Subgroups — C₁₀₉×#12613 #42130607

Order 27904.

410 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2115	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2108	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#1757	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1378	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₁₀₉	C₂²	1	1	yes	characteristic
7	#504×C₁₀₉	—	2	2	no	—
8	#504×C₁₀₉	—	2	2	no	—
9	C₂×M₃₂×C₁₀₉	C₄	1	2	yes	complemented
10	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
11	C₂×M₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₁₀₉	C₄	1	2	yes	complemented
13	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	#413×C₁₀₉	C₄	1	1	yes	characteristic
15	#410×C₁₀₉	C₄	1	1	yes	characteristic
16	#405×C₁₀₉	C₂²	1	2	yes	—
17	#349×C₁₀₉	C₄	1	4	yes	complemented
18	#346×C₁₀₉	—	2	2	no	—
19	C₄×C₁₆×C₁₀₉	—	2	2	no	—
20	#344×C₁₀₉	C₄	1	2	yes	—
21	C₂×#41×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	#131×C₁₀₉	—	2	2	no	—
23	#131×C₁₀₉	—	2	2	no	—
24	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
29	#117×C₁₀₉	C₂×C₄	1	2	yes	—
30	M₃₂×C₁₀₉	—	2	2	no	—
31	M₃₂×C₁₀₉	—	2	2	no	—
32	M₃₂×C₁₀₉	—	2	4	no	—
33	M₃₂×C₁₀₉	C₂×C₄	1	4	yes	—
34	C₂×C₁₆×C₁₀₉	—	2	2	no	—
35	C₂×C₁₆×C₁₀₉	—	2	2	no	—
36	C₂×C₁₆×C₁₀₉	—	2	2	no	—
37	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
41	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
42	#101×C₁₀₉	C₈	1	2	yes	—
43	#100×C₁₀₉	C₈	1	2	yes	—
44	#99×C₁₀₉	C₈	1	4	yes	—
45	#97×C₁₀₉	—	2	2	no	—
46	C₄×C₈×C₁₀₉	—	2	2	no	—
47	#41×C₁₀₉	—	2	2	no	—
48	#41×C₁₀₉	—	2	2	no	—
49	C₂×Q₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
50	C₂×D₄×C₁₀₉	C₂×C₈	1	2	yes	—
51	C₂×D₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₉	C₄²	1	2	yes	—
53	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
54	M₁₆×C₁₀₉	—	2	2	no	—
55	M₁₆×C₁₀₉	—	2	2	no	—
56	M₁₆×C₁₀₉	—	2	4	no	—
57	M₁₆×C₁₀₉	C₂×C₈	1	4	yes	—
58	C₂×C₈×C₁₀₉	—	2	2	no	—
59	C₂×C₈×C₁₀₉	—	2	2	no	—
60	C₂×C₈×C₁₀₉	—	2	2	no	—
61	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
62	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₁₀₉	#41	1	2	yes	—
66	C₄⋊C₄×C₁₀₉	—	2	2	no	—
67	C₂²⋊C₄×C₁₀₉	C₂×C₈	1	4	yes	—
68	C₄²×C₁₀₉	—	2	2	no	—
69	C₁₆×C₁₀₉	—	2	2	no	—
70	C₁₆×C₁₀₉	—	2	4	no	—
71	C₁₆×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₁₆×C₁₀₉	C₂²⋊C₄	1	2	yes	—
73	C₁₆×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂³×C₁₀₉	C₄×C₈	1	2	yes	—
75	C₂³×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
76	Q₈×C₁₀₉	—	2	2	no	—
77	D₄×C₁₀₉	—	2	4	no	—
78	D₄×C₁₀₉	—	2	2	no	—
79	C₂×C₄×C₁₀₉	—	2	2	no	—
80	C₂×C₄×C₁₀₉	—	2	2	no	—
81	C₂×C₄×C₁₀₉	—	2	4	no	—
82	C₂×C₄×C₁₀₉	C₄×C₈	1	2	yes	—
83	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₀₉	#131	1	2	yes	—
87	C₈×C₁₀₉	—	2	2	no	—
88	C₈×C₁₀₉	—	2	4	no	—
89	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₈×C₁₀₉	#117	1	1	yes	characteristic
91	C₈×C₁₀₉	C₄×D₄	1	2	yes	—
92	C₂²×C₁₀₉	—	2	2	no	—
93	C₂²×C₁₀₉	—	4	1	no	—
94	C₂²×C₁₀₉	—	4	2	no	—
95	C₂²×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
96	C₂²×C₁₀₉	C₈×D₄	1	2	yes	—
97	C₄×C₁₀₉	—	2	2	no	—
98	C₄×C₁₀₉	—	4	4	no	—
99	C₄×C₁₀₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
100	C₄×C₁₀₉	#405	1	1	yes	characteristic
101	C₄×C₁₀₉	C₈×D₄	1	2	yes	—
102	#12613	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂×C₁₀₉	—	2	2	no	—
104	C₂×C₁₀₉	—	2	1	no	—
105	C₂×C₁₀₉	—	4	2	no	—
106	C₂×C₁₀₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×#504	C₂×C₁₀₉	1	1	yes	characteristic
108	#2115	C₂×C₁₀₉	1	1	yes	characteristic
109	#2108	C₂×C₁₀₉	1	2	yes	complemented
110	#1757	C₂×C₁₀₉	1	1	yes	characteristic
111	#1378	C₂×C₁₀₉	1	2	yes	complemented
112	C₁₀₉	#12613	1	1	yes	characteristic, direct product factor, Sylow
113	C₂×#131	C₂²×C₁₀₉	1	1	yes	characteristic
114	#504	—	2	2	no	—
115	#504	—	2	2	no	—
116	C₂×M₃₂	C₄×C₁₀₉	1	2	yes	complemented
117	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
118	C₂×M₃₂	C₂²×C₁₀₉	1	1	yes	characteristic
119	C₂²×C₁₆	C₄×C₁₀₉	1	2	yes	complemented
