Subgroups — C₁₀₉×#2970 #28360224

Order 13952.

406 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#446×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	#435×C₁₀₉	C₂	1	4	yes	maximal, complemented
4	C₈×D₄×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#432×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#432×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#97×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₁₀₉	C₄	1	1	yes	characteristic
10	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₀₉	C₂²	1	4	yes	—
12	C₄×Q₈×C₁₀₉	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₁₀₉	C₄	1	2	yes	—
15	C₄×D₄×C₁₀₉	C₄	1	1	yes	characteristic
16	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
18	#117×C₁₀₉	C₄	1	1	yes	characteristic
19	#117×C₁₀₉	C₄	1	2	yes	—
20	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
23	C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	—
24	C₂²⋊C₈×C₁₀₉	C₂²	1	4	yes	—
25	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
26	#97×C₁₀₉	C₂²	1	1	yes	characteristic
27	#97×C₁₀₉	C₂²	1	2	yes	—
28	#97×C₁₀₉	C₂²	1	4	yes	—
29	#97×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
31	#41×C₁₀₉	—	2	4	no	—
32	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
41	M₁₆×C₁₀₉	—	2	4	no	—
42	C₂×C₈×C₁₀₉	—	2	8	no	—
43	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
44	C₂×C₈×C₁₀₉	C₂³	1	4	yes	—
45	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
46	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
50	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
53	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
54	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
56	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
58	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₄²×C₁₀₉	C₂³	1	2	yes	—
60	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
61	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
62	Q₈×C₁₀₉	—	2	2	no	—
63	D₄×C₁₀₉	—	2	4	no	—
64	D₄×C₁₀₉	M₁₆	1	4	yes	complemented
65	C₂×C₄×C₁₀₉	—	2	4	no	—
66	C₂×C₄×C₁₀₉	—	2	2	no	—
67	C₂×C₄×C₁₀₉	—	2	4	no	—
68	C₂×C₄×C₁₀₉	—	2	2	no	—
69	C₂×C₄×C₁₀₉	—	4	2	no	—
70	C₂×C₄×C₁₀₉	—	4	1	no	—
71	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
80	C₈×C₁₀₉	—	2	4	no	—
81	C₈×C₁₀₉	—	2	2	no	—
82	C₈×C₁₀₉	#41	1	4	yes	complemented
83	C₂²×C₁₀₉	—	2	2	no	—
84	C₂²×C₁₀₉	—	2	2	no	—
85	C₂²×C₁₀₉	—	2	1	no	—
86	C₂²×C₁₀₉	—	2	1	no	—
87	C₂²×C₁₀₉	—	2	1	no	—
88	C₂²×C₁₀₉	C₂×M₁₆	1	4	yes	—
89	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
90	C₄×C₁₀₉	—	2	2	no	—
91	C₄×C₁₀₉	—	2	1	no	—
92	C₄×C₁₀₉	—	2	1	no	—
93	C₄×C₁₀₉	—	2	2	no	—
94	C₄×C₁₀₉	—	2	1	no	—
95	C₄×C₁₀₉	—	2	2	no	—
96	C₄×C₁₀₉	—	2	1	no	—
97	C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
98	C₄×C₁₀₉	C₂×#41	1	2	yes	—
99	C₂×C₁₀₉	—	2	4	no	—
100	C₂×C₁₀₉	—	4	1	no	—
101	C₂×C₁₀₉	C₄×#41	1	1	yes	characteristic
102	C₂×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
103	C₂×C₁₀₉	#568	1	1	yes	characteristic
104	#2970	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₁₀₉	#2970	1	1	yes	characteristic, direct product factor, Sylow
106	C₄×#41	C₂×C₁₀₉	1	1	yes	characteristic
107	#446	C₂×C₁₀₉	1	2	yes	complemented
108	#435	C₂×C₁₀₉	1	4	yes	complemented
109	C₈×D₄	C₂×C₁₀₉	1	2	yes	complemented
110	#432	C₂×C₁₀₉	1	1	yes	characteristic, complemented
111	#432	C₂×C₁₀₉	1	2	yes	complemented
112	C₄×M₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
113	C₂×#97	C₂×C₁₀₉	1	2	yes	complemented
114	C₂×#41	C₄×C₁₀₉	1	1	yes	characteristic
115	C₂×M₁₆	C₂²×C₁₀₉	1	2	yes	—
116	C₂²×C₈	C₂²×C₁₀₉	1	4	yes	—
117	C₄×Q₈	C₄×C₁₀₉	1	1	yes	characteristic
118	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
119	C₄×D₄	C₄×C₁₀₉	1	2	yes	—
120	C₄×D₄	C₄×C₁₀₉	1	1	yes	characteristic
121	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
122	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
123	#117	C₄×C₁₀₉	1	1	yes	characteristic
124	#117	C₄×C₁₀₉	1	2	yes	—
125	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
