Subgroups — C₂×#20299409 #28405334

Order 14464.

1460 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813669	C₂	1	1	yes	maximal, characteristic, complemented
2	#20299409	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16813652	C₂	1	3	yes	maximal, complemented
4	C₂×#16813648	C₂	1	3	yes	maximal, complemented
5	C₂×#16813642	C₂	1	3	yes	maximal, complemented
6	C₂²×#14268635	C₂	1	3	yes	maximal, complemented
7	C₂×#16813629	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×H(5,𝔽₂)×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#16813669	—	2	4	no	—
10	#16813669	C₂²	1	8	yes	complemented
11	C₂×#14268647	C₂²	1	3	yes	complemented
12	C₂×#14268646	—	2	3	no	—
13	C₂²×Dic₂₂₆	C₂²	1	3	yes	complemented
14	C₂×Q₈×D₁₁₃	—	2	1	no	—
15	#16813652	C₂²	1	24	yes	complemented
16	#16813648	C₂²	1	24	yes	complemented
17	#16813642	C₂²	1	24	yes	complemented
18	C₂×#14268637	C₂²	1	3	yes	complemented
19	C₂×#14268635	C₂²	1	3	yes	complemented
20	C₂×#14268635	C₂²	1	6	yes	complemented
21	C₂×#14268635	C₂²	1	24	yes	complemented
22	C₂×#14268634	C₂²	1	1	yes	characteristic, complemented
23	C₂×#14268632	C₂²	1	3	yes	complemented
24	#16813629	C₂²	1	8	yes	complemented
25	C₂×#14268631	C₂²	1	3	yes	complemented
26	C₂²×#11744751	C₂²	1	3	yes	complemented
27	H(5,𝔽₂)×C₁₁₃	—	2	4	no	—
28	H(5,𝔽₂)×C₁₁₃	C₂²	1	8	yes	complemented
29	C₂×#41×C₁₁₃	—	2	1	no	—
30	C₂×#41×C₁₁₃	C₂²	1	3	yes	—
31	C₂×#41×C₁₁₃	C₂²	1	1	yes	characteristic
32	C₂²×D₄×C₁₁₃	—	2	3	no	—
33	C₂²×D₄×C₁₁₃	C₂²	1	3	yes	—
34	#14268647	—	2	6	no	—
35	#14268647	C₂³	1	12	yes	complemented
36	#14268646	—	2	24	no	—
37	C₂×Dic₂₂₆	—	2	3	no	—
38	C₂×Dic₂₂₆	—	2	6	no	—
39	C₂×Dic₂₂₆	C₂³	1	3	yes	—
40	C₂×Dic₂₂₆	C₂³	1	12	yes	complemented
41	Q₈×D₁₁₃	—	2	8	no	—
42	C₂×D₄₅₂	C₂³	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₁₃	—	2	3	no	—
44	C₂²×Dic₁₁₃	—	2	3	no	—
45	C₂×C₄×D₁₁₃	—	2	3	no	—
46	#14268637	C₂³	1	12	yes	complemented
47	#14268635	C₂³	1	12	yes	complemented
48	#14268635	C₂³	1	24	yes	complemented
49	#14268634	C₂³	1	4	yes	complemented
50	#14268632	C₂³	1	12	yes	complemented
51	#14268631	C₂³	1	12	yes	complemented
52	C₂×#11744751	C₂³	1	1	yes	characteristic
53	C₂×#11744751	C₂³	1	3	yes	—
54	C₂×#11744751	C₂³	1	12	yes	complemented
55	C₂⁴×C₁₁₃	—	2	3	no	—
56	#41×C₁₁₃	—	2	8	no	—
