Subgroups — C₂×#20265980 #27185883

Order 12928.

1512 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692301	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265980	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15692281	C₂	1	3	yes	maximal, complemented
4	C₂×#15692280	C₂	1	3	yes	maximal, complemented
5	C₂×#15692275	C₂	1	3	yes	maximal, complemented
6	C₂²×#13108059	C₂	1	3	yes	maximal, complemented
7	C₂×#15692262	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×H(5,𝔽₂)×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#15692301	—	2	4	no	—
10	#15692301	C₂²	1	8	yes	complemented
11	C₂×#13108075	—	2	1	no	—
12	C₂×#13108074	C₂²	1	3	yes	complemented
13	C₂²×D₄₀₄	C₂²	1	3	yes	complemented
14	C₂×D₄×D₁₀₁	—	2	3	no	—
15	#15692281	C₂²	1	24	yes	complemented
16	#15692280	C₂²	1	24	yes	complemented
17	#15692275	C₂²	1	24	yes	complemented
18	C₂×#13108064	C₂²	1	1	yes	characteristic, complemented
19	C₂×#13108062	C₂²	1	3	yes	complemented
20	C₂×#13108061	C₂²	1	3	yes	complemented
21	C₂×#13108059	C₂²	1	3	yes	complemented
22	C₂×#13108059	C₂²	1	6	yes	complemented
23	C₂×#13108059	C₂²	1	24	yes	complemented
24	#15692262	C₂²	1	8	yes	complemented
25	C₂×#13108058	C₂²	1	3	yes	complemented
26	C₂²×#11717579	C₂²	1	3	yes	complemented
27	H(5,𝔽₂)×C₁₀₁	—	2	4	no	—
28	H(5,𝔽₂)×C₁₀₁	C₂²	1	8	yes	complemented
29	C₂×#41×C₁₀₁	—	2	1	no	—
30	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
31	C₂×#41×C₁₀₁	C₂²	1	3	yes	complemented
32	C₂²×D₄×C₁₀₁	—	2	3	no	—
33	C₂²×D₄×C₁₀₁	C₂²	1	3	yes	complemented
34	C₂³×D₁₀₁	—	2	3	no	—
35	#13108075	—	2	8	no	—
36	#13108074	—	2	6	no	—
37	#13108074	C₂³	1	12	yes	complemented
38	C₂×Dic₂₀₂	C₂³	1	1	yes	characteristic
39	C₂×D₄₀₄	—	2	3	no	—
40	C₂×D₄₀₄	—	2	6	no	—
41	C₂×D₄₀₄	C₂³	1	3	yes	—
42	C₂×D₄₀₄	C₂³	1	12	yes	complemented
43	C₂×C₂²⋊D₁₀₁	—	2	3	no	—
44	D₄×D₁₀₁	—	2	24	no	—
45	C₂×C₄×D₁₀₁	—	2	3	no	—
46	#13108064	C₂³	1	4	yes	complemented
47	#13108062	C₂³	1	12	yes	complemented
48	#13108061	C₂³	1	12	yes	complemented
49	#13108059	C₂³	1	12	yes	complemented
50	#13108059	C₂³	1	24	yes	complemented
51	#13108058	C₂³	1	12	yes	complemented
52	C₂×#11717579	C₂³	1	1	yes	characteristic
53	C₂×#11717579	C₂³	1	3	yes	—
54	C₂×#11717579	C₂³	1	12	yes	complemented
55	C₂⁴×C₁₀₁	—	2	3	no	—
56	#41×C₁₀₁	—	2	8	no	—
57	#41×C₁₀₁	—	2	6	no	—
58	#41×C₁₀₁	D₄	1	4	yes	—
59	#41×C₁₀₁	C₂³	1	4	yes	—
60	#41×C₁₀₁	C₂³	1	12	yes	complemented
