Subgroups — C₁₀₉×#8283 #42126277

Order 27904.

414 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#106×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1770	C₂	1	2	yes	maximal
3	C₂×#334×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈²×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1279	C₂	1	2	yes	maximal
6	C₂×#106×C₁₀₉	C₄	1	2	yes	—
7	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
8	C₂×#97×C₁₀₉	—	2	2	no	—
9	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
13	#340×C₁₀₉	C₄	1	2	yes	—
14	#336×C₁₀₉	—	2	2	no	—
15	#334×C₁₀₉	C₄	1	4	yes	complemented
16	C₈²×C₁₀₉	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₀₉	—	2	1	no	—
18	C₂²×C₈×C₁₀₉	—	2	2	no	—
19	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₁₀₉	—	2	2	no	—
25	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
26	#106×C₁₀₉	—	2	4	no	—
27	C₄⋊C₈×C₁₀₉	—	2	2	no	—
28	C₄⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
29	#97×C₁₀₉	—	4	4	no	—
30	C₄×C₈×C₁₀₉	—	2	1	no	—
31	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₁₀₉	Q₈	1	2	yes	—
36	#95×C₁₀₉	—	2	2	no	—
37	C₂²×C₄×C₁₀₉	—	2	4	no	—
38	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
41	C₂×C₈×C₁₀₉	—	2	4	no	—
42	C₂×C₈×C₁₀₉	—	2	2	no	—
43	C₂×C₈×C₁₀₉	—	2	2	no	—
44	C₂×C₈×C₁₀₉	—	2	2	no	—
45	C₂×C₈×C₁₀₉	—	2	4	no	—
46	C₂×C₈×C₁₀₉	—	4	2	no	—
47	C₂×C₈×C₁₀₉	—	4	2	no	—
48	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
49	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
50	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₁₀₉	M₁₆	1	4	yes	—
53	C₂×C₈×C₁₀₉	SD₁₆	1	4	yes	—
54	C₄⋊C₄×C₁₀₉	—	2	4	no	—
55	C₄⋊C₄×C₁₀₉	—	2	4	no	—
56	C₄²×C₁₀₉	—	4	2	no	—
57	C₄²×C₁₀₉	—	4	2	no	—
58	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₁₀₉	#41	1	1	yes	characteristic
62	C₂³×C₁₀₉	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₀₉	—	2	2	no	—
64	C₂×C₄×C₁₀₉	—	4	4	no	—
65	C₂×C₄×C₁₀₉	—	4	4	no	—
66	C₂×C₄×C₁₀₉	—	4	4	no	—
67	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
70	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₁₀₉	#131	1	2	yes	—
75	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₁₀₉	#135	1	2	yes	—
77	C₈×C₁₀₉	—	2	4	no	—
78	C₈×C₁₀₉	—	2	2	no	—
79	C₈×C₁₀₉	—	2	2	no	—
80	C₈×C₁₀₉	—	4	4	no	—
81	C₈×C₁₀₉	#97	1	4	yes	complemented
82	C₈×C₁₀₉	#106	1	4	yes	complemented
83	C₂²×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₁₀₉	#436	1	2	yes	—
85	C₂²×C₁₀₉	C₄×SD₁₆	1	2	yes	—
86	C₂²×C₁₀₉	#443	1	2	yes	—
87	C₄×C₁₀₉	—	2	1	no	—
88	C₄×C₁₀₉	—	2	1	no	—
89	C₄×C₁₀₉	—	8	4	no	—
90	C₄×C₁₀₉	C₂×#97	1	1	yes	characteristic
91	C₄×C₁₀₉	#405	1	1	yes	characteristic
92	C₄×C₁₀₉	C₂×#106	1	1	yes	characteristic
93	C₄×C₁₀₉	#427	1	1	yes	characteristic
94	C₄×C₁₀₉	C₄×SD₁₆	1	2	yes	—
95	C₄×C₁₀₉	#446	1	2	yes	—
96	#8283	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂×C₁₀₉	#1590	1	4	yes	—
98	C₂×C₁₀₉	#1770	1	1	yes	characteristic
99	C₂×C₁₀₉	#1772	1	1	yes	characteristic
100	C₂×C₁₀₉	C₄×#106	1	1	yes	characteristic
101	C₄×#106	C₂×C₁₀₉	1	1	yes	characteristic
102	#1770	C₂×C₁₀₉	1	2	yes	—
103	C₂×#334	C₂×C₁₀₉	1	1	yes	characteristic
104	C₂×C₈²	C₂×C₁₀₉	1	1	yes	characteristic
105	#1279	C₂×C₁₀₉	1	2	yes	—
106	C₁₀₉	#8283	1	1	yes	characteristic, direct product factor, Sylow
107	C₂×#106	C₄×C₁₀₉	1	2	yes	—
108	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
109	C₂×#97	—	2	2	no	—
