Subgroups — C₁₀₉×#3237 #28360491

Order 13952.

322 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#501×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#491×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#485×C₁₀₉	C₂	1	4	yes	maximal
6	#477×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#475×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	#432×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	#125×C₁₀₉	—	2	1	no	—
14	#125×C₁₀₉	C₂²	1	2	yes	—
15	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
16	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
17	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
18	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
19	C₄×Q₈×C₁₀₉	—	2	1	no	—
20	C₄×Q₈×C₁₀₉	—	2	1	no	—
21	C₄×Q₈×C₁₀₉	—	2	1	no	—
22	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
23	#117×C₁₀₉	—	2	1	no	—
24	#117×C₁₀₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
26	#107×C₁₀₉	C₂²	1	4	yes	—
27	#106×C₁₀₉	C₂²	1	4	yes	—
28	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
29	#103×C₁₀₉	C₂²	1	4	yes	—
30	#103×C₁₀₉	C₂²	1	4	yes	—
31	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
32	#97×C₁₀₉	C₂²	1	2	yes	—
33	C₂×Q₈×C₁₀₉	—	2	1	no	—
34	C₂×Q₈×C₁₀₉	—	2	1	no	—
35	C₂×Q₈×C₁₀₉	—	2	1	no	—
36	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₁₀₉	C₂³	1	4	yes	—
41	C₄⋊C₄×C₁₀₉	—	2	1	no	—
42	C₄⋊C₄×C₁₀₉	—	2	1	no	—
43	C₄⋊C₄×C₁₀₉	—	2	2	no	—
44	C₄⋊C₄×C₁₀₉	—	2	2	no	—
45	C₄⋊C₄×C₁₀₉	—	2	1	no	—
46	C₄⋊C₄×C₁₀₉	—	2	1	no	—
47	C₄⋊C₄×C₁₀₉	—	2	1	no	—
48	C₄⋊C₄×C₁₀₉	—	2	2	no	—
49	C₄⋊C₄×C₁₀₉	—	2	2	no	—
50	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
51	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
52	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
53	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
54	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
55	C₄²×C₁₀₉	—	2	2	no	—
56	C₄²×C₁₀₉	—	2	2	no	—
57	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
58	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
59	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
61	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
62	Q₈×C₁₀₉	—	2	4	no	—
63	Q₈×C₁₀₉	—	4	2	no	—
64	Q₈×C₁₀₉	—	4	2	no	—
65	Q₈×C₁₀₉	—	4	2	no	—
66	C₂×C₄×C₁₀₉	—	2	2	no	—
67	C₂×C₄×C₁₀₉	—	2	2	no	—
68	C₂×C₄×C₁₀₉	—	2	2	no	—
69	C₂×C₄×C₁₀₉	—	2	2	no	—
70	C₂×C₄×C₁₀₉	—	2	2	no	—
71	C₂×C₄×C₁₀₉	—	2	2	no	—
72	C₂×C₄×C₁₀₉	—	4	1	no	—
73	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
79	C₈×C₁₀₉	—	2	4	no	—
80	C₂²×C₁₀₉	—	2	1	no	—
81	C₂²×C₁₀₉	—	2	1	no	—
82	C₂²×C₁₀₉	—	2	1	no	—
83	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
84	C₄×C₁₀₉	—	2	1	no	—
85	C₄×C₁₀₉	—	2	1	no	—
86	C₄×C₁₀₉	—	2	1	no	—
87	C₄×C₁₀₉	—	2	1	no	—
88	C₄×C₁₀₉	—	4	2	no	—
89	C₄×C₁₀₉	—	4	2	no	—
90	C₄×C₁₀₉	—	4	2	no	—
91	C₄×C₁₀₉	—	4	2	no	—
92	C₄×C₁₀₉	#137	1	2	yes	—
93	C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
94	C₂×C₁₀₉	—	4	1	no	—
95	C₂×C₁₀₉	#536	1	1	yes	characteristic
96	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
97	C₂×C₁₀₉	#575	1	1	yes	characteristic
98	#3237	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₁₀₉	#3237	1	1	yes	characteristic, direct product factor, Sylow
100	#533	C₂×C₁₀₉	1	1	yes	characteristic
101	#533	C₂×C₁₀₉	1	1	yes	characteristic
102	#501	C₂×C₁₀₉	1	2	yes	complemented
103	#491	C₂×C₁₀₉	1	2	yes	complemented
104	#485	C₂×C₁₀₉	1	4	yes	—
105	#477	C₂×C₁₀₉	1	2	yes	complemented
106	#475	C₂×C₁₀₉	1	2	yes	complemented
107	#432	C₂×C₁₀₉	1	1	yes	characteristic
108	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
109	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
110	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
111	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
112	#125	—	2	1	no	—
113	#125	C₂²×C₁₀₉	1	2	yes	—
114	C₂²⋊Q₈	—	2	1	no	—
115	C₂²⋊Q₈	—	2	1	no	—
116	C₂²⋊Q₈	—	2	1	no	—
117	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
118	C₄×Q₈	—	2	1	no	—
119	C₄×Q₈	—	2	1	no	—
120	C₄×Q₈	—	2	1	no	—
121	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
122	#117	—	2	1	no	—
123	#117	C₂²×C₁₀₉	1	2	yes	—
124	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
125	#107	C₂²×C₁₀₉	1	4	yes	—
126	#106	C₂²×C₁₀₉	1	4	yes	—
127	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
128	#103	C₂²×C₁₀₉	1	4	yes	—
129	#103	C₂²×C₁₀₉	1	4	yes	—
130	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
131	#97	C₂²×C₁₀₉	1	2	yes	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	—	2	1	no	—
135	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
136	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
137	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
138	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
139	C₂×C₈	C₂³×C₁₀₉	1	4	yes	—
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	2	1	no	—
146	C₄⋊C₄	—	2	1	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
150	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
151	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
157	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
158	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
159	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
160	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
161	Q₈	—	2	4	no	—
162	Q₈	—	4	2	no	—
163	Q₈	—	4	2	no	—
164	Q₈	—	4	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
173	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
174	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
175	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
176	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
177	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
178	C₈	—	2	4	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	2	no	—
191	C₄	#137×C₁₀₉	1	2	yes	—
192	C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
193	C₂	—	4	1	no	—
194	C₂	#536×C₁₀₉	1	1	yes	characteristic
195	C₂	C₂×#137×C₁₀₉	1	1	yes	characteristic
196	C₂	#575×C₁₀₉	1	1	yes	characteristic

	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 ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. #501×C₁₀₉	C₂	semidirect
4. #491×C₁₀₉	C₂	semidirect
6. #477×C₁₀₉	C₂	semidirect
7. #475×C₁₀₉	C₂	semidirect
98. #3237	C₁₀₉	direct (N × H)
99. C₁₀₉	#3237	direct (N × H)
102. #501	C₂×C₁₀₉	semidirect
103. #491	C₂×C₁₀₉	semidirect
105. #477	C₂×C₁₀₉	semidirect
106. #475	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#533×C₁₀₉	2
#501×C₁₀₉	2
#491×C₁₀₉	2
#485×C₁₀₉	2
#477×C₁₀₉	2
#475×C₁₀₉	2
#432×C₁₀₉	2
#3237	109

Subgroups — C109×#3237 #28360491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3237 #28360491