Subgroups — C₂×#478×C₁₀₉ #28360326

Order 13952.

448 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#125×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#478×C₁₀₉	C₂	1	8	yes	maximal, direct product factor
4	C₂×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#106×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#102×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#102×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₀₉	—	2	1	no	—
11	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	#125×C₁₀₉	—	2	2	no	—
14	#125×C₁₀₉	C₂²	1	4	yes	complemented
15	C₄×D₄×C₁₀₉	—	2	2	no	—
16	C₄×D₄×C₁₀₉	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
18	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
19	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
23	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
24	#107×C₁₀₉	C₂²	1	4	yes	complemented
25	#106×C₁₀₉	C₂²	1	4	yes	complemented
26	C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	complemented
27	#102×C₁₀₉	C₂²	1	4	yes	—
28	#102×C₁₀₉	C₂²	1	4	yes	—
29	C₂⁴×C₁₀₉	—	2	1	no	—
30	C₂×D₄×C₁₀₉	—	2	1	no	—
31	C₂×D₄×C₁₀₉	—	2	2	no	—
32	C₂×D₄×C₁₀₉	Q₈	1	4	yes	—
33	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₀₉	—	2	1	no	—
35	C₂²×C₄×C₁₀₉	—	2	1	no	—
36	C₂²×C₄×C₁₀₉	—	2	1	no	—
37	C₂²×C₄×C₁₀₉	—	2	4	no	—
38	C₂²×C₄×C₁₀₉	—	4	4	no	—
39	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₁₀₉	—	2	2	no	—
43	C₂×C₈×C₁₀₉	—	2	2	no	—
44	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
45	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
46	C₄⋊C₄×C₁₀₉	—	2	4	no	—
47	C₄⋊C₄×C₁₀₉	—	2	4	no	—
48	C₄⋊C₄×C₁₀₉	—	2	1	no	—
49	C₄⋊C₄×C₁₀₉	—	2	1	no	—
50	C₄⋊C₄×C₁₀₉	—	2	1	no	—
51	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
52	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
53	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
54	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
55	C₄²×C₁₀₉	D₄	1	2	yes	—
56	C₄²×C₁₀₉	C₂³	1	2	yes	—
57	C₂³×C₁₀₉	—	2	4	no	—
58	C₂³×C₁₀₉	—	2	2	no	—
59	C₂³×C₁₀₉	—	4	2	no	—
60	C₂³×C₁₀₉	—	4	2	no	—
61	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
62	D₄×C₁₀₉	—	2	4	no	—
63	D₄×C₁₀₉	—	2	2	no	—
64	D₄×C₁₀₉	C₂×Q₈	1	4	yes	—
65	C₂×C₄×C₁₀₉	—	2	2	no	—
66	C₂×C₄×C₁₀₉	—	2	2	no	—
67	C₂×C₄×C₁₀₉	—	2	2	no	—
68	C₂×C₄×C₁₀₉	—	2	2	no	—
69	C₂×C₄×C₁₀₉	—	2	2	no	—
70	C₂×C₄×C₁₀₉	—	4	8	no	—
71	C₂×C₄×C₁₀₉	—	4	2	no	—
72	C₂×C₄×C₁₀₉	—	4	2	no	—
73	C₂×C₄×C₁₀₉	—	4	2	no	—
74	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
78	C₂×C₄×C₁₀₉	#41	1	2	yes	—
79	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
80	C₈×C₁₀₉	—	2	2	no	—
81	C₈×C₁₀₉	—	2	2	no	—
82	C₂²×C₁₀₉	—	2	4	no	—
83	C₂²×C₁₀₉	—	4	2	no	—
84	C₂²×C₁₀₉	—	4	2	no	—
85	C₂²×C₁₀₉	—	4	8	no	—
86	C₂²×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
87	C₂²×C₁₀₉	#135	1	2	yes	—
88	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
89	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
90	C₄×C₁₀₉	—	2	2	no	—
91	C₄×C₁₀₉	—	2	2	no	—
92	C₄×C₁₀₉	—	4	2	no	—
93	C₄×C₁₀₉	—	4	2	no	—
94	C₄×C₁₀₉	—	4	2	no	—
95	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
96	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
97	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
98	C₂×C₁₀₉	—	4	4	no	—
99	C₂×C₁₀₉	#478	1	4	yes	direct product factor
100	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
102	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×#478	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₁₀₉	C₂×#478	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×#125	C₂×C₁₀₉	1	1	yes	characteristic, complemented
106	C₂×C₄×D₄	C₂×C₁₀₉	1	1	yes	characteristic
107	#478	C₂×C₁₀₉	1	8	yes	direct product factor
108	C₂×#107	C₂×C₁₀₉	1	1	yes	characteristic, complemented
