Subgroups — #26370800

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689539	C₂	1	1	yes	maximal, characteristic, complemented
3	#20253060	C₂	1	2	yes	maximal, complemented
4	#20245502	C₂	1	4	yes	maximal, complemented
5	#20245480	C₂	1	2	yes	maximal, complemented
6	#20245475	C₂	1	2	yes	maximal, complemented
7	#20245098	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₇×#366	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	C₂²	1	2	yes	—
10	C₂×#13094030	C₂²	1	1	yes	characteristic
11	#15689704	—	2	2	no	—
12	#15689555	—	2	4	no	—
13	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic
14	C₂×#13093969	C₂²	1	2	yes	—
15	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
16	#15689539	—	2	4	no	—
17	C₂×#13093964	C₂²	1	1	yes	characteristic
18	C₂×#13093963	C₂²	1	2	yes	—
19	C₂×C₁₆×D₄₇	C₂²	1	2	yes	—
20	#15689370	—	2	4	no	—
21	#15689365	C₂²	1	4	yes	—
22	#15689360	C₂²	1	4	yes	—
23	#15689122	—	2	2	no	—
24	#15688926	C₂²	1	4	yes	—
25	#15688923	C₂²	1	4	yes	—
26	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#107×C₄₇	C₂²	1	2	yes	—
28	C₄₇×#366	C₂²	1	2	yes	—
29	C₄₇×#366	C₂²	1	2	yes	—
30	C₂×#10625350	—	2	2	no	—
31	C₂×#10625350	D₄	1	1	yes	characteristic
32	#13094046	—	2	4	no	—
33	#13094046	—	2	4	no	—
34	C₂×Dic₁₈₈	C₂³	1	2	yes	—
35	C₂×#10625325	—	2	2	no	—
36	C₂×D₃₇₆	C₂³	1	2	yes	—
37	#13094030	—	2	2	no	—
38	#13094030	—	2	2	no	—
39	C₂×#10625329	D₄	1	1	yes	characteristic
40	C₂×#10625324	D₄	1	2	yes	—
41	C₂×C₈×D₄₇	C₂³	1	2	yes	—
42	#13094004	—	2	4	no	—
43	#13094004	—	4	2	no	—
44	#13093999	—	2	4	no	—
45	#13093999	—	4	2	no	—
46	#13093998	—	4	4	no	—
47	Dic₃₇₆	—	2	2	no	—
48	#13093969	—	2	4	no	—
49	D₇₅₂	—	2	2	no	—
50	#13093964	—	2	2	no	—
51	#13093963	—	2	4	no	—
52	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
53	C₁₆×D₄₇	—	2	4	no	—
54	#13093947	—	2	4	no	—
55	#13093944	—	2	4	no	—
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
58	C₂×C₁₆×C₄₇	D₄	1	4	yes	complemented
59	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
61	#107×C₄₇	—	2	2	no	—
62	#107×C₄₇	C₂³	1	4	yes	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₄₇	—	4	2	no	—
66	C₂×C₂²⋊D₄₇	D₈	1	2	yes	—
67	#10625350	—	2	2	no	—
68	#10625350	—	2	2	no	—
69	#10625350	—	4	4	no	—
70	#10625350	—	4	4	no	—
71	C₂×D₁₈₈	—	2	2	no	—
72	C₂×D₁₈₈	—	4	4	no	—
73	C₂×D₁₈₈	D₈	1	1	yes	characteristic
74	C₂×C₄×D₄₇	—	4	2	no	—
75	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
76	C₂×Dic₉₄	—	2	2	no	—
77	C₂×Dic₉₄	D₈	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₄₇	#41	1	2	yes	—
81	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₄₇	—	2	4	no	—
83	C₄⋊C₄×C₄₇	—	4	2	no	—
84	(C₂×D₄₇)⋊C₄	—	4	4	no	—
85	(C₂×D₄₇)⋊C₄	—	4	4	no	—
86	C₄⋊Dic₄₇	—	4	4	no	—
87	#10625329	—	2	2	no	—
88	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
89	Dic₁₈₈	—	2	2	no	—
90	Dic₁₈₈	—	2	2	no	—
91	D₃₇₆	—	2	2	no	—
92	D₃₇₆	—	2	2	no	—
93	#10625325	—	4	4	no	—
94	#10625324	—	2	4	no	—
95	C₈×D₄₇	—	2	2	no	—
96	C₈×D₄₇	—	2	2	no	—
97	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
98	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
99	C₄₇⋊C₁₆	—	2	2	no	—