61	C₂×Q₈×C₁₀₁	D₄	1	1	yes	characteristic
62	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₀₁	—	2	3	no	—
64	C₂×D₄×C₁₀₁	—	2	24	no	—
65	C₂×D₄×C₁₀₁	—	2	3	no	—
66	C₂×D₄×C₁₀₁	—	2	6	no	—
67	C₂×D₄×C₁₀₁	D₄	1	3	yes	—
68	C₂×D₄×C₁₀₁	C₂³	1	3	yes	—
69	C₂×D₄×C₁₀₁	C₂³	1	12	yes	complemented
70	C₂²×C₄×C₁₀₁	—	2	3	no	—
71	C₂²×C₄×C₁₀₁	C₂³	1	3	yes	—
72	C₂³×C₁₀₁	—	2	3	no	—
73	C₂³×C₁₀₁	—	2	12	no	—
74	C₂³×C₁₀₁	—	8	6	no	—
75	C₂³×C₁₀₁	C₂×D₄	1	3	yes	—
76	C₂²×D₁₀₁	—	2	3	no	—
77	C₂²×D₁₀₁	—	2	12	no	—
78	C₂²×D₁₀₁	—	8	6	no	—
79	Q₈×C₁₀₁	—	2	1	no	—
80	Q₈×C₁₀₁	C₂×D₄	1	4	yes	—
81	Q₈×C₁₀₁	C₂⁴	1	2	yes	—
82	D₄×C₁₀₁	—	2	12	no	—
83	D₄×C₁₀₁	—	2	3	no	—
84	D₄×C₁₀₁	—	2	12	no	—
85	D₄×C₁₀₁	C₂×D₄	1	12	yes	—
86	D₄×C₁₀₁	C₂⁴	1	6	yes	—
87	C₂×C₄×C₁₀₁	—	2	1	no	—
88	C₂×C₄×C₁₀₁	—	2	12	no	—
89	C₂×C₄×C₁₀₁	C₂×D₄	1	3	yes	—
90	C₂×C₄×C₁₀₁	C₂×D₄	1	6	yes	—
91	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
92	C₂×C₄×C₁₀₁	C₂⁴	1	6	yes	—
93	C₂²⋊D₁₀₁	—	2	12	no	—
94	C₂×Dic₁₀₁	—	2	1	no	—
95	D₄₀₄	—	2	12	no	—
96	D₄₀₄	—	2	3	no	—
97	D₄₀₄	C₂⁴	1	6	yes	—
98	C₄×D₁₀₁	—	2	12	no	—
99	Dic₂₀₂	—	2	1	no	—
100	Dic₂₀₂	C₂⁴	1	2	yes	—
101	#11717579	C₂⁴	1	2	yes	—
102	#11717579	C₂⁴	1	6	yes	—
103	C₂²×C₁₀₁	—	2	6	no	—
104	C₂²×C₁₀₁	—	2	6	no	—
105	C₂²×C₁₀₁	—	4	6	no	—
106	C₂²×C₁₀₁	—	8	12	no	—
107	C₂²×C₁₀₁	C₂²×D₄	1	6	yes	—
108	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×D₁₀₁	—	2	6	no	—
110	C₂×D₁₀₁	—	4	6	no	—
111	C₂×D₁₀₁	—	8	12	no	—
112	C₄×C₁₀₁	—	2	2	no	—
113	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₁₀₁	C₂²×D₄	1	6	yes	—
115	C₄×C₁₀₁	C₂⁵	1	1	yes	characteristic
116	Dic₁₀₁	—	2	2	no	—
117	C₂×C₁₀₁	—	2	6	no	—
118	C₂×C₁₀₁	—	4	6	no	—
119	C₂×C₁₀₁	#576	1	2	yes	complemented
120	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
121	D₁₀₁	—	4	6	no	—
122	C₂×#576	—	101	1	no	maximal, Sylow
123	C₁₀₁	C₂×#576	1	1	yes	characteristic, complemented, Sylow
124	C₂×H(5,𝔽₂)	—	101	1	no	—
125	C₂×H(5,𝔽₂)	D₁₀₁	1	1	yes	characteristic, complemented
126	#576	—	101	16	no	—
127	C₂×AGL(1,ℤ/8)	—	101	3	no	—
128	C₂×AGL(1,ℤ/8)	—	101	3	no	—
129	C₂×#135	—	101	3	no	—
130	C₂²×D₈	—	101	3	no	—
131	C₂×#131	—	101	1	no	—
132	H(5,𝔽₂)	—	2	4	no	—
133	H(5,𝔽₂)	—	101	8	no	—
134	H(5,𝔽₂)	—	202	4	no	—
135	H(5,𝔽₂)	C₂×D₁₀₁	1	8	yes	complemented
136	C₂×#41	—	2	1	no	—
137	C₂×#41	—	101	3	no	—
138	C₂×#41	—	202	1	no	—
139	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
140	C₂×#41	C₂×D₁₀₁	1	3	yes	complemented
141	C₂²×D₄	—	2	3	no	—
