Subgroups — C₂²×#16812659 #28376200

Order 13952.

2528 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16812659	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#13182008	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13182006	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×C₂²⋊D₁₀₉	C₄	1	16	yes	complemented
11	C₂²×C₂²⋊D₁₀₉	C₂²	1	12	yes	—
12	C₂³×Dic₁₀₉	—	2	8	no	—
13	C₂³×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₂³×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	C₂³×Dic₁₀₉	C₂²	1	1	yes	characteristic
16	C₂³×Dic₁₀₉	C₂²	1	12	yes	complemented
17	C₂²×C₄×D₁₀₉	—	2	8	no	—
18	C₂²×C₄×D₁₀₉	C₂²	1	12	yes	complemented
19	#16812659	C₂²	1	64	yes	direct product factor
20	C₂×#13182008	C₂²	1	12	yes	complemented
21	C₂×#13182006	C₂²	1	12	yes	complemented
22	C₂×C₄×Dic₁₀₉	C₂²	1	12	yes	complemented
23	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	12	yes	complemented
27	C₂³×D₁₀₉	—	2	4	no	—
28	C₂³×D₁₀₉	—	2	4	no	—
29	C₂³×D₁₀₉	C₂×C₄	1	8	yes	complemented
30	C₂³×D₁₀₉	C₂³	1	6	yes	—
31	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	96	yes	complemented
32	C₂×C₂²⋊D₁₀₉	C₂³	1	16	yes	—
33	C₂²×Dic₁₀₉	—	2	4	no	—
34	C₂²×Dic₁₀₉	—	2	48	no	—
35	C₂²×Dic₁₀₉	C₂×C₄	1	8	yes	complemented
36	C₂²×Dic₁₀₉	D₄	1	8	yes	complemented
37	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
38	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
39	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
40	C₂²×Dic₁₀₉	C₂³	1	16	yes	complemented
41	C₂×C₄×D₁₀₉	—	2	48	no	—
42	C₂×C₄×D₁₀₉	C₂³	1	16	yes	complemented
43	#13182008	C₂³	1	16	yes	complemented
44	#13182006	C₂³	1	16	yes	complemented
45	C₄×Dic₁₀₉	C₂³	1	16	yes	complemented
46	C₂⁴×C₁₀₉	—	2	4	no	—
47	C₂⁴×C₁₀₉	—	2	4	no	—
48	C₂⁴×C₁₀₉	C₂×C₄	1	8	yes	complemented
49	C₂⁴×C₁₀₉	C₂³	1	6	yes	—
50	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₉	—	2	4	no	—
52	C₂²×C₄×C₁₀₉	—	2	4	no	—
53	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
54	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
55	C₂²⋊C₄×C₁₀₉	C₂³	1	16	yes	complemented
56	C₂³×C₁₀₉	—	2	32	no	—
57	C₂³×C₁₀₉	—	2	12	no	—
58	C₂³×C₁₀₉	—	2	12	no	—
59	C₂³×C₁₀₉	C₂²×C₄	1	4	yes	—
60	C₂³×C₁₀₉	C₂²×C₄	1	24	yes	complemented
61	C₂³×C₁₀₉	C₂×D₄	1	4	yes	—
62	C₂³×C₁₀₉	#41	1	4	yes	—
63	C₂³×C₁₀₉	C₂⁴	1	4	yes	—
64	C₂³×C₁₀₉	C₂⁴	1	3	yes	—
65	C₂²×D₁₀₉	—	2	32	no	—
66	C₂²×D₁₀₉	—	2	12	no	—
67	C₂²×D₁₀₉	—	2	12	no	—
68	C₂²×D₁₀₉	C₂²×C₄	1	24	yes	complemented
69	C₂²×D₁₀₉	C₂⁴	1	4	yes	—
70	C₂×C₄×C₁₀₉	—	2	12	no	—
71	C₂×C₄×C₁₀₉	—	2	12	no	—
72	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
73	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
74	C₂²⋊D₁₀₉	C₂²×C₄	1	64	yes	complemented
75	C₂×Dic₁₀₉	—	2	12	no	—
76	C₂×Dic₁₀₉	—	2	32	no	—
77	C₂×Dic₁₀₉	C₂²×C₄	1	24	yes	complemented
78	C₂×Dic₁₀₉	C₂×D₄	1	24	yes	complemented
79	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
80	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
81	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
