Subgroups — C₁₀₉×#3170 #28360424

Order 13952.

368 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533×C₁₀₉	C₂	1	2	yes	maximal
2	#500×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#494×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#487×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#467×C₁₀₉	C₂	1	4	yes	maximal
6	C₄×Q₁₆×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×C₁₀₉	C₂²	1	4	yes	—
9	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
10	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
12	#125×C₁₀₉	—	2	2	no	—
13	C₂²⋊Q₈×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	4	yes	—
17	#117×C₁₀₉	—	2	2	no	—
18	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
19	#107×C₁₀₉	C₂²	1	4	yes	—
20	#103×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	C₂×Q₈×C₁₀₉	—	2	2	no	—
25	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
26	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
29	Q₁₆×C₁₀₉	—	2	8	no	—
30	C₂×C₈×C₁₀₉	—	2	4	no	—
31	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
32	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
33	C₄⋊C₄×C₁₀₉	—	2	2	no	—
34	C₄⋊C₄×C₁₀₉	—	2	2	no	—
35	C₄⋊C₄×C₁₀₉	—	2	2	no	—
36	C₄⋊C₄×C₁₀₉	—	2	4	no	—
37	C₄⋊C₄×C₁₀₉	—	2	4	no	—
38	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
39	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
40	C₄²×C₁₀₉	—	2	4	no	—
41	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
42	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
43	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₁₀₉	—	2	8	no	—
47	Q₈×C₁₀₉	—	4	4	no	—
48	C₂×C₄×C₁₀₉	—	2	2	no	—
49	C₂×C₄×C₁₀₉	—	2	2	no	—
50	C₂×C₄×C₁₀₉	—	2	4	no	—
51	C₂×C₄×C₁₀₉	—	2	4	no	—
52	C₂×C₄×C₁₀₉	Q₁₆	1	4	yes	complemented
53	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
59	C₈×C₁₀₉	—	2	2	no	—
60	C₈×C₁₀₉	#41	1	4	yes	—
61	C₂²×C₁₀₉	—	2	1	no	—
62	C₂²×C₁₀₉	—	2	1	no	—
63	C₂²×C₁₀₉	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₀₉	—	2	1	no	—
66	C₄×C₁₀₉	—	2	1	no	—
67	C₄×C₁₀₉	—	4	4	no	—
68	C₄×C₁₀₉	—	4	4	no	—
69	C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
70	C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
71	C₄×C₁₀₉	#135	1	2	yes	—
72	C₄×C₁₀₉	C₂×#41	1	2	yes	—
73	C₂×C₁₀₉	—	2	2	no	—
74	C₂×C₁₀₉	#532	1	1	yes	characteristic
75	C₂×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
76	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
77	#3170	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₁₀₉	#3170	1	1	yes	characteristic, direct product factor, Sylow
79	#533	C₂×C₁₀₉	1	2	yes	—
80	#500	C₂×C₁₀₉	1	1	yes	characteristic, complemented
81	#494	C₂×C₁₀₉	1	1	yes	characteristic, complemented
82	#487	C₂×C₁₀₉	1	2	yes	complemented
83	#467	C₂×C₁₀₉	1	4	yes	—
84	C₄×Q₁₆	C₂×C₁₀₉	1	4	yes	complemented
85	C₂×C₄×C₈	C₂×C₁₀₉	1	1	yes	characteristic
86	C₂×Q₁₆	C₂²×C₁₀₉	1	4	yes	—
87	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	—
88	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
89	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
90	#125	—	2	2	no	—
91	C₂²⋊Q₈	—	2	2	no	—
92	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
93	C₄×Q₈	—	2	2	no	—
94	C₄×Q₈	C₂²×C₁₀₉	1	4	yes	—
95	#117	—	2	2	no	—
96	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
97	#107	C₂²×C₁₀₉	1	4	yes	—
98	#103	C₂²×C₁₀₉	1	8	yes	—
99	C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
100	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
101	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	C₂³×C₁₀₉	1	4	yes	—
104	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
105	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
106	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
107	Q₁₆	—	2	8	no	—
108	C₂×C₈	—	2	4	no	—
109	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
110	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
117	C₂²⋊C₄	—	2	4	no	—
118	C₄²	—	2	4	no	—
119	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
120	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
121	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
122	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
123	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
124	Q₈	—	2	8	no	—
125	Q₈	—	4	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	Q₁₆×C₁₀₉	1	4	yes	complemented
131	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
132	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
133	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
134	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
135	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
136	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	#41×C₁₀₉	1	4	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×Q₁₆×C₁₀₉	1	2	yes	—
142	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	4	no	—
146	C₄	—	4	4	no	—
147	C₄	C₂×Q₁₆×C₁₀₉	1	2	yes	—
148	C₄	C₂×Q₁₆×C₁₀₉	1	2	yes	—
149	C₄	#135×C₁₀₉	1	2	yes	—
150	C₄	C₂×#41×C₁₀₉	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	#532×C₁₀₉	1	1	yes	characteristic
153	C₂	C₂²×Q₁₆×C₁₀₉	1	1	yes	characteristic
154	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
prime radical	C₂³×C₁₀₉	C₂×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
2. #500×C₁₀₉	C₂	semidirect
3. #494×C₁₀₉	C₂	semidirect
4. #487×C₁₀₉	C₂	semidirect
6. C₄×Q₁₆×C₁₀₉	C₂	semidirect
52. C₂×C₄×C₁₀₉	Q₁₆	semidirect
77. #3170	C₁₀₉	direct (N × H)
78. C₁₀₉	#3170	direct (N × H)
80. #500	C₂×C₁₀₉	semidirect
81. #494	C₂×C₁₀₉	semidirect
82. #487	C₂×C₁₀₉	semidirect
84. C₄×Q₁₆	C₂×C₁₀₉	semidirect
130. C₂×C₄	Q₁₆×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#533×C₁₀₉	2
#500×C₁₀₉	2
#494×C₁₀₉	2
#487×C₁₀₉	2
#467×C₁₀₉	2
C₄×Q₁₆×C₁₀₉	2
C₂×C₄×C₈×C₁₀₉	2
#3170	109

Subgroups — C109×#3170 #28360424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3170 #28360424