Subgroups — C₂×#20265989 #27185893

Order 12928.

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×#15692302	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692301	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265989	C₂	1	16	yes	maximal, direct product factor
4	C₂×#15692288	C₂	1	3	yes	maximal, complemented
5	C₂×#15692283	C₂	1	3	yes	maximal, complemented
6	C₂×#15692277	C₂	1	3	yes	maximal, complemented
7	C₂²×#13108063	C₂	1	3	yes	maximal, complemented
8	C₂×#131×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#15692302	—	2	4	no	—
10	#15692302	C₂²	1	8	yes	complemented
11	#15692301	—	2	4	no	—
12	#15692301	C₂²	1	8	yes	complemented
13	C₂×#13108075	—	2	1	no	—
14	C₂×#13108074	C₂²	1	3	yes	—
15	C₂×#13108074	C₂²	1	3	yes	complemented
16	C₂×#13108073	—	2	3	no	—
17	C₂²×Dic₂₀₂	C₂²	1	3	yes	complemented
18	C₂×Q₈×D₁₀₁	—	2	1	no	—
19	C₂²×D₄₀₄	C₂²	1	3	yes	—
20	C₂×D₄×D₁₀₁	—	2	3	no	—
21	#15692288	C₂²	1	24	yes	complemented
22	#15692283	C₂²	1	24	yes	complemented
23	#15692277	C₂²	1	24	yes	complemented
24	C₂×Dic₄₀₄	C₂²	1	3	yes	complemented
25	C₂×#13108063	C₂²	1	3	yes	complemented
26	C₂×#13108063	C₂²	1	6	yes	complemented
27	C₂×#13108063	C₂²	1	1	yes	characteristic, complemented
28	C₂×#13108063	C₂²	1	24	yes	complemented
29	C₂×D₈₀₈	C₂²	1	3	yes	complemented
30	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
31	#131×C₁₀₁	C₂²	1	8	yes	complemented
32	C₂×M₁₆×C₁₀₁	C₂²	1	3	yes	complemented
33	C₂²×C₈×C₁₀₁	C₂²	1	3	yes	complemented
34	C₂³×D₁₀₁	—	2	3	no	—
35	#13108075	—	2	8	no	—
36	#13108074	—	2	6	no	—
37	#13108074	—	2	6	no	—
38	#13108074	C₂³	1	12	yes	—
39	#13108074	C₂³	1	12	yes	complemented
40	#13108073	—	2	24	no	—
41	C₂×Dic₂₀₂	—	2	3	no	—
42	C₂×Dic₂₀₂	—	2	6	no	—
43	C₂×Dic₂₀₂	C₂³	1	3	yes	—
44	C₂×Dic₂₀₂	C₂³	1	1	yes	characteristic
45	C₂×Dic₂₀₂	C₂³	1	12	yes	complemented
46	Q₈×D₁₀₁	—	2	8	no	—
47	C₂×D₄₀₄	—	2	3	no	—
48	C₂×D₄₀₄	—	2	6	no	—
49	C₂×D₄₀₄	C₂³	1	3	yes	—
50	C₂×D₄₀₄	C₂³	1	12	yes	—
51	C₂×D₄₀₄	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₁₀₁	—	2	3	no	—
53	C₂×C₂²⋊D₁₀₁	—	2	3	no	—
54	D₄×D₁₀₁	—	2	24	no	—
55	C₂²×Dic₁₀₁	—	2	3	no	—
56	C₂×C₄×D₁₀₁	—	2	3	no	—
57	C₂×C₄×D₁₀₁	—	2	3	no	—
58	Dic₄₀₄	C₂³	1	12	yes	complemented
59	#13108063	C₂³	1	24	yes	complemented
60	#13108063	C₂³	1	12	yes	complemented
