Subgroups — C₁₀₉×#3568 #28360822

Order 13952.

958 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#563×C₁₀₉	C₂	1	3	yes	maximal, complemented
2	H(3,𝔽₄)×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#556×C₁₀₉	C₂	1	6	yes	maximal, complemented
4	#551×C₁₀₉	C₂	1	6	yes	maximal, complemented
5	#542×C₁₀₉	C₂	1	6	yes	maximal, complemented
6	#538×C₁₀₉	C₂	1	6	yes	maximal, complemented
7	#529×C₁₀₉	C₂	1	3	yes	maximal, complemented
8	C₄⋊Q₈×C₁₀₉	C₂²	1	6	yes	complemented
9	#126×C₁₀₉	C₂²	1	6	yes	complemented
10	#126×C₁₀₉	C₂²	1	6	yes	complemented
11	#126×C₁₀₉	C₂²	1	6	yes	complemented
12	#126×C₁₀₉	C₂²	1	2	yes	complemented
13	#126×C₁₀₉	C₂²	1	6	yes	complemented
14	#125×C₁₀₉	C₂²	1	3	yes	complemented
15	#124×C₁₀₉	C₂²	1	3	yes	—
16	#124×C₁₀₉	C₂²	1	6	yes	—
17	#124×C₁₀₉	C₂²	1	6	yes	—
18	#124×C₁₀₉	C₂²	1	6	yes	complemented
19	#123×C₁₀₉	C₂²	1	6	yes	—
20	#123×C₁₀₉	C₂²	1	6	yes	—
21	#123×C₁₀₉	C₂²	1	6	yes	complemented
22	C₂²⋊Q₈×C₁₀₉	C₂²	1	3	yes	complemented
23	C₂²⋊Q₈×C₁₀₉	C₂²	1	6	yes	complemented
24	C₂²⋊Q₈×C₁₀₉	C₂²	1	6	yes	complemented
25	C₂²⋊Q₈×C₁₀₉	C₂²	1	6	yes	complemented
26	#121×C₁₀₉	C₂²	1	3	yes	—
27	#121×C₁₀₉	C₂²	1	6	yes	—
28	#120×C₁₀₉	C₂²	1	6	yes	—
29	#120×C₁₀₉	C₂²	1	2	yes	complemented
30	C₄×Q₈×C₁₀₉	C₂²	1	3	yes	complemented
31	C₄×Q₈×C₁₀₉	C₂²	1	6	yes	complemented
32	C₄×D₄×C₁₀₉	C₂²	1	3	yes	—
33	C₄×D₄×C₁₀₉	C₂²	1	6	yes	—
34	C₄×D₄×C₁₀₉	C₂²	1	6	yes	complemented
35	#117×C₁₀₉	C₂²	1	6	yes	complemented
36	#117×C₁₀₉	C₂²	1	6	yes	complemented
37	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	6	yes	complemented
38	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
39	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
40	C₂×Q₈×C₁₀₉	C₂³	1	6	yes	—
41	C₂×Q₈×C₁₀₉	C₂³	1	3	yes	—
42	C₂×D₄×C₁₀₉	C₂³	1	3	yes	—
43	C₂×D₄×C₁₀₉	C₂³	1	6	yes	—
44	C₂×D₄×C₁₀₉	C₂³	1	6	yes	—
45	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
46	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
47	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
48	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
49	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
50	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
51	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
52	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
53	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
54	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
