Subgroups — #26109141

Order 12032.

462 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689079	C₂	1	1	yes	maximal, characteristic
2	#20242134	C₂	1	2	yes	maximal, complemented
3	#20241702	C₂	1	1	yes	maximal, characteristic
4	#20241218	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1458	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₄	1	1	yes	characteristic
7	C₂³×C₄₇⋊C₈	C₄	1	1	yes	characteristic
8	#15689176	C₂²	1	2	yes	—
9	#15689079	—	2	2	no	—
10	#15689079	—	2	2	no	—
11	C₂×#13093935	C₄	1	1	yes	characteristic
12	C₂×#13093935	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	#15688875	—	2	2	no	—
16	#15688873	—	2	2	no	—
17	#15688849	—	2	1	no	—
18	#15688847	—	2	1	no	—
19	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#431	C₂²	1	2	yes	—
21	C₄₇×#324	—	2	1	no	—
22	C₄₇×#323	—	2	1	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	C₈	1	4	yes	complemented
26	C₂²×C₄₇⋊C₈	—	2	2	no	—
27	C₂²×C₄₇⋊C₈	—	2	1	no	—
28	C₂²×C₄₇⋊C₈	—	2	1	no	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
35	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
36	#13093935	—	2	2	no	—
37	#13093935	—	2	2	no	—
38	C₄×C₄₇⋊C₈	—	2	1	no	—
39	C₄×C₄₇⋊C₈	—	2	1	no	—
40	C₄×C₄₇⋊C₈	—	2	2	no	—
41	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
42	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
43	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
44	#117×C₄₇	—	2	2	no	—
45	#117×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₄₇	Q₈	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₄₇	Q₈	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₄₇	—	2	2	no	—
53	C₂²⋊C₈×C₄₇	—	2	2	no	—
54	C₄×C₈×C₄₇	—	2	2	no	—
55	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	—	2	1	no	—
57	C₂²×C₄×C₄₇	—	2	1	no	—
58	C₂²×C₄×C₄₇	—	2	1	no	—
59	C₂²×C₄×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₄₇	—	4	2	no	—
69	C₄⋊C₄×C₄₇	—	2	2	no	—
70	C₄⋊C₄×C₄₇	—	2	2	no	—
71	C₂²⋊C₄×C₄₇	—	2	2	no	—
72	C₂²⋊C₄×C₄₇	—	2	2	no	—
73	C₄²×C₄₇	—	2	1	no	—
74	C₄²×C₄₇	—	2	1	no	—
75	C₄²×C₄₇	C₂×D₄	1	2	yes	—
76	C₄²×C₄₇	#41	1	2	yes	—
77	#10625329	—	2	8	no	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	2	1	no	—
81	C₂×C₄₇⋊C₈	—	2	1	no	—
82	C₂×C₄₇⋊C₈	—	2	1	no	—
83	C₂×C₄₇⋊C₈	—	2	1	no	—
84	C₂×C₄₇⋊C₈	—	2	1	no	—
85	C₂×C₄₇⋊C₈	—	2	1	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
90	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
91	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
92	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
93	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	2	1	no	—
96	C₂³×C₄₇	—	4	1	no	—
97	C₂³×C₄₇	—	4	1	no	—
98	C₂³×C₄₇	C₄⋊C₈	1	2	yes	—
99	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	4	1	no	—
109	C₂×C₄×C₄₇	—	4	1	no	—
110	C₂×C₄×C₄₇	—	4	1	no	—
111	C₂×C₄×C₄₇	—	4	1	no	—
112	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#121	1	2	yes	—
116	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
121	C₈×C₄₇	—	8	2	no	—
122	C₄₇⋊C₈	—	2	2	no	—
123	C₄₇⋊C₈	—	2	1	no	—
124	C₄₇⋊C₈	—	2	1	no	—
125	C₄₇⋊C₈	—	2	2	no	—
126	C₄₇⋊C₈	—	8	2	no	—
127	C₄₇⋊C₈	C₂²⋊C₈	1	4	yes	—
128	#8602	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	1	no	—
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	4	2	no	—
133	C₂²×C₄₇	—	4	2	no	—
134	C₂²×C₄₇	—	4	1	no	—
135	C₂²×C₄₇	—	4	1	no	—
136	C₂²×C₄₇	#380	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₂²×C₄₇	#423	1	1	yes	characteristic
139	C₄×C₄₇	—	2	2	no	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	4	1	no	—
143	C₄×C₄₇	—	4	1	no	—
144	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₄×C₄₇	#407	1	1	yes	characteristic
146	C₄×C₄₇	#431	1	1	yes	characteristic
147	C₄×C₄₇	#432	1	1	yes	characteristic
148	C₂×#405	—	47	1	no	—
149	#1564	—	47	2	no	—
150	#1458	—	47	1	no	—
151	#1458	D₄₇	1	1	yes	characteristic
152	#1275	—	47	2	no	—
153	C₂×C₄₇	—	2	1	no	—
154	C₂×C₄₇	—	2	1	no	—
