Subgroups — C₁₀₉×#12835 #42130829

Order 27904.

438 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#367×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₁₀₉×#1808	C₂	1	2	yes	maximal
4	C₁₀₉×#1380	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆×C₁₀₉	—	2	4	no	—
7	C₂²×C₁₆×C₁₀₉	C₄	1	4	yes	complemented
8	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×#107×C₁₀₉	C₂²	1	4	yes	—
11	#380×C₁₀₉	—	2	2	no	—
12	#367×C₁₀₉	—	2	4	no	—
13	#340×C₁₀₉	—	2	2	no	—
14	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₀₉	—	2	2	no	—
16	C₂²×C₈×C₁₀₉	—	2	2	no	—
17	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₉	Q₈	1	2	yes	—
20	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
22	C₂×C₂²⋊C₄×C₁₀₉	—	4	2	no	—
23	C₂×C₁₆×C₁₀₉	—	2	2	no	—
24	C₂×C₁₆×C₁₀₉	—	2	8	no	—
25	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₁₀₉	D₄	1	4	yes	—
28	#107×C₁₀₉	—	2	4	no	—
29	#107×C₁₀₉	—	2	4	no	—
30	#95×C₁₀₉	—	4	2	no	—
31	C₂⁴×C₁₀₉	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₁₀₉	—	2	2	no	—
33	C₂²×C₄×C₁₀₉	—	2	2	no	—
34	C₂²×C₄×C₁₀₉	—	4	4	no	—
35	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₁₀₉	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₉	Q₁₆	1	2	yes	—
38	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₀₉	—	2	2	no	—
40	C₂×C₈×C₁₀₉	—	2	2	no	—
41	C₂×C₈×C₁₀₉	—	2	2	no	—
42	C₂×C₈×C₁₀₉	—	2	4	no	—
43	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₀₉	#41	1	2	yes	—
48	C₄⋊C₄×C₁₀₉	—	4	4	no	—
49	C₄⋊C₄×C₁₀₉	—	4	4	no	—
50	C₂²⋊C₄×C₁₀₉	—	8	4	no	—
51	C₁₆×C₁₀₉	—	2	2	no	—
52	C₁₆×C₁₀₉	C₂²⋊C₄	1	4	yes	complemented
53	C₂³×C₁₀₉	—	2	2	no	—
54	C₂³×C₁₀₉	—	2	2	no	—
55	C₂³×C₁₀₉	#107	1	2	yes	—
56	C₂³×C₁₀₉	SD₃₂	1	4	yes	—
57	C₂³×C₁₀₉	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	—	2	2	no	—
59	C₂×C₄×C₁₀₉	—	2	2	no	—
60	C₂×C₄×C₁₀₉	—	2	2	no	—
61	C₂×C₄×C₁₀₉	—	2	4	no	—
62	C₂×C₄×C₁₀₉	—	8	4	no	—
63	C₂×C₄×C₁₀₉	—	8	4	no	—
64	C₂×C₄×C₁₀₉	—	8	4	no	—
65	C₂×C₄×C₁₀₉	#107	1	2	yes	—
66	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₀₉	#121	1	2	yes	—
68	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	#135	1	2	yes	—
70	C₈×C₁₀₉	—	2	2	no	—
71	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₈×C₁₀₉	#117	1	1	yes	characteristic
73	C₈×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
74	C₂²×C₁₀₉	—	2	2	no	—
75	C₂²×C₁₀₉	—	2	2	no	—
76	C₂²×C₁₀₉	—	2	8	no	—
77	C₂²×C₁₀₉	#367	1	4	yes	complemented
78	C₂²×C₁₀₉	C₂×#107	1	1	yes	characteristic
79	C₂²×C₁₀₉	#466	1	2	yes	—
80	C₂²×C₁₀₉	C₂×SD₃₂	1	2	yes	—
81	C₂²×C₁₀₉	#508	1	2	yes	—
82	C₄×C₁₀₉	—	2	2	no	—
83	C₄×C₁₀₉	—	16	4	no	—
84	C₄×C₁₀₉	#380	1	1	yes	characteristic
85	C₄×C₁₀₉	#427	1	1	yes	characteristic
86	C₄×C₁₀₉	#467	1	2	yes	—
87	#12835	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂×C₁₀₉	—	2	4	no	—
89	C₂×C₁₀₉	#1808	1	1	yes	characteristic
90	C₂×C₁₀₉	C₂×#367	1	1	yes	characteristic
91	C₂×C₁₀₉	#2158	1	1	yes	characteristic
92	C₂×C₁₀₉	#2217	1	4	yes	—
93	C₂³×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
94	C₂×#367	C₂×C₁₀₉	1	2	yes	complemented
95	#1808	C₂×C₁₀₉	1	2	yes	—
96	#1380	C₂×C₁₀₉	1	2	yes	complemented
97	C₁₀₉	#12835	1	1	yes	characteristic, direct product factor, Sylow
98	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
99	C₂²×C₁₆	—	2	4	no	—
100	C₂²×C₁₆	C₄×C₁₀₉	1	4	yes	complemented
101	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
