Subgroups — #26214431

Order 12032.

930 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689184	C₂	1	2	yes	maximal
2	#20243245	C₂	1	2	yes	maximal, complemented
3	#20243241	C₂	1	2	yes	maximal, complemented
4	#20243154	C₂	1	2	yes	maximal
5	#20243150	C₂	1	2	yes	maximal
6	#20243140	C₂	1	2	yes	maximal, complemented
7	C₄×#13093935	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13093935	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
10	#15689184	C₄	1	16	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
12	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
13	C₂×#13093935	C₂²	1	2	yes	—
14	C₂×#13093935	C₂²	1	2	yes	—
15	C₂×#13093935	C₂²	1	2	yes	—
16	C₂×#13093935	C₂²	1	2	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
19	#15688878	C₂²	1	4	yes	—
20	#15688878	C₂²	1	4	yes	—
21	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₄₇	C₄	1	4	yes	—
25	C₄×C₂²⋊C₄×C₄₇	C₄	1	4	yes	—
26	C₄₇×#376	C₂²	1	1	yes	characteristic
27	C₄₇×#376	C₂²	1	1	yes	characteristic
28	C₄³×C₄₇	C₂²	1	1	yes	characteristic
29	C₄³×C₄₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
32	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	complemented
33	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	—
34	#13093935	—	2	4	no	—
35	#13093935	—	2	4	no	—
36	#13093935	—	2	4	no	—
37	#13093935	C₂×C₄	1	8	yes	complemented
38	C₄×C₄₇⋊C₈	—	2	4	no	—
39	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
40	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	#117×C₄₇	—	2	4	no	—
43	#117×C₄₇	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₇	—	2	4	no	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	#95×C₄₇	C₂×C₄	1	4	yes	—
59	#95×C₄₇	C₂×C₄	1	4	yes	—
60	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	4	4	no	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₄⋊C₄×C₄₇	C₄²	1	8	yes	—
76	C₂²⋊C₄×C₄₇	—	2	4	no	—
77	C₂²⋊C₄×C₄₇	C₄²	1	8	yes	—
78	C₄²×C₄₇	—	2	4	no	—
79	C₄²×C₄₇	—	2	2	no	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	4	4	no	—
82	C₄²×C₄₇	—	4	4	no	—
83	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
84	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
85	C₄²×C₄₇	#41	1	4	yes	—
86	C₄²×C₄₇	#41	1	4	yes	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
92	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	2	4	no	—
96	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
97	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	8	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	4	8	no	—
110	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
111	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
112	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₄₇	#117	1	4	yes	—
114	C₂×C₄×C₄₇	#117	1	4	yes	—
115	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
118	C₄₇⋊C₈	—	2	8	no	—
119	C₄₇⋊C₈	—	2	8	no	—
120	#20465	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	4	no	—
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	—	4	2	no	—
124	C₂²×C₄₇	—	4	2	no	—
125	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
126	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
127	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
128	C₂²×C₄₇	#568	1	4	yes	—
129	C₄×C₄₇	—	2	4	no	—
130	C₄×C₄₇	—	2	4	no	—
131	C₄×C₄₇	—	2	4	no	—
132	C₄×C₄₇	—	2	4	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	4	2	no	—
135	C₄×C₄₇	—	4	2	no	—
136	C₄×C₄₇	C₄×#41	1	4	yes	—
137	C₂×#433	—	47	2	no	—
138	C₄×#117	D₄₇	1	1	yes	characteristic
139	#1768	—	47	2	no	—
140	#1767	—	47	2	no	—
141	#1750	—	47	2	no	—
142	#1749	—	47	2	no	—
143	#1746	—	47	2	no	—
144	C₄×#97	—	47	1	no	—
145	C₄×#97	—	47	1	no	—
146	C₂×C₄₇	—	4	2	no	—
147	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
148	C₂×C₄₇	#2875	1	2	yes	—
149	C₂×C₄₇	#2921	1	2	yes	—
150	C₂×C₄₇	#2923	1	2	yes	—
151	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
152	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
153	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
154	#433	—	47	16	no	—
155	C₂×C₄⋊C₈	—	47	4	no	—
156	C₂×C₂²⋊C₈	—	47	4	no	—
157	C₂×#97	—	47	2	no	—
