Subgroups — C₁₀₉×#3155 #28360409

Order 13952.

596 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×AGL(1,ℤ/8)×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	C₂×#135×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#535×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#496×C₁₀₉	C₂	1	4	yes	maximal, complemented
6	#495×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#493×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
10	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
11	AGL(1,ℤ/8)×C₁₀₉	—	2	8	no	—
12	#135×C₁₀₉	—	2	4	no	—
13	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
14	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	complemented
15	C₂×SD₁₆×C₁₀₉	C₂²	1	4	yes	complemented
16	C₂×D₈×C₁₀₉	—	2	4	no	—
17	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
18	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
19	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
20	C₂×D₈×C₁₀₉	C₂²	1	4	yes	complemented
21	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	#127×C₁₀₉	C₂²	1	4	yes	complemented
24	#124×C₁₀₉	C₂²	1	4	yes	complemented
25	#121×C₁₀₉	—	2	4	no	—
26	#120×C₁₀₉	—	2	4	no	—
27	#117×C₁₀₉	C₂²	1	1	yes	characteristic
28	#97×C₁₀₉	C₂²	1	2	yes	complemented
29	C₄×C₈×C₁₀₉	C₂²	1	2	yes	complemented
30	C₂⁴×C₁₀₉	—	2	2	no	—
31	#41×C₁₀₉	—	2	4	no	—
32	#41×C₁₀₉	—	2	4	no	—
33	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
34	C₂×D₄×C₁₀₉	—	2	2	no	—
35	C₂×D₄×C₁₀₉	—	2	4	no	—
36	C₂×D₄×C₁₀₉	—	2	4	no	—
37	C₂×D₄×C₁₀₉	—	2	2	no	—
38	C₂×D₄×C₁₀₉	—	2	8	no	—
39	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
40	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₀₉	—	2	2	no	—
42	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
43	Q₁₆×C₁₀₉	—	2	2	no	—
44	SD₁₆×C₁₀₉	—	2	4	no	—
45	SD₁₆×C₁₀₉	—	2	8	no	—
46	D₈×C₁₀₉	—	2	4	no	—
47	D₈×C₁₀₉	—	2	4	no	—
48	D₈×C₁₀₉	—	2	2	no	—
49	D₈×C₁₀₉	—	2	8	no	—
50	M₁₆×C₁₀₉	D₄	1	4	yes	complemented
51	C₂×C₈×C₁₀₉	D₄	1	4	yes	complemented
52	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
54	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
56	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
57	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
58	C₄²×C₁₀₉	C₂³	1	2	yes	—
59	C₂³×C₁₀₉	—	2	4	no	—
60	C₂³×C₁₀₉	—	2	2	no	—
61	C₂³×C₁₀₉	—	4	4	no	—
62	C₂³×C₁₀₉	—	8	4	no	—
63	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
64	Q₈×C₁₀₉	—	2	2	no	—
65	Q₈×C₁₀₉	—	2	2	no	—
66	D₄×C₁₀₉	—	2	4	no	—
67	D₄×C₁₀₉	—	2	2	no	—
68	D₄×C₁₀₉	—	2	4	no	—
69	D₄×C₁₀₉	—	2	2	no	—
70	D₄×C₁₀₉	—	4	4	no	—
71	D₄×C₁₀₉	—	4	4	no	—
72	D₄×C₁₀₉	—	8	8	no	—
73	C₂×C₄×C₁₀₉	—	2	2	no	—
74	C₂×C₄×C₁₀₉	—	4	4	no	—
75	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
76	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
80	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
