Subgroups — C₂×D₄×#14524130 #46819558

Order 34432.

616 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#14524130	C₂	1	2	yes	maximal, complemented
2	D₄×#14524130	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#14524130	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊#14524130	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#14524130	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#14524130	—	2	8	no	—
8	C₂²×#14524130	C₂²	1	2	yes	—
9	C₂²×#14524130	C₂²	1	1	yes	characteristic
10	C₂²×#14524130	C₂²	1	1	yes	characteristic
11	C₂²×#14524130	C₂²	1	8	yes	complemented
12	C₄⋊#14524130	C₂²	1	4	yes	complemented
13	C₂²⋊#14524130	C₂²	1	8	yes	complemented
14	C₄×#14524130	C₂²	1	4	yes	complemented
15	C₂²×D₄×C₂₆₉	C₄	1	1	yes	characteristic
16	C₂³×C₄×C₂₆₉	C₂²	1	2	yes	—
17	C₄×D₄×C₂₆₉	C₄	1	4	yes	—
18	C₄×D₄×C₂₆₉	C₂²	1	4	yes	—
19	C₂×C₄⋊C₄×C₂₆₉	C₄	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₆₉	C₄	1	2	yes	—
21	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
22	C₂×#14524130	—	2	16	no	—
23	C₂×#14524130	—	2	4	no	—
24	C₂×#14524130	—	2	2	no	—
25	C₂×#14524130	D₄	1	4	yes	direct product factor
26	C₂×#14524130	C₂³	1	2	yes	—
27	C₂×#14524130	C₂³	1	4	yes	—
28	C₂×#14524130	C₂³	1	2	yes	—
29	C₂⁴×C₂₆₉	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₂₆₉	C₈	1	8	yes	complemented
31	C₂×D₄×C₂₆₉	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₂₆₉	—	2	4	no	—
33	C₂²×C₄×C₂₆₉	—	2	4	no	—
34	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₂₆₉	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
39	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₂₆₉	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₂₆₉	C₂×C₄	1	8	yes	—
42	C₄²×C₂₆₉	C₂×C₄	1	2	yes	—
43	C₄²×C₂₆₉	C₂³	1	2	yes	—
44	#14524130	—	2	4	no	—
45	#14524130	—	2	2	no	—
46	#14524130	C₂×D₄	1	4	yes	direct product factor
47	C₂³×C₂₆₉	—	2	4	no	—
48	C₂³×C₂₆₉	—	2	4	no	—
49	C₂³×C₂₆₉	C₂×C₈	1	8	yes	complemented
50	C₂³×C₂₆₉	C₂²×C₄	1	4	yes	—
51	C₂³×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
52	D₄×C₂₆₉	C₂×C₈	1	16	yes	complemented
53	C₂×C₄×C₂₆₉	—	2	4	no	—
54	C₂×C₄×C₂₆₉	—	2	2	no	—
55	C₂×C₄×C₂₆₉	—	2	8	no	—
56	C₂×C₄×C₂₆₉	—	2	4	no	—
57	C₂×C₄×C₂₆₉	—	2	4	no	—
58	C₂×C₄×C₂₆₉	C₂×C₈	1	4	yes	complemented
59	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₂₆₉	#41	1	2	yes	—
65	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
66	C₂²×C₂₆₉	—	2	4	no	—
67	C₂²×C₂₆₉	—	2	4	no	—
68	C₂²×C₂₆₉	—	2	16	no	—
69	C₂²×C₂₆₉	C₄×D₄	1	2	yes	—
70	C₂²×C₂₆₉	C₂²×C₈	1	2	yes	—
71	C₂²×C₂₆₉	C₂²×C₈	1	8	yes	complemented
72	C₂²×C₂₆₉	#131	1	2	yes	—
73	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₂₆₉	—	2	4	no	—
75	C₄×C₂₆₉	—	2	2	no	—
76	C₄×C₂₆₉	C₄×D₄	1	2	yes	—
77	C₄×C₂₆₉	C₂²×C₈	1	4	yes	complemented
78	C₄×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₂₆₉	C₂×#41	1	1	yes	characteristic
80	C₂×C₂₆₉	—	2	8	no	—
81	C₂×C₂₆₉	C₈×D₄	1	4	yes	complemented
82	C₂×C₂₆₉	C₂×C₄×D₄	1	1	yes	characteristic
83	C₂×C₂₆₉	C₂³×C₈	1	1	yes	characteristic
84	C₂×C₂₆₉	C₂×#131	1	1	yes	characteristic
85	C₂₆₉	C₂×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₈×D₄	—	269	1	no	maximal, Sylow
87	C₂³×C₈	—	269	2	no	—
88	C₂×C₄×D₄	D₂₆₉	1	1	yes	characteristic
89	C₈×D₄	—	269	8	no	—
90	C₂×C₄⋊C₈	—	269	1	no	—
91	C₂×C₂²⋊C₈	—	269	2	no	—
92	C₂×C₄×C₈	—	269	1	no	—
93	C₂²×D₄	Dic₂₆₉	1	1	yes	characteristic
94	C₂³×C₄	C₂×D₂₆₉	1	2	yes	—
95	C₂²×C₈	—	269	1	no	—
96	C₂²×C₈	—	269	8	no	—
97	C₂²×C₈	—	269	2	no	—
98	C₂²×C₈	—	269	1	no	—
99	C₂²×C₈	—	538	8	no	—
100	C₄×D₄	Dic₂₆₉	1	4	yes	—
101	C₄×D₄	C₂×D₂₆₉	1	4	yes	—
102	C₂×C₄⋊C₄	Dic₂₆₉	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	Dic₂₆₉	1	2	yes	—
