Subgroups — C₁₀₉×#3174 #28360428

Order 13952.

362 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	#541×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#540×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#499×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#492×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#491×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#490×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#468×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	#465×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	#440×C₁₀₉	C₂	1	2	yes	maximal, complemented
10	C₄×SD₁₆×C₁₀₉	C₂	1	2	yes	maximal, complemented
11	#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
13	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	#127×C₁₀₉	C₂²	1	1	yes	characteristic
18	#126×C₁₀₉	—	2	2	no	—
19	#125×C₁₀₉	—	2	2	no	—
20	#125×C₁₀₉	C₂²	1	1	yes	characteristic
21	#125×C₁₀₉	C₂²	1	1	yes	characteristic
22	#123×C₁₀₉	—	2	2	no	—
23	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
24	#121×C₁₀₉	—	2	2	no	—
25	#121×C₁₀₉	C₂²	1	2	yes	—
26	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
27	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
28	#117×C₁₀₉	C₂²	1	1	yes	characteristic
29	#107×C₁₀₉	C₂²	1	2	yes	—
30	#106×C₁₀₉	C₂²	1	2	yes	—
31	#103×C₁₀₉	C₂²	1	2	yes	—
32	#103×C₁₀₉	C₂²	1	2	yes	—
33	#102×C₁₀₉	C₂²	1	2	yes	—
34	#102×C₁₀₉	C₂²	1	2	yes	—
35	#97×C₁₀₉	C₂²	1	1	yes	characteristic
36	#97×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
40	C₂×D₄×C₁₀₉	—	2	2	no	—
41	C₂×D₄×C₁₀₉	—	2	2	no	—
42	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₀₉	—	2	2	no	—
44	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
45	SD₁₆×C₁₀₉	—	2	4	no	—
46	SD₁₆×C₁₀₉	—	2	4	no	—
47	M₁₆×C₁₀₉	—	2	2	no	—
48	C₂×C₈×C₁₀₉	—	2	2	no	—
49	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₁₀₉	—	2	2	no	—
54	C₄⋊C₄×C₁₀₉	—	2	2	no	—
55	C₄⋊C₄×C₁₀₉	—	2	2	no	—
56	C₄⋊C₄×C₁₀₉	—	2	2	no	—
57	C₄⋊C₄×C₁₀₉	—	2	2	no	—
58	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
59	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
60	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
62	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
63	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
64	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
65	C₄²×C₁₀₉	—	2	2	no	—
66	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₂³×C₁₀₉	—	2	2	no	—
69	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
70	Q₈×C₁₀₉	—	2	2	no	—
71	Q₈×C₁₀₉	—	2	2	no	—
72	D₄×C₁₀₉	—	2	2	no	—
73	D₄×C₁₀₉	—	2	2	no	—
74	D₄×C₁₀₉	—	8	2	no	—
75	D₄×C₁₀₉	—	8	2	no	—
76	C₂×C₄×C₁₀₉	—	2	1	no	—
77	C₂×C₄×C₁₀₉	—	2	1	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	—	8	2	no	—
82	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
86	C₈×C₁₀₉	—	2	1	no	—
87	C₈×C₁₀₉	—	2	1	no	—
88	C₈×C₁₀₉	#41	1	2	yes	—
89	C₈×C₁₀₉	#41	1	2	yes	—
90	C₂²×C₁₀₉	—	2	1	no	—
91	C₂²×C₁₀₉	—	2	2	no	—
92	C₂²×C₁₀₉	—	4	2	no	—
93	C₂²×C₁₀₉	—	4	2	no	—
94	C₂²×C₁₀₉	—	4	2	no	—
95	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₁₀₉	—	2	1	no	—
97	C₄×C₁₀₉	—	2	2	no	—
98	C₄×C₁₀₉	—	2	2	no	—
99	C₄×C₁₀₉	—	4	2	no	—
100	C₄×C₁₀₉	—	4	2	no	—
101	C₄×C₁₀₉	—	4	2	no	—
102	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
103	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
104	C₂×C₁₀₉	—	4	1	no	—
105	C₂×C₁₀₉	—	8	2	no	—
106	C₂×C₁₀₉	#532	1	1	yes	characteristic
107	C₂×C₁₀₉	#577	1	2	yes	—
108	#3174	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₁₀₉	#3174	1	1	yes	characteristic, direct product factor, Sylow
110	#541	C₂×C₁₀₉	1	1	yes	characteristic, complemented
111	#540	C₂×C₁₀₉	1	1	yes	characteristic
112	#499	C₂×C₁₀₉	1	1	yes	characteristic, complemented
113	#492	C₂×C₁₀₉	1	1	yes	characteristic, complemented
