Subgroups — C₂×#20296093 #28380009

Order 13952.

1920 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20296093	C₂	1	16	yes	maximal, direct product factor
2	C₂×D₄×#89137	C₂	1	8	yes	maximal, complemented
3	C₂×#16812778	C₂	1	2	yes	maximal, complemented
4	C₂²×C₂²⋊#89137	C₂	1	4	yes	maximal, complemented
5	C₂²×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×#89137	C₂²	1	8	yes	complemented
7	D₄×#89137	C₂²	1	64	yes	complemented
8	#16812778	C₂²	1	16	yes	complemented
9	C₂×C₄⋊#89137	C₂²	1	4	yes	complemented
10	C₂×C₂²⋊#89137	C₂²	1	4	yes	—
11	C₂×C₂²⋊#89137	C₂²	1	32	yes	complemented
12	C₂×C₂²⋊#89137	C₂²	1	8	yes	complemented
13	C₂×C₄×#89137	C₂²	1	4	yes	complemented
14	C₂⁴×D₁₀₉	C₂²	1	2	yes	—
15	C₂²×D₄₃₆	C₄	1	1	yes	characteristic, complemented
16	C₂²×C₂²⋊D₁₀₉	C₄	1	2	yes	complemented
17	C₂×D₄×D₁₀₉	C₄	1	4	yes	complemented
18	C₂×D₄×D₁₀₉	C₄	1	8	yes	complemented
19	C₂×D₄×D₁₀₉	C₂²	1	4	yes	—
20	C₂×D₄×D₁₀₉	C₂²	1	8	yes	—
21	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
22	C₂²×D₄×C₁₀₉	C₄	1	1	yes	characteristic, complemented
23	C₂²×#89137	—	4	16	no	—
24	C₂²×#89137	C₂³	1	8	yes	—
25	C₂²×#89137	C₂³	1	32	yes	complemented
26	C₄⋊#89137	C₂³	1	16	yes	complemented
27	C₂²⋊#89137	C₂³	1	16	yes	—
28	C₂²⋊#89137	C₂³	1	32	yes	complemented
29	C₄×#89137	C₂³	1	16	yes	complemented
30	C₂³×D₁₀₉	—	4	8	no	—
31	C₂³×D₁₀₉	C₂×C₄	1	8	yes	complemented
32	C₂³×D₁₀₉	C₂×C₄	1	4	yes	complemented
33	C₂³×D₁₀₉	C₂×C₄	1	2	yes	complemented
34	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
35	C₂³×D₁₀₉	C₂³	1	4	yes	—
36	C₂³×D₁₀₉	C₂³	1	8	yes	—
37	C₂×D₄₃₆	C₂×C₄	1	4	yes	complemented
38	C₂×D₄₃₆	C₂×C₄	1	8	yes	complemented
39	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	8	yes	complemented
40	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	16	yes	complemented
41	D₄×D₁₀₉	C₂×C₄	1	16	yes	complemented
42	D₄×D₁₀₉	C₂×C₄	1	32	yes	complemented
43	D₄×D₁₀₉	C₂³	1	16	yes	—
44	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
45	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	complemented
46	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	complemented
47	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
48	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
49	C₂⁴×C₁₀₉	C₂×C₄	1	2	yes	complemented
50	C₂×D₄×C₁₀₉	C₂×C₄	1	4	yes	complemented
51	C₂×D₄×C₁₀₉	C₂×C₄	1	8	yes	complemented
52	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
53	C₂³×C₁₀₉	—	4	4	no	—
54	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
