Subgroups — #26105822

Order 12032.

614 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	#20245786	C₂	1	1	yes	maximal, characteristic
3	#20241228	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
5	#15689176	—	2	2	no	—
6	#15689176	C₄	1	4	yes	complemented
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
9	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
10	#15688991	—	2	1	no	—
11	#15688989	C₄	1	2	yes	—
12	#15688986	—	2	1	no	—
13	#15688983	C₄	1	2	yes	—
14	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
15	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
16	#15688969	—	2	1	no	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#431	—	2	2	no	—
19	C₄₇×#431	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
21	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
25	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
29	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
30	C₄×C₄₇⋊C₈	—	2	4	no	—
31	C₄₇⋊#95	—	2	2	no	—
32	C₄₇⋊#95	—	2	2	no	—
33	C₄₇⋊#95	—	2	2	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	2	yes	—
45	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₄₇	—	2	4	no	—
48	C₂²⋊C₈×C₄₇	—	2	4	no	—
49	C₄×C₈×C₄₇	—	2	2	no	—
50	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
51	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	—	2	4	no	—
69	C₂×C₈×C₄₇	—	4	4	no	—
70	C₄²×C₄₇	C₄²	1	2	yes	—
71	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
74	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
75	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
76	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
80	C₂²⋊Dic₄₇	—	4	4	no	—
81	C₄⋊Dic₄₇	—	2	8	no	—
82	C₄×Dic₄₇	—	4	4	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	4	4	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	—	2	2	no	—
88	C₂³×C₄₇	#95	1	2	yes	—
89	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	4	2	no	—
99	C₂×C₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	#95	1	2	yes	—
101	C₂×C₄×C₄₇	#95	1	2	yes	—
102	C₂×C₄×C₄₇	#95	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#131	1	4	yes	—
110	C₂×C₄×C₄₇	#135	1	4	yes	—
111	C₂×Dic₄₇	—	4	4	no	—
112	C₂×Dic₄₇	—	4	4	no	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₈×C₄₇	—	4	4	no	—
115	C₄₇⋊C₈	—	4	4	no	—
116	#7894	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	4	4	no	—
121	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
122	C₂²×C₄₇	#408	1	2	yes	—
123	C₂²×C₄₇	#416	1	2	yes	—
124	C₂²×C₄₇	#421	1	2	yes	—
125	C₄×C₄₇	—	2	2	no	—
126	C₄×C₄₇	—	2	1	no	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	—	2	1	no	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	#405	1	2	yes	—
131	C₄×C₄₇	#416	1	2	yes	—
132	C₄×C₄₇	#424	1	2	yes	—
133	C₄×C₄₇	#427	1	2	yes	—
134	Dic₄₇	—	8	4	no	—
135	C₂×#431	—	47	1	no	—
136	C₂×#431	D₄₇	1	1	yes	characteristic
137	#2283	—	47	1	no	—
138	#1279	—	47	4	no	—
139	C₂×C₄₇	—	4	2	no	—
140	C₂×C₄₇	#1482	1	4	yes	—
141	C₂×C₄₇	#1729	1	1	yes	characteristic
142	C₂×C₄₇	#1733	1	1	yes	characteristic
143	C₂×C₄₇	#1735	1	1	yes	characteristic
144	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
145	#431	—	2	2	no	—
146	#431	—	47	4	no	—
147	#431	—	94	2	no	—
148	#431	Dic₄₇	1	4	yes	complemented
149	C₂×C₄⋊C₈	—	2	1	no	—
150	C₂×C₄⋊C₈	—	47	2	no	—
151	C₂×C₂²⋊C₈	—	2	1	no	—
152	C₂×C₂²⋊C₈	—	94	1	no	—
153	C₂×C₄×C₈	—	94	1	no	—
154	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
155	#383	—	94	1	no	—
156	#382	—	47	2	no	—
157	#381	—	94	1	no	—
158	#380	—	47	2	no	—
159	C₄×C₄⋊C₄	—	47	2	no	—
160	C₄×C₂²⋊C₄	—	94	1	no	—
161	#376	—	94	1	no	—
162	C₄₇	#7894	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
164	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
165	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	94	2	no	—
168	C₂×C₄⋊C₄	—	47	4	no	—
169	C₂×C₂²⋊C₄	—	94	2	no	—
170	C₂×C₄²	—	94	2	no	—
171	C₂×C₄²	Dic₉₄	1	2	yes	—
172	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₄₇	1	2	yes	—
174	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
175	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
176	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
177	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
178	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
179	C₄⋊C₈	—	2	4	no	—
180	C₄⋊C₈	—	47	4	no	—
181	C₄⋊C₈	—	94	2	no	—
182	C₂²⋊C₈	—	2	4	no	—
183	C₂²⋊C₈	—	94	4	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	94	4	no	—
186	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
187	#95	—	94	2	no	—
188	#95	—	94	2	no	—
189	#95	—	94	2	no	—
190	#95	—	94	2	no	—
191	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	94	4	no	—
196	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
197	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
198	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
199	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
200	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
201	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
202	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
204	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
205	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	4	no	—
210	C₂×C₈	—	94	4	no	—
211	C₂×C₈	—	188	4	no	—
212	C₄⋊C₄	—	94	8	no	—
213	C₂²⋊C₄	—	188	4	no	—
214	C₄²	—	188	4	no	—
215	C₄²	C₄×Dic₄₇	1	2	yes	—
216	C₄²	Dic₄₇⋊C₄	1	2	yes	—
217	C₄²	Dic₄₇⋊C₄	1	2	yes	—
218	C₄²	C₄⋊Dic₄₇	1	2	yes	—
219	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
220	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
221	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
222	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
223	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
224	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	C₄₇⋊#95	1	2	yes	—
229	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	188	4	no	—
241	C₂×C₄	—	188	4	no	—
242	C₂×C₄	—	188	4	no	—
243	C₂×C₄	C₄₇⋊#95	1	2	yes	—
244	C₂×C₄	C₄₇⋊#95	1	2	yes	—
245	C₂×C₄	C₄₇⋊#95	1	2	yes	—
246	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
247	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
248	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
249	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
250	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
252	C₂×C₄	#13094032	1	4	yes	—
253	C₂×C₄	#13094046	1	4	yes	—
254	C₈	—	4	4	no	—
255	C₈	—	188	4	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	4	4	no	—
260	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
261	C₂²	#15689091	1	2	yes	—
262	C₂²	#15689120	1	2	yes	—
263	C₂²	#15689141	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	2	no	—
269	C₄	—	376	4	no	—
270	C₄	#15689081	1	2	yes	—
271	C₄	#15689120	1	2	yes	—
272	C₄	#15689151	1	2	yes	—
273	C₄	#15689161	1	2	yes	—
274	C₂	—	4	2	no	—
275	C₂	#20241797	1	4	yes	—
276	C₂	#20243086	1	1	yes	characteristic
277	C₂	#20243098	1	1	yes	characteristic
278	C₂	#20243105	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#431	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#431	D₄₇
hypocenter	C₄₇	#7894
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20241228	C₂	semidirect
6. #15689176	C₄	semidirect
19. C₄₇×#431	C₄	semidirect
148. #431	Dic₄₇	semidirect
162. C₄₇	#7894	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
#20245786	2
#20241228	2
C₂×C₄₇×#431	2
#7894	47