Subgroups — #26214500

Order 12032.

1078 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	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20243258	C₂	1	4	yes	maximal, complemented
4	#20243167	C₂	1	4	yes	maximal
5	#20243133	C₂	1	2	yes	maximal, complemented
6	C₄×#13093934	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₂×#13093983	C₄	1	2	yes	—
10	C₂×#13093934	C₂²	1	8	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₄	1	8	yes	complemented
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
16	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
17	C₄×C₂²⋊Dic₄₇	C₄	1	2	yes	—
18	C₄×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
19	#15688969	C₄	1	2	yes	—
20	C₄²×Dic₄₇	C₂²	1	2	yes	—
21	#15688877	C₂²	1	8	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#431	C₄	1	8	yes	complemented
24	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
27	C₂²×C₄₇⋊C₈	—	2	8	no	—
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	#13093983	—	2	8	no	—
32	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
37	#13093934	—	2	16	no	—
38	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
39	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
40	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
41	C₄₇⋊#95	C₂×C₄	1	4	yes	—
42	C₄₇⋊#95	C₂×C₄	1	4	yes	—
43	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
54	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
55	C₄×C₈×C₄₇	C₂×C₄	1	8	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	C₄²	1	4	yes	—
60	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
61	C₂²×C₄×C₄₇	M₁₆	1	8	yes	—
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	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
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	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
71	C₂×C₈×C₄₇	—	2	8	no	—
72	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
73	C₄²×C₄₇	C₄²	1	4	yes	—
74	C₄²×C₄₇	C₄²	1	4	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	2	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	C₂²⋊Dic₄₇	—	2	8	no	—
80	C₄⋊Dic₄₇	—	2	8	no	—
81	C₄×Dic₄₇	—	2	8	no	—
82	C₄×Dic₄₇	—	2	8	no	—
83	C₄×Dic₄₇	—	2	8	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	16	no	—
86	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
87	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
88	C₂×C₄₇⋊C₈	#41	1	8	yes	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
92	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
93	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	2	8	no	—
99	C₂×C₄×C₄₇	#97	1	16	yes	complemented
100	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
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×M₁₆	1	4	yes	—
106	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
107	C₂×C₄×C₄₇	#131	1	4	yes	—
108	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×#41	1	4	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	4	no	—
116	C₄₇⋊C₈	#117	1	8	yes	complemented
117	#20473	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	—	2	8	no	—
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	C₂×#97	1	4	yes	—
122	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
123	C₂²×C₄₇	C₄×#41	1	2	yes	—
124	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
125	C₂²×C₄₇	C₂×#131	1	2	yes	—
126	C₄×C₄₇	—	2	2	no	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	C₂×#97	1	4	yes	—
129	C₄×C₄₇	C₂×#97	1	4	yes	—
130	C₄×C₄₇	#405	1	4	yes	—
131	C₄×C₄₇	C₂×#117	1	2	yes	—
132	C₄×C₄₇	C₄×#41	1	2	yes	—
133	Dic₄₇	—	4	8	no	—
134	C₂×#431	D₄₇	1	1	yes	characteristic
135	C₂²×C₄×C₈	—	47	1	no	—
136	C₄×#117	—	47	1	no	—
137	#1770	—	47	4	no	—
138	#1752	—	47	4	no	—
139	#1744	—	47	2	no	—
140	C₄×#97	—	47	2	no	—
141	C₂×C₄₇	—	2	4	no	—
142	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
143	C₂×C₄₇	C₂²×#97	1	1	yes	characteristic
144	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
145	C₂×C₄₇	#2965	1	4	yes	—
146	C₂³×C₈	—	47	2	no	—
147	C₂×#117	—	47	2	no	—
148	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
149	#431	Dic₄₇	1	8	yes	complemented
150	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
151	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
