Subgroups — C₁₀₉×#3396 #28360650

Order 13952.

384 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#559×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	#495×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	#493×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#463×C₁₀₉	C₂	1	4	yes	maximal, complemented
5	#460×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	C₈×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₈×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×SD₁₆×C₁₀₉	C₂²	1	4	yes	—
11	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
12	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
13	#127×C₁₀₉	C₂²	1	2	yes	—
14	#124×C₁₀₉	—	2	2	no	—
15	#124×C₁₀₉	C₂²	1	4	yes	—
16	#123×C₁₀₉	—	2	4	no	—
17	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
18	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₁₀₉	—	2	2	no	—
20	C₄×D₄×C₁₀₉	C₂²	1	4	yes	—
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	2	yes	—
25	#102×C₁₀₉	C₂²	1	4	yes	—
26	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
27	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
29	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
30	C₂×D₄×C₁₀₉	—	2	2	no	—
31	C₂×D₄×C₁₀₉	—	2	2	no	—
32	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
33	C₂²×C₄×C₁₀₉	—	2	4	no	—
34	Q₁₆×C₁₀₉	—	2	2	no	—
35	SD₁₆×C₁₀₉	—	4	4	no	—
36	D₈×C₁₀₉	—	2	4	no	—
37	D₈×C₁₀₉	—	4	2	no	—
38	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
39	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
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	4	no	—
47	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
48	C₄²×C₁₀₉	—	2	2	no	—
49	C₄²×C₁₀₉	C₂³	1	2	yes	—
50	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₂³×C₁₀₉	—	2	4	no	—
52	Q₈×C₁₀₉	—	2	2	no	—
53	Q₈×C₁₀₉	—	2	2	no	—
54	Q₈×C₁₀₉	—	2	2	no	—
55	D₄×C₁₀₉	—	2	4	no	—
56	D₄×C₁₀₉	—	2	4	no	—
57	D₄×C₁₀₉	—	4	4	no	—
58	D₄×C₁₀₉	—	8	2	no	—
59	C₂×C₄×C₁₀₉	—	2	2	no	—
60	C₂×C₄×C₁₀₉	—	2	2	no	—
61	C₂×C₄×C₁₀₉	—	8	4	no	—
62	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
67	C₈×C₁₀₉	—	2	1	no	—
68	C₈×C₁₀₉	—	2	2	no	—
69	C₈×C₁₀₉	#41	1	2	yes	—
70	C₂²×C₁₀₉	—	4	4	no	—
71	C₂²×C₁₀₉	—	4	4	no	—
72	C₂²×C₁₀₉	—	4	4	no	—
73	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
74	C₄×C₁₀₉	—	2	2	no	—
75	C₄×C₁₀₉	—	2	2	no	—
76	C₄×C₁₀₉	—	2	1	no	—
77	C₄×C₁₀₉	—	4	2	no	—
78	C₄×C₁₀₉	—	4	2	no	—
79	C₄×C₁₀₉	#135	1	2	yes	—
80	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
81	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
82	C₂×C₁₀₉	—	8	4	no	—
83	C₂×C₁₀₉	#550	1	1	yes	characteristic
84	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
85	C₂×C₁₀₉	#576	1	1	yes	characteristic
86	#3396	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₁₀₉	#3396	1	1	yes	characteristic, direct product factor, Sylow
88	#559	C₂×C₁₀₉	1	2	yes	complemented
89	#495	C₂×C₁₀₉	1	2	yes	complemented
90	#493	C₂×C₁₀₉	1	1	yes	characteristic
91	#463	C₂×C₁₀₉	1	4	yes	complemented
92	#460	C₂×C₁₀₉	1	2	yes	complemented
93	C₈×Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
94	C₄×Q₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
95	C₄×D₈	C₂×C₁₀₉	1	2	yes	complemented
96	C₂×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
97	C₂×SD₁₆	C₂²×C₁₀₉	1	4	yes	—
98	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
99	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
100	#127	C₂²×C₁₀₉	1	2	yes	—
101	#124	—	2	2	no	—
102	#124	C₂²×C₁₀₉	1	4	yes	—
103	#123	—	2	4	no	—
104	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
105	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
106	C₄×D₄	—	2	2	no	—
107	C₄×D₄	C₂²×C₁₀₉	1	4	yes	—
108	#107	C₂²×C₁₀₉	1	1	yes	characteristic
109	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
110	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
111	#103	C₂²×C₁₀₉	1	2	yes	—
112	#102	C₂²×C₁₀₉	1	4	yes	—
113	C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
114	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
115	C₂×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
116	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	C₂³×C₁₀₉	1	4	yes	—
120	C₂²×C₄	—	2	4	no	—
121	Q₁₆	—	2	2	no	—
122	SD₁₆	—	4	4	no	—
123	D₈	—	2	4	no	—
124	D₈	—	4	2	no	—
125	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
126	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
127	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
130	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
131	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₂²⋊C₄	—	2	4	no	—
134	C₂²⋊C₄	—	2	4	no	—
135	C₄²	—	2	2	no	—
136	C₄²	C₂³×C₁₀₉	1	2	yes	—
137	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
138	C₂³	—	2	4	no	—
139	Q₈	—	2	2	no	—
140	Q₈	—	2	2	no	—
141	Q₈	—	2	2	no	—
142	D₄	—	2	4	no	—
143	D₄	—	2	4	no	—
144	D₄	—	4	4	no	—
145	D₄	—	8	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	8	4	no	—
149	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
150	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
151	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
152	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
153	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	2	2	no	—
156	C₈	#41×C₁₀₉	1	2	yes	—
157	C₂²	—	4	4	no	—
158	C₂²	—	4	4	no	—
159	C₂²	—	4	4	no	—
160	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	2	no	—
163	C₄	—	2	1	no	—
164	C₄	—	4	2	no	—
165	C₄	—	4	2	no	—
166	C₄	#135×C₁₀₉	1	2	yes	—
167	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
168	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
169	C₂	—	8	4	no	—
170	C₂	#550×C₁₀₉	1	1	yes	characteristic
171	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
172	C₂	#576×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	#493×C₁₀₉	C₂
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. #559×C₁₀₉	C₂	semidirect
2. #495×C₁₀₉	C₂	semidirect
4. #463×C₁₀₉	C₂	semidirect
5. #460×C₁₀₉	C₂	semidirect
6. C₈×Q₈×C₁₀₉	C₂	semidirect
7. C₄×Q₁₆×C₁₀₉	C₂	semidirect
8. C₄×D₈×C₁₀₉	C₂	semidirect
86. #3396	C₁₀₉	direct (N × H)
87. C₁₀₉	#3396	direct (N × H)
88. #559	C₂×C₁₀₉	semidirect
89. #495	C₂×C₁₀₉	semidirect
91. #463	C₂×C₁₀₉	semidirect
92. #460	C₂×C₁₀₉	semidirect
93. C₈×Q₈	C₂×C₁₀₉	semidirect
94. C₄×Q₁₆	C₂×C₁₀₉	semidirect
95. C₄×D₈	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#559×C₁₀₉	2
#495×C₁₀₉	2
#493×C₁₀₉	2
#463×C₁₀₉	2
#460×C₁₀₉	2
C₈×Q₈×C₁₀₉	2
C₄×Q₁₆×C₁₀₉	2
C₄×D₈×C₁₀₉	2
#3396	109

Subgroups — C109×#3396 #28360650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3396 #28360650