Subgroups — C₂×#532×C₁₀₉ #28360696

Order 13952.

1728 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	#532×C₁₀₉	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#127×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#124×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	C₂×#121×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×D₄×C₁₀₉	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄²×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₁₀₉	—	2	16	no	—
10	C₂×#41×C₁₀₉	C₂²	1	16	yes	complemented
11	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
12	C₂²×D₄×C₁₀₉	C₂²	1	4	yes	—
13	C₂²×D₄×C₁₀₉	C₂²	1	4	yes	—
14	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
16	C₂³×C₄×C₁₀₉	C₂²	1	4	yes	—
17	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₀₉	C₂²	1	8	yes	complemented
19	#127×C₁₀₉	C₂²	1	8	yes	complemented
20	#124×C₁₀₉	C₂²	1	16	yes	complemented
21	#121×C₁₀₉	C₂²	1	32	yes	complemented
22	C₄×D₄×C₁₀₉	C₂²	1	32	yes	complemented
23	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	8	yes	—
25	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₀₉	C₂²	1	8	yes	complemented
29	C₂⁴×C₁₀₉	C₂³	1	4	yes	—
30	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
31	#41×C₁₀₉	—	2	64	no	—
32	C₂×Q₈×C₁₀₉	—	2	8	no	—
33	C₂×Q₈×C₁₀₉	C₂³	1	8	yes	—
34	C₂×D₄×C₁₀₉	—	2	16	no	—
35	C₂×D₄×C₁₀₉	—	2	8	no	—
36	C₂×D₄×C₁₀₉	—	2	16	no	—
37	C₂×D₄×C₁₀₉	C₂³	1	16	yes	—
38	C₂×D₄×C₁₀₉	C₂³	1	16	yes	—
39	C₂×D₄×C₁₀₉	C₂³	1	8	yes	—
40	C₂²×C₄×C₁₀₉	—	2	16	no	—
41	C₂²×C₄×C₁₀₉	—	2	8	no	—
42	C₂²×C₄×C₁₀₉	D₄	1	8	yes	complemented
43	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₀₉	C₂³	1	8	yes	—
45	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
46	C₂²×C₄×C₁₀₉	C₂³	1	16	yes	—
47	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
49	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₁₀₉	C₂³	1	16	yes	—
52	C₂²⋊C₄×C₁₀₉	C₂³	1	32	yes	—
53	C₄²×C₁₀₉	C₂³	1	8	yes	—
54	C₄²×C₁₀₉	C₂³	1	4	yes	—
55	C₄²×C₁₀₉	C₂³	1	4	yes	—
56	C₂³×C₁₀₉	—	2	4	no	—
57	C₂³×C₁₀₉	—	2	8	no	—
58	C₂³×C₁₀₉	—	2	16	no	—
59	C₂³×C₁₀₉	C₂×D₄	1	4	yes	—
60	C₂³×C₁₀₉	C₂⁴	1	8	yes	—
61	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
62	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
63	Q₈×C₁₀₉	—	2	16	no	—
64	D₄×C₁₀₉	—	2	32	no	—
65	D₄×C₁₀₉	—	2	16	no	—
66	D₄×C₁₀₉	—	2	32	no	—
67	C₂×C₄×C₁₀₉	—	2	8	no	—
68	C₂×C₄×C₁₀₉	—	2	4	no	—
69	C₂×C₄×C₁₀₉	—	2	32	no	—
70	C₂×C₄×C₁₀₉	—	2	16	no	—
71	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
72	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
73	C₂×C₄×C₁₀₉	C₂×D₄	1	16	yes	complemented
74	C₂×C₄×C₁₀₉	#41	1	8	yes	complemented
75	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
76	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
77	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
78	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
79	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
80	C₂²×C₁₀₉	—	2	8	no	—
81	C₂²×C₁₀₉	—	2	4	no	—
