Subgroups — #26124960

Order 12032.

686 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251978	C₂	1	1	yes	maximal, characteristic
2	#20251977	C₂	1	2	yes	maximal
3	C₂×C₄₇⋊#338	C₂	1	1	yes	maximal, characteristic
4	#20243084	C₂	1	2	yes	maximal
5	C₄₇×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	2	yes	—
8	#15689166	—	2	4	no	—
9	#15689165	—	2	4	no	—
10	#15689163	—	2	4	no	—
11	#15689161	—	2	2	no	—
12	#15689159	—	2	4	no	—
13	C₄₇⋊#338	C₄	1	4	yes	complemented
14	#15688877	—	2	4	no	—
15	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#336	—	2	2	no	—
19	C₂×#10625329	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	C₂×C₄	1	4	yes	—
24	C₂×#10625329	Q₈	1	4	yes	—
25	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
26	#13093983	—	2	2	no	—
27	#13093983	—	2	2	no	—
28	#13093983	—	2	2	no	—
29	#13093983	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	D₄	1	4	yes	—
31	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄×Dic₄₇	—	2	2	no	—
33	#13093958	—	2	4	no	—
34	#13093957	—	2	8	no	—
35	#13093956	—	2	4	no	—
36	#13093955	—	2	8	no	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₄₇	—	2	2	no	—
39	C₂×M₁₆×C₄₇	—	2	2	no	—
40	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₄₇	Q₈	1	2	yes	—
42	C₂²×C₈×C₄₇	—	2	1	no	—
43	C₂²×C₈×C₄₇	—	2	2	no	—
44	C₂²×C₈×C₄₇	—	2	1	no	—
45	C₂²×C₈×C₄₇	—	2	2	no	—
46	C₂²×C₈×C₄₇	D₄	1	2	yes	—
47	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
48	#117×C₄₇	—	2	2	no	—
49	#117×C₄₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
52	C₂×C₄²×C₄₇	—	2	1	no	—
53	C₂⁴×C₄₇	#41	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
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	1	yes	characteristic
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×Dic₄₇	—	2	4	no	—
66	M₁₆×C₄₇	—	2	4	no	—
67	M₁₆×C₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	—	2	2	no	—
72	C₂×C₈×C₄₇	—	4	1	no	—
73	C₂×C₈×C₄₇	—	4	2	no	—
74	C₂×C₈×C₄₇	—	4	1	no	—
75	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
76	C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₂²⋊C₄×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	—	4	1	no	—
79	C₄²×C₄₇	—	4	1	no	—
80	C₂²⋊Dic₄₇	—	4	2	no	—
81	C₂²⋊Dic₄₇	—	4	2	no	—
82	C₄⋊Dic₄₇	—	2	4	no	—
83	C₄⋊Dic₄₇	—	2	4	no	—
84	C₄×Dic₄₇	—	4	2	no	—
85	C₄×Dic₄₇	—	4	2	no	—
86	#10625329	—	2	4	no	—
87	#10625329	—	2	4	no	—
88	#10625329	—	2	4	no	—
89	#10625329	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
93	C₂³×C₄₇	—	4	2	no	—
94	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
95	C₂³×C₄₇	#121	1	2	yes	—
96	C₂³×C₄₇	#121	1	1	yes	characteristic
97	C₂³×C₄₇	#125	1	2	yes	—
98	C₂³×C₄₇	#125	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	4	4	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#120	1	2	yes	—
107	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#123	1	2	yes	—
114	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
118	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
119	C₂×Dic₄₇	—	4	8	no	—
120	C₂×Dic₄₇	—	4	4	no	—
121	C₈×C₄₇	—	2	4	no	—
122	C₈×C₄₇	—	4	2	no	—
123	C₄₇⋊C₈	—	2	8	no	—
124	#11337	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	4	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	4	1	no	—
128	C₂²×C₄₇	—	4	2	no	—
129	C₂²×C₄₇	—	4	1	no	—
130	C₂²×C₄₇	#389	1	2	yes	—
131	C₂²×C₄₇	#389	1	1	yes	characteristic
132	C₂²×C₄₇	#392	1	1	yes	characteristic
133	C₂²×C₄₇	#398	1	1	yes	characteristic
134	C₂²×C₄₇	#398	1	2	yes	—
135	C₄×C₄₇	—	8	2	no	—
136	C₄×C₄₇	#384	1	1	yes	characteristic
137	C₄×C₄₇	#385	1	1	yes	characteristic
138	C₄×C₄₇	#385	1	2	yes	—
139	C₄×C₄₇	#395	1	1	yes	characteristic
140	C₄×C₄₇	#395	1	2	yes	—
141	C₄×C₄₇	#399	1	1	yes	characteristic
142	Dic₄₇	—	8	4	no	—
143	#2911	—	47	1	no	—
144	#2911	—	47	2	no	—
145	C₂×#338	—	47	1	no	—
146	#1729	—	47	2	no	—
147	#1729	D₄₇	1	1	yes	characteristic
148	C₂×C₄₇	—	2	2	no	—
149	C₂×C₄₇	—	2	1	no	—
150	C₂×C₄₇	—	2	1	no	—
151	C₂×C₄₇	—	2	2	no	—
152	C₂×C₄₇	#1444	1	1	yes	characteristic
153	C₂×C₄₇	#2017	1	2	yes	—
154	C₂²×M₁₆	—	47	2	no	—
155	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
