Subgroups — C₁₀₉×#12839 #42130833

Order 27904.

362 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2158	C₂	1	2	yes	maximal, complemented
3	C₁₀₉×#1812	C₂	1	2	yes	maximal
4	C₁₀₉×#1384	C₂	1	2	yes	maximal, complemented
5	C₂×#131×C₁₀₉	C₂²	1	1	yes	characteristic
6	#504×C₁₀₉	—	2	2	no	—
7	#504×C₁₀₉	—	2	2	no	—
8	C₂×M₃₂×C₁₀₉	C₄	1	2	yes	complemented
9	C₂×M₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₁₀₉	C₄	1	2	yes	complemented
11	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
12	#427×C₁₀₉	C₂²	1	4	yes	—
13	#410×C₁₀₉	—	2	2	no	—
14	#367×C₁₀₉	—	2	2	no	—
15	#366×C₁₀₉	—	2	2	no	—
16	H(3,ℤ/4)×C₁₀₉	—	2	2	no	—
17	C₂×#41×C₁₀₉	D₄	1	1	yes	characteristic
18	#131×C₁₀₉	—	2	2	no	—
19	#131×C₁₀₉	—	2	2	no	—
20	C₂×M₁₆×C₁₀₉	D₄	1	1	yes	characteristic
21	C₂×M₁₆×C₁₀₉	Q₈	1	2	yes	—
22	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
24	#117×C₁₀₉	—	2	4	no	—
25	M₃₂×C₁₀₉	—	2	2	no	—
26	M₃₂×C₁₀₉	—	2	4	no	—
27	C₂×C₁₆×C₁₀₉	—	2	2	no	—
28	C₂×C₁₆×C₁₀₉	—	2	2	no	—
29	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
33	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
34	#107×C₁₀₉	—	2	4	no	—
35	#106×C₁₀₉	—	2	4	no	—
36	#99×C₁₀₉	—	4	4	no	—
37	#41×C₁₀₉	—	2	2	no	—
38	#41×C₁₀₉	—	2	2	no	—
39	C₂×Q₈×C₁₀₉	Q₁₆	1	1	yes	characteristic
40	C₂×D₄×C₁₀₉	D₈	1	2	yes	—
41	C₂×D₄×C₁₀₉	Q₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₁₀₉	—	2	2	no	—
45	M₁₆×C₁₀₉	—	2	4	no	—
46	C₂×C₈×C₁₀₉	—	2	2	no	—
47	C₂×C₈×C₁₀₉	—	2	2	no	—
48	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₀₉	#41	1	2	yes	—
53	C₄⋊C₄×C₁₀₉	—	4	4	no	—
54	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
55	C₄²×C₁₀₉	—	4	4	no	—
56	C₁₆×C₁₀₉	—	2	2	no	—
57	C₁₆×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₁₆×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₁₆×C₁₀₉	C₂²⋊C₄	1	2	yes	—
60	C₂³×C₁₀₉	#107	1	2	yes	—
61	C₂³×C₁₀₉	C₂×D₈	1	1	yes	characteristic
62	Q₈×C₁₀₉	—	2	2	no	—
63	D₄×C₁₀₉	—	2	4	no	—
64	D₄×C₁₀₉	—	2	2	no	—
65	C₂×C₄×C₁₀₉	—	2	2	no	—
66	C₂×C₄×C₁₀₉	—	2	2	no	—
67	C₂×C₄×C₁₀₉	—	8	4	no	—
68	C₂×C₄×C₁₀₉	#107	1	2	yes	—
69	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₀₉	#121	1	2	yes	—
71	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	#135	1	2	yes	—
73	C₈×C₁₀₉	—	2	2	no	—
74	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₈×C₁₀₉	#117	1	1	yes	characteristic
76	C₈×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
77	C₂²×C₁₀₉	—	2	2	no	—
78	C₂²×C₁₀₉	—	4	1	no	—
79	C₂²×C₁₀₉	—	4	2	no	—
80	C₂²×C₁₀₉	C₂×#107	1	1	yes	characteristic
81	C₂²×C₁₀₉	#466	1	2	yes	—
82	C₄×C₁₀₉	—	2	2	no	—
83	C₄×C₁₀₉	—	16	4	no	—
84	C₄×C₁₀₉	#380	1	1	yes	characteristic
85	C₄×C₁₀₉	#427	1	1	yes	characteristic
86	C₄×C₁₀₉	#467	1	2	yes	—
87	#12839	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂×C₁₀₉	—	2	2	no	—
89	C₂×C₁₀₉	—	2	1	no	—
90	C₂×C₁₀₉	—	4	2	no	—
91	C₂×C₁₀₉	#1808	1	1	yes	characteristic
92	C₂×#504	C₂×C₁₀₉	1	1	yes	characteristic
93	#2158	C₂×C₁₀₉	1	2	yes	complemented
94	#1812	C₂×C₁₀₉	1	2	yes	—
95	#1384	C₂×C₁₀₉	1	2	yes	complemented
96	C₁₀₉	#12839	1	1	yes	characteristic, direct product factor, Sylow
97	C₂×#131	C₂²×C₁₀₉	1	1	yes	characteristic
98	#504	—	2	2	no	—
99	#504	—	2	2	no	—
100	C₂×M₃₂	C₄×C₁₀₉	1	2	yes	complemented
101	C₂×M₃₂	C₂²×C₁₀₉	1	1	yes	characteristic
102	C₂²×C₁₆	C₄×C₁₀₉	1	2	yes	complemented
103	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
