Subgroups — #26215125

Order 12032.

866 conjugacy classes of subgroups.

The subgroup lattice has 278 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	#20245786	C₂	1	1	yes	maximal, characteristic
3	#20243265	C₂	1	4	yes	maximal, complemented
4	#20243247	C₂	1	4	yes	maximal, complemented
5	#20243169	C₂	1	4	yes	maximal
6	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
7	#15689180	C₄	1	8	yes	complemented
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×#13093935	C₂²	1	2	yes	—
11	C₂×#13093934	C₂²	1	4	yes	—
12	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
13	#15688991	C₄	1	2	yes	—
14	#15688989	C₄	1	2	yes	—
15	#15688986	C₄	1	2	yes	—
16	#15688983	C₄	1	2	yes	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
18	C₄×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
19	#15688969	C₂²	1	2	yes	—
20	#15688877	C₂²	1	8	yes	—
21	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#423	C₄	1	8	yes	complemented
24	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
30	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
31	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
32	#13093935	C₂×C₄	1	8	yes	—
33	#13093934	—	4	4	no	—
34	C₈×Dic₄₇	—	4	4	no	—
35	C₄₇⋊#95	C₂×C₄	1	4	yes	—
36	C₄₇⋊#95	C₂×C₄	1	4	yes	—
37	C₄₇⋊#95	C₂×C₄	1	4	yes	—
38	C₄₇⋊#95	C₂×C₄	1	4	yes	—
39	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₄₇	—	4	2	no	—
43	C₂×M₁₆×C₄₇	—	4	2	no	—
44	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
45	C₂²×C₈×C₄₇	C₂³	1	4	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	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
53	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
54	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
60	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
70	M₁₆×C₄₇	—	4	8	no	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
74	C₄²×C₄₇	C₄²	1	4	yes	—
75	C₄²×C₄₇	C₄²	1	4	yes	—
76	C₄²×C₄₇	C₂²×C₄	1	2	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	C₂²⋊Dic₄₇	—	4	4	no	—
81	C₄⋊Dic₄₇	—	4	4	no	—
82	C₄×Dic₄₇	—	2	4	no	—
83	C₄×Dic₄₇	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
87	C₂³×C₄₇	—	2	2	no	—
88	C₂³×C₄₇	—	2	2	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
91	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
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	4	no	—
100	C₂×C₄×C₄₇	—	4	4	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
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	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
110	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
111	C₂×Dic₄₇	—	2	8	no	—
112	C₂×Dic₄₇	—	2	8	no	—
113	C₂×Dic₄₇	—	2	8	no	—
114	C₈×C₄₇	—	2	8	no	—
115	C₄₇⋊C₈	—	2	8	no	—
116	#20551	—	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	C₂²×C₄²	1	1	yes	characteristic
122	C₂²×C₄₇	#520	1	2	yes	—
123	C₂²×C₄₇	#568	1	2	yes	—
124	C₂²×C₄₇	#568	1	2	yes	—
125	C₄×C₄₇	—	2	2	no	—
126	C₄×C₄₇	—	2	2	no	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	#518	1	1	yes	characteristic
131	C₄×C₄₇	#519	1	1	yes	characteristic
132	C₄×C₄₇	#520	1	2	yes	—
133	Dic₄₇	—	4	8	no	—
134	C₂×#432	—	47	1	no	—
135	C₂×#423	D₄₇	1	1	yes	characteristic
136	#2283	—	47	1	no	—
137	#1771	—	47	4	no	—
138	#1768	—	47	4	no	—
139	#1753	—	47	4	no	—
140	C₂×C₄₇	—	4	2	no	—
141	C₂×C₄₇	#2274	1	1	yes	characteristic
142	C₂×C₄₇	#2873	1	1	yes	characteristic
143	C₂×C₄₇	#2873	1	1	yes	characteristic
144	C₂×C₄₇	#2995	1	4	yes	—
145	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
146	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
147	#432	—	47	8	no	—
148	#423	Dic₄₇	1	8	yes	complemented
149	C₂×C₄⋊C₈	—	47	2	no	—
150	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
151	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
152	C₂×C₂²⋊C₈	—	47	2	no	—
153	C₂×#97	—	47	2	no	—
