Subgroups — #26214538

Order 12032.

922 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×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
2	#20245786	C₂	1	1	yes	maximal, characteristic
3	#20243265	C₂	1	4	yes	maximal, complemented
4	#20243174	C₂	1	4	yes	maximal
5	#20243141	C₂	1	2	yes	maximal, complemented
6	#20243076	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₄×#10625329	C₄	1	8	yes	complemented
10	C₂×#13093935	C₂²	1	2	yes	—
11	C₂×#13093934	C₂²	1	4	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
15	#15688991	C₄	1	2	yes	—
16	#15688989	C₄	1	2	yes	—
17	#15688986	C₄	1	2	yes	—
18	#15688983	C₄	1	2	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
20	C₄×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
21	#15688969	C₂²	1	2	yes	—
22	#15688877	C₂²	1	8	yes	—
23	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#432	C₄	1	8	yes	complemented
25	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
27	C₂×#97×C₄₇	C₂²	1	2	yes	—
28	C₂×#10625329	—	2	8	no	—
29	C₂×#10625329	C₂×C₄	1	8	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
35	#13093935	C₂×C₄	1	8	yes	—
36	#13093934	—	2	8	no	—
37	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
38	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
39	C₈×Dic₄₇	—	2	8	no	—
40	C₄₇⋊#95	C₂×C₄	1	4	yes	—
41	C₄₇⋊#95	C₂×C₄	1	4	yes	—
42	C₄₇⋊#95	C₂×C₄	1	4	yes	—
43	C₄₇⋊#95	C₂×C₄	1	4	yes	—
44	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
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	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
55	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
56	#97×C₄₇	C₂×C₄	1	8	yes	—
57	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	—	4	2	no	—
61	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
62	C₂²×C₄×C₄₇	C₄²	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	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
75	C₄²×C₄₇	C₄²	1	4	yes	—
76	C₄²×C₄₇	C₄²	1	4	yes	—
77	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂²⋊Dic₄₇	—	4	4	no	—
82	C₄⋊Dic₄₇	—	4	4	no	—
83	C₄×Dic₄₇	—	2	4	no	—
84	C₄×Dic₄₇	—	2	4	no	—
85	#10625329	—	2	16	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
88	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
89	C₂×C₄₇⋊C₈	#41	1	8	yes	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
94	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	2	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	4	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	4	4	no	—
103	C₂×C₄×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
112	C₂×Dic₄₇	—	2	8	no	—
113	C₂×Dic₄₇	—	2	8	no	—
114	C₂×Dic₄₇	—	2	8	no	—
115	C₈×C₄₇	—	2	8	no	—
116	C₄₇⋊C₈	—	2	4	no	—
117	C₄₇⋊C₈	#117	1	8	yes	—
118	#20477	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	4	4	no	—
123	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
124	C₂²×C₄₇	C₄×#41	1	2	yes	—
125	C₂²×C₄₇	#568	1	2	yes	—
126	C₂²×C₄₇	#568	1	2	yes	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	—	2	2	no	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	C₂×#117	1	2	yes	—
133	C₄×C₄₇	C₄×#41	1	2	yes	—
134	Dic₄₇	—	4	8	no	—
135	C₂×#432	D₄₇	1	1	yes	characteristic
136	C₂×C₄×M₁₆	—	47	1	no	—
137	#2283	—	47	1	no	—
138	#1771	—	47	4	no	—
139	#1753	—	47	4	no	—
140	#1746	—	47	2	no	—
141	#1726	—	47	2	no	—
142	C₂×C₄₇	—	4	2	no	—
143	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
144	C₂×C₄₇	#2873	1	1	yes	characteristic
145	C₂×C₄₇	#2873	1	1	yes	characteristic
146	C₂×C₄₇	#2967	1	4	yes	—
147	C₂²×M₁₆	—	47	2	no	—
148	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
149	#432	Dic₄₇	1	8	yes	complemented
150	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
151	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
152	C₄×M₁₆	—	47	8	no	—