114	#491	C₂×C₁₀₉	1	1	yes	characteristic, complemented
115	#490	C₂×C₁₀₉	1	1	yes	characteristic, complemented
116	#468	C₂×C₁₀₉	1	2	yes	complemented
117	#465	C₂×C₁₀₉	1	2	yes	complemented
118	#440	C₂×C₁₀₉	1	2	yes	complemented
119	C₄×SD₁₆	C₂×C₁₀₉	1	2	yes	complemented
120	#405	C₂×C₁₀₉	1	1	yes	characteristic, complemented
121	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
122	C₂×SD₁₆	C₂²×C₁₀₉	1	2	yes	—
123	C₂×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
124	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
125	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
126	#127	C₂²×C₁₀₉	1	1	yes	characteristic
127	#126	—	2	2	no	—
128	#125	—	2	2	no	—
129	#125	C₂²×C₁₀₉	1	1	yes	characteristic
130	#125	C₂²×C₁₀₉	1	1	yes	characteristic
131	#123	—	2	2	no	—
132	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
133	#121	—	2	2	no	—
134	#121	C₂²×C₁₀₉	1	2	yes	—
135	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
136	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
137	#117	C₂²×C₁₀₉	1	1	yes	characteristic
138	#107	C₂²×C₁₀₉	1	2	yes	—
139	#106	C₂²×C₁₀₉	1	2	yes	—
140	#103	C₂²×C₁₀₉	1	2	yes	—
141	#103	C₂²×C₁₀₉	1	2	yes	—
142	#102	C₂²×C₁₀₉	1	2	yes	—
143	#102	C₂²×C₁₀₉	1	2	yes	—
144	#97	C₂²×C₁₀₉	1	1	yes	characteristic
145	#97	C₂²×C₁₀₉	1	1	yes	characteristic
146	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
147	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
148	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
154	SD₁₆	—	2	4	no	—
155	SD₁₆	—	2	4	no	—
156	M₁₆	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
159	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
160	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
161	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
168	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
169	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
174	C₄²	—	2	2	no	—
175	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
176	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
177	C₂³	—	2	2	no	—
178	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
179	Q₈	—	2	2	no	—
180	Q₈	—	2	2	no	—
181	D₄	—	2	2	no	—
182	D₄	—	2	2	no	—
183	D₄	—	8	2	no	—
184	D₄	—	8	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
192	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
193	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
194	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
195	C₈	—	2	1	no	—
196	C₈	—	2	1	no	—
197	C₈	#41×C₁₀₉	1	2	yes	—
198	C₈	#41×C₁₀₉	1	2	yes	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
212	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
213	C₂	—	4	1	no	—
214	C₂	—	8	2	no	—
215	C₂	#532×C₁₀₉	1	1	yes	characteristic
216	C₂	#577×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	#540×C₁₀₉	C₂
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. #541×C₁₀₉	C₂	semidirect
3. #499×C₁₀₉	C₂	semidirect
4. #492×C₁₀₉	C₂	semidirect
5. #491×C₁₀₉	C₂	semidirect
6. #490×C₁₀₉	C₂	semidirect
7. #468×C₁₀₉	C₂	semidirect
8. #465×C₁₀₉	C₂	semidirect
9. #440×C₁₀₉	C₂	semidirect
10. C₄×SD₁₆×C₁₀₉	C₂	semidirect
11. #405×C₁₀₉	C₂	semidirect
108. #3174	C₁₀₉	direct (N × H)
109. C₁₀₉	#3174	direct (N × H)
110. #541	C₂×C₁₀₉	semidirect
112. #499	C₂×C₁₀₉	semidirect
113. #492	C₂×C₁₀₉	semidirect
114. #491	C₂×C₁₀₉	semidirect
115. #490	C₂×C₁₀₉	semidirect
116. #468	C₂×C₁₀₉	semidirect
117. #465	C₂×C₁₀₉	semidirect
118. #440	C₂×C₁₀₉	semidirect
119. C₄×SD₁₆	C₂×C₁₀₉	semidirect
120. #405	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#541×C₁₀₉	2
#540×C₁₀₉	2
#499×C₁₀₉	2
#492×C₁₀₉	2
#491×C₁₀₉	2
#490×C₁₀₉	2
#468×C₁₀₉	2
#465×C₁₀₉	2
#440×C₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
#405×C₁₀₉	2
#3174	109

Subgroups — C109×#3174 #28360428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3174 #28360428