104	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
105	C₄⋊C₈	—	269	4	no	—
106	C₂²⋊C₈	—	269	8	no	—
107	C₄×C₈	—	269	4	no	—
108	C₂⁴	C₂×Dic₂₆₉	1	2	yes	—
109	C₂×D₄	#14524130	1	8	yes	direct product factor
110	C₂×D₄	C₂×Dic₂₆₉	1	4	yes	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	C₂×Dic₂₆₉	1	1	yes	characteristic
114	C₂²×C₄	C₂×Dic₂₆₉	1	2	yes	—
115	C₂²×C₄	C₂×Dic₂₆₉	1	4	yes	—
116	C₂²×C₄	C₂×Dic₂₆₉	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
118	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
119	C₂×C₈	—	269	4	no	—
120	C₂×C₈	—	269	2	no	—
121	C₂×C₈	—	269	2	no	—
122	C₂×C₈	—	269	4	no	—
123	C₂×C₈	—	538	16	no	—
124	C₂×C₈	—	538	4	no	—
125	C₂×C₈	—	538	2	no	—
126	C₄⋊C₄	C₂×Dic₂₆₉	1	4	yes	—
127	C₂²⋊C₄	C₂×Dic₂₆₉	1	8	yes	—
128	C₄²	C₂×Dic₂₆₉	1	2	yes	—
129	C₄²	C₂²×D₂₆₉	1	2	yes	—
130	C₂³	—	2	4	no	—
131	C₂³	—	2	4	no	—
132	C₂³	C₂×#14524130	1	8	yes	complemented
133	C₂³	C₂²×Dic₂₆₉	1	4	yes	—
134	C₂³	C₂²×Dic₂₆₉	1	1	yes	characteristic
135	D₄	C₂×#14524130	1	16	yes	direct product factor
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	C₂×#14524130	1	4	yes	complemented
142	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
143	C₂×C₄	C₂²×Dic₂₆₉	1	4	yes	—
144	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
145	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
146	C₂×C₄	D₄×D₂₆₉	1	2	yes	—
147	C₂×C₄	#16864401	1	2	yes	—
148	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
149	C₈	—	269	4	no	—
150	C₈	—	538	4	no	—
151	C₈	—	538	2	no	—
152	C₂²	—	2	16	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	D₄×Dic₂₆₉	1	2	yes	—
156	C₂²	C₂²×#14524130	1	2	yes	—
157	C₂²	C₂²×#14524130	1	8	yes	complemented
158	C₂²	#22575460	1	2	yes	—
159	C₂²	C₂³×Dic₂₆₉	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	2	2	no	—
162	C₄	D₄×Dic₂₆₉	1	2	yes	—
163	C₄	C₂²×#14524130	1	4	yes	complemented
164	C₄	C₂×D₄×D₂₆₉	1	1	yes	characteristic
165	C₄	C₂×#16864401	1	1	yes	characteristic
166	C₂	—	2	8	no	—
167	C₂	D₄×#14524130	1	4	yes	direct product factor
168	C₂	C₂×D₄×Dic₂₆₉	1	1	yes	characteristic
169	C₂	C₂³×#14524130	1	1	yes	characteristic
170	C₂	C₂×#22575460	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₆₉
commutator	C₂×C₂₆₉	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	C₂×C₄×D₄×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂²×C₄
prime radical	C₂²×D₄×C₂₆₉	C₄
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	C₂×C₄×D₄	D₂₆₉
hypocenter	C₂₆₉	C₂×C₈×D₄
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#14524130	C₂	semidirect
2. D₄×#14524130	C₂	direct (N × H)
3. C₂×C₄⋊#14524130	C₂	semidirect
4. C₂×C₂²⋊#14524130	C₂	semidirect
5. C₂×C₄×#14524130	C₂	semidirect
11. C₂²×#14524130	C₂²	semidirect
12. C₄⋊#14524130	C₂²	semidirect
13. C₂²⋊#14524130	C₂²	semidirect
14. C₄×#14524130	C₂²	semidirect
25. C₂×#14524130	D₄	direct (N × H)
30. C₂×D₄×C₂₆₉	C₈	semidirect
46. #14524130	C₂×D₄	direct (N × H)
49. C₂³×C₂₆₉	C₂×C₈	semidirect
52. D₄×C₂₆₉	C₂×C₈	semidirect
58. C₂×C₄×C₂₆₉	C₂×C₈	semidirect
71. C₂²×C₂₆₉	C₂²×C₈	semidirect
77. C₄×C₂₆₉	C₂²×C₈	semidirect
81. C₂×C₂₆₉	C₈×D₄	semidirect
85. C₂₆₉	C₂×C₈×D₄	semidirect
109. C₂×D₄	#14524130	direct (N × H)
132. C₂³	C₂×#14524130	semidirect
135. D₄	C₂×#14524130	direct (N × H)
141. C₂×C₄	C₂×#14524130	semidirect
157. C₂²	C₂²×#14524130	semidirect
163. C₄	C₂²×#14524130	semidirect
167. C₂	D₄×#14524130	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×#14524130	2
D₄×#14524130	2
C₂×C₄⋊#14524130	2
C₂×C₂²⋊#14524130	2
C₂×C₄×#14524130	2
C₂×C₄×D₄×C₂₆₉	2
C₂×C₈×D₄	269

Subgroups — C2×D4×#14524130 #46819558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×#14524130 #46819558