Subgroups — C₁₀₉×#17828 #42135822

Order 27904.

702 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2906	C₂	1	2	yes	maximal
3	C₁₀₉×#1571	C₂	1	6	yes	maximal, complemented
4	C₁₀₉×#1567	C₂	1	3	yes	maximal, complemented
5	C₂×#335×C₁₀₉	C₂	1	3	yes	maximal, complemented
6	C₂×#131×C₁₀₉	—	2	2	no	—
7	C₄×#41×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₁₀₉	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₁₀₉	C₄	1	1	yes	characteristic
10	C₈×D₄×C₁₀₉	C₄	1	3	yes	—
11	C₈×D₄×C₁₀₉	C₄	1	3	yes	—
12	#431×C₁₀₉	C₂²	1	3	yes	—
13	#424×C₁₀₉	—	2	6	no	—
14	#423×C₁₀₉	C₂²	1	6	yes	—
15	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	6	yes	—
16	C₂×C₄×C₈×C₁₀₉	C₂²	1	3	yes	—
17	#335×C₁₀₉	C₂²	1	1	yes	characteristic
18	#335×C₁₀₉	C₂²	1	3	yes	—
19	#335×C₁₀₉	C₂²	1	6	yes	—
20	#335×C₁₀₉	C₂²	1	6	yes	—
21	C₂×#41×C₁₀₉	D₄	1	1	yes	characteristic
22	#131×C₁₀₉	—	4	8	no	—
23	C₂×M₁₆×C₁₀₉	—	2	6	no	—
24	C₂²×C₈×C₁₀₉	—	2	6	no	—
25	C₂²×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
27	C₄×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₄×Q₈×C₁₀₉	Q₈	1	1	yes	characteristic
29	C₄×D₄×C₁₀₉	C₂×C₄	1	3	yes	—
30	C₄×D₄×C₁₀₉	Q₈	1	3	yes	—
31	#117×C₁₀₉	D₄	1	3	yes	—
32	C₂×C₄²×C₁₀₉	C₂³	1	3	yes	—
33	C₄⋊C₈×C₁₀₉	—	2	6	no	—
34	C₄⋊C₈×C₁₀₉	—	2	6	no	—
35	C₄⋊C₈×C₁₀₉	C₂×C₄	1	3	yes	—
36	C₄⋊C₈×C₁₀₉	C₂×C₄	1	3	yes	—
37	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
38	C₄⋊C₈×C₁₀₉	C₂³	1	6	yes	—
39	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	6	yes	—
40	C₄×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
41	C₄×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
42	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₁₀₉	C₂³	1	3	yes	—
44	#41×C₁₀₉	D₈	1	4	yes	—
45	#41×C₁₀₉	Q₁₆	1	4	yes	—
46	C₂×Q₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂×D₄×C₁₀₉	C₄⋊C₄	1	3	yes	—
48	C₂²×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
49	C₂²×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
50	C₂²×C₄×C₁₀₉	C₂×Q₈	1	3	yes	—
51	M₁₆×C₁₀₉	—	4	12	no	—
52	C₂×C₈×C₁₀₉	—	2	6	no	—
53	C₂×C₈×C₁₀₉	—	2	2	no	—
54	C₂×C₈×C₁₀₉	—	2	6	no	—
55	C₂×C₈×C₁₀₉	—	4	3	no	—
56	C₂×C₈×C₁₀₉	—	4	12	no	—
57	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₀₉	C₂²×C₄	1	3	yes	—
59	C₂×C₈×C₁₀₉	C₂²×C₄	1	3	yes	—
60	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
61	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	3	yes	—
62	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	3	yes	—
63	C₂²⋊C₄×C₁₀₉	C₄⋊C₄	1	6	yes	—
64	C₄²×C₁₀₉	C₂²×C₄	1	3	yes	—
65	C₄²×C₁₀₉	C₂×D₄	1	3	yes	—
66	C₄²×C₁₀₉	C₂×Q₈	1	3	yes	—
67	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
68	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	3	yes	—
69	Q₈×C₁₀₉	#107	1	4	yes	—
70	D₄×C₁₀₉	#107	1	12	yes	—
71	C₂×C₄×C₁₀₉	—	2	3	no	—
72	C₂×C₄×C₁₀₉	—	2	3	no	—
73	C₂×C₄×C₁₀₉	—	4	3	no	—
