Subgroups — C₂×#485×C₁₀₉ #28360342

Order 13952.

416 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	C₂×C₂²⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#485×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×#103×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#103×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
14	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
15	#117×C₁₀₉	—	2	2	no	—
16	#117×C₁₀₉	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
22	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
23	C₂×C₄²×C₁₀₉	—	2	1	no	—
24	#107×C₁₀₉	C₂²	1	4	yes	complemented
25	#106×C₁₀₉	C₂²	1	4	yes	complemented
26	#103×C₁₀₉	C₂²	1	4	yes	complemented
27	#103×C₁₀₉	C₂²	1	4	yes	complemented
28	C₂²⋊C₈×C₁₀₉	C₂²	1	4	yes	—
29	C₂⁴×C₁₀₉	D₄	1	1	yes	characteristic
30	C₂×Q₈×C₁₀₉	—	2	1	no	—
31	C₂×Q₈×C₁₀₉	—	2	2	no	—
32	C₂×Q₈×C₁₀₉	—	2	2	no	—
33	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₀₉	—	2	2	no	—
35	C₂²×C₄×C₁₀₉	—	2	2	no	—
36	C₂²×C₄×C₁₀₉	—	2	1	no	—
37	C₂²×C₄×C₁₀₉	—	2	1	no	—
38	C₂²×C₄×C₁₀₉	—	2	1	no	—
39	C₂²×C₄×C₁₀₉	—	2	1	no	—
40	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
41	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₁₀₉	—	2	2	no	—
45	C₂×C₈×C₁₀₉	—	2	2	no	—
46	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
47	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
48	C₄⋊C₄×C₁₀₉	—	2	4	no	—
49	C₄⋊C₄×C₁₀₉	—	2	1	no	—
50	C₄⋊C₄×C₁₀₉	—	2	1	no	—
51	C₄⋊C₄×C₁₀₉	—	2	1	no	—
52	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
53	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
54	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
56	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
57	C₄²×C₁₀₉	—	2	4	no	—
58	C₂³×C₁₀₉	—	2	2	no	—
59	C₂³×C₁₀₉	—	2	2	no	—
60	C₂³×C₁₀₉	—	2	2	no	—
61	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
62	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
63	Q₈×C₁₀₉	—	2	2	no	—
64	Q₈×C₁₀₉	—	2	2	no	—
65	Q₈×C₁₀₉	—	4	2	no	—
66	C₂×C₄×C₁₀₉	—	2	2	no	—
67	C₂×C₄×C₁₀₉	—	2	4	no	—
68	C₂×C₄×C₁₀₉	—	2	2	no	—
69	C₂×C₄×C₁₀₉	—	2	2	no	—
70	C₂×C₄×C₁₀₉	—	2	2	no	—
71	C₂×C₄×C₁₀₉	—	2	2	no	—
72	C₂×C₄×C₁₀₉	—	2	2	no	—
73	C₂×C₄×C₁₀₉	—	2	2	no	—
74	C₂×C₄×C₁₀₉	—	4	2	no	—
75	C₂×C₄×C₁₀₉	—	4	2	no	—
76	C₂×C₄×C₁₀₉	—	4	2	no	—
77	C₂×C₄×C₁₀₉	—	4	2	no	—
78	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₀₉	#41	1	2	yes	—
81	C₂×C₄×C₁₀₉	#41	1	2	yes	—
82	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
83	C₈×C₁₀₉	—	2	2	no	—
84	C₈×C₁₀₉	—	2	2	no	—
85	C₂²×C₁₀₉	—	2	2	no	—
86	C₂²×C₁₀₉	—	2	2	no	—
87	C₂²×C₁₀₉	—	2	2	no	—
88	C₂²×C₁₀₉	—	4	4	no	—
89	C₂²×C₁₀₉	#123	1	2	yes	—
90	C₂²×C₁₀₉	#135	1	2	yes	—
91	C₂²×C₁₀₉	#137	1	2	yes	—
92	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
93	C₄×C₁₀₉	—	2	2	no	—
94	C₄×C₁₀₉	—	4	2	no	—
95	C₄×C₁₀₉	—	4	2	no	—
96	C₄×C₁₀₉	—	4	2	no	—
97	C₄×C₁₀₉	—	4	2	no	—
98	C₄×C₁₀₉	#123	1	2	yes	—
99	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
100	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
101	C₂×C₁₀₉	—	4	2	no	—
102	C₂×C₁₀₉	#485	1	4	yes	direct product factor
103	C₂×C₁₀₉	C₂×#123	1	1	yes	characteristic
104	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
105	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
106	C₂×#485	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₁₀₉	C₂×#485	1	1	yes	characteristic, direct product factor, Sylow
108	C₂×C₂²⋊Q₈	C₂×C₁₀₉	1	1	yes	characteristic
109	C₂×#117	C₂×C₁₀₉	1	1	yes	characteristic
110	#485	C₂×C₁₀₉	1	8	yes	direct product factor
111	C₂×#107	C₂×C₁₀₉	1	1	yes	characteristic, complemented
