Subgroups — C₂×#20242213 #26185678

Order 12032.

894 conjugacy classes of subgroups.

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

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₄×C₄₇	C₂³×C₈
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂³×C₄₇	C₄×D₄
prime radical	C₂²×D₃₇₆	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	C₂×#1580
LCS	G ▷ 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. C₂×C₈×D₁₈₈	C₂	semidirect
3. #20242213	C₂	direct (N × H)
4. C₂×#15688874	C₂	semidirect
5. C₂×#15688852	C₂	semidirect
6. C₂×C₄₇×#322	C₂	semidirect
13. C₈×D₁₈₈	C₂²	semidirect
18. #15688874	C₂²	semidirect
19. #15688852	C₂²	semidirect
23. C₄₇×#322	C₂²	semidirect
26. C₂×D₃₇₆	C₈	semidirect
60. C₂×D₁₈₈	C₂×C₈	semidirect
70. C₂×C₈×C₄₇	C₂×C₈	semidirect
84. D₃₇₆	C₂×C₈	semidirect
105. D₁₈₈	C₂²×C₈	semidirect
112. C₈×C₄₇	C₂²×C₈	semidirect
136. C₂×#322	D₄₇	semidirect
137. C₂×C₄₇	#1580	semidirect
158. #322	C₂×D₄₇	semidirect
159. C₄₇	C₂×#1580	semidirect
210. C₂×C₈	C₈×D₄₇	semidirect
249. C₈	C₂×C₈×D₄₇	semidirect
267. C₂	#20242213	direct (N × H)

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
C₂×C₈×D₁₈₈	2
#20242213	2
C₂×#15688874	2
C₂×#15688852	2
C₂×C₄₇×#322	2
C₂×#1580	47

Subgroups — C2×#20242213 #26185678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242213 #26185678