Subgroups — C₁₀₉×#3176 #28360430

Order 13952.

386 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	#540×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	#499×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#493×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#490×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#469×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#466×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#442×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	C₄×D₈×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
11	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
12	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	#127×C₁₀₉	C₂²	1	2	yes	—
15	#125×C₁₀₉	C₂²	1	2	yes	—
16	#123×C₁₀₉	—	2	4	no	—
17	#121×C₁₀₉	—	2	4	no	—
18	#121×C₁₀₉	C₂²	1	4	yes	—
19	C₄×D₄×C₁₀₉	C₂²	1	4	yes	—
20	#117×C₁₀₉	C₂²	1	1	yes	characteristic
21	#107×C₁₀₉	C₂²	1	2	yes	—
22	#106×C₁₀₉	C₂²	1	2	yes	—
23	#102×C₁₀₉	C₂²	1	4	yes	—
24	#102×C₁₀₉	C₂²	1	4	yes	—
25	#97×C₁₀₉	C₂²	1	1	yes	characteristic
26	#97×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂×D₄×C₁₀₉	—	2	4	no	—
30	C₂×D₄×C₁₀₉	—	2	4	no	—
31	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
32	C₂²×C₄×C₁₀₉	—	2	4	no	—
33	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
34	D₈×C₁₀₉	—	2	4	no	—
35	D₈×C₁₀₉	—	2	4	no	—
36	M₁₆×C₁₀₉	—	2	2	no	—
37	C₂×C₈×C₁₀₉	—	2	2	no	—
38	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₁₀₉	—	2	4	no	—
43	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
44	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
45	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
46	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
47	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
48	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
50	C₂³×C₁₀₉	—	2	4	no	—
51	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
52	D₄×C₁₀₉	—	2	4	no	—
53	D₄×C₁₀₉	—	2	4	no	—
54	D₄×C₁₀₉	—	8	4	no	—
55	D₄×C₁₀₉	—	8	4	no	—
56	C₂×C₄×C₁₀₉	—	2	1	no	—
57	C₂×C₄×C₁₀₉	—	2	1	no	—
58	C₂×C₄×C₁₀₉	—	2	4	no	—
59	C₂×C₄×C₁₀₉	—	8	4	no	—
60	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
64	C₈×C₁₀₉	—	2	1	no	—
65	C₈×C₁₀₉	—	2	1	no	—
66	C₈×C₁₀₉	#41	1	2	yes	—
67	C₈×C₁₀₉	#41	1	2	yes	—
68	C₂²×C₁₀₉	—	2	1	no	—
69	C₂²×C₁₀₉	—	2	2	no	—
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	1	no	—
75	C₄×C₁₀₉	—	2	2	no	—
76	C₄×C₁₀₉	—	2	2	no	—
77	C₄×C₁₀₉	—	4	4	no	—
78	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
79	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
80	C₂×C₁₀₉	—	4	1	no	—
81	C₂×C₁₀₉	—	8	4	no	—
82	C₂×C₁₀₉	#532	1	1	yes	characteristic
83	C₂×C₁₀₉	#576	1	2	yes	—
84	#3176	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₁₀₉	#3176	1	1	yes	characteristic, direct product factor, Sylow
86	#540	C₂×C₁₀₉	1	2	yes	complemented
87	#499	C₂×C₁₀₉	1	1	yes	characteristic, complemented
88	#493	C₂×C₁₀₉	1	1	yes	characteristic, complemented
89	#490	C₂×C₁₀₉	1	2	yes	complemented
90	#469	C₂×C₁₀₉	1	2	yes	complemented
91	#466	C₂×C₁₀₉	1	2	yes	complemented
92	#442	C₂×C₁₀₉	1	2	yes	complemented
93	C₄×D₈	C₂×C₁₀₉	1	2	yes	complemented
94	#405	C₂×C₁₀₉	1	1	yes	characteristic, complemented
95	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
96	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
97	C₂×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
98	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
99	#127	C₂²×C₁₀₉	1	2	yes	—
100	#125	C₂²×C₁₀₉	1	2	yes	—
101	#123	—	2	4	no	—
102	#121	—	2	4	no	—
103	#121	C₂²×C₁₀₉	1	4	yes	—
104	C₄×D₄	C₂²×C₁₀₉	1	4	yes	—
105	#117	C₂²×C₁₀₉	1	1	yes	characteristic
106	#107	C₂²×C₁₀₉	1	2	yes	—
107	#106	C₂²×C₁₀₉	1	2	yes	—
108	#102	C₂²×C₁₀₉	1	4	yes	—
109	#102	C₂²×C₁₀₉	1	4	yes	—
110	#97	C₂²×C₁₀₉	1	1	yes	characteristic
111	#97	C₂²×C₁₀₉	1	1	yes	characteristic
112	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
113	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
114	C₂×D₄	—	2	4	no	—
115	C₂×D₄	—	2	4	no	—
116	C₂×D₄	C₂³×C₁₀₉	1	4	yes	—
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
119	D₈	—	2	4	no	—
120	D₈	—	2	4	no	—
121	M₁₆	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
124	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
125	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
126	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
129	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
130	C₂²⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
133	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
134	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
135	C₂³	—	2	4	no	—
136	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
137	D₄	—	2	4	no	—
138	D₄	—	2	4	no	—
139	D₄	—	8	4	no	—
140	D₄	—	8	4	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	8	4	no	—
145	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
146	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
147	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
148	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	#41×C₁₀₉	1	2	yes	—
152	C₈	#41×C₁₀₉	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	—	4	4	no	—
158	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	4	4	no	—
163	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
164	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
165	C₂	—	4	1	no	—
166	C₂	—	8	4	no	—
167	C₂	#532×C₁₀₉	1	1	yes	characteristic
168	C₂	#576×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ #117×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #540×C₁₀₉	C₂	semidirect
2. #499×C₁₀₉	C₂	semidirect
3. #493×C₁₀₉	C₂	semidirect
4. #490×C₁₀₉	C₂	semidirect
5. #469×C₁₀₉	C₂	semidirect
6. #466×C₁₀₉	C₂	semidirect
7. #442×C₁₀₉	C₂	semidirect
8. C₄×D₈×C₁₀₉	C₂	semidirect
9. #405×C₁₀₉	C₂	semidirect
84. #3176	C₁₀₉	direct (N × H)
85. C₁₀₉	#3176	direct (N × H)
86. #540	C₂×C₁₀₉	semidirect
87. #499	C₂×C₁₀₉	semidirect
88. #493	C₂×C₁₀₉	semidirect
89. #490	C₂×C₁₀₉	semidirect
90. #469	C₂×C₁₀₉	semidirect
91. #466	C₂×C₁₀₉	semidirect
92. #442	C₂×C₁₀₉	semidirect
93. C₄×D₈	C₂×C₁₀₉	semidirect
94. #405	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#540×C₁₀₉	2
#499×C₁₀₉	2
#493×C₁₀₉	2
#490×C₁₀₉	2
#469×C₁₀₉	2
#466×C₁₀₉	2
#442×C₁₀₉	2
C₄×D₈×C₁₀₉	2
#405×C₁₀₉	2
#3176	109

Subgroups — C109×#3176 #28360430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3176 #28360430