112	C₂×#106	C₂×C₁₀₉	1	1	yes	characteristic, complemented
113	C₂×#103	C₂×C₁₀₉	1	1	yes	characteristic, complemented
114	C₂×#103	C₂×C₁₀₉	1	1	yes	characteristic, complemented
115	C₂×C₂²⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic
116	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
117	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
118	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
119	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
120	C₂²⋊Q₈	—	2	2	no	—
121	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
122	#117	—	2	2	no	—
123	#117	C₂²×C₁₀₉	1	4	yes	—
124	C₂×C₄⋊C₄	—	2	1	no	—
125	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
127	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
128	C₂×C₂²⋊C₄	—	2	1	no	—
129	C₂×C₂²⋊C₄	—	2	1	no	—
130	C₂×C₄²	—	2	1	no	—
131	#107	C₂²×C₁₀₉	1	4	yes	complemented
132	#106	C₂²×C₁₀₉	1	4	yes	complemented
133	#103	C₂²×C₁₀₉	1	4	yes	complemented
134	#103	C₂²×C₁₀₉	1	4	yes	complemented
135	C₂²⋊C₈	C₂²×C₁₀₉	1	4	yes	—
136	C₂⁴	D₄×C₁₀₉	1	1	yes	characteristic
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	—	2	2	no	—
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
148	C₂²×C₄	D₄×C₁₀₉	1	2	yes	—
149	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
150	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
154	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
160	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
161	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
162	C₂²⋊C₄	—	2	4	no	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₄²	—	2	4	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	2	no	—
168	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
169	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	Q₈	—	4	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
186	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
187	C₂×C₄	#41×C₁₀₉	1	2	yes	—
188	C₂×C₄	#41×C₁₀₉	1	2	yes	—
189	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
190	C₈	—	2	2	no	—
191	C₈	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	4	4	no	—
196	C₂²	#123×C₁₀₉	1	2	yes	—
197	C₂²	#135×C₁₀₉	1	2	yes	—
198	C₂²	#137×C₁₀₉	1	2	yes	—
199	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	2	no	—
205	C₄	#123×C₁₀₉	1	2	yes	—
206	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
207	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
208	C₂	—	4	2	no	—
209	C₂	#485×C₁₀₉	1	4	yes	direct product factor
210	C₂	C₂×#123×C₁₀₉	1	1	yes	characteristic
211	C₂	C₂×#135×C₁₀₉	1	1	yes	characteristic
212	C₂	C₂×#137×C₁₀₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. #485×C₁₀₉	C₂	direct (N × H)
4. C₂×#107×C₁₀₉	C₂	semidirect
5. C₂×#106×C₁₀₉	C₂	semidirect
6. C₂×#103×C₁₀₉	C₂	semidirect
7. C₂×#103×C₁₀₉	C₂	semidirect
24. #107×C₁₀₉	C₂²	semidirect
25. #106×C₁₀₉	C₂²	semidirect
26. #103×C₁₀₉	C₂²	semidirect
27. #103×C₁₀₉	C₂²	semidirect
102. C₂×C₁₀₉	#485	direct (N × H)
106. C₂×#485	C₁₀₉	direct (N × H)
107. C₁₀₉	C₂×#485	direct (N × H)
110. #485	C₂×C₁₀₉	direct (N × H)
111. C₂×#107	C₂×C₁₀₉	semidirect
112. C₂×#106	C₂×C₁₀₉	semidirect
113. C₂×#103	C₂×C₁₀₉	semidirect
114. C₂×#103	C₂×C₁₀₉	semidirect
131. #107	C₂²×C₁₀₉	semidirect
132. #106	C₂²×C₁₀₉	semidirect
133. #103	C₂²×C₁₀₉	semidirect
134. #103	C₂²×C₁₀₉	semidirect
209. C₂	#485×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×C₂²⋊Q₈×C₁₀₉	2
C₂×#117×C₁₀₉	2
#485×C₁₀₉	2
C₂×#107×C₁₀₉	2
C₂×#106×C₁₀₉	2
C₂×#103×C₁₀₉	2
C₂×C₂²⋊C₈×C₁₀₉	2
C₂×#485	109

Subgroups — C2×#485×C109 #28360342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#485×C₁₀₉ #28360342