152	C₂×#97	—	47	8	no	—
153	C₂×C₄×C₈	—	47	8	no	—
154	C₂×C₄×C₈	—	47	2	no	—
155	C₂×C₄×C₈	—	47	1	no	—
156	C₂×C₄×C₈	—	47	1	no	—
157	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
158	C₄×C₄⋊C₄	—	47	2	no	—
159	C₄×C₄⋊C₄	—	47	2	no	—
160	C₄×C₂²⋊C₄	—	47	2	no	—
161	C₄×C₂²⋊C₄	—	47	2	no	—
162	#376	—	47	2	no	—
163	C₄³	—	47	2	no	—
164	#336	—	47	8	no	—
165	C₄₇	#20473	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
167	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
168	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
169	C₂²×C₈	—	47	2	no	—
170	C₂²×C₈	—	47	8	no	—
171	C₂²×C₈	—	47	2	no	—
172	C₂²×C₈	—	94	8	no	—
173	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
174	#117	—	94	8	no	—
175	C₂×C₄⋊C₄	—	47	4	no	—
176	C₂×C₂²⋊C₄	—	47	4	no	—
177	C₂×C₄²	—	47	4	no	—
178	C₂×C₄²	—	47	4	no	—
179	C₂×C₄²	—	47	4	no	—
180	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
181	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
182	C₂×C₄²	C₄×D₄₇	1	4	yes	—
183	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
184	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
186	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
187	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
188	#97	—	94	16	no	—
189	C₄×C₈	—	47	8	no	—
190	C₄×C₈	—	47	4	no	—
191	C₄×C₈	—	47	4	no	—
192	C₄×C₈	C₂×Dic₄₇	1	8	yes	—
193	#95	—	47	4	no	—
194	#95	—	47	4	no	—
195	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	47	8	no	—
199	C₂²×C₄	#10625324	1	8	yes	—
200	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
201	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
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	2	yes	—
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	C₂²×Dic₄₇	1	2	yes	—
209	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	47	4	no	—
212	C₂×C₈	—	47	8	no	—
213	C₂×C₈	—	47	4	no	—
214	C₂×C₈	—	94	16	no	—
215	C₂×C₈	—	94	4	no	—
216	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
217	C₄⋊C₄	—	94	8	no	—
218	C₂²⋊C₄	—	94	8	no	—
219	C₄²	—	94	8	no	—
220	C₄²	—	94	8	no	—
221	C₄²	—	94	8	no	—
222	C₄²	C₄×Dic₄₇	1	4	yes	—
223	C₄²	C₄×Dic₄₇	1	4	yes	—
224	C₄²	C₂×C₄×D₄₇	1	2	yes	—
225	C₄²	C₂×C₄×D₄₇	1	2	yes	—
226	C₄²	C₂²×Dic₄₇	1	2	yes	—
227	C₄²	C₂²×Dic₄₇	1	2	yes	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
231	C₂³	C₂×#10625324	1	4	yes	—
232	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	94	8	no	—
239	C₂×C₄	—	94	8	no	—
240	C₂×C₄	—	94	8	no	—
241	C₂×C₄	#13093934	1	16	yes	complemented
242	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
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×#10625324	1	4	yes	—
248	C₂×C₄	C₂×#10625324	1	4	yes	—
249	C₂×C₄	#13094032	1	4	yes	—
250	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
251	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
252	C₂×C₄	#41×D₄₇	1	4	yes	—
253	C₈	—	2	8	no	—
254	C₈	—	47	8	no	—
255	C₈	—	94	4	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	8	no	—
258	C₂²	—	2	2	no	—
259	C₂²	C₂×#13093934	1	4	yes	—
260	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
261	C₂²	#15689591	1	2	yes	—
262	C₂²	C₂²×#10625324	1	2	yes	—
263	C₂²	C₂×#13094032	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	188	8	no	—
267	C₄	C₂×#13093934	1	4	yes	—
268	C₄	C₂×#13093934	1	4	yes	—
269	C₄	#15689081	1	4	yes	—
270	C₄	#117×D₄₇	1	2	yes	—
271	C₄	#41×Dic₄₇	1	2	yes	—
272	C₂	—	2	4	no	—
273	C₂	#117×Dic₄₇	1	1	yes	characteristic
274	C₂	C₂²×#13093934	1	1	yes	characteristic
275	C₂	C₂×#15689081	1	1	yes	characteristic
276	C₂	#20252349	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#431	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#431	D₄₇
hypocenter	C₄₇	#20473
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20243258	C₂	semidirect
5. #20243133	C₂	semidirect
6. C₄×#13093934	C₂	semidirect
11. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
23. C₄₇×#431	C₄	semidirect
99. C₂×C₄×C₄₇	#97	semidirect
116. C₄₇⋊C₈	#117	semidirect
149. #431	Dic₄₇	semidirect
165. C₄₇	#20473	semidirect
241. C₂×C₄	#13093934	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₄₇⋊C₈	2
C₄×#13093983	2
#20243258	2
#20243167	2
#20243133	2
C₄×#13093934	2
C₂×C₄₇×#431	2
#20473	47