Subgroups — C₁₀₉×#2586 #28359840

Order 13952.

412 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#398×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#395×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#395×C₁₀₉	C₂	1	2	yes	maximal
4	#391×C₁₀₉	C₂	1	2	yes	maximal
5	#391×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#389×C₁₀₉	C₂	1	2	yes	maximal
7	#387×C₁₀₉	C₂	1	2	yes	maximal
8	#387×C₁₀₉	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₄×C₁₀₉	C₂	1	2	yes	maximal
10	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
22	#95×C₁₀₉	C₂²	1	2	yes	—
23	#95×C₁₀₉	C₂²	1	2	yes	—
24	#95×C₁₀₉	C₂²	1	1	yes	characteristic
25	#95×C₁₀₉	C₂²	1	2	yes	—
26	#95×C₁₀₉	C₂²	1	1	yes	characteristic
27	#95×C₁₀₉	C₂²	1	2	yes	—
28	#95×C₁₀₉	C₂²	1	2	yes	—
29	#95×C₁₀₉	C₂²	1	2	yes	—
30	#95×C₁₀₉	C₂²	1	2	yes	—
31	C₂×Q₈×C₁₀₉	—	2	4	no	—
32	C₂×Q₈×C₁₀₉	—	2	2	no	—
33	C₂×Q₈×C₁₀₉	—	4	1	no	—
34	C₂×Q₈×C₁₀₉	—	4	1	no	—
35	C₂×Q₈×C₁₀₉	—	4	1	no	—
36	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₁₀₉	—	2	4	no	—
46	C₄⋊C₄×C₁₀₉	—	2	4	no	—
47	C₄⋊C₄×C₁₀₉	—	4	2	no	—
48	C₄⋊C₄×C₁₀₉	—	4	2	no	—
49	C₄⋊C₄×C₁₀₉	—	4	1	no	—
50	C₄⋊C₄×C₁₀₉	—	4	1	no	—
51	C₄⋊C₄×C₁₀₉	Q₈	1	8	yes	complemented
52	C₄²×C₁₀₉	—	2	4	no	—
53	C₄²×C₁₀₉	—	4	1	no	—
54	C₄²×C₁₀₉	—	4	2	no	—
55	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
56	Q₈×C₁₀₉	—	4	4	no	—
57	Q₈×C₁₀₉	—	4	4	no	—
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	—	2	2	no	—
65	C₂×C₄×C₁₀₉	—	2	1	no	—
66	C₂×C₄×C₁₀₉	—	2	1	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	—	2	2	no	—
73	C₂×C₄×C₁₀₉	—	2	2	no	—
74	C₂×C₄×C₁₀₉	—	2	1	no	—
75	C₂×C₄×C₁₀₉	—	2	2	no	—
76	C₂×C₄×C₁₀₉	—	2	2	no	—
77	C₂×C₄×C₁₀₉	—	2	1	no	—
78	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
79	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
80	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
81	C₂×C₄×C₁₀₉	#41	1	4	yes	—
82	C₂×C₄×C₁₀₉	#41	1	2	yes	—
83	C₂²×C₁₀₉	C₂²×Q₈	1	2	yes	—
84	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
85	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
86	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
87	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
88	C₄×C₁₀₉	—	2	4	no	—
89	C₄×C₁₀₉	—	2	2	no	—
90	C₄×C₁₀₉	—	2	4	no	—
91	C₄×C₁₀₉	—	2	4	no	—
92	C₄×C₁₀₉	—	2	4	no	—
93	C₄×C₁₀₉	—	4	2	no	—
94	C₄×C₁₀₉	—	4	2	no	—
95	C₄×C₁₀₉	—	4	1	no	—
96	C₄×C₁₀₉	—	4	1	no	—
97	C₂×C₁₀₉	#533	1	2	yes	—
98	C₂×C₁₀₉	#538	1	1	yes	characteristic
99	C₂×C₁₀₉	#551	1	1	yes	characteristic
100	C₂×C₁₀₉	#554	1	2	yes	—
101	C₂×C₁₀₉	Q₈²	1	1	yes	characteristic
102	#2586	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₁₀₉	#2586	1	1	yes	characteristic, direct product factor, Sylow
104	#398	C₂×C₁₀₉	1	1	yes	characteristic
105	#395	C₂×C₁₀₉	1	1	yes	characteristic
106	#395	C₂×C₁₀₉	1	2	yes	—
107	#391	C₂×C₁₀₉	1	2	yes	—
108	#391	C₂×C₁₀₉	1	1	yes	characteristic
