Subgroups — C₁₀₉×#12658 #42130652

Order 27904.

350 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2150	C₂	1	2	yes	maximal
2	C₂×#347×C₁₀₉	C₂	1	2	yes	maximal
3	C₄×#97×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1375	C₂	1	2	yes	maximal
5	C₂×M₃₂×C₁₀₉	C₂²	1	4	yes	—
6	C₂×#97×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂×#97×C₁₀₉	C₄	1	1	yes	characteristic
8	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄³×C₁₀₉	C₄	1	1	yes	characteristic
13	C₄⋊C₁₆×C₁₀₉	C₄	1	8	yes	complemented
14	#347×C₁₀₉	—	2	4	no	—
15	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₀₉	—	2	2	no	—
20	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	M₃₂×C₁₀₉	—	2	4	no	—
24	M₃₂×C₁₀₉	—	2	4	no	—
25	C₂×C₁₆×C₁₀₉	C₂×C₄	1	8	yes	—
26	#97×C₁₀₉	—	2	2	no	—
27	#97×C₁₀₉	—	2	2	no	—
28	#97×C₁₀₉	—	2	2	no	—
29	#97×C₁₀₉	D₄	1	2	yes	—
30	#97×C₁₀₉	Q₈	1	2	yes	—
31	C₄×C₈×C₁₀₉	—	2	1	no	—
32	C₄×C₈×C₁₀₉	—	2	1	no	—
33	C₄×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₁₀₉	—	2	1	no	—
35	C₂²×C₄×C₁₀₉	—	2	1	no	—
36	C₂²×C₄×C₁₀₉	M₁₆	1	2	yes	—
37	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₀₉	—	2	1	no	—
39	C₂×C₈×C₁₀₉	—	2	1	no	—
40	C₂×C₈×C₁₀₉	—	2	1	no	—
41	C₂×C₈×C₁₀₉	—	2	1	no	—
42	C₂×C₈×C₁₀₉	—	2	1	no	—
43	C₂×C₈×C₁₀₉	—	2	1	no	—
44	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
45	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₀₉	M₁₆	1	2	yes	—
47	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₁₀₉	#41	1	2	yes	—
51	C₄²×C₁₀₉	—	2	1	no	—
52	C₄²×C₁₀₉	—	2	1	no	—
53	C₄²×C₁₀₉	—	4	4	no	—
54	C₄²×C₁₀₉	C₄²	1	2	yes	—
55	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
56	C₄²×C₁₀₉	M₁₆	1	2	yes	—
57	C₄²×C₁₀₉	M₁₆	1	1	yes	characteristic
58	C₄²×C₁₀₉	M₁₆	1	1	yes	characteristic
59	C₁₆×C₁₀₉	—	2	8	no	—
60	C₂³×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
61	C₂×C₄×C₁₀₉	—	2	4	no	—
62	C₂×C₄×C₁₀₉	—	2	2	no	—
63	C₂×C₄×C₁₀₉	—	2	2	no	—
64	C₂×C₄×C₁₀₉	—	4	2	no	—
65	C₂×C₄×C₁₀₉	—	4	2	no	—
66	C₂×C₄×C₁₀₉	#97	1	2	yes	—
67	C₂×C₄×C₁₀₉	#97	1	2	yes	—
68	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
74	C₈×C₁₀₉	—	2	2	no	—
75	C₈×C₁₀₉	—	2	2	no	—
76	C₈×C₁₀₉	—	2	2	no	—
77	C₈×C₁₀₉	C₄×D₄	1	2	yes	—
78	C₈×C₁₀₉	C₄×Q₈	1	2	yes	—
79	C₂²×C₁₀₉	—	2	2	no	—
80	C₂²×C₁₀₉	C₂×#97	1	1	yes	characteristic
81	C₂²×C₁₀₉	#423	1	1	yes	characteristic
82	C₂²×C₁₀₉	#432	1	1	yes	characteristic
83	C₄×C₁₀₉	—	2	2	no	—
84	C₄×C₁₀₉	—	4	2	no	—
85	C₄×C₁₀₉	—	4	2	no	—
86	C₄×C₁₀₉	#347	1	4	yes	complemented
87	C₄×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₄×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
89	C₄×C₁₀₉	#423	1	1	yes	characteristic
90	C₄×C₁₀₉	#432	1	1	yes	characteristic
91	#12658	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂×C₁₀₉	—	2	1	no	—
93	C₂×C₁₀₉	—	2	1	no	—
94	C₂×C₁₀₉	#1767	1	1	yes	characteristic
95	C₂×C₁₀₉	C₂×#347	1	2	yes	—
96	#2150	C₂×C₁₀₉	1	2	yes	—
97	C₂×#347	C₂×C₁₀₉	1	2	yes	—
98	C₄×#97	C₂×C₁₀₉	1	1	yes	characteristic
99	#1375	C₂×C₁₀₉	1	2	yes	—
100	C₁₀₉	#12658	1	1	yes	characteristic, direct product factor, Sylow
101	C₂×M₃₂	C₂²×C₁₀₉	1	4	yes	—
102	C₂×#97	C₄×C₁₀₉	1	1	yes	characteristic
103	C₂×#97	C₄×C₁₀₉	1	1	yes	characteristic
104	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