100	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
101	C₂²×D₄₇	—	4	2	no	—
102	C₂²×D₄₇	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	—	16	2	no	—
105	C₂×C₄×C₄₇	#121	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#135	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₂²⋊D₄₇	—	2	4	no	—
110	C₂²⋊D₄₇	—	8	4	no	—
111	C₂×Dic₄₇	—	4	2	no	—
112	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
113	D₁₈₈	—	2	2	no	—
114	D₁₈₈	—	4	2	no	—
115	D₁₈₈	—	4	2	no	—
116	D₁₈₈	—	16	4	no	—
117	C₄×D₄₇	—	2	2	no	—
118	C₄×D₄₇	—	2	2	no	—
119	C₄×D₄₇	—	8	4	no	—
120	Dic₉₄	—	2	2	no	—
121	Dic₉₄	—	4	2	no	—
122	Dic₉₄	—	4	2	no	—
123	C₈×C₄₇	#121	1	2	yes	—
124	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
126	C₄₇⋊C₈	—	2	2	no	—
127	#33610	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	1	no	—
129	C₂²×C₄₇	—	2	1	no	—
130	C₂²×C₄₇	#466	1	2	yes	—
131	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
132	C₂×D₄₇	—	2	2	no	—
133	C₂×D₄₇	—	2	2	no	—
134	C₂×D₄₇	—	2	2	no	—
135	C₂×D₄₇	—	8	2	no	—
136	C₂×D₄₇	—	8	2	no	—
137	C₂×D₄₇	—	8	2	no	—
138	C₄×C₄₇	—	8	4	no	—
139	C₄×C₄₇	#466	1	2	yes	—
140	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
141	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
142	Dic₄₇	—	2	2	no	—
143	Dic₄₇	—	8	2	no	—
144	C₂×#508	—	47	1	no	—
145	C₂×#504	—	47	1	no	—
146	#3061	—	47	2	no	—
147	#2220	—	47	4	no	—
148	#2216	—	47	2	no	—
149	#2215	—	47	2	no	—
150	C₂×#366	D₄₇	1	1	yes	characteristic, complemented
151	#2139	—	47	2	no	—
152	C₂×C₄₇	—	2	2	no	—
153	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
154	C₂×C₄₇	#3415	1	1	yes	characteristic
155	C₂×C₄₇	#3417	1	1	yes	characteristic
156	D₄₇	—	4	2	no	—
157	D₄₇	—	16	2	no	—
158	C₂×#135	—	47	2	no	—
159	C₂×#131	—	47	1	no	—
160	#537	—	94	2	no	—
161	#508	—	94	4	no	—
162	C₂×Q₃₂	—	47	1	no	—
163	C₂×SD₃₂	—	47	2	no	—
164	C₂×D₁₆	—	47	1	no	—
165	#504	—	94	4	no	—
166	C₂×M₃₂	—	47	1	no	—
167	C₂×M₃₂	—	47	2	no	—
168	C₂²×C₁₆	—	47	2	no	—
169	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
170	#468	—	94	4	no	—
171	#467	—	47	4	no	—
172	#466	—	47	4	no	—
173	C₂×#107	C₂×D₄₇	1	2	yes	—
174	#417	—	94	2	no	—
175	#366	C₂×D₄₇	1	2	yes	—
176	#366	C₂×D₄₇	1	2	yes	—
177	#358	—	47	4	no	—
178	#357	—	47	4	no	—
179	C₄₇	#33610	1	1	yes	characteristic, complemented, Sylow
180	C₂×#41	—	47	1	no	—
181	C₂×#41	—	94	2	no	—
182	#135	—	94	4	no	—
183	#135	—	94	4	no	—
184	C₂×Q₁₆	—	47	2	no	—
185	C₂×SD₁₆	—	94	2	no	—
186	C₂×D₈	—	47	2	no	—
187	#131	—	94	2	no	—
188	#131	—	94	2	no	—
189	C₂×M₁₆	—	47	1	no	—
190	C₂×M₁₆	—	47	2	no	—
191	C₂²×C₈	—	47	2	no	—
192	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
193	C₂²⋊Q₈	—	94	4	no	—
194	C₂²⋊Q₈	—	188	2	no	—
195	#121	—	94	4	no	—
196	#121	—	188	2	no	—
197	#121	—	188	4	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	Q₃₂	—	94	2	no	—
200	SD₃₂	—	94	4	no	—
201	D₁₆	—	94	2	no	—
202	M₃₂	—	94	2	no	—
203	M₃₂	—	94	4	no	—
204	C₂×C₁₆	—	47	2	no	—
205	C₂×C₁₆	—	94	4	no	—
206	C₂×C₁₆	D₁₈₈	1	4	yes	complemented
207	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
208	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