57	#41×C₁₁₃	—	2	6	no	—
58	#41×C₁₁₃	D₄	1	4	yes	—
59	#41×C₁₁₃	C₂³	1	12	yes	—
60	#41×C₁₁₃	C₂³	1	4	yes	—
61	C₂×Q₈×C₁₁₃	D₄	1	1	yes	characteristic
62	C₂×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₁₃	—	2	3	no	—
64	C₂×D₄×C₁₁₃	—	2	24	no	—
65	C₂×D₄×C₁₁₃	—	2	3	no	—
66	C₂×D₄×C₁₁₃	—	2	6	no	—
67	C₂×D₄×C₁₁₃	D₄	1	3	yes	—
68	C₂×D₄×C₁₁₃	C₂³	1	3	yes	—
69	C₂×D₄×C₁₁₃	C₂³	1	12	yes	—
70	C₂²×C₄×C₁₁₃	—	2	3	no	—
71	C₂²×C₄×C₁₁₃	C₂³	1	3	yes	—
72	C₂³×C₁₁₃	—	2	3	no	—
73	C₂³×C₁₁₃	—	2	12	no	—
74	C₂³×C₁₁₃	—	8	6	no	—
75	C₂³×C₁₁₃	C₂×D₄	1	3	yes	—
76	C₂²×D₁₁₃	—	2	1	no	—
77	Q₈×C₁₁₃	—	2	1	no	—
78	Q₈×C₁₁₃	C₂×D₄	1	4	yes	—
79	Q₈×C₁₁₃	C₂⁴	1	2	yes	—
80	D₄×C₁₁₃	—	2	12	no	—
81	D₄×C₁₁₃	—	2	3	no	—
82	D₄×C₁₁₃	—	2	12	no	—
83	D₄×C₁₁₃	C₂×D₄	1	12	yes	—
84	D₄×C₁₁₃	C₂⁴	1	6	yes	—
85	C₂×C₄×C₁₁₃	—	2	1	no	—
86	C₂×C₄×C₁₁₃	—	2	12	no	—
87	C₂×C₄×C₁₁₃	C₂×D₄	1	3	yes	—
88	C₂×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
89	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₄×C₁₁₃	C₂⁴	1	6	yes	—
91	C₂²⋊D₁₁₃	—	2	12	no	—
92	C₂×Dic₁₁₃	—	2	3	no	—
93	C₂×Dic₁₁₃	—	2	12	no	—
94	D₄₅₂	—	2	1	no	—
95	D₄₅₂	C₂⁴	1	2	yes	—
96	C₄×D₁₁₃	—	2	12	no	—
97	Dic₂₂₆	—	2	12	no	—
98	Dic₂₂₆	—	2	3	no	—
99	Dic₂₂₆	C₂⁴	1	6	yes	—
100	#11744751	C₂⁴	1	2	yes	—
101	#11744751	C₂⁴	1	6	yes	—
102	C₂²×C₁₁₃	—	2	6	no	—
103	C₂²×C₁₁₃	—	2	6	no	—
104	C₂²×C₁₁₃	—	4	6	no	—
105	C₂²×C₁₁₃	—	8	12	no	—
106	C₂²×C₁₁₃	C₂²×D₄	1	6	yes	—
107	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×D₁₁₃	—	2	2	no	—
109	C₂×D₁₁₃	—	4	2	no	—
110	C₄×C₁₁₃	—	2	2	no	—
111	C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₄×C₁₁₃	C₂²×D₄	1	6	yes	—
113	C₄×C₁₁₃	C₂⁵	1	1	yes	characteristic
114	Dic₁₁₃	—	2	6	no	—
115	C₂×C₁₁₃	—	2	6	no	—
116	C₂×C₁₁₃	—	4	6	no	—
117	C₂×C₁₁₃	#577	1	2	yes	complemented
118	C₂×C₁₁₃	C₂³×D₄	1	1	yes	characteristic
119	D₁₁₃	—	4	2	no	—
120	C₂×#577	—	113	1	no	maximal, Sylow
121	C₁₁₃	C₂×#577	1	1	yes	characteristic, complemented, Sylow
122	C₂×2₋¹⁺⁴	—	113	1	no	—
123	C₂×H(5,𝔽₂)	D₁₁₃	1	1	yes	characteristic, complemented
124	#577	—	113	16	no	—
125	C₂×#137	—	113	3	no	—
126	C₂×AGL(1,ℤ/8)	—	113	3	no	—
127	C₂×#135	—	113	3	no	—
128	C₂²×SD₁₆	—	113	3	no	—
129	C₂×#131	—	113	1	no	—
130	2₋¹⁺⁴	—	113	8	no	—
131	2₋¹⁺⁴	—	226	4	no	—
132	H(5,𝔽₂)	—	2	4	no	—
133	H(5,𝔽₂)	C₂×D₁₁₃	1	8	yes	complemented
134	C₂×#41	—	2	1	no	—
