Subgroups — C₁₀₉×#20569 #42138563

Order 27904.

990 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₁₀₉	C₂	1	2	yes	maximal
2	C₄²×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄×C₁₀₉	C₂	1	6	yes	maximal
4	C₄×C₄⋊C₈×C₁₀₉	C₂	1	3	yes	maximal
5	C₁₀₉×#1744	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₁₀₉	C₄	1	2	yes	—
7	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₁₀₉	C₄	1	16	yes	complemented
9	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	6	yes	—
10	C₂×C₄×C₈×C₁₀₉	C₂²	1	6	yes	—
11	C₂×C₄×C₈×C₁₀₉	C₂²	1	6	yes	—
12	C₄×C₄⋊C₄×C₁₀₉	C₄	1	6	yes	—
13	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	—
14	C₄³×C₁₀₉	C₂²	1	3	yes	—
15	#336×C₁₀₉	C₂²	1	1	yes	characteristic
16	#336×C₁₀₉	C₂²	1	3	yes	—
17	#336×C₁₀₉	C₂²	1	6	yes	—
18	C₂²×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
20	C₂²×C₈×C₁₀₉	C₂³	1	6	yes	—
21	C₄×Q₈×C₁₀₉	C₈	1	16	yes	complemented
22	C₄×Q₈×C₁₀₉	C₂×C₄	1	8	yes	—
23	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	3	yes	—
24	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	6	yes	—
25	C₂×C₄²×C₁₀₉	C₂×C₄	1	6	yes	—
26	C₂×C₄²×C₁₀₉	C₂×C₄	1	6	yes	—
27	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₀₉	C₂³	1	3	yes	—
29	C₂×C₄²×C₁₀₉	C₂³	1	3	yes	—
30	C₄⋊C₈×C₁₀₉	C₂×C₄	1	24	yes	—
31	C₄×C₈×C₁₀₉	—	4	6	no	—
32	C₄×C₈×C₁₀₉	C₂×C₄	1	24	yes	—
33	#95×C₁₀₉	C₂×C₄	1	6	yes	—
34	C₂×Q₈×C₁₀₉	C₄²	1	4	yes	—
35	C₂×Q₈×C₁₀₉	C₂×C₈	1	8	yes	—
36	C₂²×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
37	C₂²×C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
38	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
39	C₂²×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
40	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
41	C₂×C₈×C₁₀₉	—	2	6	no	—
42	C₂×C₈×C₁₀₉	—	2	2	no	—
43	C₂×C₈×C₁₀₉	—	2	2	no	—
44	C₂×C₈×C₁₀₉	—	2	6	no	—
45	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
46	C₂×C₈×C₁₀₉	C₂²×C₄	1	12	yes	—
47	C₄⋊C₄×C₁₀₉	C₄²	1	12	yes	—
48	C₄⋊C₄×C₁₀₉	C₂×C₈	1	24	yes	—
49	C₄²×C₁₀₉	—	2	3	no	—
50	C₄²×C₁₀₉	—	2	2	no	—
51	C₄²×C₁₀₉	—	2	3	no	—
52	C₄²×C₁₀₉	—	4	6	no	—
53	C₄²×C₁₀₉	C₂×C₈	1	24	yes	—
54	C₄²×C₁₀₉	C₂²×C₄	1	12	yes	—
55	C₂³×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
56	Q₈×C₁₀₉	C₄×C₈	1	16	yes	complemented
57	C₂×C₄×C₁₀₉	—	2	2	no	—
58	C₂×C₄×C₁₀₉	—	2	6	no	—
59	C₂×C₄×C₁₀₉	—	2	2	no	—
60	C₂×C₄×C₁₀₉	—	2	6	no	—
61	C₂×C₄×C₁₀₉	—	2	6	no	—
62	C₂×C₄×C₁₀₉	C₂×C₄²	1	6	yes	—
63	C₂×C₄×C₁₀₉	C₂×C₄²	1	6	yes	—
64	C₂×C₄×C₁₀₉	C₂²×C₈	1	4	yes	—
65	C₂×C₄×C₁₀₉	C₂²×C₈	1	12	yes	—
66	C₂×C₄×C₁₀₉	C₂²×C₈	1	12	yes	—
67	C₂×C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
68	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
70	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
71	C₈×C₁₀₉	—	2	4	no	—
72	C₈×C₁₀₉	—	2	12	no	—
73	C₂²×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
74	C₂²×C₁₀₉	#519	1	1	yes	characteristic
75	C₂²×C₁₀₉	#520	1	1	yes	characteristic
76	C₂²×C₁₀₉	C₂³×C₈	1	2	yes	—
77	C₂²×C₁₀₉	#568	1	2	yes	—
78	C₄×C₁₀₉	—	2	6	no	—
79	C₄×C₁₀₉	—	2	4	no	—
80	C₄×C₁₀₉	—	2	12	no	—
81	C₄×C₁₀₉	C₂×C₄×C₈	1	12	yes	—
82	C₄×C₁₀₉	#519	1	2	yes	—
83	C₄×C₁₀₉	#520	1	2	yes	—
84	#20569	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂×C₁₀₉	#2276	1	1	yes	characteristic
86	C₂×C₁₀₉	C₂²×C₄×C₈	1	1	yes	characteristic
87	C₂×C₁₀₉	#2875	1	1	yes	characteristic
88	C₂×C₁₀₉	#2982	1	2	yes	—
89	C₂×C₁₀₉	#2988	1	2	yes	—
90	C₂×C₈×Q₈	C₂×C₁₀₉	1	2	yes	—
91	C₄²×Q₈	C₂×C₁₀₉	1	1	yes	characteristic
92	C₈×C₄⋊C₄	C₂×C₁₀₉	1	6	yes	—
93	C₄×C₄⋊C₈	C₂×C₁₀₉	1	3	yes	—
94	#1744	C₂×C₁₀₉	1	3	yes	—
95	C₁₀₉	#20569	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×C₄×Q₈	C₄×C₁₀₉	1	2	yes	—
97	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