104	#427	C₂²×C₁₀₉	1	4	yes	—
105	#410	—	2	2	no	—
106	#367	—	2	2	no	—
107	#366	—	2	2	no	—
108	H(3,ℤ/4)	—	2	2	no	—
109	C₂×#41	D₄×C₁₀₉	1	1	yes	characteristic
110	#131	—	2	2	no	—
111	#131	—	2	2	no	—
112	C₂×M₁₆	D₄×C₁₀₉	1	1	yes	characteristic
113	C₂×M₁₆	Q₈×C₁₀₉	1	2	yes	—
114	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
115	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
116	#117	—	2	4	no	—
117	M₃₂	—	2	2	no	—
118	M₃₂	—	2	4	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
122	C₂×C₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
123	C₂×C₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
124	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
125	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
126	#107	—	2	4	no	—
127	#106	—	2	4	no	—
128	#99	—	4	4	no	—
129	#41	—	2	2	no	—
130	#41	—	2	2	no	—
131	C₂×Q₈	Q₁₆×C₁₀₉	1	1	yes	characteristic
132	C₂×D₄	D₈×C₁₀₉	1	2	yes	—
133	C₂×D₄	Q₁₆×C₁₀₉	1	1	yes	characteristic
134	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
135	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
136	M₁₆	—	2	2	no	—
137	M₁₆	—	2	4	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
141	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
142	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
143	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
144	C₂×C₈	#41×C₁₀₉	1	2	yes	—
145	C₄⋊C₄	—	4	4	no	—
146	C₂²⋊C₄	—	4	4	no	—
147	C₄²	—	4	4	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
150	C₁₆	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
151	C₁₆	C₂²⋊C₄×C₁₀₉	1	2	yes	—
152	C₂³	#107×C₁₀₉	1	2	yes	—
153	C₂³	C₂×D₈×C₁₀₉	1	1	yes	characteristic
154	Q₈	—	2	2	no	—
155	D₄	—	2	4	no	—
156	D₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	8	4	no	—
160	C₂×C₄	#107×C₁₀₉	1	2	yes	—
161	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
162	C₂×C₄	#121×C₁₀₉	1	2	yes	—
163	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
164	C₂×C₄	#135×C₁₀₉	1	2	yes	—
165	C₈	—	2	2	no	—
166	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
167	C₈	#117×C₁₀₉	1	1	yes	characteristic
168	C₈	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	4	2	no	—
172	C₂²	C₂×#107×C₁₀₉	1	1	yes	characteristic
173	C₂²	#466×C₁₀₉	1	2	yes	—
174	C₄	—	2	2	no	—
175	C₄	—	16	4	no	—
176	C₄	#380×C₁₀₉	1	1	yes	characteristic
177	C₄	#427×C₁₀₉	1	1	yes	characteristic
178	C₄	#467×C₁₀₉	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	—	2	1	no	—
181	C₂	—	4	2	no	—
182	C₂	C₁₀₉×#1808	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₀₉	#427
commutator	C₂×C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂×C₁₀₉	#1808
prime radical	C₂×#41×C₁₀₉	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂×M₁₆×C₁₀₉ ◁ C₂×#131×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₉×#2158	C₂	semidirect
4. C₁₀₉×#1384	C₂	semidirect
8. C₂×M₃₂×C₁₀₉	C₄	semidirect
10. C₂²×C₁₆×C₁₀₉	C₄	semidirect
87. #12839	C₁₀₉	direct (N × H)
93. #2158	C₂×C₁₀₉	semidirect
95. #1384	C₂×C₁₀₉	semidirect
96. C₁₀₉	#12839	direct (N × H)
100. C₂×M₃₂	C₄×C₁₀₉	semidirect
102. C₂²×C₁₆	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#504×C₁₀₉	2
C₁₀₉×#2158	2
C₁₀₉×#1812	2
C₁₀₉×#1384	2
#12839	109

Subgroups — C109×#12839 #42130833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12839 #42130833