135	C₂×#41	—	113	3	no	—
136	C₂×#41	—	226	3	no	—
137	C₂×#41	C₂×D₁₁₃	1	3	yes	—
138	C₂×#41	C₂×D₁₁₃	1	1	yes	characteristic
139	C₂²×Q₈	—	113	3	no	—
140	C₂²×Q₈	—	226	1	no	—
141	C₂²×D₄	—	2	3	no	—
142	C₂²×D₄	C₂×D₁₁₃	1	3	yes	—
143	#137	—	113	24	no	—
144	AGL(1,ℤ/8)	—	113	24	no	—
145	#135	—	113	24	no	—
146	C₂×Q₁₆	—	113	3	no	—
147	C₂×SD₁₆	—	113	1	no	—
148	C₂×SD₁₆	—	113	3	no	—
149	C₂×SD₁₆	—	113	6	no	—
150	C₂×SD₁₆	—	113	24	no	—
151	C₂×D₈	—	113	3	no	—
152	#131	—	113	8	no	—
153	C₂×M₁₆	—	113	3	no	—
154	C₂²×C₈	—	113	3	no	—
155	C₂⁴	—	2	3	no	—
156	#41	—	2	8	no	—
157	#41	—	2	6	no	—
158	#41	—	113	12	no	—
159	#41	—	226	24	no	—
160	#41	—	226	6	no	—
161	#41	C₂²⋊D₁₁₃	1	4	yes	—
162	#41	C₂²×D₁₁₃	1	12	yes	—
163	#41	C₂²×D₁₁₃	1	4	yes	—
164	C₂×Q₈	—	113	3	no	—
165	C₂×Q₈	—	113	12	no	—
166	C₂×Q₈	—	226	8	no	—
167	C₂×Q₈	—	226	3	no	—
168	C₂×Q₈	—	226	6	no	—
169	C₂×Q₈	C₂²⋊D₁₁₃	1	1	yes	characteristic
170	C₂×Q₈	C₂²×D₁₁₃	1	1	yes	characteristic
171	C₂×D₄	—	2	3	no	—
172	C₂×D₄	—	2	24	no	—
173	C₂×D₄	—	2	3	no	—
174	C₂×D₄	—	2	6	no	—
175	C₂×D₄	—	113	1	no	—
176	C₂×D₄	—	226	3	no	—
177	C₂×D₄	C₂²⋊D₁₁₃	1	3	yes	—
178	C₂×D₄	C₂²×D₁₁₃	1	3	yes	—
179	C₂×D₄	C₂²×D₁₁₃	1	12	yes	—
180	C₂²×C₄	—	2	3	no	—
181	C₂²×C₄	—	226	3	no	—
182	C₂²×C₄	—	226	3	no	—
183	C₂²×C₄	C₂²×D₁₁₃	1	3	yes	—
184	Q₁₆	—	113	12	no	—
185	SD₁₆	—	113	4	no	—
186	SD₁₆	—	113	12	no	—
187	SD₁₆	—	113	24	no	—
188	D₈	—	113	12	no	—
189	M₁₆	—	113	12	no	—
190	C₂×C₈	—	113	1	no	—
191	C₂×C₈	—	113	3	no	—
192	C₂×C₈	—	113	12	no	—
193	C₂³	—	2	3	no	—
194	C₂³	—	2	12	no	—
195	C₂³	—	8	6	no	—
196	C₂³	—	226	1	no	—
197	C₂³	C₂×C₂²⋊D₁₁₃	1	3	yes	—
198	Q₈	—	2	1	no	—
199	Q₈	—	113	6	no	—
200	Q₈	—	226	12	no	—
201	Q₈	—	226	3	no	—
202	Q₈	C₂×C₂²⋊D₁₁₃	1	4	yes	—
203	Q₈	C₂³×D₁₁₃	1	2	yes	—
204	D₄	—	2	12	no	—
205	D₄	—	2	12	no	—
206	D₄	—	2	3	no	—
207	D₄	—	113	2	no	—
208	D₄	—	226	12	no	—
209	D₄	—	226	1	no	—
210	D₄	C₂×C₂²⋊D₁₁₃	1	12	yes	—
211	D₄	C₂³×D₁₁₃	1	6	yes	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	12	no	—
214	C₂×C₄	—	226	12	no	—
215	C₂×C₄	—	226	3	no	—
216	C₂×C₄	—	226	12	no	—
217	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	6	yes	—
218	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	3	yes	—
219	C₂×C₄	C₂³×D₁₁₃	1	6	yes	—
220	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
221	C₈	—	113	2	no	—
