Subgroups — C₁₀₉×#3338 #28360592

Order 13952.

356 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#554×C₁₀₉	C₂	1	2	yes	maximal
2	#500×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#484×C₁₀₉	C₂	1	4	yes	maximal, complemented
4	#477×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#467×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	C₄×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#107×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
12	#125×C₁₀₉	—	2	2	no	—
13	C₂²⋊Q₈×C₁₀₉	—	2	4	no	—
14	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
15	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
16	C₄×D₄×C₁₀₉	—	2	2	no	—
17	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
19	#107×C₁₀₉	C₂²	1	2	yes	—
20	#107×C₁₀₉	C₂²	1	2	yes	—
21	#107×C₁₀₉	C₂²	1	1	yes	characteristic
22	#107×C₁₀₉	C₂²	1	4	yes	—
23	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	#103×C₁₀₉	C₂²	1	2	yes	—
25	#103×C₁₀₉	C₂²	1	4	yes	—
26	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
27	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₁₀₉	—	2	2	no	—
29	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
30	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₀₉	—	2	4	no	—
32	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
33	Q₁₆×C₁₀₉	—	2	2	no	—
34	C₂×C₈×C₁₀₉	—	2	4	no	—
35	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
37	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₀₉	—	2	4	no	—
39	C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₄⋊C₄×C₁₀₉	—	2	2	no	—
41	C₄⋊C₄×C₁₀₉	—	2	4	no	—
42	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
43	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
44	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
45	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
46	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
47	C₄²×C₁₀₉	—	2	2	no	—
48	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
50	Q₈×C₁₀₉	—	2	2	no	—
51	Q₈×C₁₀₉	—	2	2	no	—
52	Q₈×C₁₀₉	—	8	2	no	—
53	D₄×C₁₀₉	Q₁₆	1	4	yes	complemented
54	C₂×C₄×C₁₀₉	—	2	2	no	—
55	C₂×C₄×C₁₀₉	—	2	2	no	—
56	C₂×C₄×C₁₀₉	—	2	2	no	—
57	C₂×C₄×C₁₀₉	—	2	2	no	—
58	C₂×C₄×C₁₀₉	—	2	4	no	—
59	C₂×C₄×C₁₀₉	—	8	4	no	—
60	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
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	2	no	—
65	C₈×C₁₀₉	—	2	1	no	—
66	C₈×C₁₀₉	#41	1	2	yes	—
67	C₂²×C₁₀₉	—	2	2	no	—
68	C₂²×C₁₀₉	—	2	2	no	—
69	C₂²×C₁₀₉	C₂×Q₁₆	1	4	yes	—
70	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
71	C₄×C₁₀₉	—	2	2	no	—
72	C₄×C₁₀₉	—	2	1	no	—
73	C₄×C₁₀₉	—	4	2	no	—
74	C₄×C₁₀₉	—	4	2	no	—
75	C₄×C₁₀₉	—	4	4	no	—
76	C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
77	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
78	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
79	C₂×C₁₀₉	—	2	4	no	—
80	C₂×C₁₀₉	#547	1	1	yes	characteristic
81	C₂×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
82	C₂×C₁₀₉	#576	1	1	yes	characteristic
83	#3338	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₁₀₉	#3338	1	1	yes	characteristic, direct product factor, Sylow
85	#554	C₂×C₁₀₉	1	2	yes	—
86	#500	C₂×C₁₀₉	1	1	yes	characteristic, complemented
87	#484	C₂×C₁₀₉	1	4	yes	complemented
88	#477	C₂×C₁₀₉	1	2	yes	complemented
89	#467	C₂×C₁₀₉	1	2	yes	complemented
90	C₄×Q₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
91	C₈×D₄	C₂×C₁₀₉	1	1	yes	characteristic
92	C₂×#107	C₂×C₁₀₉	1	2	yes	complemented
93	C₂×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
94	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	—
95	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
96	#125	—	2	2	no	—
97	C₂²⋊Q₈	—	2	4	no	—
98	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
99	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
100	C₄×D₄	—	2	2	no	—
101	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
102	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
103	#107	C₂²×C₁₀₉	1	1	yes	characteristic
104	#107	C₂²×C₁₀₉	1	2	yes	—
105	#107	C₂²×C₁₀₉	1	4	yes	—
106	#107	C₂²×C₁₀₉	1	2	yes	—
107	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
108	#103	C₂²×C₁₀₉	1	2	yes	—
109	#103	C₂²×C₁₀₉	1	4	yes	—
110	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
111	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
112	C₂×Q₈	—	2	2	no	—
113	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
114	C₂×D₄	D₄×C₁₀₉	1	1	yes	characteristic
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
117	Q₁₆	—	2	2	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
120	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
121	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
127	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
128	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
129	C₂²⋊C₄	—	2	4	no	—
130	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
131	C₄²	—	2	2	no	—
132	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
133	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
134	Q₈	—	2	2	no	—
135	Q₈	—	2	2	no	—
136	Q₈	—	8	2	no	—
137	D₄	Q₁₆×C₁₀₉	1	4	yes	complemented
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
145	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
146	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
147	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	2	1	no	—
150	C₈	#41×C₁₀₉	1	2	yes	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	2	no	—
153	C₂²	C₂×Q₁₆×C₁₀₉	1	4	yes	—
154	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
155	C₄	—	2	2	no	—
156	C₄	—	2	1	no	—
157	C₄	—	4	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	4	4	no	—
160	C₄	C₂×Q₁₆×C₁₀₉	1	2	yes	—
161	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
162	C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
163	C₂	—	2	4	no	—
164	C₂	#547×C₁₀₉	1	1	yes	characteristic
165	C₂	C₂²×Q₁₆×C₁₀₉	1	1	yes	characteristic
166	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	C₂×D₄×C₁₀₉	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄×D₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #500×C₁₀₉	C₂	semidirect
3. #484×C₁₀₉	C₂	semidirect
4. #477×C₁₀₉	C₂	semidirect
5. #467×C₁₀₉	C₂	semidirect
6. C₄×Q₁₆×C₁₀₉	C₂	semidirect
8. C₂×#107×C₁₀₉	C₂	semidirect
53. D₄×C₁₀₉	Q₁₆	semidirect
83. #3338	C₁₀₉	direct (N × H)
84. C₁₀₉	#3338	direct (N × H)
86. #500	C₂×C₁₀₉	semidirect
87. #484	C₂×C₁₀₉	semidirect
88. #477	C₂×C₁₀₉	semidirect
89. #467	C₂×C₁₀₉	semidirect
90. C₄×Q₁₆	C₂×C₁₀₉	semidirect
92. C₂×#107	C₂×C₁₀₉	semidirect
137. D₄	Q₁₆×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#554×C₁₀₉	2
#500×C₁₀₉	2
#484×C₁₀₉	2
#477×C₁₀₉	2
#467×C₁₀₉	2
C₄×Q₁₆×C₁₀₉	2
C₈×D₄×C₁₀₉	2
C₂×#107×C₁₀₉	2
#3338	109

Subgroups — C109×#3338 #28360592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3338 #28360592