155	C₂×C₄₇	—	4	2	no	—
156	C₂×C₄₇	#1797	1	1	yes	characteristic
157	C₂×C₄₇	#1812	1	1	yes	characteristic
158	C₂×C₄₇	#1813	1	1	yes	characteristic
159	C₂²×M₁₆	—	47	1	no	—
160	C₂³×C₈	—	47	1	no	—
161	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
162	#431	—	47	2	no	—
163	#431	C₂×D₄₇	1	2	yes	—
164	#405	—	94	2	no	—
165	#405	—	94	2	no	—
166	C₂×#97	—	47	1	no	—
167	C₂×#97	—	47	1	no	—
168	C₂×C₄×C₈	—	47	1	no	—
169	C₂×C₄×C₈	—	47	1	no	—
170	#335	—	94	2	no	—
171	#334	—	94	2	no	—
172	#324	—	2	1	no	—
173	#324	—	94	1	no	—
174	#323	—	2	1	no	—
175	#323	—	94	1	no	—
176	C₄₇	#8602	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
178	C₂×M₁₆	—	47	4	no	—
179	C₂×M₁₆	—	94	2	no	—
180	C₂×M₁₆	—	94	2	no	—
181	C₂²×C₈	—	47	1	no	—
182	C₂²×C₈	—	47	1	no	—
183	C₂²×C₈	—	47	1	no	—
184	C₂²×C₈	—	47	1	no	—
185	C₂²×C₈	—	94	2	no	—
186	C₂²×C₈	—	94	1	no	—
187	C₂²×C₈	—	94	1	no	—
188	C₂²×C₈	—	94	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	C₂×C₄⋊C₄	Dic₉₄	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
193	C₂×C₂²⋊C₄	Dic₉₄	1	1	yes	characteristic
194	C₂×C₂²⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
196	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	94	2	no	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	—	94	2	no	—
201	#97	—	94	2	no	—
202	#97	—	94	2	no	—
203	C₄×C₈	—	2	2	no	—
204	C₄×C₈	—	47	2	no	—
205	C₄×C₈	—	47	2	no	—
206	C₄×C₈	—	94	1	no	—
207	C₄×C₈	—	94	1	no	—
208	C₄×C₈	—	94	2	no	—
209	C₂⁴	Dic₄₇⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₈×D₄₇	1	2	yes	—
215	C₂²×C₄	#10625324	1	2	yes	—
216	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
217	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
222	M₁₆	—	94	8	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	47	2	no	—
225	C₂×C₈	—	47	2	no	—
226	C₂×C₈	—	47	2	no	—
227	C₂×C₈	—	47	1	no	—
228	C₂×C₈	—	47	1	no	—
229	C₂×C₈	—	47	2	no	—
230	C₂×C₈	—	47	2	no	—
231	C₂×C₈	—	94	4	no	—
232	C₂×C₈	—	94	4	no	—
233	C₂×C₈	—	94	1	no	—
234	C₂×C₈	—	94	1	no	—
235	C₂×C₈	—	94	1	no	—
236	C₂×C₈	—	94	1	no	—
237	C₂×C₈	—	94	1	no	—
238	C₂×C₈	—	94	1	no	—
239	C₂×C₈	—	188	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₄²	D₄×D₄₇	1	2	yes	—
247	C₄²	#10625352	1	2	yes	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	Dic₄₇⋊C₈	1	2	yes	—
253	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
267	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
268	C₂×C₄	#13093980	1	1	yes	characteristic
269	C₂×C₄	#13093996	1	2	yes	—
270	C₂×C₄	#13094001	1	2	yes	—
271	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
272	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
273	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
275	C₈	—	8	2	no	—
276	C₈	—	47	4	no	—
277	C₈	—	94	2	no	—
278	C₈	—	94	1	no	—
279	C₈	—	94	1	no	—
280	C₈	—	94	2	no	—
281	C₈	—	376	2	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	#15688981	1	1	yes	characteristic
290	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
291	C₂²	#15689146	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
298	C₄	#15689085	1	1	yes	characteristic
299	C₄	#15689173	1	1	yes	characteristic
300	C₄	#15689177	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	4	2	no	—
304	C₂	#20243366	1	1	yes	characteristic
305	C₂	#20243432	1	1	yes	characteristic
306	C₂	#20243436	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093980
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1458	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1458	D₄₇
hypocenter	C₄₇	#8602
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1458

Complements

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

N	H ≅ G/N	type
2. #20242134	C₂	semidirect
4. #20241218	C₂	semidirect
25. C₂×#10625329	C₈	semidirect
176. C₄₇	#8602	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689079	2
#20242134	2
#20241702	2
#20241218	2
C₄₇×#1458	2
#8602	47