222	C₈	—	113	6	no	—
223	C₂²	—	2	6	no	—
224	C₂²	—	2	6	no	—
225	C₂²	—	4	6	no	—
226	C₂²	—	8	12	no	—
227	C₂²	—	452	2	no	—
228	C₂²	—	226	2	no	—
229	C₂²	C₂²×C₂²⋊D₁₁₃	1	6	yes	—
230	C₂²	C₂²×C₂²⋊D₁₁₃	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	226	6	no	—
233	C₄	C₂²×C₂²⋊D₁₁₃	1	6	yes	—
234	C₄	C₂²×C₂²⋊D₁₁₃	1	1	yes	characteristic
235	C₄	C₂⁴×D₁₁₃	1	1	yes	characteristic
236	C₂	—	2	6	no	—
237	C₂	—	4	6	no	—
238	C₂	—	452	2	no	—
239	C₂	#20299409	1	2	yes	direct product factor
240	C₂	C₂³×C₂²⋊D₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₁₃
commutator, cosocle	C₄×C₁₁₃	C₂⁵
Frattini	C₄	C₂⁴×D₁₁₃
Fitting	C₂×H(5,𝔽₂)×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂²×D₄
hypercenter	C₂×H(5,𝔽₂)	D₁₁₃
hypocenter	C₁₁₃	C₂×#577
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#16813669	C₂	semidirect
2. #20299409	C₂	direct (N × H)
3. C₂×#16813652	C₂	semidirect
4. C₂×#16813648	C₂	semidirect
5. C₂×#16813642	C₂	semidirect
6. C₂²×#14268635	C₂	semidirect
7. C₂×#16813629	C₂	semidirect
8. C₂×H(5,𝔽₂)×C₁₁₃	C₂	semidirect
10. #16813669	C₂²	semidirect
11. C₂×#14268647	C₂²	semidirect
13. C₂²×Dic₂₂₆	C₂²	semidirect
15. #16813652	C₂²	semidirect
16. #16813648	C₂²	semidirect
17. #16813642	C₂²	semidirect
18. C₂×#14268637	C₂²	semidirect
19. C₂×#14268635	C₂²	semidirect
20. C₂×#14268635	C₂²	semidirect
21. C₂×#14268635	C₂²	semidirect
22. C₂×#14268634	C₂²	semidirect
23. C₂×#14268632	C₂²	semidirect
24. #16813629	C₂²	semidirect
25. C₂×#14268631	C₂²	semidirect
26. C₂²×#11744751	C₂²	semidirect
28. H(5,𝔽₂)×C₁₁₃	C₂²	semidirect
35. #14268647	C₂³	semidirect
40. C₂×Dic₂₂₆	C₂³	semidirect
46. #14268637	C₂³	semidirect
47. #14268635	C₂³	semidirect
48. #14268635	C₂³	semidirect
49. #14268634	C₂³	semidirect
50. #14268632	C₂³	semidirect
51. #14268631	C₂³	semidirect
54. C₂×#11744751	C₂³	semidirect
117. C₂×C₁₁₃	#577	semidirect
121. C₁₁₃	C₂×#577	semidirect
123. C₂×H(5,𝔽₂)	D₁₁₃	semidirect
133. H(5,𝔽₂)	C₂×D₁₁₃	semidirect
239. C₂	#20299409	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×#16813669	2
#20299409	2
C₂×#16813652	2
C₂×#16813648	2
C₂×#16813642	2
C₂²×#14268635	2
C₂×#16813629	2
C₂×H(5,𝔽₂)×C₁₁₃	2
C₂×#577	113

Subgroups — C2×#20299409 #28405334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20299409 #28405334