Subgroups — C₁₀₉×#9899 #42127893

Order 27904.

430 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2859	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1296	C₂	1	4	yes	maximal
5	C₂×#125×C₁₀₉	—	2	1	no	—
6	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₀₉	—	2	1	no	—
9	#389×C₁₀₉	—	2	2	no	—
10	#389×C₁₀₉	C₄	1	4	yes	—
11	#384×C₁₀₉	C₂²	1	2	yes	—
12	#382×C₁₀₉	—	2	1	no	—
13	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
14	#336×C₁₀₉	—	2	1	no	—
15	#336×C₁₀₉	—	2	2	no	—
16	C₂²×C₈×C₁₀₉	—	2	2	no	—
17	C₂²×C₈×C₁₀₉	—	2	2	no	—
18	#125×C₁₀₉	—	4	2	no	—
19	#125×C₁₀₉	—	4	2	no	—
20	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
21	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
27	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₁₀₉	—	2	2	no	—
30	C₄⋊C₈×C₁₀₉	—	2	4	no	—
31	C₄⋊C₈×C₁₀₉	Q₈	1	4	yes	—
32	C₄×C₈×C₁₀₉	—	4	2	no	—
33	#95×C₁₀₉	—	2	4	no	—
34	C₂²×C₄×C₁₀₉	—	2	4	no	—
35	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₀₉	—	2	4	no	—
41	C₂×C₈×C₁₀₉	—	4	2	no	—
42	C₂×C₈×C₁₀₉	—	4	2	no	—
43	C₂×C₈×C₁₀₉	—	4	2	no	—
44	C₂×C₈×C₁₀₉	—	4	4	no	—
45	C₄⋊C₄×C₁₀₉	—	4	4	no	—
46	C₄⋊C₄×C₁₀₉	—	4	8	no	—
47	C₄⋊C₄×C₁₀₉	—	4	2	no	—
48	C₄⋊C₄×C₁₀₉	—	4	2	no	—
49	C₄⋊C₄×C₁₀₉	—	4	4	no	—
50	C₄²×C₁₀₉	—	2	4	no	—
51	C₄²×C₁₀₉	—	2	2	no	—
52	C₄²×C₁₀₉	—	4	2	no	—
53	C₄²×C₁₀₉	D₈	1	2	yes	—
54	C₄²×C₁₀₉	SD₁₆	1	2	yes	—
55	C₄²×C₁₀₉	C₂×Q₈	1	2	yes	—
56	C₄²×C₁₀₉	#41	1	2	yes	—
57	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	—	2	2	no	—
59	C₂×C₄×C₁₀₉	—	2	2	no	—
60	C₂×C₄×C₁₀₉	—	2	2	no	—
61	C₂×C₄×C₁₀₉	—	2	2	no	—
62	C₂×C₄×C₁₀₉	—	2	2	no	—
63	C₂×C₄×C₁₀₉	—	2	2	no	—
64	C₂×C₄×C₁₀₉	—	4	4	no	—
65	C₂×C₄×C₁₀₉	—	4	4	no	—
66	C₂×C₄×C₁₀₉	—	4	4	no	—
67	C₂×C₄×C₁₀₉	#102	1	4	yes	—
68	C₂×C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
70	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₀₉	#135	1	4	yes	—
75	C₂×C₄×C₁₀₉	#137	1	2	yes	—
76	C₈×C₁₀₉	—	4	4	no	—
77	C₈×C₁₀₉	—	8	2	no	—
78	C₂²×C₁₀₉	#391	1	1	yes	characteristic
79	C₂²×C₁₀₉	#413	1	2	yes	—
80	C₂²×C₁₀₉	C₂×#102	1	1	yes	characteristic
81	C₂²×C₁₀₉	#416	1	2	yes	—
82	C₂²×C₁₀₉	#419	1	1	yes	characteristic
83	C₄×C₁₀₉	—	2	4	no	—
84	C₄×C₁₀₉	—	2	2	no	—
85	C₄×C₁₀₉	—	4	2	no	—
86	C₄×C₁₀₉	—	8	4	no	—
87	C₄×C₁₀₉	#474	1	2	yes	—
88	C₄×C₁₀₉	#475	1	2	yes	—
89	C₄×C₁₀₉	#488	1	2	yes	—
90	C₄×C₁₀₉	#499	1	2	yes	—
91	#9899	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂×C₁₀₉	#1540	1	2	yes	—
93	C₂×C₁₀₉	#1554	1	2	yes	—
94	C₂×C₁₀₉	#1988	1	1	yes	characteristic
95	C₂×C₁₀₉	#1991	1	1	yes	characteristic
96	C₂×C₁₀₉	#1998	1	1	yes	characteristic
97	#2859	C₂×C₁₀₉	1	1	yes	characteristic
98	#1831	C₂×C₁₀₉	1	1	yes	characteristic
99	C₄×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic
100	#1296	C₂×C₁₀₉	1	4	yes	—
101	C₁₀₉	#9899	1	1	yes	characteristic, direct product factor, Sylow
102	C₂×#125	—	2	1	no	—