109	#389	C₂×C₁₀₉	1	2	yes	—
110	#387	C₂×C₁₀₉	1	2	yes	—
111	#387	C₂×C₁₀₉	1	2	yes	—
112	C₄×C₄⋊C₄	C₂×C₁₀₉	1	2	yes	—
113	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
114	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
116	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
117	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
118	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
122	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
123	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
124	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
125	#95	C₂²×C₁₀₉	1	2	yes	—
126	#95	C₂²×C₁₀₉	1	2	yes	—
127	#95	C₂²×C₁₀₉	1	1	yes	characteristic
128	#95	C₂²×C₁₀₉	1	2	yes	—
129	#95	C₂²×C₁₀₉	1	1	yes	characteristic
130	#95	C₂²×C₁₀₉	1	2	yes	—
131	#95	C₂²×C₁₀₉	1	2	yes	—
132	#95	C₂²×C₁₀₉	1	2	yes	—
133	#95	C₂²×C₁₀₉	1	2	yes	—
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	—	2	2	no	—
136	C₂×Q₈	—	4	1	no	—
137	C₂×Q₈	—	4	1	no	—
138	C₂×Q₈	—	4	1	no	—
139	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
140	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
141	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
143	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
144	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
145	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
146	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
147	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	1	no	—
153	C₄⋊C₄	—	4	1	no	—
154	C₄⋊C₄	Q₈×C₁₀₉	1	8	yes	complemented
155	C₄²	—	2	4	no	—
156	C₄²	—	4	1	no	—
157	C₄²	—	4	2	no	—
158	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
159	Q₈	—	4	4	no	—
160	Q₈	—	4	4	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	C₂×Q₈×C₁₀₉	1	4	yes	—
182	C₂×C₄	C₂×Q₈×C₁₀₉	1	4	yes	—
183	C₂×C₄	C₂×Q₈×C₁₀₉	1	4	yes	—
184	C₂×C₄	#41×C₁₀₉	1	4	yes	—
185	C₂×C₄	#41×C₁₀₉	1	2	yes	—
186	C₂²	C₂²×Q₈×C₁₀₉	1	2	yes	—
187	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
188	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
189	C₂²	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
190	C₂²	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	4	no	—
194	C₄	—	2	4	no	—
195	C₄	—	2	4	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	1	no	—
199	C₄	—	4	1	no	—
200	C₂	#533×C₁₀₉	1	2	yes	—
201	C₂	#538×C₁₀₉	1	1	yes	characteristic
202	C₂	#551×C₁₀₉	1	1	yes	characteristic
203	C₂	#554×C₁₀₉	1	2	yes	—
204	C₂	Q₈²×C₁₀₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
51. C₄⋊C₄×C₁₀₉	Q₈	semidirect
102. #2586	C₁₀₉	direct (N × H)
103. C₁₀₉	#2586	direct (N × H)
154. C₄⋊C₄	Q₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#398×C₁₀₉	2
#395×C₁₀₉	2
#391×C₁₀₉	2
#389×C₁₀₉	2
#387×C₁₀₉	2
C₄×C₄⋊C₄×C₁₀₉	2
#2586	109

Subgroups — C109×#2586 #28359840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2586 #28359840