no	—
162	Dic₄₇	—	4	2	no	—
163	Dic₄₇	—	8	2	no	—
164	Dic₄₇	—	8	2	no	—
165	#2647	—	47	1	no	—
166	#2330	—	47	1	no	—
167	#2087	—	47	2	no	—
168	#2086	—	47	2	no	—
169	#1992	D₄₇	1	1	yes	characteristic
170	#1991	—	47	1	no	—
171	#1921	—	47	2	no	—
172	#1918	—	47	2	no	—
173	#1845	—	47	1	no	—
174	#1836	—	47	1	no	—
175	#1768	—	47	1	no	—
176	C₂×C₄₇	#2565	1	1	yes	characteristic
177	C₂×C₄₇	#3083	1	2	yes	—
178	C₂×C₄₇	#3176	1	1	yes	characteristic
179	C₂×C₄₇	#3177	1	1	yes	characteristic
180	C₂×C₄₇	#3243	1	2	yes	—
181	C₂×#125	C₂×D₄₇	1	1	yes	characteristic
182	C₂×#107	—	47	2	no	—
183	C₂×#107	—	47	2	no	—
184	C₂×C₄⋊C₈	—	47	1	no	—
185	C₂×C₄⋊C₈	—	47	1	no	—
186	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
187	C₂×C₄×C₈	—	47	1	no	—
188	#400	—	94	2	no	—
189	#398	—	94	2	no	—
190	#389	—	47	2	no	—
191	#389	—	47	2	no	—
192	#389	—	47	2	no	—
193	#389	—	94	2	no	—
194	#387	—	94	2	no	—
195	#384	—	47	1	no	—
196	#384	C₂×D₄₇	1	1	yes	characteristic
197	#382	—	47	1	no	—
198	C₄×C₄⋊C₄	—	47	2	no	—
199	#376	—	47	1	no	—
200	#340	—	47	2	no	—
201	#340	—	47	2	no	—
202	#340	—	47	2	no	—
203	#340	—	47	2	no	—
204	#340	C₂×D₄₇	1	2	yes	—
205	#340	C₂×D₄₇	1	2	yes	—
206	#336	—	47	1	no	—
207	#336	—	47	1	no	—
208	C₄₇	#29186	1	1	yes	characteristic, complemented, Sylow
209	C₂²×C₈	—	47	1	no	—
210	C₂²×C₈	—	47	1	no	—
211	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
213	#125	—	2	2	no	—
214	#125	Dic₉₄	1	4	yes	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	47	2	no	—
218	C₂×C₄⋊C₄	—	47	2	no	—
219	C₂×C₄⋊C₄	—	94	2	no	—
220	C₂×C₄⋊C₄	—	94	2	no	—
221	C₂×C₄⋊C₄	—	94	2	no	—
222	C₂×C₄⋊C₄	—	94	2	no	—
223	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
224	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
225	C₂×C₄²	—	47	1	no	—
226	C₂×C₄²	—	47	1	no	—
227	C₂×C₄²	—	94	2	no	—
228	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
229	#107	—	94	4	no	—
230	#107	—	94	4	no	—
231	C₄⋊C₈	—	94	2	no	—
232	C₄⋊C₈	—	94	2	no	—
233	#97	—	2	2	no	—
234	C₄×C₈	—	94	2	no	—
235	#95	—	47	2	no	—
236	#95	—	47	2	no	—
237	#95	—	94	2	no	—
238	#95	—	94	2	no	—
239	#95	—	94	2	no	—
240	#95	—	94	2	no	—
241	#95	—	94	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	47	1	no	—
245	C₂²×C₄	—	47	2	no	—
246	C₂²×C₄	—	47	1	no	—
247	C₂²×C₄	—	94	2	no	—
248	C₂²×C₄	—	94	2	no	—
249	C₂²×C₄	D₄×D₄₇	1	2	yes	—
250	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	47	2	no	—
258	C₂×C₈	—	47	2	no	—
259	C₂×C₈	—	94	1	no	—
260	C₂×C₈	—	94	1	no	—
261	C₂×C₈	—	94	1	no	—
262	C₂×C₈	—	94	1	no	—
263	C₄⋊C₄	—	2	8	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	8	2	no	—
268	C₄⋊C₄	—	94	2	no	—
269	C₄⋊C₄	—	94	2	no	—
270	C₄⋊C₄	—	94	2	no	—
271	C₄⋊C₄	—	94	2	no	—
272	C₄⋊C₄	—	376	2	no	—
273	C₄⋊C₄	—	376	2	no	—
274	C₄⋊C₄	—	376	2	no	—
275	C₄⋊C₄	—	376	2	no	—
276	C₄⋊C₄	C₂×Dic₉₄	1	4	yes	—
277	C₄²	—	94	1	no	—
278	C₄²	—	94	1	no	—
279	C₄²	—	94	1	no	—
280	C₄²	—	94	1	no	—
281	C₄²	—	376	2	no	—
282	C₄²	C₂×Dic₉₄	1	2	yes	—
283	C₄²	#10625352	1	2	yes	—
284	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	94	2	no	—
294	C₂×C₄	—	94	2	no	—
295	C₂×C₄	—	94	1	no	—
296	C₂×C₄	—	94	1	no	—
297	C₂×C₄	—	94	2	no	—
298	C₂×C₄	—	94	2	no	—
299	C₂×C₄	—	94	2	no	—
300	C₂×C₄	—	188	2	no	—
301	C₂×C₄	—	188	2	no	—
302	C₂×C₄	—	188	2	no	—
303	C₂×C₄	—	188	2	no	—
304	C₂×C₄	—	188	2	no	—
305	C₂×C₄	—	188	2	no	—
306	C₂×C₄	#13094001	1	4	yes	—
307	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
308	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
309	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
310	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
311	C₂×C₄	#13094072	1	1	yes	characteristic
312	C₂×C₄	#13094072	1	1	yes	characteristic
313	C₈	—	4	1	no	—
314	C₈	—	4	1	no	—
315	C₈	—	94	2	no	—
316	C₈	—	94	2	no	—
317	C₂²	C₂×#13094001	1	1	yes	characteristic
318	C₂²	#15689672	1	1	yes	characteristic
319	C₂²	#15689698	1	1	yes	characteristic
320	C₂²	#15689927	1	2	yes	—
321	C₂²	#15689939	1	2	yes	—
322	C₄	—	2	4	no	—
323	C₄	—	4	4	no	—
324	C₄	—	8	2	no	—
325	C₄	—	376	2	no	—
326	C₄	—	376	2	no	—
327	C₄	—	188	1	no	—
328	C₄	—	188	1	no	—
329	C₄	—	188	2	no	—
330	C₄	#15689807	1	1	yes	characteristic
331	C₄	#15689807	1	1	yes	characteristic
332	C₄	#15689849	1	1	yes	characteristic
333	C₄	#15689849	1	1	yes	characteristic
334	C₂	#20248582	1	1	yes	characteristic
335	C₂	#20253243	1	2	yes	—
336	C₂	#20254037	1	1	yes	characteristic
337	C₂	#20254046	1	1	yes	characteristic
338	C₂	#20254672	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1992	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#1992	D₄₇
hypocenter	C₄₇	#29186
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1992

Complements

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

N	H ≅ G/N	type
208. C₄₇	#29186	semidirect

Maximal subgroups

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

subgroup	index
#20249368	2
#20246130	2
#20244885	2
#20244880	2
#20244354	2
#20243983	2
#20243962	2
#20243547	2
#20243522	2
#20243248	2
C₄₇×#1992	2
#29186	47