Subgroups — C₂×#20294783 #28362733

Order 13952.

404 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812658	C₂	1	2	yes	maximal
2	#20294783	C₂	1	4	yes	maximal, direct product factor
3	C₂²×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812658	—	2	2	no	—
5	#16812658	—	2	2	no	—
6	#16812658	C₄	1	4	yes	complemented
7	#16812658	C₂²	1	4	yes	—
8	C₂×C₄⋊Dic₁₀₉	—	2	2	no	—
9	C₂×C₂²⋊Dic₁₀₉	—	2	2	no	—
10	C₂×C₄×Dic₁₀₉	C₄	1	4	yes	complemented
11	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₁₀₉	—	2	2	no	—
13	C₂×M₁₆×C₁₀₉	—	2	2	no	—
14	C₂×M₁₆×C₁₀₉	C₄	1	2	yes	complemented
15	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₀₉	C₄	1	2	yes	complemented
17	C₂²×Dic₁₀₉	—	2	4	no	—
18	C₄⋊Dic₁₀₉	—	2	4	no	—
19	C₄⋊Dic₁₀₉	—	2	4	no	—
20	C₂²⋊Dic₁₀₉	—	2	4	no	—
21	C₂²⋊Dic₁₀₉	—	2	4	no	—
22	C₄×Dic₁₀₉	—	2	4	no	—
23	C₄×Dic₁₀₉	C₂×C₄	1	8	yes	complemented
24	C₂⁴×C₁₀₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
31	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
32	C₂²×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₀₉	—	2	4	no	—
36	M₁₆×C₁₀₉	—	2	4	no	—
37	C₂×C₈×C₁₀₉	—	2	2	no	—
38	C₂×C₈×C₁₀₉	—	2	2	no	—
39	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
40	C₂³×C₁₀₉	—	4	2	no	—
41	C₂³×C₁₀₉	C₂²⋊C₄	1	2	yes	—
42	C₂³×C₁₀₉	C₂²⋊C₄	1	2	yes	—
43	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
47	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
64	C₂×Dic₁₀₉	—	2	8	no	—
65	C₂×Dic₁₀₉	—	4	8	no	—
66	C₈×C₁₀₉	—	2	4	no	—
67	C₂²×C₁₀₉	—	2	4	no	—
68	C₂²×C₁₀₉	—	2	2	no	—
69	C₂²×C₁₀₉	—	4	2	no	—
70	C₂²×C₁₀₉	—	4	1	no	—
71	C₂²×C₁₀₉	—	4	1	no	—
72	C₂²×C₁₀₉	#95	1	1	yes	characteristic
73	C₂²×C₁₀₉	#95	1	2	yes	—
74	C₂²×C₁₀₉	#95	1	1	yes	characteristic
75	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
76	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₁₀₉	#95	1	2	yes	—
78	C₄×C₁₀₉	#95	1	1	yes	characteristic
79	C₄×C₁₀₉	#95	1	1	yes	characteristic
80	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
81	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
83	Dic₁₀₉	—	4	8	no	—
84	C₂×C₁₀₉	—	2	2	no	—
85	C₂×C₁₀₉	—	2	2	no	—
86	C₂×C₁₀₉	—	2	1	no	—
87	C₂×C₁₀₉	—	2	1	no	—
88	C₂×C₁₀₉	H(3,ℤ/4)	1	2	yes	complemented
89	C₂×C₁₀₉	C₂×#95	1	1	yes	characteristic
90	C₂×H(3,ℤ/4)	—	109	1	no	maximal, Sylow
91	C₁₀₉	C₂×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
92	C₂²×M₁₆	D₁₀₉	1	1	yes	characteristic
93	C₂×#117	—	109	2	no	—
94	H(3,ℤ/4)	—	109	4	no	—
95	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	Dic₁₀₉	1	2	yes	complemented
99	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
100	C₂²×C₈	Dic₁₀₉	1	2	yes	complemented
101	#117	—	109	4	no	—
102	#117	—	109	4	no	—
103	#117	—	218	2	no	—
104	#117	—	218	2	no	—
105	C₂×C₄⋊C₄	—	218	2	no	—
106	C₂×C₂²⋊C₄	—	218	2	no	—
107	C₂×C₄²	—	109	4	no	—
108	C₂⁴	D₄₃₆	1	1	yes	characteristic
109	C₂²×C₄	—	218	4	no	—
110	C₂²×C₄	Dic₂₁₈	1	1	yes	characteristic
111	C₂²×C₄	Dic₂₁₈	1	1	yes	characteristic
112	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
113	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
114	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
115	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
116	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