82	C₄×D₁₀₉	—	2	32	no	—
83	C₂²×C₁₀₉	—	2	48	no	—
84	C₂²×C₁₀₉	—	2	4	no	—
85	C₂²×C₁₀₉	—	2	4	no	—
86	C₂²×C₁₀₉	C₄×D₄	1	16	yes	complemented
87	C₂²×C₁₀₉	C₂³×C₄	1	6	yes	—
88	C₂²×C₁₀₉	C₂³×C₄	1	8	yes	complemented
89	C₂²×C₁₀₉	C₂²×D₄	1	6	yes	—
90	C₂²×C₁₀₉	C₂×#41	1	6	yes	—
91	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
92	C₂×D₁₀₉	—	2	48	no	—
93	C₂×D₁₀₉	—	2	4	no	—
94	C₂×D₁₀₉	—	2	4	no	—
95	C₂×D₁₀₉	C₂³×C₄	1	8	yes	complemented
96	C₄×C₁₀₉	—	2	4	no	—
97	C₄×C₁₀₉	—	2	4	no	—
98	Dic₁₀₉	—	2	4	no	—
99	Dic₁₀₉	C₂³×C₄	1	8	yes	complemented
100	Dic₁₀₉	C₂²×D₄	1	8	yes	complemented
101	C₂×C₁₀₉	—	2	8	no	—
102	C₂×C₁₀₉	C₂×C₄×D₄	1	12	yes	complemented
103	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
104	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
105	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
106	D₁₀₉	—	2	8	no	—
107	C₂²×C₄×D₄	—	109	1	no	maximal, Sylow
108	C₁₀₉	C₂²×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
109	C₂³×D₄	—	109	1	no	—
110	C₂⁴×C₄	—	109	1	no	—
111	C₂⁴×C₄	—	109	1	no	—
112	C₂×C₄×D₄	—	109	24	no	—
113	C₂²×C₄⋊C₄	—	109	1	no	—
114	C₂²×C₂²⋊C₄	—	109	1	no	—
115	C₂²×C₂²⋊C₄	D₁₀₉	1	1	yes	characteristic, complemented
116	C₂²×C₄²	—	109	1	no	—
117	C₂⁵	—	109	1	no	—
118	C₂⁵	C₂×D₁₀₉	1	1	yes	characteristic
119	C₂²×D₄	—	109	16	no	—
120	C₂²×D₄	—	109	12	no	—
121	C₂³×C₄	—	109	1	no	—
122	C₂³×C₄	—	109	1	no	—
123	C₂³×C₄	—	109	12	no	—
124	C₂³×C₄	—	109	1	no	—
125	C₂³×C₄	—	109	12	no	—
126	C₂³×C₄	—	218	8	no	—
127	C₂³×C₄	—	218	8	no	—
128	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
129	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
130	C₄×D₄	—	109	64	no	—
131	C₂×C₄⋊C₄	—	109	12	no	—
132	C₂×C₂²⋊C₄	—	109	12	no	—
133	C₂×C₂²⋊C₄	C₂×D₁₀₉	1	12	yes	complemented
134	C₂×C₄²	—	109	12	no	—
135	C₂⁴	—	2	4	no	—
136	C₂⁴	—	2	4	no	—
137	C₂⁴	—	109	8	no	—
138	C₂⁴	—	109	6	no	—
139	C₂⁴	—	218	4	no	—
140	C₂⁴	—	218	4	no	—
141	C₂⁴	C₄×D₁₀₉	1	8	yes	complemented
142	C₂⁴	C₂²×D₁₀₉	1	6	yes	—
143	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
144	C₂×D₄	—	109	96	no	—
145	C₂×D₄	—	109	16	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	109	8	no	—
149	C₂²×C₄	—	109	8	no	—
150	C₂²×C₄	—	109	6	no	—
151	C₂²×C₄	—	109	6	no	—
152	C₂²×C₄	—	109	6	no	—
153	C₂²×C₄	—	109	16	no	—
154	C₂²×C₄	—	109	16	no	—
155	C₂²×C₄	—	218	48	no	—
156	C₂²×C₄	—	218	48	no	—
157	C₂²×C₄	—	218	4	no	—
158	C₂²×C₄	C₂²×D₁₀₉	1	6	yes	—
159	C₂²×C₄	C₂²×D₁₀₉	1	6	yes	—
160	C₄⋊C₄	—	109	16	no	—
161	C₂²⋊C₄	—	109	16	no	—
162	C₂²⋊C₄	C₂²×D₁₀₉	1	16	yes	complemented
163	C₄²	—	109	16	no	—
164	C₂³	—	2	32	no	—
165	C₂³	—	2	12	no	—
166	C₂³	—	2	12	no	—
167	C₂³	—	109	24	no	—
168	C₂³	—	109	4	no	—
169	C₂³	—	218	32	no	—
170	C₂³	—	218	12	no	—
171	C₂³	—	218	12	no	—
172	C₂³	C₂×C₄×D₁₀₉	1	4	yes	—
173	C₂³	C₂×C₄×D₁₀₉	1	24	yes	complemented
174	C₂³	D₄×D₁₀₉	1	4	yes	—
175	C₂³	#13182028	1	4	yes	—
176	C₂³	C₂³×D₁₀₉	1	4	yes	—
177	C₂³	C₂³×D₁₀₉	1	3	yes	—
178	D₄	—	109	64	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	2	12	no	—
181	C₂×C₄	—	109	24	no	—
182	C₂×C₄	—	109	24	no	—