55	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
56	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
57	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
58	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
59	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
60	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
61	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
62	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
63	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
64	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
65	C₄²×C₁₀₉	C₂³	1	3	yes	—
66	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₄²×C₁₀₉	C₂³	1	6	yes	—
68	C₄²×C₁₀₉	C₂³	1	6	yes	—
69	C₄²×C₁₀₉	C₂³	1	2	yes	—
70	C₂³×C₁₀₉	—	4	6	no	—
71	C₂³×C₁₀₉	C₂⁴	1	6	yes	—
72	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
73	Q₈×C₁₀₉	—	4	6	no	—
74	Q₈×C₁₀₉	—	4	3	no	—
75	D₄×C₁₀₉	—	4	6	no	—
76	D₄×C₁₀₉	—	4	3	no	—
77	D₄×C₁₀₉	—	4	6	no	—
78	C₂×C₄×C₁₀₉	—	4	6	no	—
79	C₂×C₄×C₁₀₉	—	4	3	no	—
80	C₂×C₄×C₁₀₉	—	4	6	no	—
81	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
82	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
83	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
84	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
85	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
86	C₂²×C₁₀₉	—	2	3	no	—
87	C₂²×C₁₀₉	—	2	6	no	—
88	C₂²×C₁₀₉	—	2	6	no	—
89	C₂²×C₁₀₉	—	2	6	no	—
90	C₂²×C₁₀₉	—	8	4	no	—
91	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
92	C₄×C₁₀₉	—	2	3	no	—
93	C₄×C₁₀₉	—	2	6	no	—
94	C₄×C₁₀₉	—	2	3	no	—
95	C₄×C₁₀₉	—	2	6	no	—
96	C₄×C₁₀₉	—	2	6	no	—
97	C₂×C₁₀₉	—	4	1	no	—
98	C₂×C₁₀₉	—	4	6	no	—
99	C₂×C₁₀₉	#585	1	3	yes	—
100	#3568	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₁₀₉	#3568	1	1	yes	characteristic, direct product factor, Sylow
102	#563	C₂×C₁₀₉	1	3	yes	complemented
103	H(3,𝔽₄)	C₂×C₁₀₉	1	1	yes	characteristic, complemented
104	#556	C₂×C₁₀₉	1	6	yes	complemented
105	#551	C₂×C₁₀₉	1	6	yes	complemented
106	#542	C₂×C₁₀₉	1	6	yes	complemented
107	#538	C₂×C₁₀₉	1	6	yes	complemented
108	#529	C₂×C₁₀₉	1	3	yes	complemented
109	C₄⋊Q₈	C₂²×C₁₀₉	1	6	yes	complemented
110	#126	C₂²×C₁₀₉	1	6	yes	complemented
111	#126	C₂²×C₁₀₉	1	6	yes	complemented
112	#126	C₂²×C₁₀₉	1	6	yes	complemented
113	#126	C₂²×C₁₀₉	1	2	yes	complemented
114	#126	C₂²×C₁₀₉	1	6	yes	complemented
115	#125	C₂²×C₁₀₉	1	3	yes	complemented
