Subgroups — C₁₀₉×#9859 #42127853

Order 27904.

390 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	C₁₀₉×#3160	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1837	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1493	C₂	1	2	yes	maximal
4	C₁₀₉×#1493	C₂	1	2	yes	maximal
5	C₂×#322×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#533×C₁₀₉	C₂²	1	2	yes	—
7	#500×C₁₀₉	C₄	1	1	yes	characteristic
8	#500×C₁₀₉	C₄	1	1	yes	characteristic
9	#499×C₁₀₉	—	2	2	no	—
10	#498×C₁₀₉	C₄	1	2	yes	—
11	#427×C₁₀₉	—	2	2	no	—
12	#423×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	#341×C₁₀₉	—	2	2	no	—
17	Q₈⋊C₈×C₁₀₉	—	2	4	no	—
18	Q₈⋊C₈×C₁₀₉	—	2	4	no	—
19	#322×C₁₀₉	—	2	2	no	—
20	C₂×M₁₆×C₁₀₉	—	2	2	no	—
21	C₂²×C₈×C₁₀₉	—	2	1	no	—
22	C₂²×C₈×C₁₀₉	—	2	1	no	—
23	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
24	C₄⋊Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₄⋊Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
26	#125×C₁₀₉	—	2	2	no	—
27	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
28	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
29	C₄×Q₈×C₁₀₉	—	2	2	no	—
30	C₄×Q₈×C₁₀₉	—	2	2	no	—
31	#117×C₁₀₉	—	2	2	no	—
32	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
33	#107×C₁₀₉	—	2	2	no	—
34	#107×C₁₀₉	—	2	2	no	—
35	#106×C₁₀₉	—	2	4	no	—
36	C₄⋊C₈×C₁₀₉	—	2	2	no	—
37	C₄⋊C₈×C₁₀₉	—	2	2	no	—
38	C₄×C₈×C₁₀₉	—	2	1	no	—
39	C₄×C₈×C₁₀₉	—	2	1	no	—
40	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
43	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
44	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
45	C₂×Q₈×C₁₀₉	—	2	2	no	—
46	C₂×Q₈×C₁₀₉	—	2	2	no	—
47	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
50	M₁₆×C₁₀₉	—	4	4	no	—
51	C₂×C₈×C₁₀₉	—	2	4	no	—
52	C₂×C₈×C₁₀₉	—	2	2	no	—
53	C₂×C₈×C₁₀₉	—	2	4	no	—
54	C₂×C₈×C₁₀₉	—	2	2	no	—
55	C₂×C₈×C₁₀₉	—	4	2	no	—
56	C₂×C₈×C₁₀₉	—	4	2	no	—
57	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
59	C₄⋊C₄×C₁₀₉	—	2	2	no	—
60	C₄⋊C₄×C₁₀₉	—	2	2	no	—
61	C₄⋊C₄×C₁₀₉	—	2	2	no	—
62	C₄⋊C₄×C₁₀₉	—	2	2	no	—
63	C₄⋊C₄×C₁₀₉	—	4	2	no	—
64	C₄⋊C₄×C₁₀₉	—	4	2	no	—
65	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
66	C₄²×C₁₀₉	—	4	2	no	—
67	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₁₀₉	#124	1	1	yes	characteristic
72	Q₈×C₁₀₉	—	4	4	no	—
73	Q₈×C₁₀₉	—	4	4	no	—
74	C₂×C₄×C₁₀₉	—	2	2	no	—
75	C₂×C₄×C₁₀₉	—	2	2	no	—
76	C₂×C₄×C₁₀₉	—	4	2	no	—
77	C₂×C₄×C₁₀₉	—	4	2	no	—
78	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
84	C₂×C₄×C₁₀₉	#137	1	4	yes	—
85	C₈×C₁₀₉	—	2	4	no	—
86	C₈×C₁₀₉	—	2	2	no	—
87	C₈×C₁₀₉	—	2	2	no	—
88	C₈×C₁₀₉	—	4	2	no	—
89	C₈×C₁₀₉	—	8	2	no	—
90	C₂²×C₁₀₉	—	2	1	no	—
91	C₂²×C₁₀₉	—	2	1	no	—
92	C₂²×C₁₀₉	#390	1	1	yes	characteristic
93	C₂²×C₁₀₉	#491	1	2	yes	—
94	C₄×C₁₀₉	—	2	1	no	—
95	C₄×C₁₀₉	—	2	1	no	—
96	C₄×C₁₀₉	—	8	2	no	—
97	C₄×C₁₀₉	—	8	2	no	—
98	C₄×C₁₀₉	#419	1	2	yes	—
99	C₄×C₁₀₉	#443	1	2	yes	—
100	C₄×C₁₀₉	#444	1	2	yes	—
101	C₄×C₁₀₉	#497	1	2	yes	—
102	#9859	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂×C₁₀₉	—	2	2	no	—
104	C₂×C₁₀₉	#1960	1	1	yes	characteristic
105	C₂×C₁₀₉	#1974	1	1	yes	characteristic
106	C₂×C₁₀₉	#1975	1	1	yes	characteristic
107	#3160	C₂×C₁₀₉	1	1	yes	characteristic
108	#1837	C₂×C₁₀₉	1	1	yes	characteristic
109	#1493	C₂×C₁₀₉	1	2	yes	—
110	#1493	C₂×C₁₀₉	1	2	yes	—
