Subgroups — C₂×#563×C₁₀₉ #28360782

Order 13952.

1024 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	#563×C₁₀₉	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₂×#126×C₁₀₉	C₂	1	4	yes	maximal, complemented
4	C₂×#125×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	C₂×#124×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#123×C₁₀₉	C₂	1	2	yes	maximal
7	C₂×C₂²⋊Q₈×C₁₀₉	C₂	1	4	yes	maximal, complemented
8	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
9	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₀₉	C₂²	1	16	yes	complemented
13	#126×C₁₀₉	C₂²	1	32	yes	complemented
14	#125×C₁₀₉	C₂²	1	16	yes	complemented
15	#124×C₁₀₉	C₂²	1	8	yes	—
16	#123×C₁₀₉	C₂²	1	16	yes	—
17	C₂²⋊Q₈×C₁₀₉	C₂²	1	32	yes	complemented
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
25	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
28	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
29	C₂×Q₈×C₁₀₉	—	4	4	no	—
30	C₂×Q₈×C₁₀₉	—	4	2	no	—
31	C₂×Q₈×C₁₀₉	C₂³	1	8	yes	—
32	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
33	C₂×D₄×C₁₀₉	—	4	2	no	—
34	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
35	C₂²×C₄×C₁₀₉	—	4	4	no	—
36	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
38	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
41	C₂²×C₄×C₁₀₉	C₂³	1	8	yes	—
42	C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
43	C₄⋊C₄×C₁₀₉	C₂³	1	16	yes	—
44	C₄⋊C₄×C₁₀₉	C₂³	1	16	yes	—
45	C₄⋊C₄×C₁₀₉	C₂³	1	16	yes	—
46	C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
47	C₂²⋊C₄×C₁₀₉	C₂³	1	16	yes	—
48	C₂²⋊C₄×C₁₀₉	C₂³	1	16	yes	—
49	C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
50	C₄²×C₁₀₉	C₂³	1	4	yes	—
51	C₄²×C₁₀₉	C₂³	1	8	yes	—
52	C₂³×C₁₀₉	—	2	4	no	—
53	C₂³×C₁₀₉	—	2	8	no	—
54	C₂³×C₁₀₉	C₂⁴	1	4	yes	—
55	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
56	Q₈×C₁₀₉	—	4	8	no	—
57	Q₈×C₁₀₉	—	4	4	no	—
58	D₄×C₁₀₉	—	4	4	no	—
59	C₂×C₄×C₁₀₉	—	2	8	no	—
60	C₂×C₄×C₁₀₉	—	2	2	no	—
61	C₂×C₄×C₁₀₉	—	2	4	no	—
62	C₂×C₄×C₁₀₉	—	2	4	no	—
63	C₂×C₄×C₁₀₉	—	2	8	no	—
64	C₂×C₄×C₁₀₉	—	4	8	no	—
65	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
66	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
67	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
68	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
69	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
70	C₂²×C₁₀₉	—	2	4	no	—
71	C₂²×C₁₀₉	—	2	8	no	—
72	C₂²×C₁₀₉	—	4	8	no	—
73	C₂²×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
74	C₂²×C₁₀₉	2₋¹⁺⁴	1	4	yes	—
75	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
76	C₄×C₁₀₉	—	2	8	no	—
77	C₄×C₁₀₉	—	2	2	no	—
78	C₄×C₁₀₉	—	2	4	no	—
79	C₄×C₁₀₉	—	2	4	no	—
80	C₄×C₁₀₉	—	2	8	no	—
81	C₂×C₁₀₉	—	4	4	no	—
82	C₂×C₁₀₉	#563	1	4	yes	direct product factor
83	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
84	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	2	yes	—
85	C₂×#563	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₁₀₉	C₂×#563	1	1	yes	characteristic, direct product factor, Sylow
87	#563	C₂×C₁₀₉	1	16	yes	direct product factor
88	C₂×C₄⋊Q₈	C₂×C₁₀₉	1	2	yes	complemented
89	C₂×#126	C₂×C₁₀₉	1	4	yes	complemented
90	C₂×#125	C₂×C₁₀₉	1	2	yes	complemented
91	C₂×#124	C₂×C₁₀₉	1	1	yes	characteristic
92	C₂×#123	C₂×C₁₀₉	1	2	yes	—
93	C₂×C₂²⋊Q₈	C₂×C₁₀₉	1	4	yes	complemented
94	C₂²×Q₈	C₂²×C₁₀₉	1	2	yes	—
95	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
96	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
97	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
98	C₄⋊Q₈	C₂²×C₁₀₉	1	16	yes	complemented
99	#126	C₂²×C₁₀₉	1	32	yes	complemented
100	#125	C₂²×C₁₀₉	1	16	yes	complemented
