Subgroups — C₁₀₉×#3403 #28360657

Order 13952.

380 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	#559×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	#496×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	#495×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#464×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#463×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#459×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#440×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₄×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
12	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
13	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
16	#127×C₁₀₉	C₂²	1	2	yes	—
17	#124×C₁₀₉	—	2	2	no	—
18	#124×C₁₀₉	C₂²	1	2	yes	—
19	#124×C₁₀₉	C₂²	1	2	yes	—
20	#123×C₁₀₉	—	2	2	no	—
21	#123×C₁₀₉	—	2	2	no	—
22	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₀₉	—	2	2	no	—
25	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
26	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
27	#107×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
29	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
30	#103×C₁₀₉	C₂²	1	2	yes	—
31	#102×C₁₀₉	C₂²	1	2	yes	—
32	#102×C₁₀₉	C₂²	1	2	yes	—
33	#97×C₁₀₉	C₂²	1	2	yes	—
34	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
36	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
37	C₂×D₄×C₁₀₉	—	2	2	no	—
38	C₂×D₄×C₁₀₉	—	2	2	no	—
39	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
40	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₀₉	—	2	2	no	—
42	C₂²×C₄×C₁₀₉	—	2	2	no	—
43	Q₁₆×C₁₀₉	—	4	1	no	—
44	SD₁₆×C₁₀₉	—	2	4	no	—
45	SD₁₆×C₁₀₉	—	4	2	no	—
46	D₈×C₁₀₉	—	2	2	no	—
47	D₈×C₁₀₉	—	4	2	no	—
48	D₈×C₁₀₉	—	4	1	no	—
49	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
51	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₁₀₉	—	2	2	no	—
53	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
54	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
55	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
57	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
58	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
59	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
60	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
61	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
62	C₄²×C₁₀₉	—	2	2	no	—
63	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
64	C₄²×C₁₀₉	C₂³	1	2	yes	—
65	C₂³×C₁₀₉	—	2	2	no	—
66	C₂³×C₁₀₉	—	2	2	no	—
67	Q₈×C₁₀₉	—	2	2	no	—
68	Q₈×C₁₀₉	—	2	2	no	—
69	Q₈×C₁₀₉	—	4	1	no	—
70	D₄×C₁₀₉	—	2	2	no	—
71	D₄×C₁₀₉	—	2	2	no	—
72	D₄×C₁₀₉	—	2	2	no	—
73	D₄×C₁₀₉	—	2	2	no	—
74	D₄×C₁₀₉	—	4	4	no	—
75	D₄×C₁₀₉	—	8	2	no	—
76	C₂×C₄×C₁₀₉	—	2	2	no	—
77	C₂×C₄×C₁₀₉	—	2	2	no	—
78	C₂×C₄×C₁₀₉	—	8	2	no	—
79	C₂×C₄×C₁₀₉	—	8	2	no	—
80	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
85	C₈×C₁₀₉	—	2	2	no	—
86	C₈×C₁₀₉	—	2	1	no	—
87	C₈×C₁₀₉	#41	1	2	yes	—
88	C₂²×C₁₀₉	—	4	2	no	—
89	C₂²×C₁₀₉	—	4	2	no	—
90	C₂²×C₁₀₉	—	4	2	no	—
91	C₂²×C₁₀₉	—	4	2	no	—
92	C₂²×C₁₀₉	—	4	2	no	—
93	C₂²×C₁₀₉	—	4	2	no	—
94	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
95	C₄×C₁₀₉	—	2	2	no	—
96	C₄×C₁₀₉	—	2	1	no	—
97	C₄×C₁₀₉	—	2	1	no	—
98	C₄×C₁₀₉	—	2	2	no	—
99	C₄×C₁₀₉	—	4	2	no	—
100	C₄×C₁₀₉	—	4	2	no	—
101	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
102	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂×C₁₀₉	—	8	2	no	—
104	C₂×C₁₀₉	—	8	2	no	—
105	C₂×C₁₀₉	#550	1	1	yes	characteristic
106	C₂×C₁₀₉	#575	1	1	yes	characteristic
107	C₂×C₁₀₉	#576	1	1	yes	characteristic
108	#3403	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₁₀₉	#3403	1	1	yes	characteristic, direct product factor, Sylow
110	#559	C₂×C₁₀₉	1	2	yes	complemented
111	#496	C₂×C₁₀₉	1	2	yes	complemented
112	#495	C₂×C₁₀₉	1	1	yes	characteristic, complemented
113	#464	C₂×C₁₀₉	1	2	yes	complemented