111	C₂×#322	C₂×C₁₀₉	1	1	yes	characteristic
112	C₁₀₉	#9859	1	1	yes	characteristic, direct product factor, Sylow
113	#533	C₂²×C₁₀₉	1	2	yes	—
114	#500	C₄×C₁₀₉	1	1	yes	characteristic
115	#500	C₄×C₁₀₉	1	1	yes	characteristic
116	#499	—	2	2	no	—
117	#498	C₄×C₁₀₉	1	2	yes	—
118	#427	—	2	2	no	—
119	#423	C₂²×C₁₀₉	1	2	yes	—
120	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
123	#341	—	2	2	no	—
124	Q₈⋊C₈	—	2	4	no	—
125	Q₈⋊C₈	—	2	4	no	—
126	#322	—	2	2	no	—
127	C₂×M₁₆	—	2	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
131	C₄⋊Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
132	C₄⋊Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
133	#125	—	2	2	no	—
134	C₂²⋊Q₈	—	2	2	no	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	C₄×Q₈	—	2	2	no	—
138	#117	—	2	2	no	—
139	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
140	#107	—	2	2	no	—
141	#107	—	2	2	no	—
142	#106	—	2	4	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
148	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
149	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
150	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
151	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	2	2	no	—
154	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
155	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
156	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
157	M₁₆	—	4	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
165	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	4	2	no	—
173	C₄²	—	4	2	no	—
174	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
175	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
176	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
177	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
178	C₂³	#124×C₁₀₉	1	1	yes	characteristic
179	Q₈	—	4	4	no	—
180	Q₈	—	4	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
186	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
187	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
188	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
189	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
190	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
191	C₂×C₄	#137×C₁₀₉	1	4	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	2	2	no	—
194	C₈	—	2	2	no	—
195	C₈	—	4	2	no	—
196	C₈	—	8	2	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#390×C₁₀₉	1	1	yes	characteristic
200	C₂²	#491×C₁₀₉	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	8	2	no	—
204	C₄	—	8	2	no	—
205	C₄	#419×C₁₀₉	1	2	yes	—
206	C₄	#443×C₁₀₉	1	2	yes	—
207	C₄	#444×C₁₀₉	1	2	yes	—
208	C₄	#497×C₁₀₉	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	C₁₀₉×#1960	1	1	yes	characteristic
211	C₂	C₁₀₉×#1974	1	1	yes	characteristic
212	C₂	C₁₀₉×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#127
commutator	C₄²	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#390
prime radical	C₂³×C₁₀₉	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
102. #9859	C₁₀₉	direct (N × H)
112. C₁₀₉	#9859	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#3160	2
C₁₀₉×#1837	2
C₁₀₉×#1493	2
C₂×#322×C₁₀₉	2
#9859	109

Subgroups — C109×#9859 #42127853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#9859 #42127853