116	#124	C₂²×C₁₀₉	1	3	yes	—
117	#124	C₂²×C₁₀₉	1	6	yes	—
118	#124	C₂²×C₁₀₉	1	6	yes	—
119	#124	C₂²×C₁₀₉	1	6	yes	complemented
120	#123	C₂²×C₁₀₉	1	6	yes	—
121	#123	C₂²×C₁₀₉	1	6	yes	—
122	#123	C₂²×C₁₀₉	1	6	yes	complemented
123	C₂²⋊Q₈	C₂²×C₁₀₉	1	3	yes	complemented
124	C₂²⋊Q₈	C₂²×C₁₀₉	1	6	yes	complemented
125	C₂²⋊Q₈	C₂²×C₁₀₉	1	6	yes	complemented
126	C₂²⋊Q₈	C₂²×C₁₀₉	1	6	yes	complemented
127	#121	C₂²×C₁₀₉	1	3	yes	—
128	#121	C₂²×C₁₀₉	1	6	yes	—
129	#120	C₂²×C₁₀₉	1	6	yes	—
130	#120	C₂²×C₁₀₉	1	2	yes	complemented
131	C₄×Q₈	C₂²×C₁₀₉	1	3	yes	complemented
132	C₄×Q₈	C₂²×C₁₀₉	1	6	yes	complemented
133	C₄×D₄	C₂²×C₁₀₉	1	3	yes	—
134	C₄×D₄	C₂²×C₁₀₉	1	6	yes	—
135	C₄×D₄	C₂²×C₁₀₉	1	6	yes	complemented
136	#117	C₂²×C₁₀₉	1	6	yes	complemented
137	#117	C₂²×C₁₀₉	1	6	yes	complemented
138	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	6	yes	complemented
139	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
140	C₂⁴	C₂³×C₁₀₉	1	2	yes	—
141	C₂×Q₈	C₂³×C₁₀₉	1	6	yes	—
142	C₂×Q₈	C₂³×C₁₀₉	1	3	yes	—
143	C₂×D₄	C₂³×C₁₀₉	1	3	yes	—
144	C₂×D₄	C₂³×C₁₀₉	1	6	yes	—
145	C₂×D₄	C₂³×C₁₀₉	1	6	yes	—
146	C₂²×C₄	C₂³×C₁₀₉	1	3	yes	—
147	C₂²×C₄	C₂³×C₁₀₉	1	6	yes	—
148	C₂²×C₄	C₂³×C₁₀₉	1	6	yes	—
149	C₄⋊C₄	C₂³×C₁₀₉	1	6	yes	—
150	C₄⋊C₄	C₂³×C₁₀₉	1	6	yes	—
151	C₄⋊C₄	C₂³×C₁₀₉	1	6	yes	—
152	C₄⋊C₄	C₂³×C₁₀₉	1	6	yes	—
153	C₄⋊C₄	C₂³×C₁₀₉	1	3	yes	—
154	C₄⋊C₄	C₂³×C₁₀₉	1	6	yes	—
155	C₄⋊C₄	C₂³×C₁₀₉	1	3	yes	—
156	C₄⋊C₄	C₂³×C₁₀₉	1	6	yes	—
157	C₂²⋊C₄	C₂³×C₁₀₉	1	6	yes	—
158	C₂²⋊C₄	C₂³×C₁₀₉	1	6	yes	—
159	C₂²⋊C₄	C₂³×C₁₀₉	1	6	yes	—
160	C₂²⋊C₄	C₂³×C₁₀₉	1	6	yes	—
161	C₂²⋊C₄	C₂³×C₁₀₉	1	6	yes	—
162	C₂²⋊C₄	C₂³×C₁₀₉	1	6	yes	—
163	C₂²⋊C₄	C₂³×C₁₀₉	1	6	yes	—
164	C₂²⋊C₄	C₂³×C₁₀₉	1	6	yes	—
165	C₂²⋊C₄	C₂³×C₁₀₉	1	6	yes	—
166	C₄²	C₂³×C₁₀₉	1	3	yes	—
167	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
168	C₄²	C₂³×C₁₀₉	1	6	yes	—
169	C₄²	C₂³×C₁₀₉	1	6	yes	—
170	C₄²	C₂³×C₁₀₉	1	2	yes	—
171	C₂³	—	4	6	no	—
172	C₂³	C₂⁴×C₁₀₉	1	6	yes	—
173	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
174	Q₈	—	4	6	no	—
175	Q₈	—	4	3	no	—
176	D₄	—	4	6	no	—
177	D₄	—	4	3	no	—
178	D₄	—	4	6	no	—
179	C₂×C₄	—	4	6	no	—
180	C₂×C₄	—	4	3	no	—
181	C₂×C₄	—	4	6	no	—
182	C₂×C₄	C₂⁴×C₁₀₉	1	3	yes	—
183	C₂×C₄	C₂⁴×C₁₀₉	1	6	yes	—
184	C₂×C₄	C₂⁴×C₁₀₉	1	3	yes	—
185	C₂×C₄	C₂⁴×C₁₀₉	1	6	yes	—
186	C₂×C₄	C₂⁴×C₁₀₉	1	6	yes	—
187	C₂²	—	2	3	no	—
188	C₂²	—	2	6	no	—
189	C₂²	—	2	6	no	—
190	C₂²	—	2	6	no	—
191	C₂²	—	8	4	no	—
192	C₂²	C₂⁵×C₁₀₉	1	1	yes	characteristic