81	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
82	C₈×C₁₀₉	C₂×D₄	1	4	yes	—
83	C₂²×C₁₀₉	—	2	2	no	—
84	C₂²×C₁₀₉	—	4	2	no	—
85	C₂²×C₁₀₉	—	4	4	no	—
86	C₂²×C₁₀₉	—	4	4	no	—
87	C₂²×C₁₀₉	—	4	8	no	—
88	C₂²×C₁₀₉	—	8	8	no	—
89	C₂²×C₁₀₉	#127	1	2	yes	—
90	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
91	C₄×C₁₀₉	—	2	4	no	—
92	C₄×C₁₀₉	—	4	2	no	—
93	C₄×C₁₀₉	#127	1	2	yes	—
94	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
95	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₁₀₉	—	2	2	no	—
97	C₂×C₁₀₉	—	8	2	no	—
98	C₂×C₁₀₉	—	8	4	no	—
99	C₂×C₁₀₉	C₂×#127	1	1	yes	characteristic
100	C₂×C₁₀₉	#576	1	2	yes	—
101	#3155	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₁₀₉	#3155	1	1	yes	characteristic, direct product factor, Sylow
103	C₂×AGL(1,ℤ/8)	C₂×C₁₀₉	1	2	yes	complemented
104	C₂×#135	C₂×C₁₀₉	1	1	yes	characteristic, complemented
105	C₂²×D₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
106	#535	C₂×C₁₀₉	1	2	yes	complemented
107	#496	C₂×C₁₀₉	1	4	yes	complemented
108	#495	C₂×C₁₀₉	1	2	yes	complemented
109	#493	C₂×C₁₀₉	1	2	yes	complemented
110	#405	C₂×C₁₀₉	1	1	yes	characteristic, complemented
111	C₂×#41	C₂²×C₁₀₉	1	2	yes	—
112	C₂²×D₄	C₂²×C₁₀₉	1	2	yes	—
113	AGL(1,ℤ/8)	—	2	8	no	—
114	#135	—	2	4	no	—
115	C₂×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
116	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	complemented
117	C₂×SD₁₆	C₂²×C₁₀₉	1	4	yes	complemented
118	C₂×D₈	—	2	4	no	—
119	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
120	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
121	C₂×D₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
122	C₂×D₈	C₂²×C₁₀₉	1	4	yes	complemented
123	C₂×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
124	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
125	#127	C₂²×C₁₀₉	1	4	yes	complemented
126	#124	C₂²×C₁₀₉	1	4	yes	complemented
127	#121	—	2	4	no	—
128	#120	—	2	4	no	—
129	#117	C₂²×C₁₀₉	1	1	yes	characteristic
130	#97	C₂²×C₁₀₉	1	2	yes	complemented
131	C₄×C₈	C₂²×C₁₀₉	1	2	yes	complemented
132	C₂⁴	—	2	2	no	—
133	#41	—	2	4	no	—
134	#41	—	2	4	no	—
135	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	—	2	4	no	—
138	C₂×D₄	—	2	4	no	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	—	2	8	no	—
141	C₂×D₄	C₂³×C₁₀₉	1	4	yes	—
142	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
145	Q₁₆	—	2	2	no	—
146	SD₁₆	—	2	4	no	—
147	SD₁₆	—	2	8	no	—
148	D₈	—	2	4	no	—
149	D₈	—	2	4	no	—
150	D₈	—	2	2	no	—
151	D₈	—	2	8	no	—
152	M₁₆	D₄×C₁₀₉	1	4	yes	complemented
153	C₂×C₈	D₄×C₁₀₉	1	4	yes	complemented
154	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
155	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
156	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
157	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
158	C₂²⋊C₄	—	2	8	no	—
159	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
