Subgroups — #26115151

Order 12032.

590 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic
2	#20244182	C₂	1	2	yes	maximal, complemented
3	C₂×#15688874	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15688852	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₇₆	C₄	1	1	yes	characteristic
7	C₂×C₄×D₁₈₈	C₂²	1	2	yes	—
8	C₄×D₃₇₆	—	2	2	no	—
9	C₄×D₃₇₆	—	2	2	no	—
10	C₂×#13093958	C₄	1	1	yes	characteristic
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×#13093946	C₄	1	2	yes	—
13	C₂×(C₂×D₄₇)⋊C₈	—	2	4	no	—
14	#15688874	—	2	2	no	—
15	#15688852	—	2	4	no	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#322	—	2	2	no	—
20	C₂²×D₁₈₈	C₂×C₄	1	2	yes	—
21	C₂²×C₄×D₄₇	—	2	2	no	—
22	C₂×D₃₇₆	—	2	2	no	—
23	C₂×D₃₇₆	—	2	2	no	—
24	C₂×D₃₇₆	C₈	1	4	yes	complemented
25	C₂²×C₄₇⋊C₈	—	2	2	no	—
26	C₄×D₁₈₈	—	2	2	no	—
27	C₄×D₁₈₈	—	2	2	no	—
28	C₄×D₁₈₈	—	2	2	no	—
29	C₄×D₁₈₈	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
31	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
32	#13093958	—	2	2	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
35	#13093946	—	2	4	no	—
36	(C₂×D₄₇)⋊C₈	—	4	8	no	—
37	C₂²×C₈×C₄₇	—	2	1	no	—
38	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	D₄	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	C₄×C₈×C₄₇	—	2	1	no	—
43	C₄×C₈×C₄₇	—	2	1	no	—
44	C₄×C₈×C₄₇	—	2	1	no	—
45	C₄×C₈×C₄₇	D₄	1	2	yes	—
46	C₄×C₈×C₄₇	D₄	1	2	yes	—
47	C₄×C₈×C₄₇	D₄	1	2	yes	—
48	C₂³×D₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
52	C₂²×Dic₄₇	—	2	2	no	—
53	C₂×D₁₈₈	—	2	2	no	—
54	C₂×D₁₈₈	—	2	2	no	—
55	C₂×D₁₈₈	—	2	2	no	—
56	C₂×D₁₈₈	—	2	2	no	—
57	C₂×D₁₈₈	—	2	2	no	—
58	C₂×D₁₈₈	C₂×C₈	1	4	yes	—
59	C₂×C₄×D₄₇	—	4	2	no	—
60	C₂×C₄×D₄₇	—	4	2	no	—
61	C₂×C₄×D₄₇	—	4	4	no	—
62	C₂×C₄×D₄₇	—	4	4	no	—
63	C₂×C₈×C₄₇	—	2	1	no	—
64	C₂×C₈×C₄₇	—	2	1	no	—
65	C₂×C₈×C₄₇	—	2	2	no	—
66	C₂×C₈×C₄₇	—	2	2	no	—
67	C₂×C₈×C₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	—	4	2	no	—
69	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
73	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₄₇	#41	1	1	yes	characteristic
77	(C₂×D₄₇)⋊C₄	—	4	2	no	—
78	(C₂×D₄₇)⋊C₄	—	4	2	no	—
79	C₄⋊Dic₄₇	—	2	2	no	—
80	C₄⋊Dic₄₇	—	2	2	no	—
81	C₂×C₄₇⋊C₈	—	4	2	no	—
82	C₂×C₄₇⋊C₈	—	4	4	no	—
83	D₃₇₆	—	2	8	no	—
84	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
85	C₂²×D₄₇	—	2	4	no	—
86	C₂²×D₄₇	—	4	2	no	—
87	C₂²×D₄₇	—	4	2	no	—
88	C₂²×D₄₇	—	4	4	no	—
89	C₂²×D₄₇	—	4	4	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
94	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
95	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
96	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
100	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
101	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
102	C₂×Dic₄₇	—	4	2	no	—
103	C₂×Dic₄₇	—	4	4	no	—
104	D₁₈₈	—	2	4	no	—
105	D₁₈₈	—	2	4	no	—
