Subgroups — C₁₀₉×#12670 #42130664

Order 27904.

318 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#368×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2158	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1772	C₂	1	2	yes	maximal
5	C₁₀₉×#1382	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×#108×C₁₀₉	C₂²	1	2	yes	—
9	#427×C₁₀₉	C₂²	1	2	yes	—
10	#405×C₁₀₉	—	2	2	no	—
11	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	#368×C₁₀₉	C₄	1	4	yes	complemented
13	#367×C₁₀₉	C₄	1	2	yes	complemented
14	#366×C₁₀₉	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₁₀₉	C₄	1	4	yes	complemented
16	#339×C₁₀₉	—	2	2	no	—
17	C₂×M₁₆×C₁₀₉	—	2	2	no	—
18	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
20	#117×C₁₀₉	—	2	2	no	—
21	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₀₉	—	2	1	no	—
23	C₂×C₁₆×C₁₀₉	—	2	1	no	—
24	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
29	#108×C₁₀₉	—	2	2	no	—
30	#108×C₁₀₉	C₂×C₄	1	4	yes	—
31	#107×C₁₀₉	C₂×C₄	1	4	yes	—
32	#106×C₁₀₉	—	2	2	no	—
33	#97×C₁₀₉	—	4	2	no	—
34	C₄×C₈×C₁₀₉	—	4	2	no	—
35	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₁₀₉	Q₈	1	2	yes	—
37	C₂²×C₄×C₁₀₉	—	2	1	no	—
38	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	M₁₆×C₁₀₉	—	2	4	no	—
40	M₁₆×C₁₀₉	—	4	2	no	—
41	C₂×C₈×C₁₀₉	—	4	2	no	—
42	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₀₉	D₈	1	1	yes	characteristic
44	C₂×C₈×C₁₀₉	D₈	1	1	yes	characteristic
45	C₂×C₈×C₁₀₉	Q₁₆	1	2	yes	—
46	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
51	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
52	C₄⋊C₄×C₁₀₉	—	2	4	no	—
53	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
54	C₄²×C₁₀₉	—	2	1	no	—
55	C₄²×C₁₀₉	—	4	2	no	—
56	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
57	C₁₆×C₁₀₉	—	2	2	no	—
58	C₁₆×C₁₀₉	C₄²	1	2	yes	—
59	C₁₆×C₁₀₉	C₄²	1	2	yes	—
60	C₂³×C₁₀₉	#135	1	1	yes	characteristic
61	C₂×C₄×C₁₀₉	—	2	4	no	—
62	C₂×C₄×C₁₀₉	—	2	2	no	—
63	C₂×C₄×C₁₀₉	—	4	4	no	—
64	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	#135	1	1	yes	characteristic
70	C₈×C₁₀₉	—	4	4	no	—
71	C₈×C₁₀₉	#107	1	2	yes	—
72	C₈×C₁₀₉	#107	1	2	yes	—
73	C₈×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
74	C₈×C₁₀₉	#117	1	1	yes	characteristic
75	C₈×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
76	C₈×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
77	C₂²×C₁₀₉	—	2	2	no	—
78	C₂²×C₁₀₉	#427	1	1	yes	characteristic
79	C₂²×C₁₀₉	C₄×D₈	1	1	yes	characteristic
80	C₂²×C₁₀₉	C₄×Q₁₆	1	1	yes	characteristic
81	C₄×C₁₀₉	—	2	4	no	—
82	C₄×C₁₀₉	—	8	4	no	—
83	C₄×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₁₀₉	C₂×#107	1	1	yes	characteristic
85	C₄×C₁₀₉	C₄×D₈	1	1	yes	characteristic
86	C₄×C₁₀₉	C₄×Q₁₆	1	1	yes	characteristic
87	#12670	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂×C₁₀₉	—	2	1	no	—
89	C₂×C₁₀₉	—	2	1	no	—
90	C₂×C₁₀₉	C₄×#107	1	1	yes	characteristic
91	C₂×C₁₀₉	#2178	1	2	yes	—
92	C₂×#368	C₂×C₁₀₉	1	1	yes	characteristic
93	#2158	C₂×C₁₀₉	1	1	yes	characteristic
94	C₂×C₄×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
95	#1772	C₂×C₁₀₉	1	2	yes	—
96	#1382	C₂×C₁₀₉	1	2	yes	—
97	C₁₀₉	#12670	1	1	yes	characteristic, direct product factor, Sylow
98	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
99	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
100	C₂×#108	C₂²×C₁₀₉	1	2	yes	—
101	#427	C₂²×C₁₀₉	1	2	yes	—
102	#405	—	2	2	no	—
103	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
104	#368	C₄×C₁₀₉	1	4	yes	complemented
105	#367	C₄×C₁₀₉	1	2	yes	complemented
106	#366	C₄×C₁₀₉	1	2	yes	complemented
107	C₄×C₁₆	C₄×C₁₀₉	1	4	yes	complemented
108	#339	—	2	2	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
