Subgroups — #26124928

Order 12032.

722 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689165	C₂	1	2	yes	maximal
2	C₂²×#13093958	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688891	C₂	1	2	yes	maximal
4	#20243084	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	1	yes	characteristic
7	C₂×#13093983	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	#15689165	—	2	4	no	—
10	#15689165	—	2	4	no	—
11	C₂×#13093958	—	2	4	no	—
12	C₂×#13093958	—	2	2	no	—
13	C₂×#13093958	—	2	2	no	—
14	C₂×#13093956	—	2	2	no	—
15	#15688891	—	2	4	no	—
16	#15688877	—	2	2	no	—
17	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#341	C₄	1	4	yes	complemented
20	C₂³×Dic₄₇	—	2	2	no	—
21	C₂×#10625329	—	2	1	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	Q₈	1	2	yes	—
25	C₂²×C₄₇⋊C₈	—	2	1	no	—
26	#13093983	—	2	2	no	—
27	#13093983	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	D₄	1	4	yes	—
34	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
36	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
37	C₂×C₄×Dic₄₇	—	2	1	no	—
38	#13093958	—	2	8	no	—
39	#13093958	—	2	8	no	—
40	#13093956	—	2	8	no	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂×M₁₆×C₄₇	—	2	2	no	—
44	C₂×M₁₆×C₄₇	—	2	2	no	—
45	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
47	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
48	C₂²×C₈×C₄₇	—	2	2	no	—
49	C₂²×C₈×C₄₇	—	2	2	no	—
50	C₂²×C₈×C₄₇	—	2	2	no	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₄₇	D₄	1	2	yes	—
53	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
54	C₂⁴×C₄₇	#41	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
65	C₂²×Dic₄₇	—	2	1	no	—
66	C₂²×Dic₄₇	—	2	1	no	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	—	4	2	no	—
69	C₂²×Dic₄₇	—	4	2	no	—
70	C₂²×Dic₄₇	—	8	4	no	—
71	M₁₆×C₄₇	—	2	4	no	—
72	M₁₆×C₄₇	—	2	4	no	—
73	M₁₆×C₄₇	—	2	4	no	—
74	M₁₆×C₄₇	C₄⋊C₄	1	8	yes	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	2	4	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
83	C₂²⋊Dic₄₇	—	4	2	no	—
84	C₄⋊Dic₄₇	—	2	2	no	—
85	C₄⋊Dic₄₇	—	2	4	no	—
86	C₄⋊Dic₄₇	—	2	4	no	—
87	C₄⋊Dic₄₇	—	2	2	no	—
88	C₄⋊Dic₄₇	—	4	2	no	—
89	C₄⋊Dic₄₇	—	4	2	no	—
90	C₄×Dic₄₇	—	4	1	no	—
91	C₄×Dic₄₇	—	4	1	no	—
92	#10625329	—	2	4	no	—
93	#10625329	—	4	2	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	4	1	no	—
96	C₂×C₄₇⋊C₈	—	4	1	no	—
97	C₂³×C₄₇	—	2	2	no	—
98	C₂³×C₄₇	—	2	2	no	—
99	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
100	C₂³×C₄₇	#121	1	2	yes	—
101	C₂³×C₄₇	#121	1	1	yes	characteristic
102	C₂³×C₄₇	#125	1	2	yes	—
103	C₂³×C₄₇	#125	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#120	1	2	yes	—
110	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#123	1	2	yes	—
117	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
122	C₂×Dic₄₇	—	4	1	no	—
123	C₂×Dic₄₇	—	4	1	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	8	4	no	—
128	C₂×Dic₄₇	—	8	4	no	—
