Subgroups — C₁₀₉×#12908 #42130902

Order 27904.

350 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#367×C₁₀₉	C₂	1	2	yes	maximal
2	C₂×#366×C₁₀₉	C₂	1	2	yes	maximal
3	C₂×#346×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1846	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₆×C₁₀₉	C₂²	1	2	yes	—
7	C₂×#107×C₁₀₉	—	2	1	no	—
8	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
9	C₂×#107×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×#107×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#106×C₁₀₉	—	2	1	no	—
12	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	#384×C₁₀₉	—	2	1	no	—
14	#382×C₁₀₉	—	2	1	no	—
15	#367×C₁₀₉	—	2	4	no	—
16	#366×C₁₀₉	—	2	4	no	—
17	#346×C₁₀₉	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
21	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
22	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₀₉	—	4	2	no	—
24	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
26	C₂×C₁₆×C₁₀₉	D₄	1	4	yes	—
27	C₂×C₁₆×C₁₀₉	Q₈	1	4	yes	—
28	#107×C₁₀₉	—	2	2	no	—
29	#107×C₁₀₉	—	2	2	no	—
30	#107×C₁₀₉	—	2	2	no	—
31	#107×C₁₀₉	—	2	2	no	—
32	#107×C₁₀₉	—	4	2	no	—
33	#106×C₁₀₉	—	4	2	no	—
34	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₁₀₉	Q₈	1	2	yes	—
36	#95×C₁₀₉	—	4	2	no	—
37	C₂²×C₄×C₁₀₉	—	4	1	no	—
38	C₂²×C₄×C₁₀₉	—	4	1	no	—
39	C₂²×C₄×C₁₀₉	—	4	2	no	—
40	C₂²×C₄×C₁₀₉	D₈	1	2	yes	—
41	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₀₉	D₈	1	2	yes	—
44	C₂×C₈×C₁₀₉	Q₁₆	1	2	yes	—
45	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
49	C₄⋊C₄×C₁₀₉	—	4	1	no	—
50	C₄⋊C₄×C₁₀₉	—	4	2	no	—
51	C₄⋊C₄×C₁₀₉	—	4	1	no	—
52	C₄⋊C₄×C₁₀₉	—	4	1	no	—
53	C₄⋊C₄×C₁₀₉	—	4	1	no	—
54	C₄⋊C₄×C₁₀₉	—	4	2	no	—
55	C₄⋊C₄×C₁₀₉	—	8	4	no	—
56	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
57	C₄²×C₁₀₉	Q₁₆	1	2	yes	—
58	C₁₆×C₁₀₉	C₄⋊C₄	1	8	yes	complemented
59	C₂³×C₁₀₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₁₀₉	—	2	2	no	—
61	C₂×C₄×C₁₀₉	—	2	2	no	—
62	C₂×C₄×C₁₀₉	—	8	1	no	—
63	C₂×C₄×C₁₀₉	—	8	2	no	—
64	C₂×C₄×C₁₀₉	—	8	1	no	—
65	C₂×C₄×C₁₀₉	—	8	4	no	—
66	C₂×C₄×C₁₀₉	—	8	2	no	—
67	C₂×C₄×C₁₀₉	—	8	2	no	—
68	C₂×C₄×C₁₀₉	#107	1	4	yes	—
69	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
75	C₈×C₁₀₉	#107	1	4	yes	—
76	C₈×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
77	C₈×C₁₀₉	C₄⋊Q₈	1	2	yes	—
78	C₂²×C₁₀₉	C₂×#107	1	1	yes	characteristic
79	C₂²×C₁₀₉	#493	1	1	yes	characteristic
80	C₂²×C₁₀₉	#500	1	1	yes	characteristic
81	C₂²×C₁₀₉	#509	1	2	yes	—
82	C₂²×C₁₀₉	#510	1	2	yes	—
83	C₄×C₁₀₉	—	2	4	no	—
84	C₄×C₁₀₉	—	16	2	no	—
85	C₄×C₁₀₉	—	16	1	no	—
86	C₄×C₁₀₉	—	16	1	no	—
87	C₄×C₁₀₉	#384	1	1	yes	characteristic
88	C₄×C₁₀₉	C₂×#107	1	1	yes	characteristic
89	C₄×C₁₀₉	#494	1	1	yes	characteristic
90	C₄×C₁₀₉	#500	1	1	yes	characteristic
91	#12908	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂×C₁₀₉	#1845	1	1	yes	characteristic
93	C₂×C₁₀₉	#2159	1	2	yes	—
94	C₂×C₁₀₉	#2250	1	1	yes	characteristic
95	C₂×C₁₀₉	#2251	1	1	yes	characteristic
96	C₂×C₁₀₉	#2255	1	2	yes	—
97	C₂×#367	C₂×C₁₀₉	1	2	yes	—
98	C₂×#366	C₂×C₁₀₉	1	2	yes	—
99	C₂×#346	C₂×C₁₀₉	1	1	yes	characteristic
100	#1846	C₂×C₁₀₉	1	1	yes	characteristic
101	#1845	C₂×C₁₀₉	1	1	yes	characteristic
102	C₁₀₉	#12908	1	1	yes	characteristic, direct product factor, Sylow
103	C₂²×C₁₆	C₂²×C₁₀₉	1	2	yes	—
104	C₂×#107	—	2	1	no	—
105	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
