Subgroups — C₁₀₉×#3122 #28360376

Order 13952.

418 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	#540×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#518×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#482×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#478×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#459×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#447×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#442×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	C₄×D₈×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	#433×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
12	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
13	#127×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
14	#125×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
15	#123×C₁₀₉	—	2	2	no	—
16	#121×C₁₀₉	—	2	2	no	—
17	#121×C₁₀₉	C₂²	1	2	yes	—
18	C₄×D₄×C₁₀₉	—	2	2	no	—
19	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
20	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
21	C₄×D₄×C₁₀₉	C₂²	1	2	yes	complemented
22	#117×C₁₀₉	C₂²	1	1	yes	characteristic
23	#117×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
25	#107×C₁₀₉	C₂²	1	2	yes	complemented
26	#106×C₁₀₉	C₂²	1	2	yes	complemented
27	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	complemented
28	#102×C₁₀₉	C₂²	1	2	yes	—
29	#102×C₁₀₉	C₂²	1	2	yes	—
30	#102×C₁₀₉	C₂²	1	2	yes	complemented
31	#102×C₁₀₉	C₂²	1	2	yes	complemented
32	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
33	#97×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
34	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
35	C₂⁴×C₁₀₉	—	2	1	no	—
36	C₂×D₄×C₁₀₉	—	2	4	no	—
37	C₂×D₄×C₁₀₉	—	2	2	no	—
38	C₂×D₄×C₁₀₉	—	2	1	no	—
39	C₂×D₄×C₁₀₉	—	2	2	no	—
40	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
41	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₀₉	—	2	2	no	—
43	C₂²×C₄×C₁₀₉	—	2	2	no	—
44	C₂²×C₄×C₁₀₉	—	2	2	no	—
45	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
46	D₈×C₁₀₉	—	2	4	no	—
47	D₈×C₁₀₉	—	2	4	no	—
48	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
49	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
50	C₄⋊C₄×C₁₀₉	—	2	2	no	—
51	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
52	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
53	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
54	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
55	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
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	D₄	1	2	yes	—
60	C₄²×C₁₀₉	—	2	1	no	—
61	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
62	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
63	C₂³×C₁₀₉	—	2	4	no	—
64	C₂³×C₁₀₉	—	2	2	no	—
65	C₂³×C₁₀₉	—	2	2	no	—
66	C₂³×C₁₀₉	—	8	2	no	—
67	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
68	D₄×C₁₀₉	—	2	4	no	—
69	D₄×C₁₀₉	—	2	2	no	—
70	D₄×C₁₀₉	—	2	2	no	—
71	D₄×C₁₀₉	—	2	2	no	—
72	D₄×C₁₀₉	—	8	2	no	—
73	D₄×C₁₀₉	—	8	2	no	—
74	D₄×C₁₀₉	#41	1	4	yes	—
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	—	8	2	no	—
79	C₂×C₄×C₁₀₉	—	8	2	no	—
80	C₂×C₄×C₁₀₉	—	8	2	no	—
81	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
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	2	no	—
87	C₂²×C₁₀₉	—	2	1	no	—
88	C₂²×C₁₀₉	—	2	4	no	—
89	C₂²×C₁₀₉	—	2	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	—	4	2	no	—
95	C₂²×C₁₀₉	—	8	4	no	—
96	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₁₀₉	—	2	1	no	—
98	C₄×C₁₀₉	—	2	2	no	—
99	C₄×C₁₀₉	—	2	2	no	—
100	C₄×C₁₀₉	—	4	2	no	—
101	C₄×C₁₀₉	—	4	2	no	—
102	C₄×C₁₀₉	C₂×#41	1	2	yes	—
103	C₂×C₁₀₉	—	4	1	no	—
104	C₂×C₁₀₉	—	4	4	no	—
105	C₂×C₁₀₉	—	8	2	no	—
106	C₂×C₁₀₉	#525	1	1	yes	characteristic
107	C₂×C₁₀₉	#576	1	2	yes	—
108	#3122	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₁₀₉	#3122	1	1	yes	characteristic, direct product factor, Sylow
110	#540	C₂×C₁₀₉	1	1	yes	characteristic, complemented
111	#518	C₂×C₁₀₉	1	1	yes	characteristic, complemented
112	#482	C₂×C₁₀₉	1	2	yes	complemented
113	#478	C₂×C₁₀₉	1	2	yes	complemented
114	#459	C₂×C₁₀₉	1	2	yes	complemented
115	#447	C₂×C₁₀₉	1	2	yes	complemented
116	#442	C₂×C₁₀₉	1	2	yes	complemented
