Subgroups — #26216439

Order 12032.

914 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689150	C₂	1	2	yes	maximal
2	#20243508	C₂	1	4	yes	maximal, complemented
3	#20243505	C₂	1	4	yes	maximal, complemented
4	#20243085	C₂	1	2	yes	maximal
5	#20243082	C₂	1	1	yes	maximal, characteristic
6	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
7	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
10	#15689150	—	2	8	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
12	#15688878	C₂²	1	2	yes	—
13	#15688878	C₂²	1	2	yes	—
14	#15688878	C₂²	1	8	yes	—
15	#15688878	C₂²	1	2	yes	—
16	#15688878	C₂²	1	2	yes	—
17	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
22	C₄×C₂²⋊C₄×C₄₇	C₄	1	4	yes	—
23	C₄₇×#376	C₂²	1	2	yes	—
24	C₄³×C₄₇	C₂²	1	2	yes	—
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	—	2	4	no	—
27	C₂×#10625329	—	2	4	no	—
28	C₂²×C₄₇⋊C₈	—	2	4	no	—
29	C₂²×C₄₇⋊C₈	—	2	4	no	—
30	C₂²×C₄₇⋊C₈	—	2	4	no	—
31	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
33	C₄⋊(C₄₇⋊C₈)	—	2	16	no	—
34	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	#117×C₄₇	—	2	4	no	—
37	#117×C₄₇	D₄	1	4	yes	—
38	#117×C₄₇	Q₈	1	4	yes	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	—	2	4	no	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
52	#95×C₄₇	C₂×C₄	1	4	yes	—
53	#95×C₄₇	C₂×C₄	1	4	yes	—
54	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₄₇	#41	1	4	yes	—
67	C₂²×C₄×C₄₇	#41	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₄₇	—	2	2	no	—
70	C₄⋊C₄×C₄₇	—	2	2	no	—
71	C₄⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
72	C₂²⋊C₄×C₄₇	—	2	4	no	—
73	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
74	C₄²×C₄₇	—	2	4	no	—
75	C₄²×C₄₇	—	2	8	no	—
76	C₄²×C₄₇	—	2	2	no	—
77	C₄²×C₄₇	—	2	2	no	—
78	C₄²×C₄₇	—	2	4	no	—
79	C₄²×C₄₇	—	4	4	no	—
80	C₄²×C₄₇	C₂×D₄	1	4	yes	—
81	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
82	#10625329	—	4	8	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	4	8	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₄₇	#117	1	2	yes	—
94	C₂³×C₄₇	#117	1	2	yes	—
95	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	4	8	no	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₇	#117	1	2	yes	—
111	C₂×C₄×C₄₇	#117	1	2	yes	—
112	C₂×C₄×C₄₇	#124	1	8	yes	—
113	C₂×C₄×C₄₇	#125	1	8	yes	—
114	C₂×C₄×C₄₇	#131	1	4	yes	—
115	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
119	C₄₇⋊C₈	—	4	4	no	—
120	C₄₇⋊C₈	—	4	4	no	—
121	#20754	—	47	1	no	maximal, Sylow
122	C₂²×C₄₇	—	2	4	no	—
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	2	4	no	—
126	C₂²×C₄₇	#382	1	4	yes	—
127	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
129	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×#131	1	2	yes	—
131	C₂²×C₄₇	#568	1	2	yes	—
132	C₄×C₄₇	—	2	8	no	—
133	C₄×C₄₇	—	2	4	no	—
134	C₄×C₄₇	—	4	2	no	—
135	C₄×C₄₇	—	4	2	no	—
136	C₄×C₄₇	—	4	2	no	—
137	C₄×C₄₇	#382	1	4	yes	—
138	C₄×C₄₇	C₂×#124	1	2	yes	—
139	C₄×C₄₇	C₂×#125	1	2	yes	—
140	C₂×#424	—	47	2	no	—
141	C₄×#117	D₄₇	1	1	yes	characteristic
142	#1832	—	47	4	no	—
143	#1831	—	47	4	no	—
144	#1729	—	47	2	no	—
145	#1728	—	47	1	no	—
146	C₂×#336	—	47	1	no	—
147	C₂×C₄₇	—	2	4	no	—
148	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
149	C₂×C₄₇	#2906	1	2	yes	—
150	C₂×C₄₇	#2922	1	1	yes	characteristic
151	C₂×C₄₇	#2923	1	1	yes	characteristic
152	C₂×C₄₇	#2924	1	2	yes	—
153	C₂²×M₁₆	—	47	2	no	—
154	C₂³×C₈	—	47	2	no	—
155	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
156	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
157	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
158	#424	—	94	8	no	—
159	C₂×C₄⋊C₈	—	47	8	no	—
160	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
161	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