110	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
113	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
114	#340	C₄×C₁₀₉	1	2	yes	—
115	#336	—	2	2	no	—
116	#334	C₄×C₁₀₉	1	4	yes	complemented
117	C₈²	C₄×C₁₀₉	1	4	yes	complemented
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
121	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
122	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
123	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
127	#106	—	2	4	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
130	#97	—	4	4	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
133	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
134	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
135	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
136	C₄×C₈	Q₈×C₁₀₉	1	2	yes	—
137	#95	—	2	2	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
141	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
150	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
151	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
152	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
153	C₂×C₈	M₁₆×C₁₀₉	1	4	yes	—
154	C₂×C₈	SD₁₆×C₁₀₉	1	4	yes	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄²	—	4	2	no	—
158	C₄²	—	4	2	no	—
159	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
160	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
161	C₄²	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
162	C₄²	#41×C₁₀₉	1	1	yes	characteristic
163	C₂³	C₄×D₄×C₁₀₉	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
170	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
171	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
172	C₂×C₄	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
173	C₂×C₄	C₄×Q₈×C₁₀₉	1	1	yes	characteristic
174	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
175	C₂×C₄	#131×C₁₀₉	1	2	yes	—
176	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
177	C₂×C₄	#135×C₁₀₉	1	2	yes	—
178	C₈	—	2	4	no	—
179	C₈	—	2	2	no	—
180	C₈	—	2	2	no	—
181	C₈	—	4	4	no	—
182	C₈	#97×C₁₀₉	1	4	yes	complemented
183	C₈	#106×C₁₀₉	1	4	yes	complemented
184	C₂²	C₄×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
185	C₂²	#436×C₁₀₉	1	2	yes	—
186	C₂²	C₄×SD₁₆×C₁₀₉	1	2	yes	—
187	C₂²	#443×C₁₀₉	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	8	4	no	—
191	C₄	C₂×#97×C₁₀₉	1	1	yes	characteristic
192	C₄	#405×C₁₀₉	1	1	yes	characteristic
193	C₄	C₂×#106×C₁₀₉	1	1	yes	characteristic
194	C₄	#427×C₁₀₉	1	1	yes	characteristic
195	C₄	C₄×SD₁₆×C₁₀₉	1	2	yes	—
196	C₄	#446×C₁₀₉	1	2	yes	—
197	C₂	C₁₀₉×#1590	1	4	yes	—
198	C₂	C₁₀₉×#1770	1	1	yes	characteristic
199	C₂	C₁₀₉×#1772	1	1	yes	characteristic
200	C₂	C₄×#106×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle, prime radical	C₂³×C₁₀₉	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
15. #334×C₁₀₉	C₄	semidirect
16. C₈²×C₁₀₉	C₄	semidirect
81. C₈×C₁₀₉	#97	semidirect
82. C₈×C₁₀₉	#106	semidirect
96. #8283	C₁₀₉	direct (N × H)
106. C₁₀₉	#8283	direct (N × H)
116. #334	C₄×C₁₀₉	semidirect
117. C₈²	C₄×C₁₀₉	semidirect
182. C₈	#97×C₁₀₉	semidirect
183. C₈	#106×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#106×C₁₀₉	2
C₁₀₉×#1770	2
C₂×#334×C₁₀₉	2
C₂×C₈²×C₁₀₉	2
C₁₀₉×#1279	2
#8283	109

Subgroups — C109×#8283 #42126277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8283 #42126277