129	C₂×Dic₄₇	—	8	4	no	—
130	C₈×C₄₇	—	2	4	no	—
131	C₈×C₄₇	—	2	8	no	—
132	C₄₇⋊C₈	—	4	2	no	—
133	#11331	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	4	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	#389	1	2	yes	—
140	C₂²×C₄₇	#389	1	1	yes	characteristic
141	C₂²×C₄₇	#392	1	1	yes	characteristic
142	C₂²×C₄₇	#398	1	2	yes	—
143	C₂²×C₄₇	#398	1	1	yes	characteristic
144	C₂²×C₄₇	#472	1	2	yes	—
145	C₂²×C₄₇	#473	1	2	yes	—
146	C₄×C₄₇	#384	1	1	yes	characteristic
147	C₄×C₄₇	#385	1	1	yes	characteristic
148	C₄×C₄₇	#385	1	2	yes	—
149	C₄×C₄₇	#395	1	1	yes	characteristic
150	C₄×C₄₇	#395	1	2	yes	—
151	C₄×C₄₇	#399	1	1	yes	characteristic
152	Dic₄₇	—	8	1	no	—
153	Dic₄₇	—	8	4	no	—
154	Dic₄₇	—	8	1	no	—
155	C₂×#428	—	47	2	no	—
156	C₂²×#107	—	47	1	no	—
157	C₂×#343	—	47	2	no	—
158	C₂×#341	D₄₇	1	1	yes	characteristic
159	#1729	—	47	1	no	—
160	C₂×C₄₇	—	2	2	no	—
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	#1444	1	1	yes	characteristic
163	C₂×C₄₇	#1952	1	2	yes	—
164	C₂×C₄₇	#2018	1	1	yes	characteristic
165	C₂×C₄₇	#2020	1	1	yes	characteristic
166	C₂×C₄₇	#2089	1	2	yes	—
167	C₂²×M₁₆	—	47	1	no	—
168	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
169	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
170	C₂×#117	—	47	1	no	—
171	C₂²×C₄⋊C₄	—	47	2	no	—
172	#428	—	94	4	no	—
173	#428	—	94	4	no	—
174	C₂×#107	—	94	4	no	—
175	C₂×#107	—	94	2	no	—
176	C₂×#107	—	94	2	no	—
177	C₂×#106	—	94	2	no	—
178	#343	—	94	4	no	—
179	#341	Dic₄₇	1	4	yes	complemented
180	#336	—	94	2	no	—
181	C₄₇	#11331	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₄	—	94	2	no	—
183	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	47	2	no	—
188	C₂×M₁₆	—	94	1	no	—
189	C₂×M₁₆	—	94	2	no	—
190	C₂×M₁₆	—	94	2	no	—
191	C₂×M₁₆	Dic₉₄	1	4	yes	—
192	C₂×M₁₆	D₁₈₈	1	4	yes	—
193	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	94	1	no	—
198	C₂²×C₈	Dic₉₄	1	2	yes	—
199	C₂²×C₈	D₁₈₈	1	2	yes	—
200	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
201	#117	—	94	2	no	—
202	#117	—	94	2	no	—
203	C₂×C₄⋊C₄	—	47	4	no	—
204	C₂×C₄⋊C₄	—	47	1	no	—
205	C₂×C₄⋊C₄	—	47	1	no	—
206	C₂×C₄⋊C₄	—	94	2	no	—
207	C₂×C₄⋊C₄	—	94	2	no	—
208	C₂×C₄⋊C₄	—	94	2	no	—
209	C₂×C₄⋊C₄	—	94	2	no	—
210	C₂×C₄⋊C₄	—	94	2	no	—
211	C₂×C₂²⋊C₄	—	94	1	no	—
212	C₂×C₄²	—	94	1	no	—
213	#107	—	94	8	no	—
214	#107	—	94	8	no	—
215	#106	—	94	8	no	—
216	C₂⁴	#10625354	1	1	yes	characteristic
217	C₂²×C₄	—	94	4	no	—
218	C₂²×C₄	—	94	1	no	—
219	C₂²×C₄	—	94	1	no	—
220	C₂²×C₄	—	376	4	no	—
221	C₂²×C₄	—	188	2	no	—
222	C₂²×C₄	—	188	2	no	—
223	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
224	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
225	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
226	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
227	C₂²×C₄	D₄×D₄₇	1	2	yes	—
228	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	#10625352	1	2	yes	—
