Subgroups — C₁₀₉×#7162 #42125156

Order 27904.

200 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#364×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#364×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₁₀₉	—	2	1	no	—
5	C₂×M₃₂×C₁₀₉	—	2	1	no	—
6	C₂×M₃₂×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₁₀₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	#364×C₁₀₉	—	2	1	no	—
10	#364×C₁₀₉	—	2	1	no	—
11	#364×C₁₀₉	C₄	1	2	yes	—
12	#364×C₁₀₉	C₄	1	2	yes	—
13	#346×C₁₀₉	—	2	1	no	—
14	C₄×C₁₆×C₁₀₉	—	2	1	no	—
15	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₀₉	Q₈	1	1	yes	characteristic
18	M₃₂×C₁₀₉	—	2	1	no	—
19	M₃₂×C₁₀₉	—	2	1	no	—
20	M₃₂×C₁₀₉	—	2	2	no	—
21	M₃₂×C₁₀₉	—	2	2	no	—
22	M₃₂×C₁₀₉	—	4	1	no	—
23	M₃₂×C₁₀₉	—	4	1	no	—
24	M₃₂×C₁₀₉	C₈	1	2	yes	—
25	M₃₂×C₁₀₉	C₈	1	2	yes	—
26	C₂×C₁₆×C₁₀₉	—	2	1	no	—
27	C₂×C₁₆×C₁₀₉	—	2	1	no	—
28	C₂×C₁₆×C₁₀₉	—	4	1	no	—
29	C₂×C₁₆×C₁₀₉	—	4	1	no	—
30	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
33	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₀₉	—	2	1	no	—
35	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
43	C₂×C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
44	C₂×C₈×C₁₀₉	D₈	1	1	yes	characteristic
45	C₂×C₈×C₁₀₉	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₁₀₉	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₁₀₉	Q₁₆	1	1	yes	characteristic
48	C₄²×C₁₀₉	—	2	1	no	—
49	C₄²×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
51	C₁₆×C₁₀₉	—	2	2	no	—
52	C₁₆×C₁₀₉	—	2	2	no	—
53	C₁₆×C₁₀₉	—	4	2	no	—
54	C₁₆×C₁₀₉	—	4	2	no	—
55	C₂³×C₁₀₉	#108	1	1	yes	characteristic
56	C₂×C₄×C₁₀₉	—	2	1	no	—
57	C₂×C₄×C₁₀₉	—	2	1	no	—
58	C₂×C₄×C₁₀₉	—	4	2	no	—
59	C₂×C₄×C₁₀₉	#95	1	1	yes	characteristic
60	C₂×C₄×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	#106	1	1	yes	characteristic
63	C₂×C₄×C₁₀₉	#107	1	1	yes	characteristic
64	C₂×C₄×C₁₀₉	#108	1	1	yes	characteristic
65	C₈×C₁₀₉	C₄×C₈	1	1	yes	characteristic
66	C₈×C₁₀₉	#97	1	1	yes	characteristic
67	C₈×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
68	C₈×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
69	C₈×C₁₀₉	#102	1	1	yes	characteristic
70	C₈×C₁₀₉	#102	1	1	yes	characteristic
71	C₈×C₁₀₉	#103	1	1	yes	characteristic
72	C₈×C₁₀₉	#103	1	1	yes	characteristic
73	C₂²×C₁₀₉	—	4	1	no	—
74	C₂²×C₁₀₉	#334	1	1	yes	characteristic
75	C₂²×C₁₀₉	#335	1	1	yes	characteristic
76	C₂²×C₁₀₉	#341	1	1	yes	characteristic
77	C₄×C₁₀₉	—	2	2	no	—
78	C₄×C₁₀₉	—	4	1	no	—
79	C₄×C₁₀₉	#334	1	1	yes	characteristic
80	C₄×C₁₀₉	#335	1	1	yes	characteristic
81	C₄×C₁₀₉	#336	1	1	yes	characteristic
82	C₄×C₁₀₉	#340	1	1	yes	characteristic
83	#7162	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂×C₁₀₉	—	2	1	no	—
85	C₂×C₁₀₉	—	2	1	no	—
86	C₂×C₁₀₉	—	2	1	no	—
87	C₂×C₁₀₉	#1277	1	1	yes	characteristic
88	C₂×#364	C₂×C₁₀₉	1	1	yes	characteristic
89	C₂×#364	C₂×C₁₀₉	1	1	yes	characteristic
90	C₄×M₃₂	C₂×C₁₀₉	1	1	yes	characteristic
91	C₁₀₉	#7162	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×M₃₂	—	2	1	no	—
93	C₂×M₃₂	—	2	1	no	—
94	C₂×M₃₂	C₄×C₁₀₉	1	1	yes	characteristic
95	C₂×M₃₂	C₄×C₁₀₉	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
97	#364	—	2	1	no	—
98	#364	—	2	1	no	—