106	D₁₈₈	—	4	2	no	—
107	D₁₈₈	—	4	2	no	—
108	C₄×D₄₇	—	8	4	no	—
109	C₄×D₄₇	—	8	2	no	—
110	C₄×D₄₇	—	8	2	no	—
111	C₈×C₄₇	—	2	2	no	—
112	C₈×C₄₇	—	4	2	no	—
113	C₈×C₄₇	—	4	2	no	—
114	C₈×C₄₇	C₂²⋊C₈	1	4	yes	complemented
115	C₄₇⋊C₈	—	8	2	no	—
116	#9659	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	#390	1	1	yes	characteristic
118	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
119	C₂²×C₄₇	#436	1	1	yes	characteristic
120	C₂²×C₄₇	#442	1	2	yes	—
121	C₂²×C₄₇	#444	1	2	yes	—
122	C₂×D₄₇	—	4	8	no	—
123	C₂×D₄₇	—	4	4	no	—
124	C₂×D₄₇	—	8	4	no	—
125	C₂×D₄₇	—	8	2	no	—
126	C₂×D₄₇	—	8	2	no	—
127	C₄×C₄₇	—	2	1	no	—
128	C₄×C₄₇	—	2	1	no	—
129	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₄×C₄₇	#408	1	1	yes	characteristic
131	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
132	C₄×C₄₇	#436	1	1	yes	characteristic
133	C₄×C₄₇	#469	1	2	yes	—
134	C₄×C₄₇	#496	1	2	yes	—
135	Dic₄₇	—	8	2	no	—
136	C₂×C₄×D₈	—	47	1	no	—
137	#1954	—	47	2	no	—
138	C₂×#334	—	47	1	no	—
139	C₂×D₄⋊C₈	—	47	2	no	—
140	C₂×#322	D₄₇	1	1	yes	characteristic, complemented
141	C₂×C₄₇	#1580	1	2	yes	—
142	C₂×C₄₇	#1594	1	2	yes	—
143	C₂×C₄₇	#1954	1	1	yes	characteristic
144	C₂×C₄₇	#1972	1	1	yes	characteristic
145	C₂×C₄₇	#1975	1	1	yes	characteristic
146	D₄₇	—	8	4	no	—
147	C₂²×D₈	—	47	1	no	—
148	C₂×C₄×D₄	—	47	2	no	—
149	C₄×D₈	—	94	2	no	—
150	C₄×D₈	—	94	2	no	—
151	C₂×#107	—	47	1	no	—
152	C₂×C₄⋊C₈	—	47	2	no	—
153	C₂×#102	—	47	2	no	—
154	C₂×C₂²⋊C₈	—	94	4	no	—
155	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
158	#334	—	94	2	no	—
159	D₄⋊C₈	—	94	4	no	—
160	#322	—	2	2	no	—
161	C₄₇	#9659	1	1	yes	characteristic, complemented, Sylow
162	C₂²×D₄	—	47	2	no	—
163	C₂³×C₄	—	94	2	no	—
164	C₂×D₈	—	47	4	no	—
165	C₂×D₈	—	94	2	no	—
166	C₂×D₈	—	94	2	no	—
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	—	94	2	no	—
169	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
170	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
171	C₂²×C₈	D₁₈₈	1	2	yes	—
172	C₄×D₄	—	94	2	no	—
173	C₄×D₄	—	94	2	no	—
174	C₄×D₄	—	94	2	no	—
175	C₄×D₄	—	94	2	no	—
176	C₂×C₄⋊C₄	—	47	2	no	—
177	C₂×C₂²⋊C₄	—	94	2	no	—
178	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
179	#107	—	94	2	no	—
180	C₄⋊C₈	—	94	2	no	—
181	C₄⋊C₈	—	94	2	no	—
182	#102	—	94	4	no	—
183	C₂²⋊C₈	—	188	8	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	D₁₈₈	1	2	yes	—
188	C₄×C₈	D₁₈₈	1	2	yes	—
189	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
190	C₂⁴	—	94	2	no	—
191	C₂×D₄	—	47	4	no	—
192	C₂×D₄	—	94	2	no	—
193	C₂×D₄	—	94	2	no	—
194	C₂×D₄	—	94	2	no	—
195	C₂×D₄	—	94	2	no	—
196	C₂×D₄	—	94	2	no	—
197	C₂²×C₄	—	94	2	no	—
198	C₂²×C₄	—	188	2	no	—
199	C₂²×C₄	—	188	2	no	—
200	C₂²×C₄	—	188	4	no	—
201	C₂²×C₄	—	188	4	no	—
202	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
204	C₂²×C₄	#10625350	1	1	yes	characteristic
205	D₈	—	94	8	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	188	2	no	—
