Subgroups — C₁₀₉×#2285 #28359539

Order 13952.

656 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#117×C₁₀₉	C₂	1	2	yes	maximal
2	C₂²×C₄²×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#384×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#382×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#380×C₁₀₉	C₂	1	4	yes	maximal
6	C₄×C₄⋊C₄×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
8	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
9	#117×C₁₀₉	—	2	8	no	—
10	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	8	yes	—
12	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
13	C₂×C₄²×C₁₀₉	C₄	1	8	yes	complemented
14	C₂×C₄²×C₁₀₉	C₂²	1	4	yes	—
15	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
18	#95×C₁₀₉	C₂²	1	8	yes	—
19	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₁₀₉	—	2	8	no	—
21	C₂²×C₄×C₁₀₉	C₂×C₄	1	8	yes	—
22	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₁₀₉	D₄	1	4	yes	—
24	C₂²×C₄×C₁₀₉	Q₈	1	4	yes	—
25	C₂²×C₄×C₁₀₉	C₂³	1	8	yes	—
26	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
29	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₀₉	—	2	8	no	—
31	C₄⋊C₄×C₁₀₉	—	2	16	no	—
32	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
33	C₄²×C₁₀₉	—	2	8	no	—
34	C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
35	C₄²×C₁₀₉	C₂×C₄	1	8	yes	—
36	C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
37	C₂³×C₁₀₉	—	2	4	no	—
38	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
39	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
40	C₂³×C₁₀₉	C₂×Q₈	1	2	yes	—
41	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	—	2	4	no	—
43	C₂×C₄×C₁₀₉	—	2	16	no	—
44	C₂×C₄×C₁₀₉	—	2	8	no	—
45	C₂×C₄×C₁₀₉	—	2	16	no	—
46	C₂×C₄×C₁₀₉	C₄⋊C₄	1	16	yes	complemented
47	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
51	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₁₀₉	#41	1	8	yes	—
56	C₂²×C₁₀₉	—	2	4	no	—
57	C₂²×C₁₀₉	—	2	4	no	—
58	C₂²×C₁₀₉	—	2	4	no	—
59	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
60	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
61	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
62	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
63	C₂²×C₁₀₉	C₂×#41	1	4	yes	—
64	C₄×C₁₀₉	—	2	4	no	—
65	C₄×C₁₀₉	—	4	8	no	—
66	C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
67	C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
68	C₄×C₁₀₉	#117	1	8	yes	complemented
69	C₂×C₁₀₉	—	2	4	no	—
70	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₁₀₉	C₂×#117	1	2	yes	—
72	C₂×C₁₀₉	#532	1	2	yes	—
73	C₂×C₁₀₉	#533	1	2	yes	—
74	#2285	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₁₀₉	#2285	1	1	yes	characteristic, direct product factor, Sylow
76	C₂×#117	C₂×C₁₀₉	1	2	yes	—
77	C₂²×C₄²	C₂×C₁₀₉	1	1	yes	characteristic
78	#384	C₂×C₁₀₉	1	2	yes	complemented
79	#382	C₂×C₁₀₉	1	2	yes	complemented
80	#380	C₂×C₁₀₉	1	4	yes	—
81	C₄×C₄⋊C₄	C₂×C₁₀₉	1	4	yes	complemented
82	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
83	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
84	#117	—	2	8	no	—
85	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
86	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	8	yes	—
87	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
88	C₂×C₄²	C₄×C₁₀₉	1	8	yes	complemented
89	C₂×C₄²	C₂²×C₁₀₉	1	4	yes	—
90	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
91	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
92	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
93	#95	C₂²×C₁₀₉	1	8	yes	—
94	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
95	C₂²×C₄	—	2	8	no	—
96	C₂²×C₄	C₂×C₄×C₁₀₉	1	8	yes	—
97	C₂²×C₄	C₂×C₄×C₁₀₉	1	4	yes	—
98	C₂²×C₄	D₄×C₁₀₉	1	4	yes	—
99	C₂²×C₄	Q₈×C₁₀₉	1	4	yes	—
100	C₂²×C₄	C₂³×C₁₀₉	1	8	yes	—
101	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
102	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
103	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
104	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
105	C₄⋊C₄	—	2	8	no	—
106	C₄⋊C₄	—	2	16	no	—
107	C₂²⋊C₄	—	2	8	no	—
108	C₄²	—	2	8	no	—
109	C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
110	C₄²	C₂×C₄×C₁₀₉	1	8	yes	—
111	C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
112	C₂³	—	2	4	no	—
113	C₂³	C₂²×C₄×C₁₀₉	1	2	yes	—
114	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
115	C₂³	C₂×Q₈×C₁₀₉	1	2	yes	—
116	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	16	no	—
119	C₂×C₄	—	2	8	no	—
120	C₂×C₄	—	2	16	no	—
121	C₂×C₄	C₄⋊C₄×C₁₀₉	1	16	yes	complemented
122	C₂×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
123	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
124	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
125	C₂×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
126	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
127	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
128	C₂×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
129	C₂×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
130	C₂×C₄	#41×C₁₀₉	1	8	yes	—
131	C₂²	—	2	4	no	—
132	C₂²	—	2	4	no	—
133	C₂²	—	2	4	no	—
134	C₂²	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
135	C₂²	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
136	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
137	C₂²	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
138	C₂²	C₂×#41×C₁₀₉	1	4	yes	—
139	C₄	—	2	4	no	—
140	C₄	—	4	8	no	—
141	C₄	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
142	C₄	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
143	C₄	#117×C₁₀₉	1	8	yes	complemented
144	C₂	—	2	4	no	—
145	C₂	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
146	C₂	C₂×#117×C₁₀₉	1	2	yes	—
147	C₂	#532×C₁₀₉	1	2	yes	—
148	C₂	#533×C₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂³
commutator	C₂²	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₂³	C₂⁴×C₁₀₉
socle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂⁴×C₁₀₉	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. #384×C₁₀₉	C₂	semidirect
4. #382×C₁₀₉	C₂	semidirect
6. C₄×C₄⋊C₄×C₁₀₉	C₂	semidirect
13. C₂×C₄²×C₁₀₉	C₄	semidirect
46. C₂×C₄×C₁₀₉	C₄⋊C₄	semidirect
68. C₄×C₁₀₉	#117	semidirect
74. #2285	C₁₀₉	direct (N × H)
75. C₁₀₉	#2285	direct (N × H)
78. #384	C₂×C₁₀₉	semidirect
79. #382	C₂×C₁₀₉	semidirect
81. C₄×C₄⋊C₄	C₂×C₁₀₉	semidirect
88. C₂×C₄²	C₄×C₁₀₉	semidirect
121. C₂×C₄	C₄⋊C₄×C₁₀₉	semidirect
143. C₄	#117×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#117×C₁₀₉	2
C₂²×C₄²×C₁₀₉	2
#384×C₁₀₉	2
#382×C₁₀₉	2
#380×C₁₀₉	2
C₄×C₄⋊C₄×C₁₀₉	2
#2285	109

Subgroups — C109×#2285 #28359539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2285 #28359539