99	#364	C₄×C₁₀₉	1	2	yes	—
100	#364	C₄×C₁₀₉	1	2	yes	—
101	#346	—	2	1	no	—
102	C₄×C₁₆	—	2	1	no	—
103	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
104	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
105	C₂×C₄²	Q₈×C₁₀₉	1	1	yes	characteristic
106	M₃₂	—	2	1	no	—
107	M₃₂	—	2	1	no	—
108	M₃₂	—	2	2	no	—
109	M₃₂	—	2	2	no	—
110	M₃₂	—	4	1	no	—
111	M₃₂	—	4	1	no	—
112	M₃₂	C₈×C₁₀₉	1	2	yes	—
113	M₃₂	C₈×C₁₀₉	1	2	yes	—
114	C₂×C₁₆	—	2	1	no	—
115	C₂×C₁₆	—	2	1	no	—
116	C₂×C₁₆	—	4	1	no	—
117	C₂×C₁₆	—	4	1	no	—
118	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
119	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
120	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
121	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
124	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
127	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
128	C₂×C₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
129	C₂×C₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
130	C₂×C₈	M₁₆×C₁₀₉	1	1	yes	characteristic
131	C₂×C₈	M₁₆×C₁₀₉	1	1	yes	characteristic
132	C₂×C₈	D₈×C₁₀₉	1	1	yes	characteristic
133	C₂×C₈	SD₁₆×C₁₀₉	1	1	yes	characteristic
134	C₂×C₈	SD₁₆×C₁₀₉	1	1	yes	characteristic
135	C₂×C₈	Q₁₆×C₁₀₉	1	1	yes	characteristic
136	C₄²	—	2	1	no	—
137	C₄²	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
138	C₄²	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
139	C₁₆	—	2	2	no	—
140	C₁₆	—	2	2	no	—
141	C₁₆	—	4	2	no	—
142	C₁₆	—	4	2	no	—
143	C₂³	#108×C₁₀₉	1	1	yes	characteristic
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	#95×C₁₀₉	1	1	yes	characteristic
148	C₂×C₄	C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
149	C₂×C₄	C₄⋊C₈×C₁₀₉	1	1	yes	characteristic
150	C₂×C₄	#106×C₁₀₉	1	1	yes	characteristic
151	C₂×C₄	#107×C₁₀₉	1	1	yes	characteristic
152	C₂×C₄	#108×C₁₀₉	1	1	yes	characteristic
153	C₈	C₄×C₈×C₁₀₉	1	1	yes	characteristic
154	C₈	#97×C₁₀₉	1	1	yes	characteristic
155	C₈	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
156	C₈	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
157	C₈	#102×C₁₀₉	1	1	yes	characteristic
158	C₈	#102×C₁₀₉	1	1	yes	characteristic
159	C₈	#103×C₁₀₉	1	1	yes	characteristic
160	C₈	#103×C₁₀₉	1	1	yes	characteristic
161	C₂²	—	4	1	no	—
162	C₂²	#334×C₁₀₉	1	1	yes	characteristic
163	C₂²	#335×C₁₀₉	1	1	yes	characteristic
164	C₂²	#341×C₁₀₉	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	4	1	no	—
167	C₄	#334×C₁₀₉	1	1	yes	characteristic
168	C₄	#335×C₁₀₉	1	1	yes	characteristic
169	C₄	#336×C₁₀₉	1	1	yes	characteristic
170	C₄	#340×C₁₀₉	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	C₁₀₉×#1277	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₀₉	#102
commutator	C₈	C₄×C₈×C₁₀₉
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₀₉
socle	C₂×C₁₀₉	#1277
prime radical	C₂³×C₁₀₉	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₀₉ ◁ C₂²×C₈×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
83. #7162	C₁₀₉	direct (N × H)
91. C₁₀₉	#7162	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#364×C₁₀₉	2
C₄×M₃₂×C₁₀₉	2
#7162	109

Subgroups — C109×#7162 #42125156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#7162 #42125156