142	C₂²×D₄	—	101	3	no	—
143	C₂²×D₄	—	202	3	no	—
144	C₂²×D₄	C₂×D₁₀₁	1	3	yes	complemented
145	AGL(1,ℤ/8)	—	101	24	no	—
146	AGL(1,ℤ/8)	—	101	24	no	—
147	#135	—	101	24	no	—
148	C₂×Q₁₆	—	101	1	no	—
149	C₂×SD₁₆	—	101	3	no	—
150	C₂×SD₁₆	—	101	3	no	—
151	C₂×D₈	—	101	3	no	—
152	C₂×D₈	—	101	6	no	—
153	C₂×D₈	—	101	24	no	—
154	#131	—	101	8	no	—
155	C₂×M₁₆	—	101	3	no	—
156	C₂²×C₈	—	101	3	no	—
157	C₂⁴	—	2	3	no	—
158	C₂⁴	—	202	3	no	—
159	#41	—	2	8	no	—
160	#41	—	2	6	no	—
161	#41	—	101	12	no	—
162	#41	—	202	8	no	—
163	#41	—	202	6	no	—
164	#41	C₂²⋊D₁₀₁	1	4	yes	—
165	#41	C₂²×D₁₀₁	1	4	yes	—
166	#41	C₂²×D₁₀₁	1	12	yes	complemented
167	C₂×Q₈	—	101	1	no	—
168	C₂×Q₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
169	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
170	C₂×D₄	—	2	3	no	—
171	C₂×D₄	—	2	24	no	—
172	C₂×D₄	—	2	3	no	—
173	C₂×D₄	—	2	6	no	—
174	C₂×D₄	—	101	3	no	—
175	C₂×D₄	—	101	12	no	—
176	C₂×D₄	—	202	3	no	—
177	C₂×D₄	—	202	24	no	—
178	C₂×D₄	—	202	3	no	—
179	C₂×D₄	—	202	6	no	—
180	C₂×D₄	C₂²⋊D₁₀₁	1	3	yes	—
181	C₂×D₄	C₂²×D₁₀₁	1	3	yes	—
182	C₂×D₄	C₂²×D₁₀₁	1	12	yes	complemented
183	C₂²×C₄	—	2	3	no	—
184	C₂²×C₄	—	202	3	no	—
185	C₂²×C₄	C₂²×D₁₀₁	1	3	yes	—
186	Q₁₆	—	101	4	no	—
187	SD₁₆	—	101	12	no	—
188	SD₁₆	—	101	12	no	—
189	D₈	—	101	12	no	—
190	D₈	—	101	24	no	—
191	M₁₆	—	101	12	no	—
192	C₂×C₈	—	101	1	no	—
193	C₂×C₈	—	101	3	no	—
194	C₂×C₈	—	101	12	no	—
195	C₂³	—	2	3	no	—
196	C₂³	—	2	12	no	—
197	C₂³	—	8	6	no	—
198	C₂³	—	808	6	no	—
199	C₂³	—	202	12	no	—
200	C₂³	—	202	3	no	—
201	C₂³	C₂×C₂²⋊D₁₀₁	1	3	yes	—
202	Q₈	—	2	1	no	—
203	Q₈	—	101	2	no	—
204	Q₈	—	202	1	no	—
205	Q₈	C₂×C₂²⋊D₁₀₁	1	4	yes	—
206	Q₈	C₂³×D₁₀₁	1	2	yes	—
207	D₄	—	2	12	no	—
208	D₄	—	2	3	no	—
209	D₄	—	2	12	no	—
210	D₄	—	101	6	no	—
211	D₄	—	202	12	no	—
212	D₄	—	202	12	no	—
213	D₄	—	202	3	no	—
214	D₄	C₂×C₂²⋊D₁₀₁	1	12	yes	—
215	D₄	C₂³×D₁₀₁	1	6	yes	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	12	no	—
218	C₂×C₄	—	202	1	no	—
219	C₂×C₄	—	202	12	no	—
220	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	6	yes	—
221	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	3	yes	—
222	C₂×C₄	C₂³×D₁₀₁	1	6	yes	—
223	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
224	C₈	—	101	2	no	—
225	C₈	—	101	6	no	—
226	C₂²	—	2	6	no	—
227	C₂²	—	2	6	no	—
228	C₂²	—	4	6	no	—
229	C₂²	—	8	12	no	—
230	C₂²	—	808	12	no	—
231	C₂²	—	404	6	no	—
232	C₂²	—	202	6	no	—
233	C₂²	C₂²×C₂²⋊D₁₀₁	1	6	yes	—