61	#13108063	C₂³	1	4	yes	complemented
62	D₈₀₈	C₂³	1	12	yes	complemented
63	#41×C₁₀₁	D₄	1	4	yes	—
64	#41×C₁₀₁	C₂³	1	4	yes	—
65	C₂×Q₈×C₁₀₁	D₄	1	1	yes	characteristic
66	C₂×D₄×C₁₀₁	D₄	1	3	yes	—
67	C₂²×C₄×C₁₀₁	C₂³	1	3	yes	—
68	M₁₆×C₁₀₁	C₂³	1	12	yes	complemented
69	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₀₁	C₂³	1	3	yes	—
71	C₂×C₈×C₁₀₁	C₂³	1	12	yes	complemented
72	C₂³×C₁₀₁	C₂×D₄	1	3	yes	—
73	C₂²×D₁₀₁	—	2	1	no	—
74	C₂²×D₁₀₁	—	2	3	no	—
75	C₂²×D₁₀₁	—	2	12	no	—
76	C₂²×D₁₀₁	—	8	6	no	—
77	Q₈×C₁₀₁	C₂×D₄	1	4	yes	—
78	D₄×C₁₀₁	C₂×D₄	1	12	yes	—
79	C₂×C₄×C₁₀₁	C₂×D₄	1	3	yes	—
80	C₂×C₄×C₁₀₁	C₂×D₄	1	6	yes	—
81	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
82	C₂×C₄×C₁₀₁	C₂⁴	1	6	yes	—
83	C₂²⋊D₁₀₁	—	2	12	no	—
84	C₂²⋊D₁₀₁	—	2	12	no	—
85	C₂×Dic₁₀₁	—	2	1	no	—
86	C₂×Dic₁₀₁	—	2	3	no	—
87	C₂×Dic₁₀₁	—	2	12	no	—
88	D₄₀₄	—	2	12	no	—
89	D₄₀₄	—	2	3	no	—
90	D₄₀₄	—	2	1	no	—
91	D₄₀₄	C₂⁴	1	2	yes	—
92	D₄₀₄	C₂⁴	1	6	yes	—
93	C₄×D₁₀₁	—	2	12	no	—
94	C₄×D₁₀₁	—	2	12	no	—
95	Dic₂₀₂	—	2	12	no	—
96	Dic₂₀₂	—	2	3	no	—
97	Dic₂₀₂	—	2	1	no	—
98	Dic₂₀₂	C₂⁴	1	6	yes	—
99	Dic₂₀₂	C₂⁴	1	2	yes	—
100	C₈×C₁₀₁	C₂⁴	1	2	yes	—
101	C₈×C₁₀₁	C₂⁴	1	6	yes	—
102	C₂²×C₁₀₁	—	2	6	no	—
103	C₂²×C₁₀₁	C₂²×D₄	1	6	yes	—
104	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
105	C₂×D₁₀₁	—	2	2	no	—
106	C₂×D₁₀₁	—	2	6	no	—
107	C₂×D₁₀₁	—	4	2	no	—
108	C₂×D₁₀₁	—	4	6	no	—
109	C₂×D₁₀₁	—	8	12	no	—
110	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₁₀₁	C₂²×D₄	1	6	yes	—
112	C₄×C₁₀₁	C₂⁵	1	1	yes	characteristic
113	Dic₁₀₁	—	2	6	no	—
114	Dic₁₀₁	—	2	2	no	—
115	C₂×C₁₀₁	—	2	6	no	—
116	C₂×C₁₀₁	#577	1	2	yes	complemented
117	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
118	D₁₀₁	—	4	2	no	—
119	D₁₀₁	—	4	6	no	—
120	C₂×#577	—	101	1	no	maximal, Sylow
121	C₁₀₁	C₂×#577	1	1	yes	characteristic, complemented, Sylow
122	C₂×2₋¹⁺⁴	—	101	1	no	—
123	C₂×H(5,𝔽₂)	—	101	1	no	—
124	#577	—	101	16	no	—
125	C₂×#137	—	101	3	no	—
126	C₂×AGL(1,ℤ/8)	—	101	3	no	—
127	C₂×#135	—	101	3	no	—
128	C₂²×SD₁₆	—	101	3	no	—
129	C₂×#131	D₁₀₁	1	1	yes	characteristic, complemented
130	2₋¹⁺⁴	—	101	8	no	—
131	2₋¹⁺⁴	—	202	4	no	—
132	H(5,𝔽₂)	—	101	8	no	—
133	H(5,𝔽₂)	—	202	4	no	—
134	C₂×#41	—	101	3	no	—
135	C₂×#41	—	101	3	no	—
136	C₂×#41	—	202	1	no	—
137	C₂×#41	—	202	3	no	—