162	C₄×C₂²⋊C₄	Dic₄₇	1	4	yes	—
163	#376	C₂×D₄₇	1	2	yes	—
164	C₄³	C₂×D₄₇	1	2	yes	—
165	#336	—	47	2	no	—
166	#336	—	47	2	no	—
167	#336	—	47	8	no	—
168	#336	—	47	2	no	—
169	#336	—	47	2	no	—
170	C₄₇	#20754	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
172	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
173	C₂×M₁₆	—	94	4	no	—
174	C₂×M₁₆	—	94	4	no	—
175	C₂×M₁₆	—	94	4	no	—
176	C₂²×C₈	—	47	4	no	—
177	C₂²×C₈	—	47	4	no	—
178	C₂²×C₈	—	94	4	no	—
179	C₂²×C₈	—	94	4	no	—
180	C₂²×C₈	—	94	4	no	—
181	#117	—	2	4	no	—
182	#117	Dic₉₄	1	4	yes	—
183	#117	D₁₈₈	1	4	yes	—
184	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
186	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
188	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
189	C₂×C₄²	—	2	4	no	—
190	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
191	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
192	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
193	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
194	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
195	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
196	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
197	C₄⋊C₈	—	94	16	no	—
198	#95	C₂×Dic₄₇	1	4	yes	—
199	#95	C₂×Dic₄₇	1	4	yes	—
200	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
205	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
206	C₂²×C₄	#10625350	1	4	yes	—
207	C₂²×C₄	#10625350	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	4	yes	—
212	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
213	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
214	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
215	M₁₆	—	188	8	no	—
216	C₂×C₈	—	94	4	no	—
217	C₂×C₈	—	94	4	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	C₂×C₈	—	188	8	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
228	C₄²	—	2	4	no	—
229	C₄²	—	2	8	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	4	4	no	—
234	C₄²	C₂×Dic₉₄	1	4	yes	—
235	C₄²	C₂×D₁₈₈	1	4	yes	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
239	C₂³	#13093983	1	2	yes	—
240	C₂³	#13093983	1	2	yes	—
241	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
255	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
256	C₂×C₄	#13093983	1	2	yes	—
257	C₂×C₄	#13093983	1	2	yes	—
258	C₂×C₄	#13094012	1	8	yes	—
259	C₂×C₄	#13094016	1	8	yes	—
260	C₂×C₄	#13094031	1	4	yes	—
261	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
262	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
263	C₂×C₄	C₂×#10625350	1	2	yes	—
264	C₂×C₄	C₂×#10625350	1	2	yes	—
265	C₈	—	188	4	no	—
266	C₈	—	188	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	#15688989	1	4	yes	—
272	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
273	C₂²	C₂×#13093983	1	1	yes	characteristic
274	C₂²	C₂×#13093983	1	1	yes	characteristic
275	C₂²	C₂×#13094031	1	2	yes	—
276	C₂²	#15689890	1	2	yes	—
277	C₄	—	2	8	no	—
278	C₄	—	2	4	no	—
279	C₄	—	4	2	no	—
280	C₄	—	4	2	no	—
281	C₄	—	4	2	no	—
282	C₄	#15688989	1	4	yes	—
283	C₄	C₂×#13094012	1	2	yes	—
284	C₄	C₂×#13094016	1	2	yes	—
285	C₂	—	2	4	no	—
286	C₂	C₂×#15688989	1	1	yes	characteristic
287	C₂	#20251956	1	2	yes	—
288	C₂	#20252032	1	1	yes	characteristic
289	C₂	#20252039	1	1	yes	characteristic
290	C₂	#20252046	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄×#117×C₄₇	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#20754
LCS	G ▷ C₂³×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. #20243508	C₂	semidirect
3. #20243505	C₂	semidirect
170. C₄₇	#20754	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689150	2
#20243508	2
#20243505	2
#20243085	2
#20243082	2
C₂×#15688878	2
C₄×#117×C₄₇	2
#20754	47