156	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
157	C₂×#117	—	47	2	no	—
158	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
159	#428	—	94	4	no	—
160	#428	—	94	4	no	—
161	#428	—	94	4	no	—
162	#427	—	94	4	no	—
163	#427	—	94	2	no	—
164	#338	—	47	4	no	—
165	#336	—	2	2	no	—
166	#336	—	94	4	no	—
167	C₄₇	#11337	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	47	4	no	—
172	C₂×M₁₆	—	47	4	no	—
173	C₂×M₁₆	—	94	2	no	—
174	C₂×M₁₆	—	94	2	no	—
175	C₂×M₁₆	—	94	2	no	—
176	C₂×M₁₆	—	94	2	no	—
177	C₂×M₁₆	Dic₉₄	1	2	yes	—
178	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	47	4	no	—
184	C₂²×C₈	Dic₉₄	1	2	yes	—
185	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	—	94	2	no	—
189	#117	—	94	2	no	—
190	#117	—	94	2	no	—
191	#117	—	94	2	no	—
192	C₂×C₄⋊C₄	—	47	4	no	—
193	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
194	C₂×C₂²⋊C₄	—	2	1	no	—
195	C₂×C₂²⋊C₄	—	94	2	no	—
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	—	94	2	no	—
198	#107	—	94	8	no	—
199	#107	—	94	4	no	—
200	#106	—	94	8	no	—
201	#106	—	94	4	no	—
202	C₂⁴	#10625350	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	94	4	no	—
205	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
206	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
209	C₂²×C₄	#10625350	1	1	yes	characteristic
210	C₂²×C₄	D₄×D₄₇	1	2	yes	—
211	C₂²×C₄	D₄×D₄₇	1	2	yes	—
212	C₂²×C₄	#10625352	1	2	yes	—
213	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	94	4	no	—
218	M₁₆	—	94	4	no	—
219	M₁₆	—	94	4	no	—
220	M₁₆	—	94	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	47	8	no	—
229	C₂×C₈	—	94	4	no	—
230	C₂×C₈	—	94	4	no	—
231	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	94	4	no	—
234	C₄⋊C₄	—	94	4	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	188	2	no	—
237	C₂²⋊C₄	—	188	2	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	188	2	no	—
241	C₄²	—	188	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	C₄×Dic₉₄	1	1	yes	characteristic
244	C₂³	#13093997	1	2	yes	—
245	C₂³	#13093998	1	1	yes	characteristic
246	C₂³	#13094014	1	2	yes	—
247	C₂³	#13094016	1	1	yes	characteristic
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	188	8	no	—
251	C₂×C₄	—	188	4	no	—
252	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
253	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
254	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
255	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
256	C₂×C₄	#13093992	1	1	yes	characteristic
257	C₂×C₄	#13093994	1	2	yes	—
258	C₂×C₄	#13094001	1	2	yes	—
259	C₂×C₄	#13094001	1	2	yes	—
260	C₂×C₄	#13094002	1	2	yes	—
261	C₂×C₄	#13094003	1	1	yes	characteristic
262	C₂×C₄	#13094004	1	2	yes	—
263	C₂×C₄	#13094005	1	1	yes	characteristic
264	C₂×C₄	#13094006	1	2	yes	—
265	C₂×C₄	#13094020	1	1	yes	characteristic
266	C₂×C₄	#13094022	1	1	yes	characteristic
267	C₂×C₄	#13094023	1	2	yes	—
268	C₂×C₄	#13094023	1	2	yes	—
269	C₂×C₄	#13094024	1	1	yes	characteristic
270	C₈	—	2	4	no	—
271	C₈	—	4	2	no	—
272	C₈	—	94	8	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	#15689020	1	2	yes	—
279	C₂²	#15689021	1	1	yes	characteristic
280	C₂²	#15689032	1	1	yes	characteristic
281	C₂²	#15689053	1	1	yes	characteristic
282	C₂²	#15689057	1	2	yes	—
283	C₄	—	8	2	no	—
284	C₄	—	376	4	no	—
285	C₄	#15688992	1	1	yes	characteristic
286	C₄	#15688995	1	1	yes	characteristic
287	C₄	#15688997	1	2	yes	—
288	C₄	#15689043	1	1	yes	characteristic
289	C₄	#15689044	1	2	yes	—
290	C₄	#15689059	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	#20241652	1	1	yes	characteristic
296	C₂	#20244492	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13094020
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1729	C₂
socle	C₂²×C₄₇	#392
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1729	D₄₇
hypocenter	C₄₇	#11337
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
13. C₄₇⋊#338	C₄	semidirect
167. C₄₇	#11337	semidirect

Maximal subgroups

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

subgroup	index
#20251978	2
#20251977	2
C₂×C₄₇⋊#338	2
#20243084	2
C₄₇×#1729	2
#11337	47