102	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
103	C₂×#107	C₂²×C₁₀₉	1	4	yes	—
104	#380	—	2	2	no	—
105	#367	—	2	4	no	—
106	#340	—	2	2	no	—
107	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
108	C₂²×C₈	—	2	2	no	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
111	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
112	C₂²×C₈	Q₈×C₁₀₉	1	2	yes	—
113	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₂²⋊C₄	—	4	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	8	no	—
118	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
119	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
120	C₂×C₁₆	D₄×C₁₀₉	1	4	yes	—
121	#107	—	2	4	no	—
122	#107	—	2	4	no	—
123	#95	—	4	2	no	—
124	C₂⁴	D₈×C₁₀₉	1	1	yes	characteristic
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	4	4	no	—
128	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
129	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
130	C₂²×C₄	Q₁₆×C₁₀₉	1	2	yes	—
131	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
137	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
138	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
139	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
140	C₂×C₈	#41×C₁₀₉	1	2	yes	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₂²⋊C₄	—	8	4	no	—
144	C₁₆	—	2	2	no	—
145	C₁₆	C₂²⋊C₄×C₁₀₉	1	4	yes	complemented
146	C₂³	—	2	2	no	—
147	C₂³	—	2	2	no	—
148	C₂³	#107×C₁₀₉	1	2	yes	—
149	C₂³	SD₃₂×C₁₀₉	1	4	yes	—
150	C₂³	C₂×D₈×C₁₀₉	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	8	4	no	—
156	C₂×C₄	—	8	4	no	—
157	C₂×C₄	—	8	4	no	—
158	C₂×C₄	#107×C₁₀₉	1	2	yes	—
159	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
160	C₂×C₄	#121×C₁₀₉	1	2	yes	—
161	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
162	C₂×C₄	#135×C₁₀₉	1	2	yes	—
163	C₈	—	2	2	no	—
164	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
165	C₈	#117×C₁₀₉	1	1	yes	characteristic
166	C₈	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
167	C₂²	—	2	8	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	#367×C₁₀₉	1	4	yes	complemented
171	C₂²	C₂×#107×C₁₀₉	1	1	yes	characteristic
172	C₂²	#466×C₁₀₉	1	2	yes	—
173	C₂²	C₂×SD₃₂×C₁₀₉	1	2	yes	—
174	C₂²	#508×C₁₀₉	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	16	4	no	—
177	C₄	#380×C₁₀₉	1	1	yes	characteristic
178	C₄	#427×C₁₀₉	1	1	yes	characteristic
179	C₄	#467×C₁₀₉	1	2	yes	—
180	C₂	—	2	4	no	—
181	C₂	C₁₀₉×#1808	1	1	yes	characteristic
182	C₂	C₂×#367×C₁₀₉	1	1	yes	characteristic
183	C₂	C₁₀₉×#2158	1	1	yes	characteristic
184	C₂	C₁₀₉×#2217	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
prime radical	C₂⁴×C₁₀₉	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ C₂³×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#367×C₁₀₉	C₂	semidirect
4. C₁₀₉×#1380	C₂	semidirect
7. C₂²×C₁₆×C₁₀₉	C₄	semidirect
52. C₁₆×C₁₀₉	C₂²⋊C₄	semidirect
77. C₂²×C₁₀₉	#367	semidirect
87. #12835	C₁₀₉	direct (N × H)
94. C₂×#367	C₂×C₁₀₉	semidirect
96. #1380	C₂×C₁₀₉	semidirect
97. C₁₀₉	#12835	direct (N × H)
100. C₂²×C₁₆	C₄×C₁₀₉	semidirect
145. C₁₆	C₂²⋊C₄×C₁₀₉	semidirect
170. C₂²	#367×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₁₀₉	2
C₂×#367×C₁₀₉	2
C₁₀₉×#1808	2
C₁₀₉×#1380	2
#12835	109

Subgroups — C109×#12835 #42130829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12835 #42130829