154	C₂×#97	—	47	4	no	—
155	C₂×C₄×C₈	—	47	4	no	—
156	#383	—	47	2	no	—
157	#382	—	47	2	no	—
158	#381	—	47	2	no	—
159	#380	—	47	2	no	—
160	C₄×C₄⋊C₄	—	47	2	no	—
161	C₄×C₂²⋊C₄	—	47	2	no	—
162	#376	—	47	2	no	—
163	#336	—	47	8	no	—
164	C₄₇	#20551	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
166	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
167	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
168	C₂×M₁₆	—	4	2	no	—
169	C₂×M₁₆	—	4	2	no	—
170	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
171	C₂²×C₈	—	47	4	no	—
172	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
173	C₂×C₄⋊C₄	—	47	4	no	—
174	C₂×C₂²⋊C₄	—	47	4	no	—
175	C₂×C₄²	—	47	4	no	—
176	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
177	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₄₇	1	4	yes	—
179	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
180	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
181	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
182	C₄⋊C₈	—	47	8	no	—
183	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
184	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
185	C₂²⋊C₈	—	47	8	no	—
186	#97	—	47	8	no	—
187	#97	—	188	4	no	—
188	C₄×C₈	—	188	4	no	—
189	#95	—	47	4	no	—
190	#95	—	47	4	no	—
191	#95	—	47	4	no	—
192	#95	—	47	4	no	—
193	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	47	8	no	—
199	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
200	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
201	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
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	2	yes	—
207	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
208	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
209	M₁₆	—	4	8	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	47	8	no	—
213	C₂×C₈	—	94	4	no	—
214	C₂×C₈	—	94	4	no	—
215	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
216	C₄⋊C₄	—	188	4	no	—
217	C₂²⋊C₄	—	188	4	no	—
218	C₄²	—	94	4	no	—
219	C₄²	—	94	4	no	—
220	C₄²	C₄×Dic₄₇	1	4	yes	—
221	C₄²	C₄×Dic₄₇	1	4	yes	—
222	C₄²	C₂×C₄×D₄₇	1	2	yes	—
223	C₄²	C₂×C₄×D₄₇	1	2	yes	—
224	C₄²	C₂²×Dic₄₇	1	2	yes	—
225	C₄²	C₂²×Dic₄₇	1	2	yes	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
230	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	94	8	no	—
242	C₂×C₄	—	94	8	no	—
243	C₂×C₄	—	94	8	no	—
244	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
246	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
247	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
248	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
249	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
250	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
251	C₂×C₄	#13094069	1	2	yes	—
252	C₂×C₄	#13094072	1	2	yes	—
253	C₈	—	2	8	no	—
254	C₈	—	94	8	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	4	no	—
259	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
260	C₂²	#15689604	1	2	yes	—
261	C₂²	#15689889	1	2	yes	—
262	C₂²	#15689889	1	2	yes	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	188	8	no	—
269	C₄	#15689598	1	1	yes	characteristic
270	C₄	#15689601	1	1	yes	characteristic
271	C₄	#15689604	1	2	yes	—
272	C₂	—	4	2	no	—
273	C₂	#20245745	1	1	yes	characteristic
274	C₂	#20251806	1	1	yes	characteristic
275	C₂	#20251806	1	1	yes	characteristic
276	C₂	#20252555	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄₇×#423	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	#20551
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #20243265	C₂	semidirect
4. #20243247	C₂	semidirect
7. #15689180	C₄	semidirect
23. C₄₇×#423	C₄	semidirect
148. #423	Dic₄₇	semidirect
164. C₄₇	#20551	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
#20245786	2
#20243265	2
#20243247	2
#20243169	2
C₂×C₄₇×#423	2
#20551	47