55	C₂³×C₁₀₉	C₂²×C₄	1	4	yes	—
56	C₂³×C₁₀₉	C₂²×C₄	1	8	yes	complemented
57	C₂²×D₁₀₉	—	2	4	no	—
58	C₂²×D₁₀₉	—	2	16	no	—
59	C₂²×D₁₀₉	—	4	16	no	—
60	C₂²×D₁₀₉	—	4	16	no	—
61	C₂²×D₁₀₉	—	4	8	no	—
62	C₂²×D₁₀₉	—	4	4	no	—
63	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
64	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×D₁₀₉	C₂²×C₄	1	8	yes	—
66	C₂²×D₁₀₉	C₂²×C₄	1	4	yes	—
67	C₂²×D₁₀₉	C₂²×C₄	1	16	yes	complemented
68	C₂²×D₁₀₉	C₂²×C₄	1	8	yes	complemented
69	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
70	C₂²×D₁₀₉	C₂⁴	1	8	yes	—
71	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
72	C₂×#89137	—	2	16	no	—
73	C₂×#89137	—	4	32	no	—
74	C₂×#89137	C₂⁴	1	16	yes	—
75	D₄×C₁₀₉	C₂²×C₄	1	16	yes	complemented
76	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	complemented
78	C₂²⋊D₁₀₉	C₂²×C₄	1	32	yes	complemented
79	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
80	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	complemented
81	D₄₃₆	C₂²×C₄	1	16	yes	complemented
82	C₄×D₁₀₉	C₂²×C₄	1	4	yes	—
83	C₄×D₁₀₉	C₂²×C₄	1	8	yes	complemented
84	C₄×D₁₀₉	C₂⁴	1	4	yes	—
85	C₂²×C₁₀₉	—	2	2	no	—
86	C₂²×C₁₀₉	—	2	8	no	—
87	C₂²×C₁₀₉	—	4	8	no	—
88	C₂²×C₁₀₉	C₂³×C₄	1	2	yes	—
89	C₂²×C₁₀₉	C₂³×C₄	1	8	yes	—
90	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
91	C₂×D₁₀₉	—	2	2	no	—
92	C₂×D₁₀₉	—	2	8	no	—
93	C₂×D₁₀₉	—	2	4	no	—
94	C₂×D₁₀₉	—	2	16	no	—
95	C₂×D₁₀₉	—	2	16	no	—
96	C₂×D₁₀₉	—	2	4	no	—
97	C₂×D₁₀₉	—	4	16	no	—
98	C₂×D₁₀₉	—	4	8	no	—
99	C₂×D₁₀₉	C₂³×C₄	1	2	yes	—
100	C₂×D₁₀₉	C₂³×C₄	1	8	yes	—
101	C₂×D₁₀₉	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₁₀₉	H(5,𝔽₂)	1	4	yes	—
103	C₂×D₁₀₉	C₂⁵	1	1	yes	characteristic
104	#89137	—	2	16	no	—
105	C₄×C₁₀₉	C₂³×C₄	1	4	yes	—
106	Dic₁₀₉	C₂³×C₄	1	4	yes	—
107	C₂×C₁₀₉	—	2	2	no	—
108	C₂×C₁₀₉	—	2	8	no	—
109	C₂×C₁₀₉	#518	1	2	yes	complemented
110	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
111	D₁₀₉	—	2	2	no	—
112	D₁₀₉	—	2	8	no	—
113	D₁₀₉	#518	1	2	yes	complemented
114	D₁₀₉	C₂×H(5,𝔽₂)	1	2	yes	—
115	C₂×#518	—	109	1	no	maximal, Sylow
116	C₁₀₉	C₂×#518	1	1	yes	characteristic, complemented, Sylow
117	C₂³×D₄	—	109	1	no	—
118	#518	—	109	16	no	—
119	C₂×C₄×D₄	—	109	8	no	—
120	C₂×#117	—	109	2	no	—
121	C₂²×C₂²⋊C₄	—	109	4	no	—
122	C₂⁵	—	109	2	no	—
123	C₂²×D₄	—	109	2	no	—
124	C₂²×D₄	—	109	4	no	—
125	C₂²×D₄	—	109	8	no	—
126	C₂²×D₄	—	109	4	no	—
127	C₂²×D₄	—	109	8	no	—
128	C₂²×D₄	—	109	1	no	—
129	C₂²×D₄	#89137	1	1	yes	characteristic, complemented
130	C₂³×C₄	—	109	1	no	—