230	C₂²×C₄	#10625352	1	1	yes	characteristic
231	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	94	4	no	—
237	M₁₆	—	188	2	no	—
238	M₁₆	C₄⋊Dic₄₇	1	8	yes	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	94	2	no	—
247	C₂×C₈	—	188	1	no	—
248	C₂×C₈	—	188	1	no	—
249	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
250	C₄⋊C₄	—	94	2	no	—
251	C₄⋊C₄	—	94	4	no	—
252	C₄⋊C₄	—	94	4	no	—
253	C₄⋊C₄	—	94	2	no	—
254	C₄⋊C₄	—	188	2	no	—
255	C₄⋊C₄	—	188	2	no	—
256	C₂²⋊C₄	—	188	2	no	—
257	C₄²	—	188	1	no	—
258	C₄²	—	188	1	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	Q₈×Dic₄₇	1	1	yes	characteristic
262	C₂³	#13093999	1	2	yes	—
263	C₂³	#13093999	1	1	yes	characteristic
264	C₂³	#13094015	1	2	yes	—
265	C₂³	#13094015	1	1	yes	characteristic
266	C₂×C₄	—	376	4	no	—
267	C₂×C₄	—	376	4	no	—
268	C₂×C₄	—	376	4	no	—
269	C₂×C₄	—	188	1	no	—
270	C₂×C₄	—	188	1	no	—
271	C₂×C₄	—	188	4	no	—
272	C₂×C₄	—	188	2	no	—
273	C₂×C₄	—	188	2	no	—
274	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
276	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
277	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
278	C₂×C₄	#13093994	1	1	yes	characteristic
279	C₂×C₄	#13093994	1	2	yes	—
280	C₂×C₄	#13094001	1	2	yes	—
281	C₂×C₄	#13094001	1	2	yes	—
282	C₂×C₄	#13094001	1	2	yes	—
283	C₂×C₄	#13094004	1	1	yes	characteristic
284	C₂×C₄	#13094004	1	2	yes	—
285	C₂×C₄	#13094008	1	1	yes	characteristic
286	C₂×C₄	#13094008	1	2	yes	—
287	C₂×C₄	#13094021	1	1	yes	characteristic
288	C₂×C₄	#13094023	1	2	yes	—
289	C₂×C₄	#13094023	1	1	yes	characteristic
290	C₂×C₄	#13094024	1	2	yes	—
291	C₂×C₄	#13094024	1	1	yes	characteristic
292	C₈	—	2	4	no	—
293	C₈	—	2	8	no	—
294	C₈	—	188	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	#15689022	1	2	yes	—
301	C₂²	#15689022	1	1	yes	characteristic
302	C₂²	#15689033	1	1	yes	characteristic
303	C₂²	#15689055	1	2	yes	—
304	C₂²	#15689055	1	1	yes	characteristic
305	C₂²	#15689394	1	2	yes	—
306	C₂²	#15689399	1	2	yes	—
307	C₄	—	376	4	no	—
308	C₄	—	376	1	no	—
309	C₄	—	376	1	no	—
310	C₄	#15688993	1	1	yes	characteristic
311	C₄	#15688998	1	1	yes	characteristic
312	C₄	#15688998	1	2	yes	—
313	C₄	#15689044	1	1	yes	characteristic
314	C₄	#15689044	1	2	yes	—
315	C₄	#15689060	1	1	yes	characteristic
316	C₂	—	2	2	no	—
317	C₂	—	2	2	no	—
318	C₂	#20241653	1	1	yes	characteristic
319	C₂	#20244174	1	2	yes	—
320	C₂	#20244498	1	1	yes	characteristic
321	C₂	#20244510	1	1	yes	characteristic
322	C₂	#20244895	1	2	yes	—

	subgroup	G/H
center	C₂³	#13093999
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#341	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#341	D₄₇
hypocenter	C₄₇	#11331
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
19. C₄₇×#341	C₄	semidirect
179. #341	Dic₄₇	semidirect
181. C₄₇	#11331	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689165	2
C₂²×#13093958	2
C₂×#15688891	2
#20243084	2
C₂×C₄₇×#341	2
#11331	47