82	C₂²×C₁₀₉	—	2	8	no	—
83	C₂²×C₁₀₉	—	2	16	no	—
84	C₂²×C₁₀₉	—	4	16	no	—
85	C₂²×C₁₀₉	C₂²×D₄	1	4	yes	—
86	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
87	C₂²×C₁₀₉	C₂×#41	1	4	yes	—
88	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
89	C₄×C₁₀₉	—	2	8	no	—
90	C₄×C₁₀₉	—	2	4	no	—
91	C₄×C₁₀₉	C₂²×D₄	1	4	yes	—
92	C₄×C₁₀₉	C₂²×D₄	1	4	yes	—
93	C₄×C₁₀₉	C₂×#41	1	8	yes	complemented
94	C₂×C₁₀₉	—	2	4	no	—
95	C₂×C₁₀₉	—	4	8	no	—
96	C₂×C₁₀₉	#532	1	4	yes	direct product factor
97	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
98	C₂×C₁₀₉	C₂²×#41	1	2	yes	—
99	C₂×#532	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₁₀₉	C₂×#532	1	1	yes	characteristic, direct product factor, Sylow
101	C₂²×#41	C₂×C₁₀₉	1	2	yes	complemented
102	#532	C₂×C₁₀₉	1	16	yes	direct product factor
103	C₂×C₄⋊Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
104	C₂×#127	C₂×C₁₀₉	1	1	yes	characteristic, complemented
105	C₂×#124	C₂×C₁₀₉	1	2	yes	complemented
106	C₂×#121	C₂×C₁₀₉	1	4	yes	complemented
107	C₂×C₄×D₄	C₂×C₁₀₉	1	4	yes	complemented
108	C₂²×C₄²	C₂×C₁₀₉	1	1	yes	characteristic, complemented
109	C₂×#41	—	2	16	no	—
110	C₂×#41	C₂²×C₁₀₉	1	16	yes	complemented
111	C₂²×Q₈	C₂²×C₁₀₉	1	2	yes	—
112	C₂²×D₄	C₂²×C₁₀₉	1	4	yes	—
113	C₂²×D₄	C₂²×C₁₀₉	1	4	yes	—
114	C₂²×D₄	C₂²×C₁₀₉	1	2	yes	—
115	C₂³×C₄	C₂²×C₁₀₉	1	4	yes	—
116	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
117	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
118	C₄⋊Q₈	C₂²×C₁₀₉	1	8	yes	complemented
119	#127	C₂²×C₁₀₉	1	8	yes	complemented
120	#124	C₂²×C₁₀₉	1	16	yes	complemented
121	#121	C₂²×C₁₀₉	1	32	yes	complemented
122	C₄×D₄	C₂²×C₁₀₉	1	32	yes	complemented
123	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
124	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	8	yes	—
125	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
126	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
127	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
128	C₂×C₄²	C₂²×C₁₀₉	1	8	yes	complemented
129	C₂⁴	C₂³×C₁₀₉	1	4	yes	—
130	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
131	#41	—	2	64	no	—
132	C₂×Q₈	—	2	8	no	—
133	C₂×Q₈	C₂³×C₁₀₉	1	8	yes	—
134	C₂×D₄	—	2	16	no	—
135	C₂×D₄	—	2	8	no	—
136	C₂×D₄	—	2	16	no	—
137	C₂×D₄	C₂³×C₁₀₉	1	16	yes	—
138	C₂×D₄	C₂³×C₁₀₉	1	16	yes	—
139	C₂×D₄	C₂³×C₁₀₉	1	8	yes	—
140	C₂²×C₄	—	2	16	no	—
141	C₂²×C₄	—	2	8	no	—
142	C₂²×C₄	D₄×C₁₀₉	1	8	yes	complemented
143	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
144	C₂²×C₄	C₂³×C₁₀₉	1	8	yes	—
145	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
146	C₂²×C₄	C₂³×C₁₀₉	1	16	yes	—
147	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
150	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
151	C₄⋊C₄	C₂³×C₁₀₉	1	16	yes	—
152	C₂²⋊C₄	C₂³×C₁₀₉	1	32	yes	—
153	C₄²	C₂³×C₁₀₉	1	8	yes	—
154	C₄²	C₂³×C₁₀₉	1	4	yes	—
155	C₄²	C₂³×C₁₀₉	1	4	yes	—
156	C₂³	—	2	4	no	—
157	C₂³	—	2	8	no	—
158	C₂³	—	2	16	no	—
159	C₂³	C₂×D₄×C₁₀₉	1	4	yes	—
160	C₂³	C₂⁴×C₁₀₉	1	8	yes	—