131	C₂³×C₄	—	109	8	no	—
132	C₄×D₄	—	109	64	no	—
133	#117	—	109	16	no	—
134	C₂×C₄⋊C₄	—	109	4	no	—
135	C₂×C₂²⋊C₄	—	109	4	no	—
136	C₂×C₂²⋊C₄	—	109	8	no	—
137	C₂×C₂²⋊C₄	—	109	32	no	—
138	C₂×C₄²	—	109	4	no	—
139	C₂⁴	—	109	4	no	—
140	C₂⁴	—	109	2	no	—
141	C₂⁴	—	109	8	no	—
142	C₂⁴	—	109	8	no	—
143	C₂⁴	—	109	1	no	—
144	C₂⁴	—	109	4	no	—
145	C₂⁴	—	436	8	no	—
146	C₂⁴	C₂×#89137	1	2	yes	complemented
147	C₂×D₄	—	109	8	no	—
148	C₂×D₄	—	109	16	no	—
149	C₂×D₄	—	109	16	no	—
150	C₂×D₄	—	109	32	no	—
151	C₂×D₄	—	109	16	no	—
152	C₂×D₄	—	109	4	no	—
153	C₂×D₄	—	109	8	no	—
154	C₂×D₄	C₂×#89137	1	4	yes	complemented
155	C₂×D₄	C₂×#89137	1	8	yes	complemented
156	C₂²×C₄	—	109	1	no	—
157	C₂²×C₄	—	109	2	no	—
158	C₂²×C₄	—	109	2	no	—
159	C₂²×C₄	—	109	4	no	—
160	C₂²×C₄	—	109	4	no	—
161	C₂²×C₄	—	109	32	no	—
162	C₂²×C₄	—	109	8	no	—
163	C₂²×C₄	—	436	16	no	—
164	C₂²×C₄	C₂×#89137	1	1	yes	characteristic, complemented
165	C₄⋊C₄	—	109	16	no	—
166	C₂²⋊C₄	—	109	16	no	—
167	C₂²⋊C₄	—	109	32	no	—
168	C₄²	—	109	16	no	—
169	C₂³	—	4	4	no	—
170	C₂³	—	109	1	no	—
171	C₂³	—	109	16	no	—
172	C₂³	—	109	8	no	—
173	C₂³	—	109	2	no	—
174	C₂³	—	109	8	no	—
175	C₂³	—	109	2	no	—
176	C₂³	—	109	8	no	—
177	C₂³	—	109	1	no	—
178	C₂³	—	109	4	no	—
179	C₂³	—	436	4	no	—
180	C₂³	—	436	8	no	—
181	C₂³	—	436	16	no	—
182	C₂³	—	436	16	no	—
183	C₂³	—	218	16	no	—
184	C₂³	—	218	4	no	—
185	C₂³	C₂²×#89137	1	1	yes	characteristic
186	C₂³	C₂²×#89137	1	4	yes	—
187	C₂³	C₂²×#89137	1	8	yes	complemented
188	D₄	—	109	32	no	—
189	D₄	—	109	16	no	—
190	D₄	C₂²×#89137	1	16	yes	complemented
191	C₂×C₄	—	109	4	no	—
192	C₂×C₄	—	109	2	no	—
193	C₂×C₄	—	109	4	no	—
194	C₂×C₄	—	109	4	no	—
195	C₂×C₄	—	109	8	no	—
196	C₂×C₄	—	109	16	no	—
197	C₂×C₄	—	436	32	no	—
198	C₂×C₄	—	218	16	no	—
199	C₂×C₄	C₂²×#89137	1	2	yes	—
200	C₂×C₄	C₂²×#89137	1	4	yes	complemented
201	C₂²	—	2	2	no	—
202	C₂²	—	2	8	no	—
203	C₂²	—	4	8	no	—
204	C₂²	—	109	4	no	—
205	C₂²	—	109	1	no	—
206	C₂²	—	109	1	no	—
207	C₂²	—	109	2	no	—
208	C₂²	—	109	8	no	—
209	C₂²	—	436	8	no	—
210	C₂²	—	436	16	no	—
211	C₂²	—	218	16	no	—
212	C₂²	—	218	4	no	—
213	C₂²	—	218	4	no	—
214	C₂²	—	218	16	no	—
215	C₂²	—	218	2	no	—
216	C₂²	—	218	8	no	—
217	C₂²	C₂³×#89137	1	2	yes	—
218	C₂²	C₂³×#89137	1	8	yes	—
219	C₂²	C₂³×#89137	1	1	yes	characteristic
220	C₄	—	109	4	no	—
221	C₄	—	218	16	no	—
222	C₄	C₂³×#89137	1	4	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	2	8	no	—
225	C₂	—	109	2	no	—
226	C₂	—	109	2	no	—
227	C₂	—	218	2	no	—
228	C₂	—	218	8	no	—
