Subgroups — #26221093

Order 12032.

938 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689180	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
3	#20244112	C₂	1	4	yes	maximal, complemented
4	#20243568	C₂	1	4	yes	maximal
5	#20243241	C₂	1	4	yes	maximal, complemented
6	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#15689180	—	2	4	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×#13093935	—	2	4	no	—
13	C₂×#13093935	C₂²	1	2	yes	—
14	C₂×#13093935	C₂²	1	2	yes	—
15	C₂×#13093935	C₂²	1	2	yes	—
16	#15688878	C₂²	1	8	yes	—
17	C₂×#117×C₄₇	C₄	1	2	yes	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₄×C₂²⋊C₄×C₄₇	C₄	1	1	yes	characteristic
22	C₄×C₂²⋊C₄×C₄₇	C₄	1	1	yes	characteristic
23	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
24	C₄₇×#376	C₄	1	2	yes	—
25	C₄³×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄₇⋊C₈	—	2	4	no	—
27	C₂²×C₄₇⋊C₈	—	2	8	no	—
28	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
30	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	16	no	—
32	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
33	#13093935	—	2	4	no	—
34	#13093935	—	2	4	no	—
35	#13093935	—	2	4	no	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	#117×C₄₇	—	2	8	no	—
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	4	yes	—
49	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	#95×C₄₇	C₂×C₄	1	2	yes	—
54	#95×C₄₇	C₂×C₄	1	2	yes	—
55	#95×C₄₇	C₂×C₄	1	4	yes	—
56	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₄₇	—	2	8	no	—
73	C₂²⋊C₄×C₄₇	—	2	4	no	—
74	C₂²⋊C₄×C₄₇	M₁₆	1	8	yes	—
75	C₄²×C₄₇	—	2	8	no	—
76	C₄²×C₄₇	—	2	8	no	—
77	C₄²×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	—	4	2	no	—
79	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
80	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
81	C₄²×C₄₇	C₂×D₄	1	2	yes	—
82	C₄²×C₄₇	C₂×D₄	1	2	yes	—
83	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
84	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
85	C₂×C₄₇⋊C₈	—	2	16	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	—	2	8	no	—
90	C₂×C₄₇⋊C₈	#41	1	8	yes	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
95	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	8	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
116	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
119	C₄₇⋊C₈	—	2	8	no	—
120	C₄₇⋊C₈	—	4	4	no	—
121	#21355	—	47	1	no	maximal, Sylow
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	—	2	8	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	#423	1	4	yes	—
126	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₄₇	C₄×#41	1	2	yes	—
128	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
129	C₂²×C₄₇	#568	1	2	yes	—
130	C₄×C₄₇	—	2	2	no	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	2	8	no	—
135	C₄×C₄₇	—	2	2	no	—
136	C₄×C₄₇	—	4	4	no	—
137	C₄×C₄₇	#525	1	2	yes	—
138	C₄×C₄₇	#533	1	2	yes	—
139	C₂×#432	—	47	1	no	—
140	C₂²×#97	—	47	1	no	—
141	C₄×#117	D₄₇	1	1	yes	characteristic
142	#1940	—	47	4	no	—
143	#1850	—	47	4	no	—
144	#1767	—	47	4	no	—
145	C₂×C₄₇	—	2	4	no	—
146	C₂×C₄₇	#2296	1	1	yes	characteristic
147	C₂×C₄₇	C₂×#423	1	1	yes	characteristic
148	C₂×C₄₇	#2905	1	1	yes	characteristic
149	C₂×C₄₇	#2970	1	4	yes	—
150	C₂³×C₈	—	47	2	no	—
151	C₂×#117	Dic₄₇	1	2	yes	—
152	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
153	#432	—	94	4	no	—
154	C₂×C₄⋊C₈	—	47	8	no	—
155	C₂×C₄⋊C₈	—	47	2	no	—
156	C₂×C₂²⋊C₈	—	47	2	no	—
157	C₂×#97	—	47	2	no	—
158	C₂×#97	—	47	2	no	—
159	C₂×#97	—	47	2	no	—
160	C₂×#97	—	94	4	no	—
161	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
162	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
163	C₄×C₂²⋊C₄	Dic₄₇	1	1	yes	characteristic
164	C₄×C₂²⋊C₄	Dic₄₇	1	1	yes	characteristic