161	C₂³	C₂⁴×C₁₀₉	1	2	yes	—
162	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
163	Q₈	—	2	16	no	—
164	D₄	—	2	32	no	—
165	D₄	—	2	16	no	—
166	D₄	—	2	32	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	32	no	—
170	C₂×C₄	—	2	16	no	—
171	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
172	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
173	C₂×C₄	C₂×D₄×C₁₀₉	1	16	yes	complemented
174	C₂×C₄	#41×C₁₀₉	1	8	yes	complemented
175	C₂×C₄	C₂⁴×C₁₀₉	1	4	yes	—
176	C₂×C₄	C₂⁴×C₁₀₉	1	8	yes	—
177	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
178	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
179	C₂×C₄	C₂⁴×C₁₀₉	1	4	yes	—
180	C₂²	—	2	8	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	2	8	no	—
183	C₂²	—	2	16	no	—
184	C₂²	—	4	16	no	—
185	C₂²	C₂²×D₄×C₁₀₉	1	4	yes	—
186	C₂²	C₂²×D₄×C₁₀₉	1	2	yes	—
187	C₂²	C₂×#41×C₁₀₉	1	4	yes	—
188	C₂²	C₂⁵×C₁₀₉	1	1	yes	characteristic
189	C₄	—	2	8	no	—
190	C₄	—	2	4	no	—
191	C₄	C₂²×D₄×C₁₀₉	1	4	yes	—
192	C₄	C₂²×D₄×C₁₀₉	1	4	yes	—
193	C₄	C₂×#41×C₁₀₉	1	8	yes	complemented
194	C₂	—	2	4	no	—
195	C₂	—	4	8	no	—
196	C₂	#532×C₁₀₉	1	4	yes	direct product factor
197	C₂	C₂³×D₄×C₁₀₉	1	1	yes	characteristic
198	C₂	C₂²×#41×C₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
socle	C₂³×C₁₀₉	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₁₀₉	C₂	semidirect
2. #532×C₁₀₉	C₂	direct (N × H)
3. C₂×C₄⋊Q₈×C₁₀₉	C₂	semidirect
4. C₂×#127×C₁₀₉	C₂	semidirect
5. C₂×#124×C₁₀₉	C₂	semidirect
6. C₂×#121×C₁₀₉	C₂	semidirect
7. C₂×C₄×D₄×C₁₀₉	C₂	semidirect
8. C₂²×C₄²×C₁₀₉	C₂	semidirect
10. C₂×#41×C₁₀₉	C₂²	semidirect
18. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
19. #127×C₁₀₉	C₂²	semidirect
20. #124×C₁₀₉	C₂²	semidirect
21. #121×C₁₀₉	C₂²	semidirect
22. C₄×D₄×C₁₀₉	C₂²	semidirect
28. C₂×C₄²×C₁₀₉	C₂²	semidirect
42. C₂²×C₄×C₁₀₉	D₄	semidirect
73. C₂×C₄×C₁₀₉	C₂×D₄	semidirect
74. C₂×C₄×C₁₀₉	#41	semidirect
93. C₄×C₁₀₉	C₂×#41	semidirect
96. C₂×C₁₀₉	#532	direct (N × H)
99. C₂×#532	C₁₀₉	direct (N × H)
100. C₁₀₉	C₂×#532	direct (N × H)
101. C₂²×#41	C₂×C₁₀₉	semidirect
102. #532	C₂×C₁₀₉	direct (N × H)
103. C₂×C₄⋊Q₈	C₂×C₁₀₉	semidirect
104. C₂×#127	C₂×C₁₀₉	semidirect
105. C₂×#124	C₂×C₁₀₉	semidirect
106. C₂×#121	C₂×C₁₀₉	semidirect
107. C₂×C₄×D₄	C₂×C₁₀₉	semidirect
108. C₂²×C₄²	C₂×C₁₀₉	semidirect
110. C₂×#41	C₂²×C₁₀₉	semidirect
118. C₄⋊Q₈	C₂²×C₁₀₉	semidirect
119. #127	C₂²×C₁₀₉	semidirect
120. #124	C₂²×C₁₀₉	semidirect
121. #121	C₂²×C₁₀₉	semidirect
122. C₄×D₄	C₂²×C₁₀₉	semidirect
128. C₂×C₄²	C₂²×C₁₀₉	semidirect
142. C₂²×C₄	D₄×C₁₀₉	semidirect
173. C₂×C₄	C₂×D₄×C₁₀₉	semidirect
174. C₂×C₄	#41×C₁₀₉	semidirect
193. C₄	C₂×#41×C₁₀₉	semidirect
196. C₂	#532×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
C₂²×#41×C₁₀₉	2
#532×C₁₀₉	2
C₂×C₄⋊Q₈×C₁₀₉	2
C₂×#127×C₁₀₉	2
C₂×#124×C₁₀₉	2
C₂×#121×C₁₀₉	2
C₂×C₄×D₄×C₁₀₉	2
C₂²×C₄²×C₁₀₉	2
C₂×#532	109

Subgroups — C2×#532×C109 #28360696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#532×C₁₀₉ #28360696