74	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	3	yes	—
77	C₂×C₄×C₁₀₉	C₂×D₈	1	6	yes	—
78	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	6	yes	—
79	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
82	C₈×C₁₀₉	—	2	3	no	—
83	C₈×C₁₀₉	—	2	1	no	—
84	C₈×C₁₀₉	—	2	3	no	—
85	C₈×C₁₀₉	—	4	2	no	—
86	C₈×C₁₀₉	—	4	6	no	—
87	C₈×C₁₀₉	#131	1	2	yes	complemented
88	C₂²×C₁₀₉	—	2	3	no	—
89	C₂²×C₁₀₉	—	2	3	no	—
90	C₂²×C₁₀₉	C₂×#107	1	6	yes	—
91	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₁₀₉	—	2	3	no	—
93	C₄×C₁₀₉	—	2	1	no	—
94	C₄×C₁₀₉	—	2	3	no	—
95	C₄×C₁₀₉	C₂×#107	1	6	yes	—
96	C₄×C₁₀₉	C₂×#131	1	1	yes	characteristic
97	C₄×C₁₀₉	#568	1	1	yes	characteristic
98	C₄×C₁₀₉	C₂²×D₈	1	1	yes	characteristic
99	C₄×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
100	#17828	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂×C₁₀₉	—	2	6	no	—
102	C₂×C₁₀₉	#2906	1	1	yes	characteristic
103	C₂×C₁₀₉	C₂²×#107	1	1	yes	characteristic
104	C₂×C₁₀₉	#2916	1	1	yes	characteristic
105	C₈×#41	C₂×C₁₀₉	1	1	yes	characteristic
106	#2906	C₂×C₁₀₉	1	2	yes	—
107	#1571	C₂×C₁₀₉	1	6	yes	complemented
108	#1567	C₂×C₁₀₉	1	3	yes	complemented
109	C₂×#335	C₂×C₁₀₉	1	3	yes	complemented
110	C₁₀₉	#17828	1	1	yes	characteristic, direct product factor, Sylow
111	C₂×#131	—	2	2	no	—
112	C₄×#41	C₂²×C₁₀₉	1	1	yes	characteristic
113	C₈×Q₈	C₄×C₁₀₉	1	1	yes	characteristic
114	C₈×Q₈	C₄×C₁₀₉	1	1	yes	characteristic
115	C₈×D₄	C₄×C₁₀₉	1	3	yes	—
116	C₈×D₄	C₄×C₁₀₉	1	3	yes	—
117	#431	C₂²×C₁₀₉	1	3	yes	—
118	#424	—	2	6	no	—
119	#423	C₂²×C₁₀₉	1	6	yes	—
120	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	6	yes	—
121	C₂×C₄×C₈	C₂²×C₁₀₉	1	3	yes	—
122	#335	C₂²×C₁₀₉	1	1	yes	characteristic
123	#335	C₂²×C₁₀₉	1	3	yes	—
124	#335	C₂²×C₁₀₉	1	6	yes	—
125	#335	C₂²×C₁₀₉	1	6	yes	—
126	C₂×#41	D₄×C₁₀₉	1	1	yes	characteristic
127	#131	—	4	8	no	—
128	C₂×M₁₆	—	2	6	no	—
129	C₂²×C₈	—	2	6	no	—
130	C₂²×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
131	C₂²×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
132	C₄×Q₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
133	C₄×Q₈	Q₈×C₁₀₉	1	1	yes	characteristic
134	C₄×D₄	C₂×C₄×C₁₀₉	1	3	yes	—
135	C₄×D₄	Q₈×C₁₀₉	1	3	yes	—
136	#117	D₄×C₁₀₉	1	3	yes	—
137	C₂×C₄²	C₂³×C₁₀₉	1	3	yes	—
138	C₄⋊C₈	—	2	6	no	—
139	C₄⋊C₈	—	2	6	no	—
140	C₄⋊C₈	C₂×C₄×C₁₀₉	1	3	yes	—
141	C₄⋊C₈	C₂×C₄×C₁₀₉	1	3	yes	—
142	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
143	C₄⋊C₈	C₂³×C₁₀₉	1	6	yes	—
144	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	6	yes	—
145	C₄×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
146	C₄×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
147	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
148	C₄×C₈	C₂³×C₁₀₉	1	3	yes	—
149	#41	D₈×C₁₀₉	1	4	yes	—
150	#41	Q₁₆×C₁₀₉	1	4	yes	—
151	C₂×Q₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