165	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
166	#376	Dic₄₇	1	2	yes	—
167	C₄³	C₂×D₄₇	1	2	yes	—
168	#336	—	47	8	no	—
169	C₄₇	#21355	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
171	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
172	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
173	C₂²×C₈	—	47	4	no	—
174	C₂²×C₈	—	47	4	no	—
175	C₂²×C₈	—	94	8	no	—
176	C₂²×C₈	—	94	4	no	—
177	#117	—	2	8	no	—
178	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
179	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
180	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
181	C₂×C₄²	Dic₉₄	1	4	yes	—
182	C₂×C₄²	D₁₈₈	1	4	yes	—
183	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
184	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
185	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
186	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
187	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
188	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
189	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
191	C₄⋊C₈	—	94	16	no	—
192	C₄⋊C₈	—	94	4	no	—
193	C₂²⋊C₈	—	94	4	no	—
194	#97	—	94	4	no	—
195	#97	—	94	4	no	—
196	#97	—	94	4	no	—
197	#95	C₂×Dic₄₇	1	2	yes	—
198	#95	C₂×Dic₄₇	1	2	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	C₄⋊Dic₄₇	1	4	yes	—
205	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
206	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
207	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
208	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
209	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
210	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
211	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
213	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
214	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
215	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
216	C₂×C₈	—	47	8	no	—
217	C₂×C₈	—	94	16	no	—
218	C₂×C₈	—	94	4	no	—
219	C₂×C₈	—	94	4	no	—
220	C₂×C₈	—	94	4	no	—
221	C₂×C₈	—	94	8	no	—
222	C₄⋊C₄	—	2	8	no	—
223	C₂²⋊C₄	—	2	4	no	—
224	C₂²⋊C₄	#10625329	1	8	yes	—
225	C₄²	—	2	8	no	—
226	C₄²	—	2	8	no	—
227	C₄²	—	4	2	no	—
228	C₄²	—	4	2	no	—
229	C₄²	C₄⋊Dic₄₇	1	4	yes	—
230	C₄²	C₄⋊Dic₄₇	1	4	yes	—
231	C₄²	C₂×Dic₉₄	1	2	yes	—
232	C₄²	C₂×Dic₉₄	1	2	yes	—
233	C₄²	C₂×D₁₈₈	1	2	yes	—
234	C₄²	C₂×D₁₈₈	1	2	yes	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
238	C₂³	C₂×#10625329	1	4	yes	—
239	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
255	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
256	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
257	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
259	C₂×C₄	C₂×#10625329	1	8	yes	—
260	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
261	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
262	C₂×C₄	#41×D₄₇	1	4	yes	—
263	C₈	—	94	8	no	—
264	C₈	—	188	4	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	8	no	—
268	C₂²	#15689147	1	4	yes	—
269	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
270	C₂²	#41×Dic₄₇	1	2	yes	—
271	C₂²	C₂²×#10625329	1	2	yes	—
272	C₂²	#15689890	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	8	no	—
279	C₄	—	4	4	no	—
280	C₄	#15689635	1	2	yes	—
281	C₄	#15689676	1	2	yes	—
282	C₂	—	2	4	no	—
283	C₂	#20245872	1	1	yes	characteristic
284	C₂	C₂×#15689147	1	1	yes	characteristic
285	C₂	#20251952	1	1	yes	characteristic
286	C₂	#20252402	1	4	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	#21355
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
3. #20244112	C₂	semidirect
5. #20243241	C₂	semidirect
169. C₄₇	#21355	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
C₂²×#13093935	2
#20244112	2
#20243568	2
#20243241	2
C₄×#117×C₄₇	2
#21355	47