209	#107	—	2	2	no	—
210	#107	C₂²×D₄₇	1	4	yes	—
211	#103	—	94	4	no	—
212	#102	—	94	4	no	—
213	#41	—	94	2	no	—
214	#41	—	94	2	no	—
215	#41	—	188	4	no	—
216	#41	—	188	4	no	—
217	C₂×Q₈	—	47	1	no	—
218	C₂×Q₈	—	94	2	no	—
219	C₂×D₄	—	47	2	no	—
220	C₂×D₄	—	47	1	no	—
221	C₂×D₄	—	94	2	no	—
222	C₂×D₄	—	188	4	no	—
223	C₂×D₄	—	188	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	47	2	no	—
226	C₂²×C₄	—	188	2	no	—
227	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
228	Q₁₆	—	94	2	no	—
229	Q₁₆	—	94	2	no	—
230	SD₁₆	—	188	4	no	—
231	D₈	—	94	2	no	—
232	D₈	—	94	2	no	—
233	M₁₆	—	94	2	no	—
234	M₁₆	—	94	4	no	—
235	C₂×C₈	—	47	2	no	—
236	C₂×C₈	—	94	2	no	—
237	C₂×C₈	—	94	2	no	—
238	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
239	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
240	C₂×C₈	#10625354	1	2	yes	—
241	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	188	4	no	—
245	C₂²⋊C₄	—	188	4	no	—
246	C₂²⋊C₄	—	188	4	no	—
247	C₁₆	—	94	2	no	—
248	C₁₆	C₂×D₁₈₈	1	2	yes	—
249	C₁₆	C₂×D₁₈₈	1	2	yes	—
250	C₂³	—	47	2	no	—
251	C₂³	—	188	2	no	—
252	C₂³	D₈×D₄₇	1	1	yes	characteristic
253	Q₈	—	94	2	no	—
254	Q₈	—	188	2	no	—
255	Q₈	—	188	2	no	—
256	D₄	—	94	4	no	—
257	D₄	—	94	2	no	—
258	D₄	—	376	4	no	—
259	D₄	—	188	2	no	—
260	D₄	—	188	2	no	—
261	D₄	—	752	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	16	2	no	—
264	C₂×C₄	—	47	2	no	—
265	C₂×C₄	—	94	2	no	—
266	C₂×C₄	—	94	2	no	—
267	C₂×C₄	—	376	4	no	—
268	C₂×C₄	—	188	2	no	—
269	C₂×C₄	#13093999	1	2	yes	—
270	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
271	C₂×C₄	#13094047	1	2	yes	—
272	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
273	C₈	—	94	2	no	—
274	C₈	#13093999	1	2	yes	—
275	C₈	C₂²×D₁₈₈	1	1	yes	characteristic
276	C₈	C₂×#10625354	1	1	yes	characteristic
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	94	2	no	—
280	C₂²	—	94	2	no	—
281	C₂²	—	94	2	no	—
282	C₂²	—	376	2	no	—
283	C₂²	—	376	2	no	—
284	C₂²	—	376	2	no	—
285	C₂²	#15689360	1	2	yes	—
286	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
287	C₄	—	8	4	no	—
288	C₄	—	94	2	no	—
289	C₄	—	376	2	no	—
290	C₄	#15689360	1	2	yes	—
291	C₄	C₂×#13093999	1	1	yes	characteristic
292	C₄	C₂×#13094047	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	188	2	no	—
295	C₂	—	752	2	no	—
296	C₂	C₂×#15689360	1	1	yes	characteristic
297	C₂	#20256525	1	1	yes	characteristic
298	C₂	#20256537	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#366	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	C₂×#366	D₄₇
hypocenter	C₄₇	#33610
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
1. C₂×#15689555	C₂	semidirect
2. C₂×#15689539	C₂	semidirect
3. #20253060	C₂	semidirect
4. #20245502	C₂	semidirect
5. #20245480	C₂	semidirect
6. #20245475	C₂	semidirect
7. #20245098	C₂	semidirect
8. C₂×C₄₇×#366	C₂	semidirect
58. C₂×C₁₆×C₄₇	D₄	semidirect
150. C₂×#366	D₄₇	semidirect
179. C₄₇	#33610	semidirect
206. C₂×C₁₆	D₁₈₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689555	2
C₂×#15689539	2
#20253060	2
#20245502	2
#20245480	2
#20245475	2
#20245098	2
C₂×C₄₇×#366	2
#33610	47