213	C₂×C₈	—	188	4	no	—
214	C₂×C₈	C₈×D₄₇	1	2	yes	—
215	C₂×C₈	#10625324	1	2	yes	—
216	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
217	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
218	C₄⋊C₄	—	94	2	no	—
219	C₄⋊C₄	—	94	2	no	—
220	C₂²⋊C₄	—	188	2	no	—
221	C₂²⋊C₄	—	188	2	no	—
222	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
223	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
224	C₄²	#10625350	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
226	C₂³	—	94	4	no	—
227	C₂³	—	188	4	no	—
228	C₂³	—	188	4	no	—
229	C₂³	—	188	2	no	—
230	C₂³	—	188	2	no	—
231	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
232	D₄	—	94	4	no	—
233	D₄	—	94	4	no	—
234	D₄	—	188	2	no	—
235	D₄	—	188	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	376	2	no	—
238	C₂×C₄	—	376	2	no	—
239	C₂×C₄	—	376	4	no	—
240	C₂×C₄	—	188	4	no	—
241	C₂×C₄	—	188	2	no	—
242	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
244	C₂×C₄	#13093998	1	1	yes	characteristic
245	C₂×C₄	#13093998	1	1	yes	characteristic
246	C₂×C₄	#13094012	1	1	yes	characteristic
247	C₂×C₄	#13094021	1	1	yes	characteristic
248	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
249	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
250	C₂×C₄	#13094030	1	1	yes	characteristic
251	C₂×C₄	#13094030	1	1	yes	characteristic
252	C₂×C₄	#13094052	1	4	yes	—
253	C₈	—	2	2	no	—
254	C₈	—	4	2	no	—
255	C₈	—	4	2	no	—
256	C₈	—	376	2	no	—
257	C₈	(C₂×D₄₇)⋊C₈	1	4	yes	complemented
258	C₂²	—	376	2	no	—
259	C₂²	—	376	2	no	—
260	C₂²	—	376	4	no	—
261	C₂²	—	188	8	no	—
262	C₂²	—	188	4	no	—
263	C₂²	#15689025	1	1	yes	characteristic
264	C₂²	C₈×D₁₈₈	1	1	yes	characteristic
265	C₂²	#15689202	1	1	yes	characteristic
266	C₂²	#15689236	1	2	yes	—
267	C₂²	#15689246	1	2	yes	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	376	2	no	—
271	C₄	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
272	C₄	#15689089	1	1	yes	characteristic
273	C₄	C₈×D₁₈₈	1	1	yes	characteristic
274	C₄	#15689202	1	1	yes	characteristic
275	C₄	#15689376	1	2	yes	—
276	C₄	#15689512	1	2	yes	—
277	C₂	—	376	4	no	—
278	C₂	#20242213	1	2	yes	—
279	C₂	#20242303	1	2	yes	—
280	C₂	#20244182	1	1	yes	characteristic
281	C₂	#20244264	1	1	yes	characteristic
282	C₂	#20244281	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂³×C₄₇	C₄×D₄
prime radical	C₂²×D₃₇₆	C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#9659
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #20244182	C₂	semidirect
3. C₂×#15688874	C₂	semidirect
4. C₂×#15688852	C₂	semidirect
5. C₂×C₄₇×#322	C₂	semidirect
24. C₂×D₃₇₆	C₈	semidirect
114. C₈×C₄₇	C₂²⋊C₈	semidirect
140. C₂×#322	D₄₇	semidirect
161. C₄₇	#9659	semidirect
257. C₈	(C₂×D₄₇)⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₃₇₆	2
#20244182	2
C₂×#15688874	2
C₂×#15688852	2
C₂×C₄₇×#322	2
#9659	47