126	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
127	C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
128	C₄⋊C₈	C₂²×C₁₀₉	1	4	yes	—
129	C₂²⋊C₈	C₂²×C₁₀₉	1	4	yes	—
130	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
131	#97	C₂²×C₁₀₉	1	1	yes	characteristic
132	#97	C₂²×C₁₀₉	1	2	yes	—
133	#97	C₂²×C₁₀₉	1	4	yes	—
134	#97	C₂²×C₁₀₉	1	1	yes	characteristic
135	C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
136	#41	—	2	4	no	—
137	C₂×Q₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
138	C₂×D₄	C₂×C₄×C₁₀₉	1	2	yes	—
139	C₂×D₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₄×C₁₀₉	1	2	yes	—
141	C₂²×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₄×C₁₀₉	1	2	yes	—
144	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
145	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
146	M₁₆	—	2	4	no	—
147	C₂×C₈	—	2	8	no	—
148	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
149	C₂×C₈	C₂³×C₁₀₉	1	4	yes	—
150	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
151	C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
152	C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
153	C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
154	C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
155	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
156	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
157	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
158	C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
159	C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
160	C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
161	C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
162	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
163	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
164	C₄²	C₂³×C₁₀₉	1	2	yes	—
165	C₂³	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
166	C₂³	C₂²×C₄×C₁₀₉	1	2	yes	—
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	M₁₆×C₁₀₉	1	4	yes	complemented
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
177	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
178	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
179	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
180	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
181	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
182	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
183	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
184	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	2	2	no	—
187	C₈	#41×C₁₀₉	1	4	yes	complemented
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	C₂×M₁₆×C₁₀₉	1	4	yes	—
194	C₂²	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	1	no	—
202	C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
203	C₄	C₂×#41×C₁₀₉	1	2	yes	—
204	C₂	—	2	4	no	—
205	C₂	—	4	1	no	—
206	C₂	C₄×#41×C₁₀₉	1	1	yes	characteristic
207	C₂	C₂²×M₁₆×C₁₀₉	1	1	yes	characteristic
208	C₂	#568×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
socle	C₂²×C₁₀₉	C₂³×C₄
prime radical	C₂×#41×C₁₀₉	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #446×C₁₀₉	C₂	semidirect
3. #435×C₁₀₉	C₂	semidirect
4. C₈×D₄×C₁₀₉	C₂	semidirect
5. #432×C₁₀₉	C₂	semidirect
6. #432×C₁₀₉	C₂	semidirect
7. C₄×M₁₆×C₁₀₉	C₂	semidirect
8. C₂×#97×C₁₀₉	C₂	semidirect
64. D₄×C₁₀₉	M₁₆	semidirect
82. C₈×C₁₀₉	#41	semidirect
104. #2970	C₁₀₉	direct (N × H)
105. C₁₀₉	#2970	direct (N × H)
107. #446	C₂×C₁₀₉	semidirect
108. #435	C₂×C₁₀₉	semidirect
109. C₈×D₄	C₂×C₁₀₉	semidirect
110. #432	C₂×C₁₀₉	semidirect
111. #432	C₂×C₁₀₉	semidirect
112. C₄×M₁₆	C₂×C₁₀₉	semidirect
113. C₂×#97	C₂×C₁₀₉	semidirect
169. D₄	M₁₆×C₁₀₉	semidirect
187. C₈	#41×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#41×C₁₀₉	2
#446×C₁₀₉	2
#435×C₁₀₉	2
C₈×D₄×C₁₀₉	2
#432×C₁₀₉	2
C₄×M₁₆×C₁₀₉	2
C₂×#97×C₁₀₉	2
#2970	109

Subgroups — C109×#2970 #28360224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2970 #28360224