Subgroups — C₂×#31771082 #46824075

Order 34432.

1488 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic
2	#31771082	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Dic₅₃₈	C₂	1	4	yes	maximal, complemented
4	C₂×#22575418	C₂	1	4	yes	maximal, complemented
5	C₂×#22575412	C₂	1	1	yes	maximal, characteristic
6	C₂×#22575409	C₂	1	4	yes	maximal
7	C₂×#117×C₂₆₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₅₃₈	C₄	1	16	yes	complemented
9	C₂²×Dic₅₃₈	C₂²	1	2	yes	—
10	C₂²×Dic₅₃₈	C₂²	1	2	yes	—
11	C₂²×Dic₅₃₈	C₂²	1	8	yes	—
12	C₂³×Dic₂₆₉	C₂²	1	2	yes	—
13	C₄×Dic₅₃₈	C₂²	1	32	yes	complemented
14	#22575418	C₂²	1	32	yes	complemented
15	#22575412	C₂²	1	8	yes	—
16	#22575409	C₂²	1	32	yes	—
17	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
18	C₂×#16864381	C₂²	1	8	yes	—
19	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
20	C₂×C₄×Dic₂₆₉	C₂²	1	8	yes	—
21	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
22	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
23	#117×C₂₆₉	C₂²	1	8	yes	—
24	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₂₆₉	C₂²	1	2	yes	—
27	C₂×Dic₅₃₈	C₂×C₄	1	16	yes	—
28	C₂×Dic₅₃₈	C₂×C₄	1	16	yes	—
29	C₂×Dic₅₃₈	C₂×C₄	1	64	yes	complemented
30	C₂×Dic₅₃₈	C₂³	1	8	yes	—
31	C₂×Dic₅₃₈	C₂³	1	8	yes	—
32	C₂²×Dic₂₆₉	C₂×C₄	1	16	yes	—
33	C₂²×Dic₂₆₉	C₂³	1	4	yes	—
34	C₂²×Dic₂₆₉	C₂³	1	4	yes	—
35	C₂²×Dic₂₆₉	C₂³	1	8	yes	—
36	C₄⋊Dic₂₆₉	C₂³	1	8	yes	—
37	#16864381	C₂³	1	32	yes	—
38	C₂²⋊Dic₂₆₉	C₂³	1	8	yes	—
39	C₄×Dic₂₆₉	C₂³	1	32	yes	—
40	C₄×Dic₂₆₉	C₂³	1	8	yes	—
41	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₆₉	C₂×C₄	1	8	yes	—
43	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
44	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
47	C₄⋊C₄×C₂₆₉	C₂³	1	8	yes	—
48	C₂²⋊C₄×C₂₆₉	C₂³	1	8	yes	—
49	C₄²×C₂₆₉	C₂³	1	8	yes	—
50	Dic₅₃₈	C₂²×C₄	1	32	yes	—
51	Dic₅₃₈	C₂²×C₄	1	32	yes	—
52	C₂×Dic₂₆₉	—	2	8	no	—
53	C₂×Dic₂₆₉	C₂²×C₄	1	16	yes	—
54	C₂×Dic₂₆₉	C₂²×C₄	1	32	yes	—
55	C₂×Dic₂₆₉	C₂⁴	1	8	yes	—
56	C₂×Dic₂₆₉	C₂⁴	1	8	yes	—
57	C₂³×C₂₆₉	—	2	4	no	—
58	C₂³×C₂₆₉	C₂²×C₄	1	4	yes	—
59	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
60	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
61	C₂×C₄×C₂₆₉	—	2	8	no	—
62	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₂₆₉	C₂²×C₄	1	16	yes	—
65	C₂×C₄×C₂₆₉	C₂⁴	1	8	yes	—
66	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
67	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
68	C₂²×C₂₆₉	—	2	8	no	—
69	C₂²×C₂₆₉	—	2	4	no	—
70	C₂²×C₂₆₉	C₂³×C₄	1	4	yes	—
71	C₂²×C₂₆₉	C₂³×C₄	1	2	yes	—
72	C₂²×C₂₆₉	2₋¹⁺⁴	1	4	yes	—
73	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
74	C₄×C₂₆₉	—	2	8	no	—
75	C₄×C₂₆₉	C₂³×C₄	1	4	yes	—
76	C₄×C₂₆₉	C₂³×C₄	1	4	yes	—
77	Dic₂₆₉	—	2	8	no	—
78	Dic₂₆₉	C₂³×C₄	1	16	yes	—
79	C₂×C₂₆₉	—	2	4	no	—
80	C₂×C₂₆₉	#519	1	4	yes	complemented
81	C₂×C₂₆₉	C₂⁴×C₄	1	1	yes	characteristic
82	C₂×C₂₆₉	C₂×2₋¹⁺⁴	1	2	yes	—
83	C₂₆₉	C₂×#519	1	1	yes	characteristic, complemented, Sylow
84	C₂×#519	—	269	1	no	maximal, Sylow
85	C₂³×Q₈	—	269	1	no	—
86	#519	—	269	16	no	—
87	C₂×C₄×Q₈	—	269	4	no	—
88	C₂×C₄×Q₈	—	269	4	no	—
89	C₂×#117	—	269	1	no	—
90	C₂×#117	—	269	4	no	—
91	C₂×#117	D₂₆₉	1	1	yes	characteristic
