Subgroups — C₁₀₉×#3169 #28360423

Order 13952.

456 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	#532×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	#500×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#493×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#486×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#466×C₁₀₉	C₂	1	4	yes	maximal, complemented
6	C₄×D₈×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₁₀₉	—	2	2	no	—
9	C₂×D₈×C₁₀₉	C₂²	1	4	yes	—
10	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
12	#127×C₁₀₉	C₂²	1	2	yes	—
13	#124×C₁₀₉	—	2	2	no	—
14	#121×C₁₀₉	—	2	2	no	—
15	#121×C₁₀₉	C₂²	1	4	yes	—
16	C₄×D₄×C₁₀₉	—	2	2	no	—
17	C₄×D₄×C₁₀₉	C₂²	1	4	yes	—
18	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
19	#107×C₁₀₉	C₂²	1	4	yes	—
20	#102×C₁₀₉	C₂²	1	8	yes	—
21	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
23	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	#41×C₁₀₉	—	4	8	no	—
25	C₂×Q₈×C₁₀₉	—	2	2	no	—
26	C₂×D₄×C₁₀₉	—	2	4	no	—
27	C₂×D₄×C₁₀₉	—	2	2	no	—
28	C₂×D₄×C₁₀₉	—	2	2	no	—
29	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
30	C₂²×C₄×C₁₀₉	—	2	4	no	—
31	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
34	D₈×C₁₀₉	—	2	8	no	—
35	C₂×C₈×C₁₀₉	—	2	4	no	—
36	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
37	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
38	C₄⋊C₄×C₁₀₉	—	2	2	no	—
39	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
40	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
41	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
42	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
43	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
44	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂³×C₁₀₉	—	2	4	no	—
47	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	Q₈×C₁₀₉	—	4	4	no	—
49	D₄×C₁₀₉	—	2	8	no	—
50	D₄×C₁₀₉	—	4	8	no	—
51	D₄×C₁₀₉	—	4	4	no	—
52	D₄×C₁₀₉	—	4	4	no	—
53	C₂×C₄×C₁₀₉	—	2	2	no	—
54	C₂×C₄×C₁₀₉	—	2	2	no	—
55	C₂×C₄×C₁₀₉	—	2	4	no	—
56	C₂×C₄×C₁₀₉	—	4	8	no	—
57	C₂×C₄×C₁₀₉	D₈	1	4	yes	complemented
58	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
64	C₈×C₁₀₉	—	2	2	no	—
65	C₈×C₁₀₉	#41	1	4	yes	complemented
66	C₂²×C₁₀₉	—	2	1	no	—
67	C₂²×C₁₀₉	—	2	1	no	—
68	C₂²×C₁₀₉	—	4	4	no	—
69	C₂²×C₁₀₉	—	4	4	no	—
70	C₂²×C₁₀₉	—	4	4	no	—
71	C₂²×C₁₀₉	C₂×D₈	1	2	yes	—
72	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₁₀₉	—	2	1	no	—
74	C₄×C₁₀₉	—	2	1	no	—
75	C₄×C₁₀₉	—	4	4	no	—
76	C₄×C₁₀₉	C₂×D₈	1	2	yes	—
77	C₄×C₁₀₉	C₂×D₈	1	2	yes	—
78	C₄×C₁₀₉	#135	1	2	yes	—
79	C₄×C₁₀₉	C₂×#41	1	2	yes	—
80	C₂×C₁₀₉	—	2	2	no	—
81	C₂×C₁₀₉	—	8	4	no	—
82	C₂×C₁₀₉	#532	1	1	yes	characteristic
83	C₂×C₁₀₉	C₂²×D₈	1	1	yes	characteristic
84	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
85	#3169	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₁₀₉	#3169	1	1	yes	characteristic, direct product factor, Sylow
87	#532	C₂×C₁₀₉	1	2	yes	complemented
88	#500	C₂×C₁₀₉	1	1	yes	characteristic, complemented
89	#493	C₂×C₁₀₉	1	1	yes	characteristic, complemented
90	#486	C₂×C₁₀₉	1	2	yes	complemented
91	#466	C₂×C₁₀₉	1	4	yes	complemented
92	C₄×D₈	C₂×C₁₀₉	1	4	yes	complemented
93	C₂×C₄×C₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
94	C₂×#41	—	2	2	no	—
95	C₂×D₈	C₂²×C₁₀₉	1	4	yes	—
96	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	—