160	C₄²	C₂³×C₁₀₉	1	2	yes	—
161	C₂³	—	2	4	no	—
162	C₂³	—	2	2	no	—
163	C₂³	—	4	4	no	—
164	C₂³	—	8	4	no	—
165	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
166	Q₈	—	2	2	no	—
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	D₄	—	2	4	no	—
171	D₄	—	2	2	no	—
172	D₄	—	4	4	no	—
173	D₄	—	4	4	no	—
174	D₄	—	8	8	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
178	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
179	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
180	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
181	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
182	C₈	C₂×D₄×C₁₀₉	1	2	yes	—
183	C₈	C₂×D₄×C₁₀₉	1	2	yes	—
184	C₈	C₂×D₄×C₁₀₉	1	4	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	4	2	no	—
187	C₂²	—	4	4	no	—
188	C₂²	—	4	4	no	—
189	C₂²	—	4	8	no	—
190	C₂²	—	8	8	no	—
191	C₂²	#127×C₁₀₉	1	2	yes	—
192	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
193	C₄	—	2	4	no	—
194	C₄	—	4	2	no	—
195	C₄	#127×C₁₀₉	1	2	yes	—
196	C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
197	C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
198	C₂	—	2	2	no	—
199	C₂	—	8	4	no	—
200	C₂	—	8	2	no	—
201	C₂	C₂×#127×C₁₀₉	1	1	yes	characteristic
202	C₂	#576×C₁₀₉	1	2	yes	—

	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 ◁ #117×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×AGL(1,ℤ/8)×C₁₀₉	C₂	semidirect
2. C₂×#135×C₁₀₉	C₂	semidirect
3. C₂²×D₈×C₁₀₉	C₂	semidirect
4. #535×C₁₀₉	C₂	semidirect
5. #496×C₁₀₉	C₂	semidirect
6. #495×C₁₀₉	C₂	semidirect
7. #493×C₁₀₉	C₂	semidirect
8. #405×C₁₀₉	C₂	semidirect
13. C₂×Q₁₆×C₁₀₉	C₂²	semidirect
14. C₂×SD₁₆×C₁₀₉	C₂²	semidirect
15. C₂×SD₁₆×C₁₀₉	C₂²	semidirect
19. C₂×D₈×C₁₀₉	C₂²	semidirect
20. C₂×D₈×C₁₀₉	C₂²	semidirect
23. #127×C₁₀₉	C₂²	semidirect
24. #124×C₁₀₉	C₂²	semidirect
28. #97×C₁₀₉	C₂²	semidirect
29. C₄×C₈×C₁₀₉	C₂²	semidirect
50. M₁₆×C₁₀₉	D₄	semidirect
51. C₂×C₈×C₁₀₉	D₄	semidirect
101. #3155	C₁₀₉	direct (N × H)
102. C₁₀₉	#3155	direct (N × H)
103. C₂×AGL(1,ℤ/8)	C₂×C₁₀₉	semidirect
104. C₂×#135	C₂×C₁₀₉	semidirect
105. C₂²×D₈	C₂×C₁₀₉	semidirect
106. #535	C₂×C₁₀₉	semidirect
107. #496	C₂×C₁₀₉	semidirect
108. #495	C₂×C₁₀₉	semidirect
109. #493	C₂×C₁₀₉	semidirect
110. #405	C₂×C₁₀₉	semidirect
115. C₂×Q₁₆	C₂²×C₁₀₉	semidirect
116. C₂×SD₁₆	C₂²×C₁₀₉	semidirect
117. C₂×SD₁₆	C₂²×C₁₀₉	semidirect
121. C₂×D₈	C₂²×C₁₀₉	semidirect
122. C₂×D₈	C₂²×C₁₀₉	semidirect
125. #127	C₂²×C₁₀₉	semidirect
126. #124	C₂²×C₁₀₉	semidirect
130. #97	C₂²×C₁₀₉	semidirect
131. C₄×C₈	C₂²×C₁₀₉	semidirect
152. M₁₆	D₄×C₁₀₉	semidirect
153. C₂×C₈	D₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×AGL(1,ℤ/8)×C₁₀₉	2
C₂×#135×C₁₀₉	2
C₂²×D₈×C₁₀₉	2
#535×C₁₀₉	2
#496×C₁₀₉	2
#495×C₁₀₉	2
#493×C₁₀₉	2
#405×C₁₀₉	2
#3155	109

Subgroups — C109×#3155 #28360409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3155 #28360409