152	C₂×D₄	C₄⋊C₄×C₁₀₉	1	3	yes	—
153	C₂²×C₄	C₂²×C₄×C₁₀₉	1	3	yes	—
154	C₂²×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
155	C₂²×C₄	C₂×Q₈×C₁₀₉	1	3	yes	—
156	M₁₆	—	4	12	no	—
157	C₂×C₈	—	2	6	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	6	no	—
160	C₂×C₈	—	4	3	no	—
161	C₂×C₈	—	4	12	no	—
162	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
163	C₂×C₈	C₂²×C₄×C₁₀₉	1	3	yes	—
164	C₂×C₈	C₂²×C₄×C₁₀₉	1	3	yes	—
165	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
166	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	3	yes	—
167	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	3	yes	—
168	C₂²⋊C₄	C₄⋊C₄×C₁₀₉	1	6	yes	—
169	C₄²	C₂²×C₄×C₁₀₉	1	3	yes	—
170	C₄²	C₂×D₄×C₁₀₉	1	3	yes	—
171	C₄²	C₂×Q₈×C₁₀₉	1	3	yes	—
172	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
173	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	3	yes	—
174	Q₈	#107×C₁₀₉	1	4	yes	—
175	D₄	#107×C₁₀₉	1	12	yes	—
176	C₂×C₄	—	2	3	no	—
177	C₂×C₄	—	2	3	no	—
178	C₂×C₄	—	4	3	no	—
179	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	3	yes	—
180	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	3	yes	—
181	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	3	yes	—
182	C₂×C₄	C₂×D₈×C₁₀₉	1	6	yes	—
183	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	6	yes	—
184	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
185	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
186	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	2	1	no	—
189	C₈	—	2	3	no	—
190	C₈	—	4	2	no	—
191	C₈	—	4	6	no	—
192	C₈	#131×C₁₀₉	1	2	yes	complemented
193	C₂²	—	2	3	no	—
194	C₂²	—	2	3	no	—
195	C₂²	C₂×#107×C₁₀₉	1	6	yes	—
196	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	3	no	—
200	C₄	C₂×#107×C₁₀₉	1	6	yes	—
201	C₄	C₂×#131×C₁₀₉	1	1	yes	characteristic
202	C₄	#568×C₁₀₉	1	1	yes	characteristic
203	C₄	C₂²×D₈×C₁₀₉	1	1	yes	characteristic
204	C₄	C₂²×Q₁₆×C₁₀₉	1	1	yes	characteristic
205	C₂	—	2	6	no	—
206	C₂	C₁₀₉×#2906	1	1	yes	characteristic
207	C₂	C₂²×#107×C₁₀₉	1	1	yes	characteristic
208	C₂	C₁₀₉×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₄²	C₂⁴×C₁₀₉
socle	C₂²×C₁₀₉	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₁₀₉	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₄×#41×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1571	C₂	semidirect
4. C₁₀₉×#1567	C₂	semidirect
5. C₂×#335×C₁₀₉	C₂	semidirect
87. C₈×C₁₀₉	#131	semidirect
100. #17828	C₁₀₉	direct (N × H)
107. #1571	C₂×C₁₀₉	semidirect
108. #1567	C₂×C₁₀₉	semidirect
109. C₂×#335	C₂×C₁₀₉	semidirect
110. C₁₀₉	#17828	direct (N × H)
192. C₈	#131×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₈×#41×C₁₀₉	2
C₁₀₉×#2906	2
C₁₀₉×#1571	2
C₁₀₉×#1567	2
C₂×#335×C₁₀₉	2
#17828	109

Subgroups — C109×#17828 #42135822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#17828 #42135822