234	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	202	2	no	—
237	C₄	C₂²×C₂²⋊D₁₀₁	1	6	yes	—
238	C₄	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
239	C₄	C₂⁴×D₁₀₁	1	1	yes	characteristic
240	C₂	—	2	6	no	—
241	C₂	—	4	6	no	—
242	C₂	—	404	6	no	—
243	C₂	#20265980	1	2	yes	direct product factor
244	C₂	C₂³×C₂²⋊D₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₁
commutator, cosocle	C₄×C₁₀₁	C₂⁵
Frattini	C₄	C₂⁴×D₁₀₁
Fitting	C₂×H(5,𝔽₂)×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
hypercenter	C₂×H(5,𝔽₂)	D₁₀₁
hypocenter	C₁₀₁	C₂×#576
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
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×#15692301	C₂	semidirect
2. #20265980	C₂	direct (N × H)
3. C₂×#15692281	C₂	semidirect
4. C₂×#15692280	C₂	semidirect
5. C₂×#15692275	C₂	semidirect
6. C₂²×#13108059	C₂	semidirect
7. C₂×#15692262	C₂	semidirect
8. C₂×H(5,𝔽₂)×C₁₀₁	C₂	semidirect
10. #15692301	C₂²	semidirect
12. C₂×#13108074	C₂²	semidirect
13. C₂²×D₄₀₄	C₂²	semidirect
15. #15692281	C₂²	semidirect
16. #15692280	C₂²	semidirect
17. #15692275	C₂²	semidirect
18. C₂×#13108064	C₂²	semidirect
19. C₂×#13108062	C₂²	semidirect
20. C₂×#13108061	C₂²	semidirect
21. C₂×#13108059	C₂²	semidirect
22. C₂×#13108059	C₂²	semidirect
23. C₂×#13108059	C₂²	semidirect
24. #15692262	C₂²	semidirect
25. C₂×#13108058	C₂²	semidirect
26. C₂²×#11717579	C₂²	semidirect
28. H(5,𝔽₂)×C₁₀₁	C₂²	semidirect
31. C₂×#41×C₁₀₁	C₂²	semidirect
33. C₂²×D₄×C₁₀₁	C₂²	semidirect
37. #13108074	C₂³	semidirect
42. C₂×D₄₀₄	C₂³	semidirect
46. #13108064	C₂³	semidirect
47. #13108062	C₂³	semidirect
48. #13108061	C₂³	semidirect
49. #13108059	C₂³	semidirect
50. #13108059	C₂³	semidirect
51. #13108058	C₂³	semidirect
54. C₂×#11717579	C₂³	semidirect
60. #41×C₁₀₁	C₂³	semidirect
69. C₂×D₄×C₁₀₁	C₂³	semidirect
119. C₂×C₁₀₁	#576	semidirect
123. C₁₀₁	C₂×#576	semidirect
125. C₂×H(5,𝔽₂)	D₁₀₁	semidirect
135. H(5,𝔽₂)	C₂×D₁₀₁	semidirect
140. C₂×#41	C₂×D₁₀₁	semidirect
144. C₂²×D₄	C₂×D₁₀₁	semidirect
166. #41	C₂²×D₁₀₁	semidirect
182. C₂×D₄	C₂²×D₁₀₁	semidirect
243. C₂	#20265980	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×#15692301	2
#20265980	2
C₂×#15692281	2
C₂×#15692280	2
C₂×#15692275	2
C₂²×#13108059	2
C₂×#15692262	2
C₂×H(5,𝔽₂)×C₁₀₁	2
C₂×#576	101

Subgroups — C2×#20265980 #27185883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265980 #27185883