117	C₄×D₈	C₂×C₁₀₉	1	2	yes	complemented
118	#433	C₂×C₁₀₉	1	1	yes	characteristic, complemented
119	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
120	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
121	C₂×D₈	C₂²×C₁₀₉	1	2	yes	—
122	#127	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
123	#125	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
124	#123	—	2	2	no	—
125	#121	—	2	2	no	—
126	#121	C₂²×C₁₀₉	1	2	yes	—
127	C₄×D₄	—	2	2	no	—
128	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
129	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
130	C₄×D₄	C₂²×C₁₀₉	1	2	yes	complemented
131	#117	C₂²×C₁₀₉	1	1	yes	characteristic
132	#117	C₂²×C₁₀₉	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	—	2	2	no	—
134	#107	C₂²×C₁₀₉	1	2	yes	complemented
135	#106	C₂²×C₁₀₉	1	2	yes	complemented
136	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	complemented
137	#102	C₂²×C₁₀₉	1	2	yes	—
138	#102	C₂²×C₁₀₉	1	2	yes	—
139	#102	C₂²×C₁₀₉	1	2	yes	complemented
140	#102	C₂²×C₁₀₉	1	2	yes	complemented
141	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
142	#97	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
143	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
144	C₂⁴	—	2	1	no	—
145	C₂×D₄	—	2	4	no	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	—	2	1	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
150	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
155	D₈	—	2	4	no	—
156	D₈	—	2	4	no	—
157	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
158	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
161	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
162	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
163	C₂²⋊C₄	—	2	1	no	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
169	C₄²	—	2	1	no	—
170	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
171	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
172	C₂³	—	2	4	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	8	2	no	—
176	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
177	D₄	—	2	4	no	—
178	D₄	—	2	2	no	—
179	D₄	—	2	2	no	—
180	D₄	—	2	2	no	—
181	D₄	—	8	2	no	—
182	D₄	—	8	2	no	—
183	D₄	#41×C₁₀₉	1	4	yes	—
184	C₂×C₄	—	2	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	—	8	2	no	—
190	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
191	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
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	2	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	8	4	no	—
205	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	C₂×#41×C₁₀₉	1	2	yes	—
212	C₂	—	4	1	no	—
213	C₂	—	4	4	no	—
214	C₂	—	8	2	no	—
215	C₂	#525×C₁₀₉	1	1	yes	characteristic
216	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. #540×C₁₀₉	C₂	semidirect
2. #518×C₁₀₉	C₂	semidirect
3. #482×C₁₀₉	C₂	semidirect
4. #478×C₁₀₉	C₂	semidirect
5. #459×C₁₀₉	C₂	semidirect
6. #447×C₁₀₉	C₂	semidirect
7. #442×C₁₀₉	C₂	semidirect
8. C₄×D₈×C₁₀₉	C₂	semidirect
9. #433×C₁₀₉	C₂	semidirect
13. #127×C₁₀₉	C₂²	semidirect
14. #125×C₁₀₉	C₂²	semidirect
21. C₄×D₄×C₁₀₉	C₂²	semidirect
25. #107×C₁₀₉	C₂²	semidirect
26. #106×C₁₀₉	C₂²	semidirect
27. C₄⋊C₈×C₁₀₉	C₂²	semidirect
30. #102×C₁₀₉	C₂²	semidirect
31. #102×C₁₀₉	C₂²	semidirect
33. #97×C₁₀₉	C₂²	semidirect
34. C₄×C₈×C₁₀₉	C₂²	semidirect
108. #3122	C₁₀₉	direct (N × H)
109. C₁₀₉	#3122	direct (N × H)
110. #540	C₂×C₁₀₉	semidirect
111. #518	C₂×C₁₀₉	semidirect
112. #482	C₂×C₁₀₉	semidirect
113. #478	C₂×C₁₀₉	semidirect
114. #459	C₂×C₁₀₉	semidirect
115. #447	C₂×C₁₀₉	semidirect
116. #442	C₂×C₁₀₉	semidirect
117. C₄×D₈	C₂×C₁₀₉	semidirect
118. #433	C₂×C₁₀₉	semidirect
122. #127	C₂²×C₁₀₉	semidirect
123. #125	C₂²×C₁₀₉	semidirect
130. C₄×D₄	C₂²×C₁₀₉	semidirect
134. #107	C₂²×C₁₀₉	semidirect
135. #106	C₂²×C₁₀₉	semidirect
136. C₄⋊C₈	C₂²×C₁₀₉	semidirect
139. #102	C₂²×C₁₀₉	semidirect
140. #102	C₂²×C₁₀₉	semidirect
142. #97	C₂²×C₁₀₉	semidirect
143. C₄×C₈	C₂²×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#540×C₁₀₉	2
#518×C₁₀₉	2
#482×C₁₀₉	2
#478×C₁₀₉	2
#459×C₁₀₉	2
#447×C₁₀₉	2
#442×C₁₀₉	2
C₄×D₈×C₁₀₉	2
#433×C₁₀₉	2
#3122	109

Subgroups — C109×#3122 #28360376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3122 #28360376