138	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
139	C₂²×Q₈	—	101	3	no	—
140	C₂²×Q₈	—	202	1	no	—
141	C₂²×D₄	—	101	3	no	—
142	C₂²×D₄	—	202	3	no	—
143	#137	—	101	24	no	—
144	AGL(1,ℤ/8)	—	101	24	no	—
145	#135	—	101	24	no	—
146	C₂×Q₁₆	—	101	3	no	—
147	C₂×SD₁₆	—	101	3	no	—
148	C₂×SD₁₆	—	101	6	no	—
149	C₂×SD₁₆	—	101	1	no	—
150	C₂×SD₁₆	—	101	24	no	—
151	C₂×D₈	—	101	3	no	—
152	#131	C₂×D₁₀₁	1	8	yes	complemented
153	C₂×M₁₆	C₂×D₁₀₁	1	3	yes	complemented
154	C₂²×C₈	C₂×D₁₀₁	1	3	yes	complemented
155	C₂⁴	—	202	3	no	—
156	#41	—	101	12	no	—
157	#41	—	101	12	no	—
158	#41	—	202	8	no	—
159	#41	—	202	24	no	—
160	#41	—	202	6	no	—
161	#41	—	202	6	no	—
162	#41	D₄₀₄	1	4	yes	—
163	#41	C₂²×D₁₀₁	1	4	yes	—
164	C₂×Q₈	—	101	3	no	—
165	C₂×Q₈	—	101	1	no	—
166	C₂×Q₈	—	101	12	no	—
167	C₂×Q₈	—	202	8	no	—
168	C₂×Q₈	—	202	3	no	—
169	C₂×Q₈	—	202	6	no	—
170	C₂×Q₈	D₄₀₄	1	1	yes	characteristic
171	C₂×D₄	—	101	1	no	—
172	C₂×D₄	—	101	3	no	—
173	C₂×D₄	—	101	12	no	—
174	C₂×D₄	—	202	3	no	—
175	C₂×D₄	—	202	3	no	—
176	C₂×D₄	—	202	24	no	—
177	C₂×D₄	—	202	3	no	—
178	C₂×D₄	—	202	6	no	—
179	C₂×D₄	D₄₀₄	1	3	yes	—
180	C₂²×C₄	—	202	3	no	—
181	C₂²×C₄	—	202	3	no	—
182	C₂²×C₄	—	202	3	no	—
183	C₂²×C₄	C₂²×D₁₀₁	1	3	yes	—
184	Q₁₆	—	101	12	no	—
185	SD₁₆	—	101	12	no	—
186	SD₁₆	—	101	24	no	—
187	SD₁₆	—	101	4	no	—
188	D₈	—	101	12	no	—
189	M₁₆	C₂²×D₁₀₁	1	12	yes	complemented
190	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
191	C₂×C₈	C₂²×D₁₀₁	1	3	yes	—
192	C₂×C₈	C₂²×D₁₀₁	1	12	yes	complemented
193	C₂³	—	808	6	no	—
194	C₂³	—	202	12	no	—
195	C₂³	—	202	1	no	—
196	C₂³	—	202	3	no	—
197	C₂³	C₂×D₄₀₄	1	3	yes	—
198	Q₈	—	101	6	no	—
199	Q₈	—	101	2	no	—
200	Q₈	—	202	12	no	—
201	Q₈	—	202	3	no	—
202	Q₈	—	202	1	no	—
203	Q₈	C₂×D₄₀₄	1	4	yes	—
204	D₄	—	101	2	no	—
205	D₄	—	101	6	no	—
206	D₄	—	202	12	no	—
207	D₄	—	202	12	no	—
208	D₄	—	202	12	no	—
209	D₄	—	202	3	no	—
210	D₄	—	202	1	no	—
211	D₄	C₂×D₄₀₄	1	12	yes	—
212	C₂×C₄	—	202	12	no	—
213	C₂×C₄	—	202	1	no	—
214	C₂×C₄	—	202	3	no	—
215	C₂×C₄	—	202	12	no	—
216	C₂×C₄	—	202	12	no	—
217	C₂×C₄	C₂×D₄₀₄	1	6	yes	—
218	C₂×C₄	C₂×D₄₀₄	1	3	yes	—
219	C₂×C₄	C₂³×D₁₀₁	1	6	yes	—
220	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
221	C₈	C₂³×D₁₀₁	1	2	yes	—
222	C₈	C₂³×D₁₀₁	1	6	yes	—
223	C₂²	—	2	6	no	—
224	C₂²	—	808	12	no	—