120	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
121	#413	C₄×C₁₀₉	1	1	yes	characteristic
122	#410	C₄×C₁₀₉	1	1	yes	characteristic
123	#405	C₂²×C₁₀₉	1	2	yes	—
124	#349	C₄×C₁₀₉	1	4	yes	complemented
125	#346	—	2	2	no	—
126	C₄×C₁₆	—	2	2	no	—
127	#344	C₄×C₁₀₉	1	2	yes	—
128	C₂×#41	C₂×C₄×C₁₀₉	1	1	yes	characteristic
129	#131	—	2	2	no	—
130	#131	—	2	2	no	—
131	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
132	C₂×M₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
133	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
134	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
135	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
136	#117	C₂×C₄×C₁₀₉	1	2	yes	—
137	M₃₂	—	2	2	no	—
138	M₃₂	—	2	2	no	—
139	M₃₂	—	2	4	no	—
140	M₃₂	C₂×C₄×C₁₀₉	1	4	yes	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
145	C₂×C₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
146	C₂×C₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
147	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
148	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
149	#101	C₈×C₁₀₉	1	2	yes	—
150	#100	C₈×C₁₀₉	1	2	yes	—
151	#99	C₈×C₁₀₉	1	4	yes	—
152	#97	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	#41	—	2	2	no	—
155	#41	—	2	2	no	—
156	C₂×Q₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
157	C₂×D₄	C₂×C₈×C₁₀₉	1	2	yes	—
158	C₂×D₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
159	C₂²×C₄	C₄²×C₁₀₉	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	2	no	—
163	M₁₆	—	2	4	no	—
164	M₁₆	C₂×C₈×C₁₀₉	1	4	yes	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
169	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
170	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
171	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
172	C₂×C₈	#41×C₁₀₉	1	2	yes	—
173	C₄⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	C₂×C₈×C₁₀₉	1	4	yes	—
175	C₄²	—	2	2	no	—
176	C₁₆	—	2	2	no	—
177	C₁₆	—	2	4	no	—
178	C₁₆	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
179	C₁₆	C₂²⋊C₄×C₁₀₉	1	2	yes	—
180	C₁₆	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
181	C₂³	C₄×C₈×C₁₀₉	1	2	yes	—
182	C₂³	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
183	Q₈	—	2	2	no	—
184	D₄	—	2	4	no	—
185	D₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	C₄×C₈×C₁₀₉	1	2	yes	—
190	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
191	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
192	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
193	C₂×C₄	#131×C₁₀₉	1	2	yes	—
194	C₈	—	2	2	no	—
195	C₈	—	2	4	no	—
196	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
197	C₈	#117×C₁₀₉	1	1	yes	characteristic
198	C₈	C₄×D₄×C₁₀₉	1	2	yes	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	2	no	—
202	C₂²	C₂×C₄×C₈×C₁₀₉	1	1	yes	characteristic
203	C₂²	C₈×D₄×C₁₀₉	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	4	4	no	—
206	C₄	C₄×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
207	C₄	#405×C₁₀₉	1	1	yes	characteristic
208	C₄	C₈×D₄×C₁₀₉	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	2	1	no	—
211	C₂	—	4	2	no	—
212	C₂	C₈×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₁₀₉	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂×C₁₀₉	C₈×C₂²⋊C₄
prime radical	C₂×#41×C₁₀₉	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₁₀₉ ◁ C₂²×C₁₆×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#2108	C₂	semidirect
5. C₁₀₉×#1378	C₂	semidirect
9. C₂×M₃₂×C₁₀₉	C₄	semidirect
12. C₂²×C₁₆×C₁₀₉	C₄	semidirect
17. #349×C₁₀₉	C₄	semidirect
102. #12613	C₁₀₉	direct (N × H)
109. #2108	C₂×C₁₀₉	semidirect
111. #1378	C₂×C₁₀₉	semidirect
112. C₁₀₉	#12613	direct (N × H)
116. C₂×M₃₂	C₄×C₁₀₉	semidirect
119. C₂²×C₁₆	C₄×C₁₀₉	semidirect
124. #349	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#504×C₁₀₉	2
C₁₀₉×#2115	2
C₁₀₉×#2108	2
C₁₀₉×#1757	2
C₁₀₉×#1378	2
#12613	109

Subgroups — C109×#12613 #42130607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12613 #42130607