Subgroups — #26133224

Order 12032.

574 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689364	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093958	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688928	C₂	1	2	yes	maximal, complemented
4	#20241480	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
7	C₂×#13094003	—	2	1	no	—
8	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
9	#15689364	—	2	2	no	—
10	#15689364	C₄	1	4	yes	complemented
11	C₂×#13093958	—	2	4	no	—
12	C₂×#13093958	—	2	1	no	—
13	C₂×#13093958	C₂²	1	1	yes	characteristic
14	C₂×#13093958	C₂²	1	2	yes	—
15	C₂×#13093949	—	2	1	no	—
16	#15688928	—	2	4	no	—
17	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
19	C₄₇×C₂²⋊C₁₆	—	2	2	no	—
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂³×Dic₄₇	—	4	1	no	—
22	C₂×Dic₁₈₈	—	2	2	no	—
23	C₂×Dic₁₈₈	—	2	2	no	—
24	C₂×Dic₁₈₈	—	2	1	no	—
25	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	complemented
26	#13094003	—	2	4	no	—
27	#13094003	—	4	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	4	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	—	2	1	no	—
31	C₂×C₄⋊Dic₄₇	—	4	2	no	—
32	C₂×C₄⋊Dic₄₇	—	4	1	no	—
33	C₂×Dic₄₇⋊C₄	—	4	1	no	—
34	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
35	#13093958	—	2	4	no	—
36	#13093958	—	2	1	no	—
37	#13093958	—	2	2	no	—
38	#13093958	C₂×C₄	1	2	yes	complemented
39	#13093958	D₄	1	4	yes	—
40	#13093949	—	2	4	no	—
41	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	—	2	2	no	—
44	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	D₄	1	2	yes	—
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₄₇	—	2	2	no	—
49	C₂×C₁₆×C₄₇	—	2	2	no	—
50	C₂×C₁₆×C₄₇	—	2	2	no	—
51	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	1	no	—
53	C₂²×C₄×C₄₇	—	2	1	no	—
54	C₂²×C₄×C₄₇	—	2	1	no	—
55	C₂²×C₄×C₄₇	—	2	1	no	—
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₄₇	—	4	4	no	—
61	C₂²×Dic₄₇	—	4	1	no	—
62	C₂²×Dic₄₇	—	4	1	no	—
63	C₂²×Dic₄₇	—	4	2	no	—
64	C₂²×Dic₄₇	—	8	2	no	—
65	C₂²×Dic₄₇	—	8	2	no	—
66	C₂×Dic₉₄	—	2	2	no	—
67	C₂×Dic₉₄	—	4	1	no	—
68	C₂×Dic₉₄	—	4	1	no	—
69	C₂×Dic₉₄	—	4	1	no	—
70	C₂×Dic₉₄	—	4	1	no	—
71	C₂×Dic₉₄	—	4	1	no	—
72	C₂×C₈×C₄₇	—	2	1	no	—
73	C₂×C₈×C₄₇	—	2	1	no	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	#41	1	2	yes	—
81	C₂²⋊Dic₄₇	—	4	4	no	—
82	C₄⋊Dic₄₇	—	2	2	no	—
83	C₄⋊Dic₄₇	—	2	4	no	—
84	C₄⋊Dic₄₇	—	4	4	no	—
85	C₄⋊Dic₄₇	—	4	1	no	—
86	C₄⋊Dic₄₇	—	4	2	no	—
87	Dic₄₇⋊C₄	—	4	4	no	—
88	Dic₁₈₈	—	2	4	no	—
89	Dic₁₈₈	—	4	2	no	—
90	C₁₆×C₄₇	—	4	2	no	—
91	C₂³×C₄₇	—	2	1	no	—
92	C₂³×C₄₇	—	2	1	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	#102	1	2	yes	—
96	C₂³×C₄₇	SD₃₂	1	2	yes	—
97	C₂³×C₄₇	Q₃₂	1	2	yes	—
98	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	2	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	#102	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
112	C₂×Dic₄₇	—	4	2	no	—
113	C₂×Dic₄₇	—	4	2	no	—
114	C₂×Dic₄₇	—	4	4	no	—
115	C₂×Dic₄₇	—	8	8	no	—
116	C₂×Dic₄₇	—	8	1	no	—
117	C₂×Dic₄₇	—	8	1	no	—
118	C₂×Dic₄₇	—	8	2	no	—
119	C₂×Dic₄₇	—	8	1	no	—
120	C₂×Dic₄₇	—	8	2	no	—