97	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
98	#127	C₂²×C₁₀₉	1	2	yes	—
99	#124	—	2	2	no	—
100	#121	—	2	2	no	—
101	#121	C₂²×C₁₀₉	1	4	yes	—
102	C₄×D₄	—	2	2	no	—
103	C₄×D₄	C₂²×C₁₀₉	1	4	yes	—
104	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
105	#107	C₂²×C₁₀₉	1	4	yes	—
106	#102	C₂²×C₁₀₉	1	8	yes	—
107	C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
108	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
109	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
110	#41	—	4	8	no	—
111	C₂×Q₈	—	2	2	no	—
112	C₂×D₄	—	2	4	no	—
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	—	2	2	no	—
115	C₂×D₄	C₂³×C₁₀₉	1	4	yes	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
118	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
119	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
120	D₈	—	2	8	no	—
121	C₂×C₈	—	2	4	no	—
122	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
123	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
126	C₂²⋊C₄	—	2	4	no	—
127	C₂²⋊C₄	—	2	4	no	—
128	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
129	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
130	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
131	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
132	C₂³	—	2	4	no	—
133	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
134	Q₈	—	4	4	no	—
135	D₄	—	2	8	no	—
136	D₄	—	4	8	no	—
137	D₄	—	4	4	no	—
138	D₄	—	4	4	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	8	no	—
143	C₂×C₄	D₈×C₁₀₉	1	4	yes	complemented
144	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
145	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
146	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
147	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
148	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
149	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	#41×C₁₀₉	1	4	yes	complemented
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	4	4	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	C₂×D₈×C₁₀₉	1	2	yes	—
158	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	—	4	4	no	—
162	C₄	C₂×D₈×C₁₀₉	1	2	yes	—
163	C₄	C₂×D₈×C₁₀₉	1	2	yes	—
164	C₄	#135×C₁₀₉	1	2	yes	—
165	C₄	C₂×#41×C₁₀₉	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	—	8	4	no	—
168	C₂	#532×C₁₀₉	1	1	yes	characteristic
169	C₂	C₂²×D₈×C₁₀₉	1	1	yes	characteristic
170	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
socle	C₂²×C₁₀₉	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #532×C₁₀₉	C₂	semidirect
2. #500×C₁₀₉	C₂	semidirect
3. #493×C₁₀₉	C₂	semidirect
4. #486×C₁₀₉	C₂	semidirect
5. #466×C₁₀₉	C₂	semidirect
6. C₄×D₈×C₁₀₉	C₂	semidirect
7. C₂×C₄×C₈×C₁₀₉	C₂	semidirect
57. C₂×C₄×C₁₀₉	D₈	semidirect
65. C₈×C₁₀₉	#41	semidirect
85. #3169	C₁₀₉	direct (N × H)
86. C₁₀₉	#3169	direct (N × H)
87. #532	C₂×C₁₀₉	semidirect
88. #500	C₂×C₁₀₉	semidirect
89. #493	C₂×C₁₀₉	semidirect
90. #486	C₂×C₁₀₉	semidirect
91. #466	C₂×C₁₀₉	semidirect
92. C₄×D₈	C₂×C₁₀₉	semidirect
93. C₂×C₄×C₈	C₂×C₁₀₉	semidirect
143. C₂×C₄	D₈×C₁₀₉	semidirect
151. C₈	#41×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#532×C₁₀₉	2
#500×C₁₀₉	2
#493×C₁₀₉	2
#486×C₁₀₉	2
#466×C₁₀₉	2
C₄×D₈×C₁₀₉	2
C₂×C₄×C₈×C₁₀₉	2
#3169	109

Subgroups — C109×#3169 #28360423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3169 #28360423