225	C₂²	—	404	6	no	—
226	C₂²	—	404	2	no	—
227	C₂²	—	202	2	no	—
228	C₂²	—	202	6	no	—
229	C₂²	C₂²×D₄₀₄	1	6	yes	—
230	C₂²	C₂²×D₄₀₄	1	1	yes	characteristic
231	C₄	—	202	2	no	—
232	C₄	—	202	6	no	—
233	C₄	C₂²×D₄₀₄	1	6	yes	—
234	C₄	C₂²×D₄₀₄	1	1	yes	characteristic
235	C₄	C₂⁴×D₁₀₁	1	1	yes	characteristic
236	C₂	—	2	6	no	—
237	C₂	—	404	6	no	—
238	C₂	—	404	2	no	—
239	C₂	#20265989	1	2	yes	direct product factor
240	C₂	C₂³×D₄₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₀₄
commutator, cosocle	C₄×C₁₀₁	C₂⁵
Frattini	C₄	C₂⁴×D₁₀₁
Fitting	C₂×#131×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
hypercenter	C₂×#131	D₁₀₁
hypocenter	C₁₀₁	C₂×#577
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#15692302	C₂	semidirect
2. C₂×#15692301	C₂	semidirect
3. #20265989	C₂	direct (N × H)
4. C₂×#15692288	C₂	semidirect
5. C₂×#15692283	C₂	semidirect
6. C₂×#15692277	C₂	semidirect
7. C₂²×#13108063	C₂	semidirect
8. C₂×#131×C₁₀₁	C₂	semidirect
10. #15692302	C₂²	semidirect
12. #15692301	C₂²	semidirect
15. C₂×#13108074	C₂²	semidirect
17. C₂²×Dic₂₀₂	C₂²	semidirect
21. #15692288	C₂²	semidirect
22. #15692283	C₂²	semidirect
23. #15692277	C₂²	semidirect
24. C₂×Dic₄₀₄	C₂²	semidirect
25. C₂×#13108063	C₂²	semidirect
26. C₂×#13108063	C₂²	semidirect
27. C₂×#13108063	C₂²	semidirect
28. C₂×#13108063	C₂²	semidirect
29. C₂×D₈₀₈	C₂²	semidirect
31. #131×C₁₀₁	C₂²	semidirect
32. C₂×M₁₆×C₁₀₁	C₂²	semidirect
33. C₂²×C₈×C₁₀₁	C₂²	semidirect
39. #13108074	C₂³	semidirect
45. C₂×Dic₂₀₂	C₂³	semidirect
58. Dic₄₀₄	C₂³	semidirect
59. #13108063	C₂³	semidirect
60. #13108063	C₂³	semidirect
61. #13108063	C₂³	semidirect
62. D₈₀₈	C₂³	semidirect
68. M₁₆×C₁₀₁	C₂³	semidirect
71. C₂×C₈×C₁₀₁	C₂³	semidirect
116. C₂×C₁₀₁	#577	semidirect
121. C₁₀₁	C₂×#577	semidirect
129. C₂×#131	D₁₀₁	semidirect
152. #131	C₂×D₁₀₁	semidirect
153. C₂×M₁₆	C₂×D₁₀₁	semidirect
154. C₂²×C₈	C₂×D₁₀₁	semidirect
189. M₁₆	C₂²×D₁₀₁	semidirect
192. C₂×C₈	C₂²×D₁₀₁	semidirect
239. C₂	#20265989	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×#15692302	2
C₂×#15692301	2
#20265989	2
C₂×#15692288	2
C₂×#15692283	2
C₂×#15692277	2
C₂²×#13108063	2
C₂×#131×C₁₀₁	2
C₂×#577	101

Subgroups — C2×#20265989 #27185893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265989 #27185893