101	#124	C₂²×C₁₀₉	1	8	yes	—
102	#123	C₂²×C₁₀₉	1	16	yes	—
103	C₂²⋊Q₈	C₂²×C₁₀₉	1	32	yes	complemented
104	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
105	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
106	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
107	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
108	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
109	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
110	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
111	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
112	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
113	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
114	C₂⁴	C₂³×C₁₀₉	1	2	yes	—
115	C₂×Q₈	—	4	4	no	—
116	C₂×Q₈	—	4	2	no	—
117	C₂×Q₈	C₂³×C₁₀₉	1	8	yes	—
118	C₂×Q₈	C₂³×C₁₀₉	1	4	yes	—
119	C₂×D₄	—	4	2	no	—
120	C₂×D₄	C₂³×C₁₀₉	1	4	yes	—
121	C₂²×C₄	—	4	4	no	—
122	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
123	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
124	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
125	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
126	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
127	C₂²×C₄	C₂³×C₁₀₉	1	8	yes	—
128	C₄⋊C₄	C₂³×C₁₀₉	1	8	yes	—
129	C₄⋊C₄	C₂³×C₁₀₉	1	16	yes	—
130	C₄⋊C₄	C₂³×C₁₀₉	1	16	yes	—
131	C₄⋊C₄	C₂³×C₁₀₉	1	16	yes	—
132	C₄⋊C₄	C₂³×C₁₀₉	1	8	yes	—
133	C₂²⋊C₄	C₂³×C₁₀₉	1	16	yes	—
134	C₂²⋊C₄	C₂³×C₁₀₉	1	16	yes	—
135	C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
136	C₄²	C₂³×C₁₀₉	1	4	yes	—
137	C₄²	C₂³×C₁₀₉	1	8	yes	—
138	C₂³	—	2	4	no	—
139	C₂³	—	2	8	no	—
140	C₂³	C₂⁴×C₁₀₉	1	4	yes	—
141	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
142	Q₈	—	4	8	no	—
143	Q₈	—	4	4	no	—
144	D₄	—	4	4	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	8	no	—
150	C₂×C₄	—	4	8	no	—
151	C₂×C₄	C₂⁴×C₁₀₉	1	4	yes	—
152	C₂×C₄	C₂⁴×C₁₀₉	1	8	yes	—
153	C₂×C₄	C₂⁴×C₁₀₉	1	4	yes	—
154	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
155	C₂×C₄	C₂⁴×C₁₀₉	1	8	yes	—
156	C₂²	—	2	4	no	—
157	C₂²	—	2	8	no	—
158	C₂²	—	4	8	no	—
159	C₂²	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
160	C₂²	2₋¹⁺⁴×C₁₀₉	1	4	yes	—
161	C₂²	C₂⁵×C₁₀₉	1	1	yes	characteristic
162	C₄	—	2	8	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	—	2	4	no	—
166	C₄	—	2	8	no	—
167	C₂	—	4	4	no	—
168	C₂	#563×C₁₀₉	1	4	yes	direct product factor
169	C₂	C₂×H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
170	C₂	C₂×2₋¹⁺⁴×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
prime radical	C₂²×D₄×C₁₀₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #563×C₁₀₉	C₂	direct (N × H)
2. C₂×C₄⋊Q₈×C₁₀₉	C₂	semidirect
3. C₂×#126×C₁₀₉	C₂	semidirect
4. C₂×#125×C₁₀₉	C₂	semidirect
7. C₂×C₂²⋊Q₈×C₁₀₉	C₂	semidirect
12. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
13. #126×C₁₀₉	C₂²	semidirect
14. #125×C₁₀₉	C₂²	semidirect
17. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
82. C₂×C₁₀₉	#563	direct (N × H)
85. C₂×#563	C₁₀₉	direct (N × H)
86. C₁₀₉	C₂×#563	direct (N × H)
87. #563	C₂×C₁₀₉	direct (N × H)
88. C₂×C₄⋊Q₈	C₂×C₁₀₉	semidirect
89. C₂×#126	C₂×C₁₀₉	semidirect
90. C₂×#125	C₂×C₁₀₉	semidirect
93. C₂×C₂²⋊Q₈	C₂×C₁₀₉	semidirect
98. C₄⋊Q₈	C₂²×C₁₀₉	semidirect
99. #126	C₂²×C₁₀₉	semidirect
100. #125	C₂²×C₁₀₉	semidirect
103. C₂²⋊Q₈	C₂²×C₁₀₉	semidirect
168. C₂	#563×C₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
#563×C₁₀₉	2
C₂×C₄⋊Q₈×C₁₀₉	2
C₂×#126×C₁₀₉	2
C₂×#125×C₁₀₉	2
C₂×#124×C₁₀₉	2
C₂×#123×C₁₀₉	2
C₂×C₂²⋊Q₈×C₁₀₉	2
C₂×#563	109

Subgroups — C2×#563×C109 #28360782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#563×C₁₀₉ #28360782