121	C₂×Dic₄₇	—	8	1	no	—
122	Dic₉₄	—	4	2	no	—
123	Dic₉₄	—	4	2	no	—
124	Dic₉₄	—	8	1	no	—
125	Dic₉₄	—	8	1	no	—
126	C₈×C₄₇	—	2	2	no	—
127	C₈×C₄₇	C₄×D₄	1	2	yes	—
128	C₈×C₄₇	#121	1	2	yes	—
129	#12993	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	#358	1	4	yes	complemented
137	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
138	C₂²×C₄₇	#447	1	1	yes	characteristic
139	C₂²×C₄₇	#480	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
141	C₂²×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
142	C₂²×C₄₇	#509	1	1	yes	characteristic
143	C₂²×C₄₇	#510	1	1	yes	characteristic
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	#386	1	1	yes	characteristic
146	C₄×C₄₇	#417	1	1	yes	characteristic
147	C₄×C₄₇	#448	1	1	yes	characteristic
148	C₄×C₄₇	#481	1	1	yes	characteristic
149	Dic₄₇	—	8	2	no	—
150	Dic₄₇	—	8	2	no	—
151	Dic₄₇	—	8	4	no	—
152	C₂×#467	—	47	1	no	—
153	C₂²×#107	—	47	1	no	—
154	C₂×#358	—	47	2	no	—
155	C₂×C₂²⋊C₁₆	D₄₇	1	1	yes	characteristic
156	#1380	—	47	2	no	—
157	C₂×C₄₇	—	2	4	no	—
158	C₂×C₄₇	#1874	1	1	yes	characteristic
159	C₂×C₄₇	C₂×#358	1	1	yes	characteristic
160	C₂×C₄₇	#2139	1	1	yes	characteristic
161	C₂×C₄₇	#2187	1	1	yes	characteristic
162	C₂×C₄₇	#2188	1	1	yes	characteristic
163	C₂×C₄₇	#2235	1	1	yes	characteristic
164	C₂×C₄₇	#2236	1	1	yes	characteristic
165	C₂²×Q₁₆	—	47	1	no	—
166	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
167	C₂×C₂²⋊Q₈	—	94	1	no	—
168	C₂²×C₄⋊C₄	—	94	1	no	—
169	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
170	#467	—	47	4	no	—
171	#467	—	94	2	no	—
172	C₂×#107	—	47	2	no	—
173	C₂×#107	—	47	1	no	—
174	C₂×#107	—	94	4	no	—
175	C₂×#107	—	94	1	no	—
176	C₂×#103	—	94	1	no	—
177	#358	—	94	4	no	—
178	C₂²⋊C₁₆	—	2	2	no	—
179	C₄₇	#12993	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	—	94	1	no	—
181	C₂³×C₄	—	188	1	no	—
182	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
183	C₂×Q₁₆	—	47	2	no	—
184	C₂×Q₁₆	—	94	2	no	—
185	C₂×Q₁₆	—	94	2	no	—
186	C₂×Q₁₆	—	94	1	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	C₄×D₄₇	1	2	yes	—
190	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
191	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
192	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
193	C₂²⋊Q₈	—	94	4	no	—
194	C₂²⋊Q₈	—	188	2	no	—
195	C₂×C₄⋊C₄	—	94	4	no	—
196	C₂×C₄⋊C₄	—	94	2	no	—
197	C₂×C₄⋊C₄	—	94	1	no	—
198	C₂×C₄⋊C₄	—	188	2	no	—
199	C₂×C₄⋊C₄	—	188	1	no	—
200	C₂×C₄⋊C₄	—	188	1	no	—
201	C₂×C₂²⋊C₄	—	188	1	no	—
202	C₂×C₁₆	—	2	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	2	2	no	—
205	#107	—	47	2	no	—
206	#107	—	47	4	no	—
207	#107	—	94	4	no	—
208	#107	—	94	2	no	—
209	#107	—	94	1	no	—
210	#103	—	94	4	no	—
211	C₂⁴	D₃₇₆	1	1	yes	characteristic
212	C₂×Q₈	—	94	2	no	—
213	C₂×Q₈	—	188	1	no	—
214	C₂×Q₈	—	188	1	no	—
215	C₂×Q₈	—	188	1	no	—
216	C₂×Q₈	—	188	1	no	—
217	C₂×Q₈	—	188	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	376	2	no	—
223	C₂²×C₄	—	376	2	no	—
224	C₂²×C₄	—	188	4	no	—
225	C₂²×C₄	—	188	1	no	—
226	C₂²×C₄	—	188	1	no	—
227	C₂²×C₄	—	188	2	no	—
228	C₂²×C₄	#10625325	1	2	yes	—
229	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
