Subgroups — C₁₀₉×#3393 #28360647

Order 13952.

360 conjugacy classes of subgroups.

The subgroup lattice has 170 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, complemented
2	#495×C₁₀₉	C₂	1	2	yes	maximal
3	#494×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#464×C₁₀₉	C₂	1	4	yes	maximal, complemented
5	#461×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	C₈×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Q₁₆×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	C₄×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
10	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₁₀₉	C₂²	1	4	yes	—
12	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
14	#126×C₁₀₉	—	2	4	no	—
15	#124×C₁₀₉	C₂²	1	4	yes	—
16	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
17	C₄×Q₈×C₁₀₉	C₂²	1	4	yes	—
18	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
20	#117×C₁₀₉	—	2	2	no	—
21	#107×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
24	#103×C₁₀₉	C₂²	1	4	yes	—
25	#102×C₁₀₉	C₂²	1	2	yes	—
26	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
28	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
29	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
30	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₀₉	—	2	2	no	—
32	Q₁₆×C₁₀₉	—	2	4	no	—
33	Q₁₆×C₁₀₉	—	4	2	no	—
34	SD₁₆×C₁₀₉	—	4	4	no	—
35	D₈×C₁₀₉	—	2	2	no	—
36	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
38	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₄⋊C₄×C₁₀₉	—	2	4	no	—
41	C₄⋊C₄×C₁₀₉	—	2	2	no	—
42	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
43	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
44	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
45	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
46	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
47	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
48	C₄²×C₁₀₉	—	2	4	no	—
49	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
50	C₄²×C₁₀₉	C₂³	1	2	yes	—
51	C₂³×C₁₀₉	—	2	2	no	—
52	Q₈×C₁₀₉	—	2	2	no	—
53	Q₈×C₁₀₉	—	2	4	no	—
54	Q₈×C₁₀₉	—	2	4	no	—
55	D₄×C₁₀₉	—	2	2	no	—
56	D₄×C₁₀₉	—	2	2	no	—
57	C₂×C₄×C₁₀₉	—	2	2	no	—
58	C₂×C₄×C₁₀₉	—	2	4	no	—
59	C₂×C₄×C₁₀₉	—	4	4	no	—
60	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
65	C₈×C₁₀₉	—	2	2	no	—
66	C₈×C₁₀₉	—	2	1	no	—
67	C₈×C₁₀₉	#41	1	2	yes	—
68	C₂²×C₁₀₉	—	4	2	no	—
69	C₂²×C₁₀₉	—	4	2	no	—
70	C₂²×C₁₀₉	—	4	2	no	—
71	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
72	C₄×C₁₀₉	—	2	2	no	—
73	C₄×C₁₀₉	—	2	2	no	—
74	C₄×C₁₀₉	—	2	1	no	—
75	C₄×C₁₀₉	—	4	4	no	—
76	C₄×C₁₀₉	—	4	2	no	—
77	C₄×C₁₀₉	#135	1	2	yes	—
78	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
79	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂×C₁₀₉	—	8	2	no	—
81	C₂×C₁₀₉	#550	1	1	yes	characteristic
82	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
83	C₂×C₁₀₉	#578	1	1	yes	characteristic
84	#3393	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₁₀₉	#3393	1	1	yes	characteristic, direct product factor, Sylow
86	#556	C₂×C₁₀₉	1	2	yes	complemented
87	#495	C₂×C₁₀₉	1	2	yes	—
88	#494	C₂×C₁₀₉	1	1	yes	characteristic, complemented
89	#464	C₂×C₁₀₉	1	4	yes	complemented
90	#461	C₂×C₁₀₉	1	2	yes	complemented
91	C₈×Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
92	C₄×Q₁₆	C₂×C₁₀₉	1	2	yes	complemented
93	C₄×D₈	C₂×C₁₀₉	1	1	yes	characteristic
94	C₂×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
95	C₂×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
96	C₂×SD₁₆	C₂²×C₁₀₉	1	4	yes	—
97	C₂×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
98	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
99	#126	—	2	4	no	—
100	#124	C₂²×C₁₀₉	1	4	yes	—
101	C₂²⋊Q₈	—	2	2	no	—
102	C₄×Q₈	C₂²×C₁₀₉	1	4	yes	—
103	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
104	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
105	#117	—	2	2	no	—
106	#107	C₂²×C₁₀₉	1	1	yes	characteristic
107	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
108	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
109	#103	C₂²×C₁₀₉	1	4	yes	—
110	#102	C₂²×C₁₀₉	1	2	yes	—
111	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
112	C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
113	C₂×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
114	C₂×Q₈	C₂³×C₁₀₉	1	4	yes	—
115	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
116	C₂²×C₄	—	2	2	no	—
117	Q₁₆	—	2	4	no	—
118	Q₁₆	—	4	2	no	—
119	SD₁₆	—	4	4	no	—
120	D₈	—	2	2	no	—
121	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
122	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
123	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
128	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
129	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
130	C₂²⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	2	2	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₄²	—	2	4	no	—
134	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
135	C₄²	C₂³×C₁₀₉	1	2	yes	—
136	C₂³	—	2	2	no	—
137	Q₈	—	2	2	no	—
138	Q₈	—	2	4	no	—
139	Q₈	—	2	4	no	—
140	D₄	—	2	2	no	—
141	D₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	4	4	no	—
145	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
146	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
147	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
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	—	2	1	no	—
152	C₈	#41×C₁₀₉	1	2	yes	—
153	C₂²	—	4	2	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	2	no	—
156	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	1	no	—
160	C₄	—	4	4	no	—
161	C₄	—	4	2	no	—
162	C₄	#135×C₁₀₉	1	2	yes	—
163	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
164	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
165	C₂	—	8	2	no	—
166	C₂	#550×C₁₀₉	1	1	yes	characteristic
167	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
168	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×D₈×C₁₀₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄×Q₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #556×C₁₀₉	C₂	semidirect
3. #494×C₁₀₉	C₂	semidirect
4. #464×C₁₀₉	C₂	semidirect
5. #461×C₁₀₉	C₂	semidirect
6. C₈×Q₈×C₁₀₉	C₂	semidirect
7. C₄×Q₁₆×C₁₀₉	C₂	semidirect
84. #3393	C₁₀₉	direct (N × H)
85. C₁₀₉	#3393	direct (N × H)
86. #556	C₂×C₁₀₉	semidirect
88. #494	C₂×C₁₀₉	semidirect
89. #464	C₂×C₁₀₉	semidirect
90. #461	C₂×C₁₀₉	semidirect
91. C₈×Q₈	C₂×C₁₀₉	semidirect
92. C₄×Q₁₆	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
#495×C₁₀₉	2
#494×C₁₀₉	2
#464×C₁₀₉	2
#461×C₁₀₉	2
C₈×Q₈×C₁₀₉	2
C₄×Q₁₆×C₁₀₉	2
C₄×D₈×C₁₀₉	2
#3393	109

Subgroups — C109×#3393 #28360647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3393 #28360647