103	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
104	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
105	C₂×C₄×C₈	—	2	1	no	—
106	#389	—	2	2	no	—
107	#389	C₄×C₁₀₉	1	4	yes	—
108	#384	C₂²×C₁₀₉	1	2	yes	—
109	#382	—	2	1	no	—
110	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
111	#336	—	2	2	no	—
112	#336	—	2	1	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	2	2	no	—
115	#125	—	4	2	no	—
116	#125	—	4	2	no	—
117	C₂×C₄⋊C₄	—	2	4	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	—
121	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
122	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
123	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
124	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
125	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	4	no	—
128	C₄⋊C₈	Q₈×C₁₀₉	1	4	yes	—
129	C₄×C₈	—	4	2	no	—
130	#95	—	2	4	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
133	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
134	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
135	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄⋊C₄	—	4	8	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄²	—	2	4	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	4	2	no	—
150	C₄²	D₈×C₁₀₉	1	2	yes	—
151	C₄²	SD₁₆×C₁₀₉	1	2	yes	—
152	C₄²	C₂×Q₈×C₁₀₉	1	2	yes	—
153	C₄²	#41×C₁₀₉	1	2	yes	—
154	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	#102×C₁₀₉	1	4	yes	—
165	C₂×C₄	C₄×Q₈×C₁₀₉	1	2	yes	—
166	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
167	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
168	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
169	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
170	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
171	C₂×C₄	#135×C₁₀₉	1	4	yes	—
172	C₂×C₄	#137×C₁₀₉	1	2	yes	—
173	C₈	—	4	4	no	—
174	C₈	—	8	2	no	—
175	C₂²	#391×C₁₀₉	1	1	yes	characteristic
176	C₂²	#413×C₁₀₉	1	2	yes	—
177	C₂²	C₂×#102×C₁₀₉	1	1	yes	characteristic
178	C₂²	#416×C₁₀₉	1	2	yes	—
179	C₂²	#419×C₁₀₉	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	4	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	4	no	—
184	C₄	#474×C₁₀₉	1	2	yes	—
185	C₄	#475×C₁₀₉	1	2	yes	—
186	C₄	#488×C₁₀₉	1	2	yes	—
187	C₄	#499×C₁₀₉	1	2	yes	—
188	C₂	C₁₀₉×#1540	1	2	yes	—
189	C₂	C₁₀₉×#1554	1	2	yes	—
190	C₂	C₁₀₉×#1988	1	1	yes	characteristic
191	C₂	C₁₀₉×#1991	1	1	yes	characteristic
192	C₂	C₁₀₉×#1998	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₄³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
91. #9899	C₁₀₉	direct (N × H)
101. C₁₀₉	#9899	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#2859	2
C₁₀₉×#1831	2
C₄×C₄⋊C₈×C₁₀₉	2
C₁₀₉×#1296	2
#9899	109

Subgroups — C109×#9899 #42127893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#9899 #42127893