230	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
231	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
232	Q₁₆	—	94	4	no	—
233	Q₁₆	—	188	2	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
239	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂×C₈	D₄×D₄₇	1	2	yes	—
241	C₂×C₈	#10625352	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
243	C₄⋊C₄	—	94	2	no	—
244	C₄⋊C₄	—	94	4	no	—
245	C₄⋊C₄	—	188	4	no	—
246	C₄⋊C₄	—	188	4	no	—
247	C₄⋊C₄	—	188	1	no	—
248	C₄⋊C₄	—	188	2	no	—
249	C₂²⋊C₄	—	188	4	no	—
250	C₁₆	—	4	2	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	#13093946	1	2	yes	—
256	C₂³	#13093969	1	2	yes	—
257	C₂³	Dic₃₇₆	1	2	yes	—
258	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
259	Q₈	—	376	1	no	—
260	Q₈	—	376	1	no	—
261	Q₈	—	188	2	no	—
262	Q₈	—	188	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	376	8	no	—
270	C₂×C₄	—	376	1	no	—
271	C₂×C₄	—	376	2	no	—
272	C₂×C₄	—	376	2	no	—
273	C₂×C₄	—	376	1	no	—
274	C₂×C₄	—	376	1	no	—
275	C₂×C₄	—	376	1	no	—
276	C₂×C₄	—	188	4	no	—
277	C₂×C₄	—	188	2	no	—
278	C₂×C₄	—	188	2	no	—
279	C₂×C₄	#13093946	1	2	yes	—
280	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
281	C₂×C₄	#13093994	1	1	yes	characteristic
282	C₂×C₄	#13094008	1	1	yes	characteristic
283	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
284	C₂×C₄	#13094052	1	1	yes	characteristic
285	C₂×C₄	#13094057	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
288	C₈	#13093996	1	2	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	#15688928	1	4	yes	complemented
296	C₂²	C₂×#13093946	1	1	yes	characteristic
297	C₂²	#15689259	1	1	yes	characteristic
298	C₂²	#15689437	1	1	yes	characteristic
299	C₂²	C₂×#13093969	1	1	yes	characteristic
300	C₂²	C₂×Dic₃₇₆	1	1	yes	characteristic
301	C₂²	#15689561	1	1	yes	characteristic
302	C₂²	#15689566	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	376	4	no	—
305	C₄	—	376	2	no	—
306	C₄	—	376	2	no	—
307	C₄	#15689000	1	1	yes	characteristic
308	C₄	#15689125	1	1	yes	characteristic
309	C₄	#15689264	1	1	yes	characteristic
310	C₄	#15689442	1	1	yes	characteristic
311	C₂	—	2	4	no	—
312	C₂	#20243703	1	1	yes	characteristic
313	C₂	C₂×#15688928	1	1	yes	characteristic
314	C₂	#20245101	1	1	yes	characteristic
315	C₂	#20245319	1	1	yes	characteristic
316	C₂	#20245324	1	1	yes	characteristic
317	C₂	#20245586	1	1	yes	characteristic
318	C₂	#20245591	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₇₆
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×C₂²⋊C₁₆	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂⁴×C₄₇	D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₄₇
hypocenter	C₄₇	#12993
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#15688928	C₂	semidirect
4. #20241480	C₂	semidirect
10. #15689364	C₄	semidirect
25. C₂×Dic₁₈₈	C₂×C₄	semidirect
38. #13093958	C₂×C₄	semidirect
136. C₂²×C₄₇	#358	semidirect
179. C₄₇	#12993	semidirect
295. C₂²	#15688928	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689364	2
C₂²×#13093958	2
C₂×#15688928	2
#20241480	2
C₂×C₄₇×C₂²⋊C₁₆	2
#12993	47