Subgroups — C₂×#31771405 #46825632

Order 34432.

1484 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16864402	C₂	1	2	yes	maximal, complemented
2	#31771405	C₂	1	16	yes	maximal, direct product factor
3	C₂×#22575486	C₂	1	4	yes	maximal, complemented
4	C₂×#22575481	C₂	1	4	yes	maximal, complemented
5	C₂×#22575475	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16864402	—	2	2	no	—
8	C₂×#16864402	—	2	2	no	—
9	C₂×#16864402	—	2	8	no	—
10	C₂×#16864402	C₂²	1	4	yes	—
11	C₂×#16864402	C₂²	1	4	yes	complemented
12	C₂×#16864402	C₂²	1	16	yes	complemented
13	C₂²×Dic₅₃₈	C₂²	1	2	yes	complemented
14	C₂²×D₁₀₇₆	C₂²	1	2	yes	complemented
15	C₂²×C₂²⋊D₂₆₉	—	2	2	no	—
16	C₂²×C₄×D₂₆₉	—	2	2	no	—
17	#22575486	C₂²	1	32	yes	complemented
18	#22575481	C₂²	1	32	yes	complemented
19	#22575475	C₂²	1	32	yes	complemented
20	C₂×Dic₁₀₇₆	C₂²	1	4	yes	complemented
21	C₂×#16864391	C₂²	1	8	yes	complemented
22	C₂×D₂₁₅₂	C₂²	1	4	yes	complemented
23	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
24	C₂×M₁₆×C₂₆₉	C₂²	1	4	yes	complemented
25	C₂×M₁₆×C₂₆₉	C₂²	1	8	yes	complemented
26	C₂²×C₈×C₂₆₉	C₂²	1	2	yes	complemented
27	C₂³×D₂₆₉	—	2	2	no	—
28	#16864402	—	2	8	no	—
29	#16864402	—	2	16	no	—
30	#16864402	—	2	16	no	—
31	#16864402	—	2	8	no	—
32	#16864402	C₂³	1	16	yes	—
33	#16864402	C₂³	1	16	yes	complemented
34	C₂×Dic₅₃₈	—	2	2	no	—
35	C₂×Dic₅₃₈	—	2	4	no	—
36	C₂×Dic₅₃₈	C₂³	1	4	yes	—
37	C₂×Dic₅₃₈	C₂³	1	8	yes	complemented
38	C₂×D₁₀₇₆	—	2	2	no	—
39	C₂×D₁₀₇₆	—	2	4	no	—
40	C₂×D₁₀₇₆	C₂³	1	4	yes	—
41	C₂×D₁₀₇₆	C₂³	1	8	yes	complemented
42	C₂×C₂²⋊D₂₆₉	—	2	4	no	—
43	C₂×C₂²⋊D₂₆₉	—	2	4	no	—
44	C₂×C₂²⋊D₂₆₉	—	2	16	no	—
45	C₂²×Dic₂₆₉	—	2	2	no	—
46	C₂×C₄×D₂₆₉	—	2	4	no	—
47	C₂×C₄×D₂₆₉	—	2	16	no	—
48	C₂×C₄×D₂₆₉	—	2	4	no	—
49	Dic₁₀₇₆	C₂³	1	16	yes	complemented
50	#16864391	C₂³	1	32	yes	complemented
51	D₂₁₅₂	C₂³	1	16	yes	complemented
52	C₂⁴×C₂₆₉	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₆₉	D₄	1	2	yes	—
54	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₆₉	D₄	1	4	yes	—
56	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
59	M₁₆×C₂₆₉	C₂³	1	16	yes	complemented
60	C₂×C₈×C₂₆₉	C₂³	1	4	yes	—
61	C₂×C₈×C₂₆₉	C₂³	1	8	yes	complemented
62	C₂²×D₂₆₉	—	2	4	no	—
63	C₂²×D₂₆₉	—	2	8	no	—
64	C₂²×D₂₆₉	—	4	8	no	—
65	Dic₅₃₈	—	2	4	no	—
66	Dic₅₃₈	—	2	8	no	—
67	Dic₅₃₈	C₂⁴	1	8	yes	—
68	D₁₀₇₆	—	2	4	no	—
69	D₁₀₇₆	—	2	8	no	—
70	D₁₀₇₆	C₂⁴	1	8	yes	—
71	C₂²⋊D₂₆₉	—	2	16	no	—
72	C₂²⋊D₂₆₉	—	2	16	no	—
73	C₂×Dic₂₆₉	—	2	4	no	—
74	C₂×Dic₂₆₉	—	2	8	no	—
75	C₄×D₂₆₉	—	2	16	no	—
76	C₄×D₂₆₉	—	2	16	no	—
77	C₂³×C₂₆₉	—	4	2	no	—
78	C₂³×C₂₆₉	C₂×D₄	1	2	yes	—
79	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
80	C₂³×C₂₆₉	C₂×D₄	1	4	yes	—
81	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₂₆₉	C₂×D₄	1	8	yes	—
83	C₂×C₄×C₂₆₉	C₂×D₄	1	4	yes	—
84	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₂₆₉	C₂×D₄	1	4	yes	—