105	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
106	C₂×C₄×C₈	C₄×C₁₀₉	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
108	C₄³	C₄×C₁₀₉	1	1	yes	characteristic
109	C₄⋊C₁₆	C₄×C₁₀₉	1	8	yes	complemented
110	#347	—	2	4	no	—
111	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
112	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
113	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
114	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
115	C₂×C₄²	—	2	2	no	—
116	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
117	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
118	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
119	M₃₂	—	2	4	no	—
120	M₃₂	—	2	4	no	—
121	C₂×C₁₆	C₂×C₄×C₁₀₉	1	8	yes	—
122	#97	—	2	2	no	—
123	#97	—	2	2	no	—
124	#97	—	2	2	no	—
125	#97	D₄×C₁₀₉	1	2	yes	—
126	#97	Q₈×C₁₀₉	1	2	yes	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	M₁₆×C₁₀₉	1	2	yes	—
133	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
141	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
142	C₂×C₈	M₁₆×C₁₀₉	1	2	yes	—
143	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
144	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
145	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
146	C₂×C₈	#41×C₁₀₉	1	2	yes	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	4	4	no	—
150	C₄²	C₄²×C₁₀₉	1	2	yes	—
151	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
152	C₄²	M₁₆×C₁₀₉	1	2	yes	—
153	C₄²	M₁₆×C₁₀₉	1	1	yes	characteristic
154	C₄²	M₁₆×C₁₀₉	1	1	yes	characteristic
155	C₁₆	—	2	8	no	—
156	C₂³	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	#97×C₁₀₉	1	2	yes	—
163	C₂×C₄	#97×C₁₀₉	1	2	yes	—
164	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
165	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
166	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
167	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
168	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
169	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	—	2	2	no	—
173	C₈	C₄×D₄×C₁₀₉	1	2	yes	—
174	C₈	C₄×Q₈×C₁₀₉	1	2	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	C₂×#97×C₁₀₉	1	1	yes	characteristic
177	C₂²	#423×C₁₀₉	1	1	yes	characteristic
178	C₂²	#432×C₁₀₉	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	4	2	no	—
182	C₄	#347×C₁₀₉	1	4	yes	complemented
183	C₄	C₄×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
184	C₄	C₄×M₁₆×C₁₀₉	1	1	yes	characteristic
185	C₄	#423×C₁₀₉	1	1	yes	characteristic
186	C₄	#432×C₁₀₉	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	C₁₀₉×#1767	1	1	yes	characteristic
190	C₂	C₂×#347×C₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×#97
prime radical	C₂³×C₁₀₉	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄⋊C₁₆×C₁₀₉	C₄	semidirect
86. C₄×C₁₀₉	#347	semidirect
91. #12658	C₁₀₉	direct (N × H)
100. C₁₀₉	#12658	direct (N × H)
109. C₄⋊C₁₆	C₄×C₁₀₉	semidirect
182. C₄	#347×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2150	2
C₂×#347×C₁₀₉	2
C₄×#97×C₁₀₉	2
C₁₀₉×#1375	2
#12658	109

Subgroups — C109×#12658 #42130652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12658 #42130652