98	C₈×Q₈	C₄×C₁₀₉	1	16	yes	complemented
99	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	6	yes	—
100	C₂×C₄×C₈	C₂²×C₁₀₉	1	6	yes	—
101	C₂×C₄×C₈	C₂²×C₁₀₉	1	6	yes	—
102	C₄×C₄⋊C₄	C₄×C₁₀₉	1	6	yes	—
103	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	3	yes	—
104	C₄³	C₂²×C₁₀₉	1	3	yes	—
105	#336	C₂²×C₁₀₉	1	1	yes	characteristic
106	#336	C₂²×C₁₀₉	1	3	yes	—
107	#336	C₂²×C₁₀₉	1	6	yes	—
108	C₂²×Q₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
109	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
110	C₂²×C₈	C₂³×C₁₀₉	1	6	yes	—
111	C₄×Q₈	C₈×C₁₀₉	1	16	yes	complemented
112	C₄×Q₈	C₂×C₄×C₁₀₉	1	8	yes	—
113	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	3	yes	—
114	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	6	yes	—
115	C₂×C₄²	C₂×C₄×C₁₀₉	1	6	yes	—
116	C₂×C₄²	C₂×C₄×C₁₀₉	1	6	yes	—
117	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
118	C₂×C₄²	C₂³×C₁₀₉	1	3	yes	—
119	C₂×C₄²	C₂³×C₁₀₉	1	3	yes	—
120	C₄⋊C₈	C₂×C₄×C₁₀₉	1	24	yes	—
121	C₄×C₈	—	4	6	no	—
122	C₄×C₈	C₂×C₄×C₁₀₉	1	24	yes	—
123	#95	C₂×C₄×C₁₀₉	1	6	yes	—
124	C₂×Q₈	C₄²×C₁₀₉	1	4	yes	—
125	C₂×Q₈	C₂×C₈×C₁₀₉	1	8	yes	—
126	C₂²×C₄	C₂²×C₄×C₁₀₉	1	3	yes	—
127	C₂²×C₄	C₂²×C₄×C₁₀₉	1	6	yes	—
128	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
129	C₂²×C₄	C₂²×C₄×C₁₀₉	1	3	yes	—
130	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
131	C₂×C₈	—	2	6	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	6	no	—
135	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
136	C₂×C₈	C₂²×C₄×C₁₀₉	1	12	yes	—
137	C₄⋊C₄	C₄²×C₁₀₉	1	12	yes	—
138	C₄⋊C₄	C₂×C₈×C₁₀₉	1	24	yes	—
139	C₄²	—	2	3	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	3	no	—
142	C₄²	—	4	6	no	—
143	C₄²	C₂×C₈×C₁₀₉	1	24	yes	—
144	C₄²	C₂²×C₄×C₁₀₉	1	12	yes	—
145	C₂³	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
146	Q₈	C₄×C₈×C₁₀₉	1	16	yes	complemented
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	6	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	6	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	C₂×C₄²×C₁₀₉	1	6	yes	—
153	C₂×C₄	C₂×C₄²×C₁₀₉	1	6	yes	—
154	C₂×C₄	C₂²×C₈×C₁₀₉	1	4	yes	—
155	C₂×C₄	C₂²×C₈×C₁₀₉	1	12	yes	—
156	C₂×C₄	C₂²×C₈×C₁₀₉	1	12	yes	—
157	C₂×C₄	C₂³×C₄×C₁₀₉	1	2	yes	—
158	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
159	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
160	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
161	C₈	—	2	4	no	—
162	C₈	—	2	12	no	—
163	C₂²	C₂²×C₄²×C₁₀₉	1	1	yes	characteristic
164	C₂²	#519×C₁₀₉	1	1	yes	characteristic
165	C₂²	#520×C₁₀₉	1	1	yes	characteristic
166	C₂²	C₂³×C₈×C₁₀₉	1	2	yes	—
167	C₂²	#568×C₁₀₉	1	2	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	2	12	no	—
170	C₄	—	2	6	no	—
171	C₄	C₂×C₄×C₈×C₁₀₉	1	12	yes	—
172	C₄	#519×C₁₀₉	1	2	yes	—
173	C₄	#520×C₁₀₉	1	2	yes	—
174	C₂	C₁₀₉×#2276	1	1	yes	characteristic
175	C₂	C₂²×C₄×C₈×C₁₀₉	1	1	yes	characteristic
176	C₂	C₁₀₉×#2875	1	1	yes	characteristic
177	C₂	C₁₀₉×#2982	1	2	yes	—
178	C₂	C₁₀₉×#2988	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂⁴
commutator	C₂	C₂²×C₄×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
socle, prime radical	C₂³×C₁₀₉	C₂³×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₁₀₉	C₄	semidirect
21. C₄×Q₈×C₁₀₉	C₈	semidirect
56. Q₈×C₁₀₉	C₄×C₈	semidirect
84. #20569	C₁₀₉	direct (N × H)
95. C₁₀₉	#20569	direct (N × H)
98. C₈×Q₈	C₄×C₁₀₉	semidirect
111. C₄×Q₈	C₈×C₁₀₉	semidirect
146. Q₈	C₄×C₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₁₀₉	2
C₄²×Q₈×C₁₀₉	2
C₈×C₄⋊C₄×C₁₀₉	2
C₄×C₄⋊C₈×C₁₀₉	2
C₁₀₉×#1744	2
#20569	109

Subgroups — C109×#20569 #42138563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#20569 #42138563