193	C₄	—	2	3	no	—
194	C₄	—	2	6	no	—
195	C₄	—	2	3	no	—
196	C₄	—	2	6	no	—
197	C₄	—	2	6	no	—
198	C₂	—	4	6	no	—
199	C₂	—	4	1	no	—
200	C₂	#585×C₁₀₉	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #563×C₁₀₉	C₂	semidirect
2. H(3,𝔽₄)×C₁₀₉	C₂	semidirect
3. #556×C₁₀₉	C₂	semidirect
4. #551×C₁₀₉	C₂	semidirect
5. #542×C₁₀₉	C₂	semidirect
6. #538×C₁₀₉	C₂	semidirect
7. #529×C₁₀₉	C₂	semidirect
8. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
9. #126×C₁₀₉	C₂²	semidirect
10. #126×C₁₀₉	C₂²	semidirect
11. #126×C₁₀₉	C₂²	semidirect
12. #126×C₁₀₉	C₂²	semidirect
13. #126×C₁₀₉	C₂²	semidirect
14. #125×C₁₀₉	C₂²	semidirect
18. #124×C₁₀₉	C₂²	semidirect
21. #123×C₁₀₉	C₂²	semidirect
22. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
23. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
24. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
25. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
29. #120×C₁₀₉	C₂²	semidirect
30. C₄×Q₈×C₁₀₉	C₂²	semidirect
31. C₄×Q₈×C₁₀₉	C₂²	semidirect
34. C₄×D₄×C₁₀₉	C₂²	semidirect
35. #117×C₁₀₉	C₂²	semidirect
36. #117×C₁₀₉	C₂²	semidirect
37. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
38. C₂×C₄²×C₁₀₉	C₂²	semidirect
100. #3568	C₁₀₉	direct (N × H)
101. C₁₀₉	#3568	direct (N × H)
102. #563	C₂×C₁₀₉	semidirect
103. H(3,𝔽₄)	C₂×C₁₀₉	semidirect
104. #556	C₂×C₁₀₉	semidirect
105. #551	C₂×C₁₀₉	semidirect
106. #542	C₂×C₁₀₉	semidirect
107. #538	C₂×C₁₀₉	semidirect
108. #529	C₂×C₁₀₉	semidirect
109. C₄⋊Q₈	C₂²×C₁₀₉	semidirect
110. #126	C₂²×C₁₀₉	semidirect
111. #126	C₂²×C₁₀₉	semidirect
112. #126	C₂²×C₁₀₉	semidirect
113. #126	C₂²×C₁₀₉	semidirect
114. #126	C₂²×C₁₀₉	semidirect
115. #125	C₂²×C₁₀₉	semidirect
119. #124	C₂²×C₁₀₉	semidirect
122. #123	C₂²×C₁₀₉	semidirect
123. C₂²⋊Q₈	C₂²×C₁₀₉	semidirect
124. C₂²⋊Q₈	C₂²×C₁₀₉	semidirect
125. C₂²⋊Q₈	C₂²×C₁₀₉	semidirect
126. C₂²⋊Q₈	C₂²×C₁₀₉	semidirect
130. #120	C₂²×C₁₀₉	semidirect
131. C₄×Q₈	C₂²×C₁₀₉	semidirect
132. C₄×Q₈	C₂²×C₁₀₉	semidirect
135. C₄×D₄	C₂²×C₁₀₉	semidirect
136. #117	C₂²×C₁₀₉	semidirect
137. #117	C₂²×C₁₀₉	semidirect
138. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
139. C₂×C₄²	C₂²×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#563×C₁₀₉	2
H(3,𝔽₄)×C₁₀₉	2
#556×C₁₀₉	2
#551×C₁₀₉	2
#542×C₁₀₉	2
#538×C₁₀₉	2
#529×C₁₀₉	2
#3568	109

Subgroups — C109×#3568 #28360822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3568 #28360822