183	C₂×C₄	—	109	4	no	—
184	C₂×C₄	—	109	4	no	—
185	C₂×C₄	—	109	4	no	—
186	C₂×C₄	—	218	32	no	—
187	C₂×C₄	—	218	32	no	—
188	C₂×C₄	—	218	12	no	—
189	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
190	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
191	C₂²	—	2	48	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	109	8	no	—
195	C₂²	—	218	48	no	—
196	C₂²	—	218	4	no	—
197	C₂²	—	218	4	no	—
198	C₂²	#16812659	1	16	yes	direct product factor
199	C₂²	C₂²×C₄×D₁₀₉	1	6	yes	—
200	C₂²	C₂²×C₄×D₁₀₉	1	8	yes	complemented
201	C₂²	C₂×D₄×D₁₀₉	1	6	yes	—
202	C₂²	C₂×#13182028	1	6	yes	—
203	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
204	C₄	—	2	4	no	—
205	C₄	—	2	4	no	—
206	C₄	—	109	8	no	—
207	C₄	—	109	8	no	—
208	C₄	—	218	4	no	—
209	C₂	—	2	8	no	—
210	C₂	—	218	8	no	—
211	C₂	C₂×#16812659	1	12	yes	direct product factor
212	C₂	C₂³×C₄×D₁₀₉	1	1	yes	characteristic
213	C₂	C₂²×D₄×D₁₀₉	1	1	yes	characteristic
214	C₂	C₂²×#13182028	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂²×C₂²⋊C₄×C₁₀₉	C₂
socle	C₂⁴×C₁₀₉	C₂³
prime radical	C₂³×C₂²⋊D₁₀₉	C₂
cosocle	C₂²×C₁₀₉	C₂⁵
hypercenter	C₂²×C₂²⋊C₄	D₁₀₉
hypocenter	C₁₀₉	C₂²×C₄×D₄
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×Dic₁₀₉	C₂	semidirect
3. C₂³×C₄×D₁₀₉	C₂	semidirect
4. C₂×#16812659	C₂	direct (N × H)
5. C₂²×#13182008	C₂	semidirect
6. C₂²×#13182006	C₂	semidirect
7. C₂²×C₄×Dic₁₀₉	C₂	semidirect
8. C₂²×C₂²⋊C₄×C₁₀₉	C₂	semidirect
10. C₂²×C₂²⋊D₁₀₉	C₄	semidirect
16. C₂³×Dic₁₀₉	C₂²	semidirect
18. C₂²×C₄×D₁₀₉	C₂²	semidirect
19. #16812659	C₂²	direct (N × H)
20. C₂×#13182008	C₂²	semidirect
21. C₂×#13182006	C₂²	semidirect
22. C₂×C₄×Dic₁₀₉	C₂²	semidirect
26. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
29. C₂³×D₁₀₉	C₂×C₄	semidirect
31. C₂×C₂²⋊D₁₀₉	C₂×C₄	semidirect
35. C₂²×Dic₁₀₉	C₂×C₄	semidirect
36. C₂²×Dic₁₀₉	D₄	semidirect
40. C₂²×Dic₁₀₉	C₂³	semidirect
42. C₂×C₄×D₁₀₉	C₂³	semidirect
43. #13182008	C₂³	semidirect
44. #13182006	C₂³	semidirect
45. C₄×Dic₁₀₉	C₂³	semidirect
48. C₂⁴×C₁₀₉	C₂×C₄	semidirect
55. C₂²⋊C₄×C₁₀₉	C₂³	semidirect
60. C₂³×C₁₀₉	C₂²×C₄	semidirect
68. C₂²×D₁₀₉	C₂²×C₄	semidirect
74. C₂²⋊D₁₀₉	C₂²×C₄	semidirect
77. C₂×Dic₁₀₉	C₂²×C₄	semidirect
78. C₂×Dic₁₀₉	C₂×D₄	semidirect
86. C₂²×C₁₀₉	C₄×D₄	semidirect
88. C₂²×C₁₀₉	C₂³×C₄	semidirect
95. C₂×D₁₀₉	C₂³×C₄	semidirect
99. Dic₁₀₉	C₂³×C₄	semidirect
100. Dic₁₀₉	C₂²×D₄	semidirect
102. C₂×C₁₀₉	C₂×C₄×D₄	semidirect
108. C₁₀₉	C₂²×C₄×D₄	semidirect
115. C₂²×C₂²⋊C₄	D₁₀₉	semidirect
133. C₂×C₂²⋊C₄	C₂×D₁₀₉	semidirect
141. C₂⁴	C₄×D₁₀₉	semidirect
162. C₂²⋊C₄	C₂²×D₁₀₉	semidirect
173. C₂³	C₂×C₄×D₁₀₉	semidirect
198. C₂²	#16812659	direct (N × H)
200. C₂²	C₂²×C₄×D₁₀₉	semidirect
211. C₂	C₂×#16812659	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₂²⋊D₁₀₉	2
C₂⁴×Dic₁₀₉	2
C₂³×C₄×D₁₀₉	2
C₂×#16812659	2
C₂²×#13182008	2
C₂²×#13182006	2
C₂²×C₄×Dic₁₀₉	2
C₂²×C₂²⋊C₄×C₁₀₉	2
C₂²×C₄×D₄	109

Subgroups — C22×#16812659 #28376200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16812659 #28376200