109	C₂×#106	C₂×C₁₀₉	1	1	yes	characteristic, complemented
110	C₂×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
111	C₂×#102	C₂×C₁₀₉	1	1	yes	characteristic
112	C₂×#102	C₂×C₁₀₉	1	1	yes	characteristic
113	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
114	C₂³×C₄	—	2	1	no	—
115	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
116	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
117	#125	—	2	2	no	—
118	#125	C₂²×C₁₀₉	1	4	yes	complemented
119	C₄×D₄	—	2	2	no	—
120	C₄×D₄	C₂²×C₁₀₉	1	4	yes	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	—	2	1	no	—
123	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
126	C₂×C₂²⋊C₄	—	2	1	no	—
127	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
128	#107	C₂²×C₁₀₉	1	4	yes	complemented
129	#106	C₂²×C₁₀₉	1	4	yes	complemented
130	C₄⋊C₈	C₂²×C₁₀₉	1	4	yes	complemented
131	#102	C₂²×C₁₀₉	1	4	yes	—
132	#102	C₂²×C₁₀₉	1	4	yes	—
133	C₂⁴	—	2	1	no	—
134	C₂×D₄	—	2	1	no	—
135	C₂×D₄	—	2	2	no	—
136	C₂×D₄	Q₈×C₁₀₉	1	4	yes	—
137	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
144	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
145	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
149	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	2	1	no	—
155	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
156	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
157	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₄²	D₄×C₁₀₉	1	2	yes	—
160	C₄²	C₂³×C₁₀₉	1	2	yes	—
161	C₂³	—	2	4	no	—
162	C₂³	—	2	2	no	—
163	C₂³	—	4	2	no	—
164	C₂³	—	4	2	no	—
165	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
166	D₄	—	2	4	no	—
167	D₄	—	2	2	no	—
168	D₄	C₂×Q₈×C₁₀₉	1	4	yes	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	8	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
179	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
180	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
181	C₂×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
182	C₂×C₄	#41×C₁₀₉	1	2	yes	—
183	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	2	2	no	—
186	C₂²	—	2	4	no	—
187	C₂²	—	4	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	8	no	—
190	C₂²	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
191	C₂²	#135×C₁₀₉	1	2	yes	—
192	C₂²	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
193	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	2	no	—
199	C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
200	C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
201	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
202	C₂	—	4	4	no	—
203	C₂	#478×C₁₀₉	1	4	yes	direct product factor
204	C₂	C₂×C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
205	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
206	C₂	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×#125×C₁₀₉	C₂	semidirect
3. #478×C₁₀₉	C₂	direct (N × H)
4. C₂×#107×C₁₀₉	C₂	semidirect
5. C₂×#106×C₁₀₉	C₂	semidirect
6. C₂×C₄⋊C₈×C₁₀₉	C₂	semidirect
14. #125×C₁₀₉	C₂²	semidirect
24. #107×C₁₀₉	C₂²	semidirect
25. #106×C₁₀₉	C₂²	semidirect
26. C₄⋊C₈×C₁₀₉	C₂²	semidirect
99. C₂×C₁₀₉	#478	direct (N × H)
103. C₂×#478	C₁₀₉	direct (N × H)
104. C₁₀₉	C₂×#478	direct (N × H)
105. C₂×#125	C₂×C₁₀₉	semidirect
107. #478	C₂×C₁₀₉	direct (N × H)
108. C₂×#107	C₂×C₁₀₉	semidirect
109. C₂×#106	C₂×C₁₀₉	semidirect
110. C₂×C₄⋊C₈	C₂×C₁₀₉	semidirect
118. #125	C₂²×C₁₀₉	semidirect
128. #107	C₂²×C₁₀₉	semidirect
129. #106	C₂²×C₁₀₉	semidirect
130. C₄⋊C₈	C₂²×C₁₀₉	semidirect
203. C₂	#478×C₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#125×C₁₀₉	2
C₂×C₄×D₄×C₁₀₉	2
#478×C₁₀₉	2
C₂×#107×C₁₀₉	2
C₂×#106×C₁₀₉	2
C₂×C₄⋊C₈×C₁₀₉	2
C₂×#102×C₁₀₉	2
C₂×#478	109

Subgroups — C2×#478×C109 #28360326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#478×C₁₀₉ #28360326