153	C₂×#97	—	47	2	no	—
154	C₂×#97	—	47	4	no	—
155	C₂×#97	C₂×D₄₇	1	2	yes	—
156	C₂×C₄×C₈	—	47	1	no	—
157	C₂×C₄×C₈	—	47	1	no	—
158	C₂×C₄×C₈	—	47	4	no	—
159	#383	—	47	2	no	—
160	#382	—	47	2	no	—
161	#381	—	47	2	no	—
162	#380	—	47	2	no	—
163	C₄×C₄⋊C₄	—	47	2	no	—
164	C₄×C₂²⋊C₄	—	47	2	no	—
165	#376	—	47	2	no	—
166	#336	—	47	8	no	—
167	C₄₇	#20477	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
169	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
170	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
171	C₂×M₁₆	—	47	8	no	—
172	C₂×M₁₆	—	94	8	no	—
173	C₂²×C₈	—	47	2	no	—
174	C₂²×C₈	—	47	2	no	—
175	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
176	C₂×C₄⋊C₄	—	47	4	no	—
177	C₂×C₂²⋊C₄	—	47	4	no	—
178	C₂×C₄²	—	47	4	no	—
179	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
180	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
181	C₂×C₄²	C₄×D₄₇	1	4	yes	—
182	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
183	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
185	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
186	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
187	#97	—	47	8	no	—
188	#97	—	94	8	no	—
189	#97	C₂×Dic₄₇	1	8	yes	—
190	C₄×C₈	—	47	4	no	—
191	C₄×C₈	—	47	4	no	—
192	C₄×C₈	—	94	8	no	—
193	#95	—	47	4	no	—
194	#95	—	47	4	no	—
195	#95	—	47	4	no	—
196	#95	—	47	4	no	—
197	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	47	8	no	—
202	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
203	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
204	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
210	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
211	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
212	M₁₆	—	94	16	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	47	4	no	—
216	C₂×C₈	—	47	8	no	—
217	C₂×C₈	—	47	4	no	—
218	C₂×C₈	—	94	4	no	—
219	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
220	C₄⋊C₄	—	188	4	no	—
221	C₂²⋊C₄	—	188	4	no	—
222	C₄²	—	94	4	no	—
223	C₄²	—	94	4	no	—
224	C₄²	C₄×Dic₄₇	1	4	yes	—
225	C₄²	C₄×Dic₄₇	1	4	yes	—
226	C₄²	C₂×C₄×D₄₇	1	2	yes	—
227	C₄²	C₂×C₄×D₄₇	1	2	yes	—
228	C₄²	C₂²×Dic₄₇	1	2	yes	—
229	C₄²	C₂²×Dic₄₇	1	2	yes	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
234	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	94	8	no	—
245	C₂×C₄	—	94	8	no	—
246	C₂×C₄	—	94	8	no	—
247	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
248	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
249	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
250	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
251	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
252	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
253	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
254	C₂×C₄	#41×D₄₇	1	4	yes	—
255	C₈	—	2	8	no	—
256	C₈	—	47	8	no	—
257	C₈	—	94	4	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
263	C₂²	#15689591	1	2	yes	—
264	C₂²	#15689889	1	2	yes	—
265	C₂²	#15689889	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	188	8	no	—
272	C₄	#117×D₄₇	1	2	yes	—
273	C₄	#41×Dic₄₇	1	2	yes	—
274	C₂	—	4	2	no	—
275	C₂	#117×Dic₄₇	1	1	yes	characteristic
276	C₂	#20251806	1	1	yes	characteristic
277	C₂	#20251806	1	1	yes	characteristic
278	C₂	#20252368	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄₇×#432	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#20477
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #20243265	C₂	semidirect
5. #20243141	C₂	semidirect
6. #20243076	C₂	semidirect
9. C₄×#10625329	C₄	semidirect
24. C₄₇×#432	C₄	semidirect
149. #432	Dic₄₇	semidirect
167. C₄₇	#20477	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10625329	2
#20245786	2
#20243265	2
#20243174	2
#20243141	2
#20243076	2
C₂×C₄₇×#432	2
#20477	47