Subgroups — C₁₀₉×#3163 #28360417

Order 13952.

340 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#544×C₁₀₉	C₂	1	2	yes	maximal
2	#501×C₁₀₉	C₂	1	4	yes	maximal, complemented
3	#499×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#498×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#428×C₁₀₉	C₂	1	2	yes	maximal
6	#427×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#106×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
12	C₄⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
13	#125×C₁₀₉	—	2	2	no	—
14	#125×C₁₀₉	C₂²	1	4	yes	—
15	C₂²⋊Q₈×C₁₀₉	—	2	4	no	—
16	#117×C₁₀₉	C₂²	1	2	yes	—
17	#117×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
19	#107×C₁₀₉	C₂²	1	4	yes	—
20	#107×C₁₀₉	C₂²	1	2	yes	—
21	#106×C₁₀₉	C₂²	1	1	yes	characteristic
22	#106×C₁₀₉	C₂²	1	2	yes	—
23	#106×C₁₀₉	C₂²	1	4	yes	—
24	#106×C₁₀₉	C₂²	1	1	yes	characteristic
25	#106×C₁₀₉	C₂²	1	2	yes	—
26	#97×C₁₀₉	C₂²	1	2	yes	—
27	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
28	C₂×Q₈×C₁₀₉	—	2	4	no	—
29	C₂²×C₄×C₁₀₉	—	2	2	no	—
30	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	M₁₆×C₁₀₉	Q₈	1	4	yes	complemented
32	C₂×C₈×C₁₀₉	Q₈	1	4	yes	complemented
33	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
35	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₀₉	—	2	2	no	—
37	C₄⋊C₄×C₁₀₉	—	2	4	no	—
38	C₄⋊C₄×C₁₀₉	—	2	4	no	—
39	C₄⋊C₄×C₁₀₉	—	2	4	no	—
40	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
41	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
42	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
43	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
44	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
45	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
46	C₄²×C₁₀₉	—	2	2	no	—
47	C₄²×C₁₀₉	C₂³	1	2	yes	—
48	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	Q₈×C₁₀₉	—	8	4	no	—
50	C₂×C₄×C₁₀₉	—	2	2	no	—
51	C₂×C₄×C₁₀₉	—	2	2	no	—
52	C₂×C₄×C₁₀₉	—	2	4	no	—
53	C₂×C₄×C₁₀₉	—	8	2	no	—
54	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
55	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
59	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
60	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
61	C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
62	C₂²×C₁₀₉	—	2	2	no	—
63	C₂²×C₁₀₉	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₀₉	—	2	4	no	—
66	C₄×C₁₀₉	—	4	2	no	—
67	C₄×C₁₀₉	—	4	4	no	—
68	C₄×C₁₀₉	—	4	2	no	—
69	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
70	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
71	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₁₀₉	—	2	2	no	—
73	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₁₀₉	#577	1	2	yes	—
75	#3163	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₁₀₉	#3163	1	1	yes	characteristic, direct product factor, Sylow
77	#544	C₂×C₁₀₉	1	2	yes	—
78	#501	C₂×C₁₀₉	1	4	yes	complemented
79	#499	C₂×C₁₀₉	1	2	yes	complemented
80	#498	C₂×C₁₀₉	1	2	yes	complemented
81	#428	C₂×C₁₀₉	1	2	yes	—
82	#427	C₂×C₁₀₉	1	1	yes	characteristic
83	C₂×#106	C₂×C₁₀₉	1	1	yes	characteristic
84	#405	C₂×C₁₀₉	1	1	yes	characteristic
85	C₂×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
86	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
87	C₄⋊Q₈	—	2	2	no	—
88	C₄⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
89	#125	—	2	2	no	—
90	#125	C₂²×C₁₀₉	1	4	yes	—
91	C₂²⋊Q₈	—	2	4	no	—
92	#117	C₂²×C₁₀₉	1	2	yes	—
93	#117	C₂²×C₁₀₉	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
95	#107	C₂²×C₁₀₉	1	2	yes	—
96	#107	C₂²×C₁₀₉	1	4	yes	—
97	#106	C₂²×C₁₀₉	1	2	yes	—
98	#106	C₂²×C₁₀₉	1	2	yes	—
99	#106	C₂²×C₁₀₉	1	1	yes	characteristic
100	#106	C₂²×C₁₀₉	1	1	yes	characteristic
101	#106	C₂²×C₁₀₉	1	4	yes	—
102	#97	C₂²×C₁₀₉	1	2	yes	—
103	C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
104	C₂×Q₈	—	2	4	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
107	M₁₆	Q₈×C₁₀₉	1	4	yes	complemented
108	C₂×C₈	Q₈×C₁₀₉	1	4	yes	complemented
109	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
110	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
111	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
117	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
118	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
119	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
122	C₄²	—	2	2	no	—
123	C₄²	C₂³×C₁₀₉	1	2	yes	—
124	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
125	Q₈	—	8	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
131	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
132	C₂×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
133	C₂×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
134	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
135	C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
136	C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
137	C₈	C₂×Q₈×C₁₀₉	1	4	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	C₄⋊Q₈×C₁₀₉	1	2	yes	—
140	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	4	4	no	—
143	C₄	—	4	2	no	—
144	C₄	—	4	2	no	—
145	C₄	C₄⋊Q₈×C₁₀₉	1	2	yes	—
146	C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
147	C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
150	C₂	#577×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	C₂³×C₁₀₉	C₂×D₄
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
2. #501×C₁₀₉	C₂	semidirect
3. #499×C₁₀₉	C₂	semidirect
4. #498×C₁₀₉	C₂	semidirect
31. M₁₆×C₁₀₉	Q₈	semidirect
32. C₂×C₈×C₁₀₉	Q₈	semidirect
75. #3163	C₁₀₉	direct (N × H)
76. C₁₀₉	#3163	direct (N × H)
78. #501	C₂×C₁₀₉	semidirect
79. #499	C₂×C₁₀₉	semidirect
80. #498	C₂×C₁₀₉	semidirect
107. M₁₆	Q₈×C₁₀₉	semidirect
108. C₂×C₈	Q₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#544×C₁₀₉	2
#501×C₁₀₉	2
#499×C₁₀₉	2
#498×C₁₀₉	2
#428×C₁₀₉	2
#427×C₁₀₉	2
C₂×#106×C₁₀₉	2
#405×C₁₀₉	2
#3163	109

Subgroups — C109×#3163 #28360417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3163 #28360417