86	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
88	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
89	C₂×C₄×C₂₆₉	C₂⁴	1	4	yes	—
90	C₈×C₂₆₉	C₂⁴	1	8	yes	—
91	C₂²×C₂₆₉	—	2	2	no	—
92	C₂²×C₂₆₉	—	2	4	no	—
93	C₂²×C₂₆₉	—	4	4	no	—
94	C₂²×C₂₆₉	C₂²×D₄	1	4	yes	—
95	C₂²×C₂₆₉	C₂²×D₄	1	2	yes	—
96	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
97	C₂×D₂₆₉	—	2	8	no	—
98	C₂×D₂₆₉	—	4	16	no	—
99	C₂×D₂₆₉	—	4	8	no	—
100	C₄×C₂₆₉	C₂²×D₄	1	2	yes	—
101	C₄×C₂₆₉	C₂²×D₄	1	4	yes	—
102	C₄×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₂₆₉	C₂⁵	1	1	yes	characteristic
104	Dic₂₆₉	—	2	8	no	—
105	C₂×C₂₆₉	—	2	4	no	—
106	C₂×C₂₆₉	—	2	2	no	—
107	C₂×C₂₆₉	#575	1	2	yes	complemented
108	C₂×C₂₆₉	C₂³×D₄	1	1	yes	characteristic
109	D₂₆₉	—	4	8	no	—
110	C₂₆₉	C₂×#575	1	1	yes	characteristic, complemented, Sylow
111	C₂×#575	—	269	1	no	maximal, Sylow
112	C₂²×#41	—	269	2	no	—
113	#575	—	269	16	no	—
114	C₂×#137	—	269	4	no	—
115	C₂×AGL(1,ℤ/8)	—	269	4	no	—
116	C₂×#135	—	269	4	no	—
117	C₂²×M₁₆	D₂₆₉	1	1	yes	characteristic, complemented
118	C₂×#41	—	269	4	no	—
119	C₂×#41	—	269	4	no	—
120	C₂×#41	—	269	16	no	—
121	C₂×#41	—	538	2	no	—
122	C₂×#41	—	538	8	no	—
123	C₂×#41	—	538	2	no	—
124	C₂²×Q₈	—	269	2	no	—
125	C₂²×D₄	—	269	2	no	—
126	C₂²×D₄	—	538	2	no	—
127	C₂³×C₄	—	538	2	no	—
128	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
129	#137	—	269	32	no	—
130	AGL(1,ℤ/8)	—	269	32	no	—
131	#135	—	269	32	no	—
132	C₂×Q₁₆	—	269	4	no	—
133	C₂×SD₁₆	—	269	8	no	—
134	C₂×D₈	—	269	4	no	—
135	C₂×M₁₆	C₂×D₂₆₉	1	4	yes	complemented
136	C₂×M₁₆	C₂×D₂₆₉	1	8	yes	complemented
137	C₂²×C₈	C₂×D₂₆₉	1	2	yes	complemented
138	C₂⁴	—	538	2	no	—
139	C₂⁴	D₁₀₇₆	1	1	yes	characteristic
140	#41	—	269	16	no	—
141	#41	—	269	16	no	—
142	#41	—	538	16	no	—
143	#41	—	538	8	no	—
144	#41	—	538	8	no	—
145	#41	—	538	16	no	—
146	C₂×Q₈	—	269	4	no	—
147	C₂×Q₈	—	269	8	no	—
148	C₂×Q₈	—	538	2	no	—
149	C₂×Q₈	—	538	4	no	—
150	C₂×D₄	—	269	4	no	—
151	C₂×D₄	—	269	8	no	—
152	C₂×D₄	—	538	4	no	—
153	C₂×D₄	—	538	4	no	—
154	C₂×D₄	—	538	16	no	—
155	C₂×D₄	—	538	2	no	—
156	C₂×D₄	—	538	4	no	—
157	C₂²×C₄	—	538	2	no	—
158	C₂²×C₄	—	538	4	no	—
159	C₂²×C₄	—	538	16	no	—
160	C₂²×C₄	—	538	4	no	—
161	C₂²×C₄	D₁₀₇₆	1	1	yes	characteristic
162	C₂²×C₄	D₁₀₇₆	1	4	yes	—
163	C₂²×C₄	D₁₀₇₆	1	2	yes	—
164	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
165	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
166	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
167	Q₁₆	—	269	16	no	—
168	SD₁₆	—	269	32	no	—
169	D₈	—	269	16	no	—
170	M₁₆	C₂²×D₂₆₉	1	16	yes	complemented
171	C₂×C₈	C₂²×D₂₆₉	1	4	yes	—
172	C₂×C₈	C₂²×D₂₆₉	1	8	yes	complemented
173	C₂³	—	4	2	no	—
174	C₂³	—	538	8	no	—
175	C₂³	—	538	4	no	—
176	C₂³	—	1076	8	no	—
177	C₂³	C₂×D₁₀₇₆	1	2	yes	—
178	C₂³	C₂×D₁₀₇₆	1	1	yes	characteristic
