Subgroups — C₁₀₉×#3342 #28360596

Order 13952.

340 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#556×C₁₀₉	C₂	1	2	yes	maximal
2	#500×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#485×C₁₀₉	C₂	1	4	yes	maximal, complemented
4	#475×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#467×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	C₄×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#427×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
12	#126×C₁₀₉	—	2	4	no	—
13	#124×C₁₀₉	—	2	2	no	—
14	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
15	C₄×Q₈×C₁₀₉	—	2	2	no	—
16	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
17	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	#117×C₁₀₉	C₂²	1	4	yes	—
19	#107×C₁₀₉	C₂²	1	2	yes	—
20	#107×C₁₀₉	C₂²	1	2	yes	—
21	#107×C₁₀₉	C₂²	1	1	yes	characteristic
22	#106×C₁₀₉	C₂²	1	4	yes	—
23	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	#103×C₁₀₉	C₂²	1	2	yes	—
25	#103×C₁₀₉	C₂²	1	4	yes	—
26	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
27	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₁₀₉	—	2	2	no	—
29	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
30	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
32	Q₁₆×C₁₀₉	—	2	2	no	—
33	C₂×C₈×C₁₀₉	—	2	4	no	—
34	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
36	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₁₀₉	—	2	4	no	—
38	C₄⋊C₄×C₁₀₉	—	2	2	no	—
39	C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
42	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
43	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
44	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
45	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
46	C₄²×C₁₀₉	—	2	4	no	—
47	C₄²×C₁₀₉	—	2	2	no	—
48	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
50	Q₈×C₁₀₉	—	2	2	no	—
51	Q₈×C₁₀₉	—	2	2	no	—
52	Q₈×C₁₀₉	—	4	4	no	—
53	D₄×C₁₀₉	—	2	2	no	—
54	C₂×C₄×C₁₀₉	—	2	2	no	—
55	C₂×C₄×C₁₀₉	—	2	2	no	—
56	C₂×C₄×C₁₀₉	—	2	2	no	—
57	C₂×C₄×C₁₀₉	—	2	2	no	—
58	C₂×C₄×C₁₀₉	—	2	4	no	—
59	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
63	C₈×C₁₀₉	—	2	2	no	—
64	C₈×C₁₀₉	—	2	1	no	—
65	C₈×C₁₀₉	#41	1	2	yes	—
66	C₂²×C₁₀₉	—	2	2	no	—
67	C₂²×C₁₀₉	—	2	2	no	—
68	C₂²×C₁₀₉	—	2	2	no	—
69	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
70	C₄×C₁₀₉	—	2	2	no	—
71	C₄×C₁₀₉	—	2	1	no	—
72	C₄×C₁₀₉	—	4	2	no	—
73	C₄×C₁₀₉	—	4	4	no	—
74	C₄×C₁₀₉	—	4	2	no	—
75	C₄×C₁₀₉	#135	1	2	yes	—
76	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
77	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
78	C₂×C₁₀₉	—	4	2	no	—
79	C₂×C₁₀₉	#547	1	1	yes	characteristic
80	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
81	C₂×C₁₀₉	#578	1	1	yes	characteristic
82	#3342	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₁₀₉	#3342	1	1	yes	characteristic, direct product factor, Sylow
84	#556	C₂×C₁₀₉	1	2	yes	—
85	#500	C₂×C₁₀₉	1	1	yes	characteristic, complemented
86	#485	C₂×C₁₀₉	1	4	yes	complemented
87	#475	C₂×C₁₀₉	1	2	yes	complemented
88	#467	C₂×C₁₀₉	1	2	yes	complemented
89	C₄×Q₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
90	C₈×D₄	C₂×C₁₀₉	1	1	yes	characteristic
91	#427	C₂×C₁₀₉	1	2	yes	complemented
92	C₂×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
93	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	—
94	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
95	#126	—	2	4	no	—
96	#124	—	2	2	no	—
97	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
98	C₄×Q₈	—	2	2	no	—
99	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
100	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
101	#117	C₂²×C₁₀₉	1	4	yes	—
102	#107	C₂²×C₁₀₉	1	1	yes	characteristic
103	#107	C₂²×C₁₀₉	1	2	yes	—
104	#107	C₂²×C₁₀₉	1	2	yes	—
105	#106	C₂²×C₁₀₉	1	4	yes	—
106	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
107	#103	C₂²×C₁₀₉	1	2	yes	—
108	#103	C₂²×C₁₀₉	1	4	yes	—
109	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
110	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
113	C₂×D₄	D₄×C₁₀₉	1	1	yes	characteristic
114	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
115	Q₁₆	—	2	2	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
118	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
119	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
124	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
125	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
126	C₂²⋊C₄	—	2	4	no	—
127	C₂²⋊C₄	—	2	4	no	—
128	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
129	C₄²	—	2	4	no	—
130	C₄²	—	2	2	no	—
131	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
132	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
133	Q₈	—	2	2	no	—
134	Q₈	—	2	2	no	—
135	Q₈	—	4	4	no	—
136	D₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
143	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
144	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
145	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	2	1	no	—
148	C₈	#41×C₁₀₉	1	2	yes	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	2	no	—
156	C₄	—	4	4	no	—
157	C₄	—	4	2	no	—
158	C₄	#135×C₁₀₉	1	2	yes	—
159	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
160	C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
161	C₂	—	4	2	no	—
162	C₂	#547×C₁₀₉	1	1	yes	characteristic
163	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
164	C₂	#578×C₁₀₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. #500×C₁₀₉	C₂	semidirect
3. #485×C₁₀₉	C₂	semidirect
4. #475×C₁₀₉	C₂	semidirect
5. #467×C₁₀₉	C₂	semidirect
6. C₄×Q₁₆×C₁₀₉	C₂	semidirect
8. #427×C₁₀₉	C₂	semidirect
82. #3342	C₁₀₉	direct (N × H)
83. C₁₀₉	#3342	direct (N × H)
85. #500	C₂×C₁₀₉	semidirect
86. #485	C₂×C₁₀₉	semidirect
87. #475	C₂×C₁₀₉	semidirect
88. #467	C₂×C₁₀₉	semidirect
89. C₄×Q₁₆	C₂×C₁₀₉	semidirect
91. #427	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#556×C₁₀₉	2
#500×C₁₀₉	2
#485×C₁₀₉	2
#475×C₁₀₉	2
#467×C₁₀₉	2
C₄×Q₁₆×C₁₀₉	2
C₈×D₄×C₁₀₉	2
#427×C₁₀₉	2
#3342	109

Subgroups — C109×#3342 #28360596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3342 #28360596