111	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
112	#117	—	2	2	no	—
113	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
114	C₂×C₁₆	—	2	1	no	—
115	C₂×C₁₆	—	2	1	no	—
116	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
117	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
118	C₂×C₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
119	C₂×C₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
120	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
121	#108	—	2	2	no	—
122	#108	C₂×C₄×C₁₀₉	1	4	yes	—
123	#107	C₂×C₄×C₁₀₉	1	4	yes	—
124	#106	—	2	2	no	—
125	#97	—	4	2	no	—
126	C₄×C₈	—	4	2	no	—
127	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
128	C₄×C₈	Q₈×C₁₀₉	1	2	yes	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
131	M₁₆	—	2	4	no	—
132	M₁₆	—	4	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
135	C₂×C₈	D₈×C₁₀₉	1	1	yes	characteristic
136	C₂×C₈	D₈×C₁₀₉	1	1	yes	characteristic
137	C₂×C₈	Q₁₆×C₁₀₉	1	2	yes	—
138	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
139	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
140	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
141	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
142	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
143	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
144	C₄⋊C₄	—	2	4	no	—
145	C₂²⋊C₄	—	4	2	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	4	2	no	—
148	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
149	C₁₆	—	2	2	no	—
150	C₁₆	C₄²×C₁₀₉	1	2	yes	—
151	C₁₆	C₄²×C₁₀₉	1	2	yes	—
152	C₂³	#135×C₁₀₉	1	1	yes	characteristic
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	4	no	—
156	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
157	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
158	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
159	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
160	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
161	C₂×C₄	#135×C₁₀₉	1	1	yes	characteristic
162	C₈	—	4	4	no	—
163	C₈	#107×C₁₀₉	1	2	yes	—
164	C₈	#107×C₁₀₉	1	2	yes	—
165	C₈	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
166	C₈	#117×C₁₀₉	1	1	yes	characteristic
167	C₈	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
168	C₈	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
169	C₂²	—	2	2	no	—
170	C₂²	#427×C₁₀₉	1	1	yes	characteristic
171	C₂²	C₄×D₈×C₁₀₉	1	1	yes	characteristic
172	C₂²	C₄×Q₁₆×C₁₀₉	1	1	yes	characteristic
173	C₄	—	2	4	no	—
174	C₄	—	8	4	no	—
175	C₄	C₄×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
176	C₄	C₂×#107×C₁₀₉	1	1	yes	characteristic
177	C₄	C₄×D₈×C₁₀₉	1	1	yes	characteristic
178	C₄	C₄×Q₁₆×C₁₀₉	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	C₄×#107×C₁₀₉	1	1	yes	characteristic
182	C₂	C₁₀₉×#2178	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₀₉	D₈
commutator	C₈	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₄×D₈
prime radical	C₂³×C₁₀₉	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₀₉ ◁ C₂²×C₈×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
12. #368×C₁₀₉	C₄	semidirect
13. #367×C₁₀₉	C₄	semidirect
14. #366×C₁₀₉	C₄	semidirect
15. C₄×C₁₆×C₁₀₉	C₄	semidirect
87. #12670	C₁₀₉	direct (N × H)
97. C₁₀₉	#12670	direct (N × H)
104. #368	C₄×C₁₀₉	semidirect
105. #367	C₄×C₁₀₉	semidirect
106. #366	C₄×C₁₀₉	semidirect
107. C₄×C₁₆	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×#368×C₁₀₉	2
C₁₀₉×#2158	2
C₂×C₄×C₁₆×C₁₀₉	2
C₁₀₉×#1772	2
C₁₀₉×#1382	2
#12670	109

Subgroups — C109×#12670 #42130664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12670 #42130664