179	C₂³	C₂×D₁₀₇₆	1	4	yes	—
180	Q₈	—	269	8	no	—
181	Q₈	—	538	8	no	—
182	Q₈	—	538	4	no	—
183	D₄	—	269	8	no	—
184	D₄	—	538	16	no	—
185	D₄	—	538	16	no	—
186	D₄	—	538	8	no	—
187	D₄	—	538	4	no	—
188	C₂×C₄	—	538	8	no	—
189	C₂×C₄	—	538	4	no	—
190	C₂×C₄	—	538	16	no	—
191	C₂×C₄	—	538	16	no	—
192	C₂×C₄	C₂×D₁₀₇₆	1	2	yes	—
193	C₂×C₄	C₂×D₁₀₇₆	1	8	yes	—
194	C₂×C₄	C₂×D₁₀₇₆	1	4	yes	—
195	C₂×C₄	C₂×D₁₀₇₆	1	2	yes	—
196	C₂×C₄	C₂×D₁₀₇₆	1	4	yes	—
197	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
198	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
199	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
200	C₂×C₄	C₂³×D₂₆₉	1	4	yes	—
201	C₈	C₂³×D₂₆₉	1	8	yes	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	4	4	no	—
205	C₂²	—	538	8	no	—
206	C₂²	—	1076	16	no	—
207	C₂²	—	1076	8	no	—
208	C₂²	C₂²×D₁₀₇₆	1	4	yes	—
209	C₂²	C₂²×D₁₀₇₆	1	2	yes	—
210	C₂²	C₂²×D₁₀₇₆	1	1	yes	characteristic
211	C₄	—	538	8	no	—
212	C₄	C₂²×D₁₀₇₆	1	2	yes	—
213	C₄	C₂²×D₁₀₇₆	1	4	yes	—
214	C₄	C₂²×D₁₀₇₆	1	1	yes	characteristic
215	C₄	C₂⁴×D₂₆₉	1	1	yes	characteristic
216	C₂	—	2	4	no	—
217	C₂	—	2	2	no	—
218	C₂	—	1076	8	no	—
219	C₂	#31771405	1	2	yes	direct product factor
220	C₂	C₂³×D₁₀₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₇₆
commutator, cosocle	C₄×C₂₆₉	C₂⁵
Frattini	C₄	C₂⁴×D₂₆₉
Fitting	C₂²×M₁₆×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂²×D₄
hypercenter	C₂²×M₁₆	D₂₆₉
hypocenter	C₂₆₉	C₂×#575
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#16864402	C₂	semidirect
2. #31771405	C₂	direct (N × H)
3. C₂×#22575486	C₂	semidirect
4. C₂×#22575481	C₂	semidirect
5. C₂×#22575475	C₂	semidirect
6. C₂²×M₁₆×C₂₆₉	C₂	semidirect
11. C₂×#16864402	C₂²	semidirect
12. C₂×#16864402	C₂²	semidirect
13. C₂²×Dic₅₃₈	C₂²	semidirect
14. C₂²×D₁₀₇₆	C₂²	semidirect
17. #22575486	C₂²	semidirect
18. #22575481	C₂²	semidirect
19. #22575475	C₂²	semidirect
20. C₂×Dic₁₀₇₆	C₂²	semidirect
21. C₂×#16864391	C₂²	semidirect
22. C₂×D₂₁₅₂	C₂²	semidirect
24. C₂×M₁₆×C₂₆₉	C₂²	semidirect
25. C₂×M₁₆×C₂₆₉	C₂²	semidirect
26. C₂²×C₈×C₂₆₉	C₂²	semidirect
33. #16864402	C₂³	semidirect
37. C₂×Dic₅₃₈	C₂³	semidirect
41. C₂×D₁₀₇₆	C₂³	semidirect
49. Dic₁₀₇₆	C₂³	semidirect
50. #16864391	C₂³	semidirect
51. D₂₁₅₂	C₂³	semidirect
59. M₁₆×C₂₆₉	C₂³	semidirect
61. C₂×C₈×C₂₆₉	C₂³	semidirect
107. C₂×C₂₆₉	#575	semidirect
110. C₂₆₉	C₂×#575	semidirect
117. C₂²×M₁₆	D₂₆₉	semidirect
135. C₂×M₁₆	C₂×D₂₆₉	semidirect
136. C₂×M₁₆	C₂×D₂₆₉	semidirect
137. C₂²×C₈	C₂×D₂₆₉	semidirect
170. M₁₆	C₂²×D₂₆₉	semidirect
172. C₂×C₈	C₂²×D₂₆₉	semidirect
219. C₂	#31771405	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#16864402	2
#31771405	2
C₂×#22575486	2
C₂×#22575481	2
C₂×#22575475	2
C₂²×M₁₆×C₂₆₉	2
C₂×#575	269

Subgroups — C2×#31771405 #46825632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771405 #46825632