114	#463	C₂×C₁₀₉	1	2	yes	complemented
115	#459	C₂×C₁₀₉	1	2	yes	complemented
116	#446	C₂×C₁₀₉	1	1	yes	characteristic, complemented
117	#440	C₂×C₁₀₉	1	2	yes	complemented
118	C₄×D₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
119	C₂×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
120	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
121	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
122	C₂×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
123	C₂×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
124	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
125	#127	C₂²×C₁₀₉	1	2	yes	—
126	#124	—	2	2	no	—
127	#124	C₂²×C₁₀₉	1	2	yes	—
128	#124	C₂²×C₁₀₉	1	2	yes	—
129	#123	—	2	2	no	—
130	#123	—	2	2	no	—
131	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
132	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
133	C₄×D₄	—	2	2	no	—
134	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
135	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
136	#107	C₂²×C₁₀₉	1	1	yes	characteristic
137	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
138	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
139	#103	C₂²×C₁₀₉	1	2	yes	—
140	#102	C₂²×C₁₀₉	1	2	yes	—
141	#102	C₂²×C₁₀₉	1	2	yes	—
142	#97	C₂²×C₁₀₉	1	2	yes	—
143	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
144	C₂×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
145	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
149	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	Q₁₆	—	4	1	no	—
153	SD₁₆	—	2	4	no	—
154	SD₁₆	—	4	2	no	—
155	D₈	—	2	2	no	—
156	D₈	—	4	2	no	—
157	D₈	—	4	1	no	—
158	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
159	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
160	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
163	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
164	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₄²	—	2	2	no	—
172	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
173	C₄²	C₂³×C₁₀₉	1	2	yes	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	Q₈	—	2	2	no	—
177	Q₈	—	2	2	no	—
178	Q₈	—	4	1	no	—
179	D₄	—	2	2	no	—
180	D₄	—	2	2	no	—
181	D₄	—	2	2	no	—
182	D₄	—	2	2	no	—
183	D₄	—	4	4	no	—
184	D₄	—	8	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	8	2	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
190	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
191	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
192	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
193	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	2	1	no	—
196	C₈	#41×C₁₀₉	1	2	yes	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	2	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
211	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
212	C₂	—	8	2	no	—
213	C₂	—	8	2	no	—
214	C₂	#550×C₁₀₉	1	1	yes	characteristic
215	C₂	#575×C₁₀₉	1	1	yes	characteristic
216	C₂	#576×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄×Q₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #559×C₁₀₉	C₂	semidirect
2. #496×C₁₀₉	C₂	semidirect
3. #495×C₁₀₉	C₂	semidirect
4. #464×C₁₀₉	C₂	semidirect
5. #463×C₁₀₉	C₂	semidirect
6. #459×C₁₀₉	C₂	semidirect
7. #446×C₁₀₉	C₂	semidirect
8. #440×C₁₀₉	C₂	semidirect
9. C₄×D₈×C₁₀₉	C₂	semidirect
108. #3403	C₁₀₉	direct (N × H)
109. C₁₀₉	#3403	direct (N × H)
110. #559	C₂×C₁₀₉	semidirect
111. #496	C₂×C₁₀₉	semidirect
112. #495	C₂×C₁₀₉	semidirect
113. #464	C₂×C₁₀₉	semidirect
114. #463	C₂×C₁₀₉	semidirect
115. #459	C₂×C₁₀₉	semidirect
116. #446	C₂×C₁₀₉	semidirect
117. #440	C₂×C₁₀₉	semidirect
118. C₄×D₈	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#559×C₁₀₉	2
#496×C₁₀₉	2
#495×C₁₀₉	2
#464×C₁₀₉	2
#463×C₁₀₉	2
#459×C₁₀₉	2
#446×C₁₀₉	2
#440×C₁₀₉	2
C₄×D₈×C₁₀₉	2
#3403	109

Subgroups — C109×#3403 #28360657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3403 #28360657