229	C₂	#20296093	1	2	yes	direct product factor
230	C₂	C₂⁴×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×#89137
commutator	C₂×C₁₀₉	C₂⁴×C₄
Frattini	C₂	C₂⁴×#89137
Fitting	C₂²×D₄×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂³×C₄
prime radical	C₂²×D₄×D₁₀₉	C₂
cosocle	C₂×D₁₀₉	C₂⁵
hypercenter	C₂²×D₄	#89137
hypocenter	C₁₀₉	C₂×#518
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. #20296093	C₂	direct (N × H)
2. C₂×D₄×#89137	C₂	semidirect
3. C₂×#16812778	C₂	semidirect
4. C₂²×C₂²⋊#89137	C₂	semidirect
6. C₂³×#89137	C₂²	semidirect
7. D₄×#89137	C₂²	semidirect
8. #16812778	C₂²	semidirect
9. C₂×C₄⋊#89137	C₂²	semidirect
11. C₂×C₂²⋊#89137	C₂²	semidirect
12. C₂×C₂²⋊#89137	C₂²	semidirect
13. C₂×C₄×#89137	C₂²	semidirect
15. C₂²×D₄₃₆	C₄	semidirect
16. C₂²×C₂²⋊D₁₀₉	C₄	semidirect
17. C₂×D₄×D₁₀₉	C₄	semidirect
18. C₂×D₄×D₁₀₉	C₄	semidirect
22. C₂²×D₄×C₁₀₉	C₄	semidirect
25. C₂²×#89137	C₂³	semidirect
26. C₄⋊#89137	C₂³	semidirect
28. C₂²⋊#89137	C₂³	semidirect
29. C₄×#89137	C₂³	semidirect
31. C₂³×D₁₀₉	C₂×C₄	semidirect
32. C₂³×D₁₀₉	C₂×C₄	semidirect
33. C₂³×D₁₀₉	C₂×C₄	semidirect
37. C₂×D₄₃₆	C₂×C₄	semidirect
38. C₂×D₄₃₆	C₂×C₄	semidirect
39. C₂×C₂²⋊D₁₀₉	C₂×C₄	semidirect
40. C₂×C₂²⋊D₁₀₉	C₂×C₄	semidirect
41. D₄×D₁₀₉	C₂×C₄	semidirect
42. D₄×D₁₀₉	C₂×C₄	semidirect
44. C₂²×Dic₁₀₉	C₂×C₄	semidirect
45. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
46. C₂×C₄×D₁₀₉	C₂×C₄	semidirect
49. C₂⁴×C₁₀₉	C₂×C₄	semidirect
50. C₂×D₄×C₁₀₉	C₂×C₄	semidirect
51. C₂×D₄×C₁₀₉	C₂×C₄	semidirect
52. C₂²×C₄×C₁₀₉	C₂×C₄	semidirect
56. C₂³×C₁₀₉	C₂²×C₄	semidirect
67. C₂²×D₁₀₉	C₂²×C₄	semidirect
68. C₂²×D₁₀₉	C₂²×C₄	semidirect
75. D₄×C₁₀₉	C₂²×C₄	semidirect
77. C₂×C₄×C₁₀₉	C₂²×C₄	semidirect
78. C₂²⋊D₁₀₉	C₂²×C₄	semidirect
80. C₂×Dic₁₀₉	C₂²×C₄	semidirect
81. D₄₃₆	C₂²×C₄	semidirect
83. C₄×D₁₀₉	C₂²×C₄	semidirect
109. C₂×C₁₀₉	#518	semidirect
113. D₁₀₉	#518	semidirect
116. C₁₀₉	C₂×#518	semidirect
129. C₂²×D₄	#89137	semidirect
146. C₂⁴	C₂×#89137	semidirect
154. C₂×D₄	C₂×#89137	semidirect
155. C₂×D₄	C₂×#89137	semidirect
164. C₂²×C₄	C₂×#89137	semidirect
187. C₂³	C₂²×#89137	semidirect
190. D₄	C₂²×#89137	semidirect
200. C₂×C₄	C₂²×#89137	semidirect
229. C₂	#20296093	direct (N × H)

Maximal subgroups

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

subgroup	index
#20296093	2
C₂×D₄×#89137	2
C₂×#16812778	2
C₂²×C₂²⋊#89137	2
C₂²×D₄×D₁₀₉	2
C₂×#518	109

Subgroups — C2×#20296093 #28380009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20296093 #28380009