92	C₂²×Q₈	—	269	16	no	—
93	C₂²×Q₈	—	269	2	no	—
94	C₂²×Q₈	—	269	2	no	—
95	C₂²×Q₈	—	269	8	no	—
96	C₂³×C₄	—	269	2	no	—
97	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
98	C₄×Q₈	—	269	32	no	—
99	C₄×Q₈	—	269	32	no	—
100	#117	—	269	8	no	—
101	#117	—	269	32	no	—
102	#117	C₂×D₂₆₉	1	8	yes	—
103	C₂×C₄⋊C₄	—	269	8	no	—
104	C₂×C₄⋊C₄	—	269	2	no	—
105	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
106	C₂×C₂²⋊C₄	—	269	2	no	—
107	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	2	yes	—
108	C₂×C₄²	—	269	8	no	—
109	C₂×C₄²	—	269	2	no	—
110	C₂×C₄²	C₂×D₂₆₉	1	2	yes	—
111	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
112	C₂×Q₈	—	269	16	no	—
113	C₂×Q₈	—	269	16	no	—
114	C₂×Q₈	—	269	64	no	—
115	C₂×Q₈	—	269	8	no	—
116	C₂×Q₈	—	269	8	no	—
117	C₂²×C₄	—	269	16	no	—
118	C₂²×C₄	—	269	8	no	—
119	C₂²×C₄	—	269	4	no	—
120	C₂²×C₄	—	269	4	no	—
121	C₂²×C₄	C₄×D₂₆₉	1	8	yes	—
122	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
123	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
124	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
126	C₄⋊C₄	—	269	32	no	—
127	C₄⋊C₄	—	269	8	no	—
128	C₄⋊C₄	C₂²×D₂₆₉	1	8	yes	—
129	C₂²⋊C₄	—	269	8	no	—
130	C₂²⋊C₄	C₂²×D₂₆₉	1	8	yes	—
131	C₄²	—	269	32	no	—
132	C₄²	—	269	8	no	—
133	C₄²	C₂²×D₂₆₉	1	8	yes	—
134	C₂³	—	2	4	no	—
135	C₂³	C₂×C₄×D₂₆₉	1	4	yes	—
136	C₂³	C₂³×D₂₆₉	1	2	yes	—
137	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
138	Q₈	—	269	32	no	—
139	Q₈	—	269	32	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	269	32	no	—
142	C₂×C₄	—	269	16	no	—
143	C₂×C₄	—	269	8	no	—
144	C₂×C₄	—	269	8	no	—
145	C₂×C₄	—	538	8	no	—
146	C₂×C₄	C₂×C₄×D₂₆₉	1	16	yes	—
147	C₂×C₄	C₂×C₄×D₂₆₉	1	4	yes	—
148	C₂×C₄	C₂×C₄×D₂₆₉	1	4	yes	—
149	C₂×C₄	C₂³×D₂₆₉	1	8	yes	—
150	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
151	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	8	no	—
154	C₂²	C₂²×C₄×D₂₆₉	1	4	yes	—
155	C₂²	C₂²×C₄×D₂₆₉	1	2	yes	—
156	C₂²	#22575500	1	4	yes	—
157	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
158	C₄	—	2	8	no	—
159	C₄	—	269	16	no	—
160	C₄	—	538	8	no	—
161	C₄	C₂²×C₄×D₂₆₉	1	4	yes	—
162	C₄	C₂²×C₄×D₂₆₉	1	4	yes	—
163	C₂	—	2	4	no	—
164	C₂	#31771082	1	4	yes	direct product factor
165	C₂	C₂³×C₄×D₂₆₉	1	1	yes	characteristic
166	C₂	C₂×#22575500	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₂₆₉
commutator	C₂×C₂₆₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₂₆₉
Fitting	C₂×#117×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂⁴
prime radical	C₂⁴×C₂₆₉	C₂³
cosocle	C₂²×C₂₆₉	C₂⁵
hypercenter	C₂×#117	D₂₆₉
hypocenter	C₂₆₉	C₂×#519
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #31771082	C₂	direct (N × H)
3. C₂×C₄×Dic₅₃₈	C₂	semidirect
4. C₂×#22575418	C₂	semidirect
8. C₂²×Dic₅₃₈	C₄	semidirect
13. C₄×Dic₅₃₈	C₂²	semidirect
14. #22575418	C₂²	semidirect
29. C₂×Dic₅₃₈	C₂×C₄	semidirect
80. C₂×C₂₆₉	#519	semidirect
83. C₂₆₉	C₂×#519	semidirect
164. C₂	#31771082	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₅₃₈	2
#31771082	2
C₂×C₄×Dic₅₃₈	2
C₂×#22575418	2
C₂×#22575412	2
C₂×#22575409	2
C₂×#117×C₂₆₉	2
C₂×#519	269

Subgroups — C2×#31771082 #46824075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771082 #46824075