158	C₂×#97	—	47	2	no	—
159	C₂×#97	—	47	2	no	—
160	C₂×#97	—	47	2	no	—
161	C₂×C₄×C₈	—	47	2	no	—
162	C₂×C₄×C₈	—	47	2	no	—
163	C₄×C₄⋊C₄	Dic₄₇	1	4	yes	—
164	C₄×C₂²⋊C₄	Dic₄₇	1	4	yes	—
165	#376	C₂×D₄₇	1	1	yes	characteristic
166	#376	C₂×D₄₇	1	1	yes	characteristic
167	C₄³	C₂×D₄₇	1	1	yes	characteristic
168	C₄³	C₂×D₄₇	1	1	yes	characteristic
169	#336	—	47	4	no	—
170	#336	—	47	4	no	—
171	C₄₇	#20465	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
173	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
174	C₂²×C₈	—	47	4	no	—
175	C₂²×C₈	—	47	4	no	—
176	#117	—	2	4	no	—
177	#117	C₄×D₄₇	1	8	yes	—
178	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
180	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
181	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
182	C₂×C₄²	—	2	4	no	—
183	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
184	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
185	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
186	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
187	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
188	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	C₄⋊C₈	—	47	16	no	—
193	C₂²⋊C₈	—	47	16	no	—
194	#97	—	47	8	no	—
195	#97	—	94	4	no	—
196	#97	—	94	4	no	—
197	#97	—	94	4	no	—
198	C₄×C₈	—	47	8	no	—
199	C₄×C₈	—	94	4	no	—
200	#95	C₂×Dic₄₇	1	4	yes	—
201	#95	C₂×Dic₄₇	1	4	yes	—
202	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
208	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
209	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
210	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
211	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
212	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
213	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
214	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
215	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
216	C₂×C₈	—	47	8	no	—
217	C₂×C₈	—	47	8	no	—
218	C₂×C₈	—	94	4	no	—
219	C₂×C₈	—	94	4	no	—
220	C₂×C₈	—	94	4	no	—
221	C₂×C₈	—	94	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	C₄×Dic₄₇	1	8	yes	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₂²⋊C₄	C₄×Dic₄₇	1	8	yes	—
226	C₄²	—	2	4	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	4	4	no	—
230	C₄²	—	4	4	no	—
231	C₄²	C₂×C₄×D₄₇	1	4	yes	—
232	C₄²	C₂×C₄×D₄₇	1	4	yes	—
233	C₄²	#10625350	1	4	yes	—
234	C₄²	#10625350	1	4	yes	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	4	no	—
238	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
239	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
253	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
254	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
255	C₂×C₄	#13093983	1	4	yes	—
256	C₂×C₄	#13093983	1	4	yes	—
257	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
258	C₂×C₄	C₂×#10625350	1	2	yes	—
259	C₂×C₄	C₂×#10625350	1	2	yes	—
260	C₈	—	94	8	no	—
261	C₈	—	94	8	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
267	C₂²	C₂×#13093983	1	1	yes	characteristic
268	C₂²	C₂×#13093983	1	1	yes	characteristic
269	C₂²	#15689890	1	4	yes	—
270	C₄	—	2	4	no	—
271	C₄	—	2	4	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	2	2	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	2	no	—
277	C₄	C₄×#10625350	1	4	yes	—
278	C₂	—	4	2	no	—
279	C₂	C₄×#13093983	1	1	yes	characteristic
280	C₂	#20251816	1	2	yes	—
281	C₂	#20252026	1	2	yes	—
282	C₂	#20252039	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#20465
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #20243245	C₂	semidirect
3. #20243241	C₂	semidirect
6. #20243140	C₂	semidirect
7. C₄×#13093935	C₂	semidirect
8. C₄×#13093935	C₂	semidirect
10. #15689184	C₄	semidirect
32. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
37. #13093935	C₂×C₄	semidirect
39. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
171. C₄₇	#20465	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689184	2
#20243245	2
#20243241	2
#20243154	2
#20243150	2
#20243140	2
C₄×#13093935	2
C₄×#117×C₄₇	2
#20465	47