106	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
107	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
108	C₂×#106	—	2	1	no	—
109	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
110	#384	—	2	1	no	—
111	#382	—	2	1	no	—
112	#367	—	2	4	no	—
113	#366	—	2	4	no	—
114	#346	C₄×C₁₀₉	1	4	yes	complemented
115	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
116	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	1	no	—
118	C₂×C₄⋊C₄	—	2	1	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	2	no	—
121	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
122	C₂×C₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
123	C₂×C₁₆	D₄×C₁₀₉	1	4	yes	—
124	C₂×C₁₆	Q₈×C₁₀₉	1	4	yes	—
125	#107	—	2	2	no	—
126	#107	—	2	2	no	—
127	#107	—	2	2	no	—
128	#107	—	2	2	no	—
129	#107	—	4	2	no	—
130	#106	—	4	2	no	—
131	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
132	C₄×C₈	Q₈×C₁₀₉	1	2	yes	—
133	#95	—	4	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	D₈×C₁₀₉	1	2	yes	—
138	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
139	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
140	C₂×C₈	D₈×C₁₀₉	1	2	yes	—
141	C₂×C₈	Q₁₆×C₁₀₉	1	2	yes	—
142	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
143	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
144	C₂×C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
145	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	1	no	—
149	C₄⋊C₄	—	4	1	no	—
150	C₄⋊C₄	—	4	1	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	8	4	no	—
153	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
154	C₄²	Q₁₆×C₁₀₉	1	2	yes	—
155	C₁₆	C₄⋊C₄×C₁₀₉	1	8	yes	complemented
156	C₂³	C₂×D₈×C₁₀₉	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	8	4	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	#107×C₁₀₉	1	4	yes	—
166	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
167	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
168	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
169	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
170	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
171	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
172	C₈	#107×C₁₀₉	1	4	yes	—
173	C₈	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
174	C₈	C₄⋊Q₈×C₁₀₉	1	2	yes	—
175	C₂²	C₂×#107×C₁₀₉	1	1	yes	characteristic
176	C₂²	#493×C₁₀₉	1	1	yes	characteristic
177	C₂²	#500×C₁₀₉	1	1	yes	characteristic
178	C₂²	#509×C₁₀₉	1	2	yes	—
179	C₂²	#510×C₁₀₉	1	2	yes	—
180	C₄	—	2	4	no	—
181	C₄	—	16	2	no	—
182	C₄	—	16	1	no	—
183	C₄	—	16	1	no	—
184	C₄	#384×C₁₀₉	1	1	yes	characteristic
185	C₄	C₂×#107×C₁₀₉	1	1	yes	characteristic
186	C₄	#494×C₁₀₉	1	1	yes	characteristic
187	C₄	#500×C₁₀₉	1	1	yes	characteristic
188	C₂	C₁₀₉×#1845	1	1	yes	characteristic
189	C₂	C₁₀₉×#2159	1	2	yes	—
190	C₂	C₁₀₉×#2250	1	1	yes	characteristic
191	C₂	C₁₀₉×#2251	1	1	yes	characteristic
192	C₂	C₁₀₉×#2255	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
17. #346×C₁₀₉	C₄	semidirect
58. C₁₆×C₁₀₉	C₄⋊C₄	semidirect
91. #12908	C₁₀₉	direct (N × H)
102. C₁₀₉	#12908	direct (N × H)
114. #346	C₄×C₁₀₉	semidirect
155. C₁₆	C₄⋊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×#367×C₁₀₉	2
C₂×#366×C₁₀₉	2
C₂×#346×C₁₀₉	2
C₁₀₉×#1846	2
C₁₀₉×#1845	2
#12908	109

Subgroups — C109×#12908 #42130902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12908 #42130902