118	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
119	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
120	M₁₆	—	2	4	no	—
121	M₁₆	—	2	4	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	C₂×Dic₁₀₉	1	4	yes	complemented
125	C₄⋊C₄	—	218	4	no	—
126	C₄⋊C₄	—	218	4	no	—
127	C₂²⋊C₄	—	218	4	no	—
128	C₂²⋊C₄	—	218	4	no	—
129	C₄²	—	109	8	no	—
130	C₄²	—	218	4	no	—
131	C₂³	—	4	2	no	—
132	C₂³	#13182006	1	2	yes	—
133	C₂³	#13182006	1	2	yes	—
134	C₂³	C₄⋊Dic₁₀₉	1	1	yes	characteristic
135	C₂³	C₄⋊Dic₁₀₉	1	1	yes	characteristic
136	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
137	C₂×C₄	—	436	8	no	—
138	C₂×C₄	—	218	8	no	—
139	C₂×C₄	C₄×Dic₁₀₉	1	1	yes	characteristic
140	C₂×C₄	C₄×Dic₁₀₉	1	2	yes	—
141	C₂×C₄	C₄×Dic₁₀₉	1	1	yes	characteristic
142	C₂×C₄	#13182006	1	2	yes	—
143	C₂×C₄	#13182006	1	2	yes	—
144	C₂×C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
145	C₂×C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
146	C₂×C₄	C₂²⋊Dic₁₀₉	1	2	yes	—
147	C₂×C₄	#13182008	1	2	yes	—
148	C₂×C₄	#13182008	1	2	yes	—
149	C₂×C₄	#13182008	1	2	yes	—
150	C₂×C₄	#13182008	1	2	yes	—
151	C₂×C₄	C₄⋊Dic₁₀₉	1	1	yes	characteristic
152	C₂×C₄	C₄⋊Dic₁₀₉	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
154	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
155	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
156	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
157	C₈	—	2	4	no	—
158	C₂²	—	2	4	no	—
159	C₂²	—	2	2	no	—
160	C₂²	—	4	2	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	1	no	—
163	C₂²	#16812606	1	1	yes	characteristic
164	C₂²	#16812606	1	2	yes	—
165	C₂²	#16812606	1	1	yes	characteristic
166	C₂²	C₂×#13182006	1	2	yes	—
167	C₂²	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
168	C₄	—	436	8	no	—
169	C₄	#16812606	1	2	yes	—
170	C₄	#16812606	1	1	yes	characteristic
171	C₄	#16812606	1	1	yes	characteristic
172	C₄	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
173	C₄	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
174	C₄	C₂×#13182008	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	—	2	2	no	—
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#20294783	1	2	yes	direct product factor
180	C₂	C₂×#16812606	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₀₉
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂²×M₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#95
prime radical	C₂⁴×C₁₀₉	D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂²×M₁₆	D₁₀₉
hypocenter	C₁₀₉	C₂×H(3,ℤ/4)
LCS	G ▷ C₄×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20294783	C₂	direct (N × H)
6. #16812658	C₄	semidirect
10. C₂×C₄×Dic₁₀₉	C₄	semidirect
14. C₂×M₁₆×C₁₀₉	C₄	semidirect
16. C₂²×C₈×C₁₀₉	C₄	semidirect
23. C₄×Dic₁₀₉	C₂×C₄	semidirect
39. C₂×C₈×C₁₀₉	C₂×C₄	semidirect
88. C₂×C₁₀₉	H(3,ℤ/4)	semidirect
91. C₁₀₉	C₂×H(3,ℤ/4)	semidirect
98. C₂×M₁₆	Dic₁₀₉	semidirect
100. C₂²×C₈	Dic₁₀₉	semidirect
124. C₂×C₈	C₂×Dic₁₀₉	semidirect
179. C₂	#20294783	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×#16812658	2
#20294783	2
C₂²×M₁₆×C₁₀₉	2
C₂×H(3,ℤ/4)	109

Subgroups — C2×#20294783 #28362733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20294783 #28362733