Subgroups — C₁₀₉×#2807 #28360061

Order 13952.

408 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#400×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	#399×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#399×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#398×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#389×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#389×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#388×C₁₀₉	C₂	1	2	yes	maximal
8	#385×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	#385×C₁₀₉	C₂	1	2	yes	maximal
10	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
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	2	yes	—
16	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
24	#95×C₁₀₉	C₂²	1	2	yes	—
25	#95×C₁₀₉	C₂²	1	2	yes	—
26	#95×C₁₀₉	C₂²	1	2	yes	—
27	#95×C₁₀₉	C₂²	1	2	yes	—
28	#95×C₁₀₉	C₂²	1	2	yes	—
29	#95×C₁₀₉	C₂²	1	1	yes	characteristic
30	#95×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₁₀₉	—	4	2	no	—
33	C₂²×C₄×C₁₀₉	—	4	1	no	—
34	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
42	C₄⋊C₄×C₁₀₉	—	2	4	no	—
43	C₄⋊C₄×C₁₀₉	—	2	4	no	—
44	C₄⋊C₄×C₁₀₉	—	4	2	no	—
45	C₄⋊C₄×C₁₀₉	—	4	2	no	—
46	C₄⋊C₄×C₁₀₉	—	4	1	no	—
47	C₄⋊C₄×C₁₀₉	—	4	2	no	—
48	C₄⋊C₄×C₁₀₉	—	4	1	no	—
49	C₄⋊C₄×C₁₀₉	—	4	2	no	—
50	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
51	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
52	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
53	C₂²⋊C₄×C₁₀₉	Q₈	1	4	yes	—
54	C₄²×C₁₀₉	—	4	2	no	—
55	C₂³×C₁₀₉	—	2	2	no	—
56	C₂³×C₁₀₉	—	2	1	no	—
57	C₂³×C₁₀₉	—	2	2	no	—
58	C₂³×C₁₀₉	—	2	1	no	—
59	C₂³×C₁₀₉	C₂×Q₈	1	2	yes	—
60	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
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	2	no	—
66	C₂×C₄×C₁₀₉	—	2	2	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	1	no	—
72	C₂×C₄×C₁₀₉	—	2	1	no	—
73	C₂×C₄×C₁₀₉	—	2	2	no	—
74	C₂×C₄×C₁₀₉	—	2	2	no	—
75	C₂×C₄×C₁₀₉	—	2	2	no	—
76	C₂×C₄×C₁₀₉	—	2	2	no	—
77	C₂×C₄×C₁₀₉	—	2	2	no	—
78	C₂×C₄×C₁₀₉	—	2	2	no	—
79	C₂×C₄×C₁₀₉	—	2	2	no	—
80	C₂×C₄×C₁₀₉	—	2	2	no	—
81	C₂×C₄×C₁₀₉	—	4	4	no	—
82	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
83	C₂×C₄×C₁₀₉	#41	1	4	yes	—
84	C₂²×C₁₀₉	—	2	2	no	—
85	C₂²×C₁₀₉	—	2	4	no	—
86	C₂²×C₁₀₉	—	4	2	no	—
87	C₂²×C₁₀₉	—	4	1	no	—
88	C₂²×C₁₀₉	—	4	1	no	—
89	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
90	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
91	C₂²×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
92	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
93	C₂²×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
94	C₄×C₁₀₉	—	2	4	no	—
95	C₄×C₁₀₉	—	2	4	no	—
96	C₄×C₁₀₉	—	4	2	no	—
97	C₄×C₁₀₉	—	4	2	no	—
98	C₄×C₁₀₉	—	4	1	no	—
99	C₄×C₁₀₉	—	4	2	no	—
100	C₄×C₁₀₉	—	4	1	no	—
101	C₄×C₁₀₉	—	4	2	no	—
102	C₂×C₁₀₉	—	4	2	no	—
103	C₂×C₁₀₉	#539	1	2	yes	—
104	C₂×C₁₀₉	#544	1	1	yes	characteristic
105	C₂×C₁₀₉	#554	1	2	yes	—
106	C₂×C₁₀₉	#559	1	1	yes	characteristic
107	C₂×C₁₀₉	#560	1	1	yes	characteristic
108	#2807	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₁₀₉	#2807	1	1	yes	characteristic, direct product factor, Sylow
110	#400	C₂×C₁₀₉	1	2	yes	complemented
111	#399	C₂×C₁₀₉	1	1	yes	characteristic
112	#399	C₂×C₁₀₉	1	1	yes	characteristic
113	#398	C₂×C₁₀₉	1	2	yes	complemented
114	#389	C₂×C₁₀₉	1	2	yes	complemented
115	#389	C₂×C₁₀₉	1	2	yes	complemented
116	#388	C₂×C₁₀₉	1	2	yes	—
117	#385	C₂×C₁₀₉	1	1	yes	characteristic
118	#385	C₂×C₁₀₉	1	2	yes	—
119	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
121	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
124	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
127	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
128	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
129	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
133	#95	C₂²×C₁₀₉	1	2	yes	—
134	#95	C₂²×C₁₀₉	1	2	yes	—
135	#95	C₂²×C₁₀₉	1	2	yes	—
136	#95	C₂²×C₁₀₉	1	2	yes	—
137	#95	C₂²×C₁₀₉	1	2	yes	—
138	#95	C₂²×C₁₀₉	1	1	yes	characteristic
139	#95	C₂²×C₁₀₉	1	1	yes	characteristic
140	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	—	4	1	no	—
143	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
144	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
145	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
146	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
147	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
148	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
150	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	1	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	1	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₂²⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	4	2	no	—
161	C₂²⋊C₄	—	4	1	no	—
162	C₂²⋊C₄	Q₈×C₁₀₉	1	4	yes	—
163	C₄²	—	4	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	1	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	1	no	—
168	C₂³	C₂×Q₈×C₁₀₉	1	2	yes	—
169	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
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	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	C₂×Q₈×C₁₀₉	1	4	yes	—
192	C₂×C₄	#41×C₁₀₉	1	4	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	4	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	1	no	—
197	C₂²	—	4	1	no	—
198	C₂²	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
199	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
200	C₂²	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
201	C₂²	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
202	C₂²	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	4	no	—
205	C₄	—	4	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	1	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	1	no	—
211	C₂	—	4	2	no	—
212	C₂	#539×C₁₀₉	1	2	yes	—
213	C₂	#544×C₁₀₉	1	1	yes	characteristic
214	C₂	#554×C₁₀₉	1	2	yes	—
215	C₂	#559×C₁₀₉	1	1	yes	characteristic
216	C₂	#560×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₀₉
prime radical	C₂⁴×C₁₀₉	C₂³
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
1. #400×C₁₀₉	C₂	semidirect
4. #398×C₁₀₉	C₂	semidirect
5. #389×C₁₀₉	C₂	semidirect
6. #389×C₁₀₉	C₂	semidirect
108. #2807	C₁₀₉	direct (N × H)
109. C₁₀₉	#2807	direct (N × H)
110. #400	C₂×C₁₀₉	semidirect
113. #398	C₂×C₁₀₉	semidirect
114. #389	C₂×C₁₀₉	semidirect
115. #389	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#400×C₁₀₉	2
#399×C₁₀₉	2
#398×C₁₀₉	2
#389×C₁₀₉	2
#388×C₁₀₉	2
#385×C₁₀₉	2
#2807	109

Subgroups — C109×#2807 #28360061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2807 #28360061