Subgroups — C₂×#26012558 #36370540

Order 22144.

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	#26012558	C₂	1	16	yes	maximal, direct product factor
2	C₂×D₄×#10621924	C₂	1	8	yes	maximal, complemented
3	C₂×#19149758	C₂	1	2	yes	maximal, complemented
4	C₂²×C₂²⋊#10621924	C₂	1	4	yes	maximal, complemented
5	C₂²×D₄×D₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×#10621924	C₂²	1	8	yes	complemented
7	D₄×#10621924	C₂²	1	64	yes	complemented
8	#19149758	C₂²	1	16	yes	complemented
9	C₂×C₄⋊#10621924	C₂²	1	4	yes	complemented
10	C₂×C₂²⋊#10621924	C₂²	1	4	yes	—
11	C₂×C₂²⋊#10621924	C₂²	1	32	yes	complemented
12	C₂×C₂²⋊#10621924	C₂²	1	8	yes	complemented
13	C₂×C₄×#10621924	C₂²	1	4	yes	complemented
14	C₂⁴×D₁₇₃	C₂²	1	2	yes	—
15	C₂²×D₆₉₂	C₄	1	1	yes	characteristic, complemented
16	C₂²×C₂²⋊D₁₇₃	C₄	1	2	yes	complemented
17	C₂×D₄×D₁₇₃	C₄	1	4	yes	complemented
18	C₂×D₄×D₁₇₃	C₄	1	8	yes	complemented
19	C₂×D₄×D₁₇₃	C₂²	1	4	yes	—
20	C₂×D₄×D₁₇₃	C₂²	1	8	yes	—
21	C₂²×C₄×D₁₇₃	C₂²	1	1	yes	characteristic
22	C₂²×D₄×C₁₇₃	C₄	1	1	yes	characteristic, complemented
23	C₂²×#10621924	—	4	16	no	—
24	C₂²×#10621924	C₂³	1	8	yes	—
25	C₂²×#10621924	C₂³	1	32	yes	complemented
26	C₄⋊#10621924	C₂³	1	16	yes	complemented
27	C₂²⋊#10621924	C₂³	1	16	yes	—
28	C₂²⋊#10621924	C₂³	1	32	yes	complemented
29	C₄×#10621924	C₂³	1	16	yes	complemented
30	C₂³×D₁₇₃	—	4	8	no	—
31	C₂³×D₁₇₃	C₂×C₄	1	4	yes	complemented
32	C₂³×D₁₇₃	C₂×C₄	1	2	yes	complemented
33	C₂³×D₁₇₃	C₂×C₄	1	8	yes	complemented
34	C₂³×D₁₇₃	C₂³	1	1	yes	characteristic
35	C₂³×D₁₇₃	C₂³	1	4	yes	—
36	C₂³×D₁₇₃	C₂³	1	8	yes	—
37	C₂×D₆₉₂	C₂×C₄	1	4	yes	complemented
38	C₂×D₆₉₂	C₂×C₄	1	8	yes	complemented
39	C₂×C₂²⋊D₁₇₃	C₂×C₄	1	8	yes	complemented
40	C₂×C₂²⋊D₁₇₃	C₂×C₄	1	16	yes	complemented
41	D₄×D₁₇₃	C₂×C₄	1	16	yes	complemented
42	D₄×D₁₇₃	C₂×C₄	1	32	yes	complemented
43	D₄×D₁₇₃	C₂³	1	16	yes	—
44	C₂²×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic, complemented
45	C₂×C₄×D₁₇₃	C₂×C₄	1	2	yes	complemented
46	C₂×C₄×D₁₇₃	C₂×C₄	1	4	yes	complemented
47	C₂×C₄×D₁₇₃	C₂³	1	2	yes	—
48	C₂×C₄×D₁₇₃	C₂³	1	4	yes	—
49	C₂⁴×C₁₇₃	C₂×C₄	1	2	yes	complemented
50	C₂×D₄×C₁₇₃	C₂×C₄	1	4	yes	complemented
51	C₂×D₄×C₁₇₃	C₂×C₄	1	8	yes	complemented
52	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic, complemented
53	C₂×#10621924	—	2	16	no	—
54	C₂×#10621924	—	4	32	no	—
55	C₂×#10621924	C₂⁴	1	16	yes	—
56	C₂²×D₁₇₃	—	2	4	no	—
57	C₂²×D₁₇₃	—	2	16	no	—
58	C₂²×D₁₇₃	—	4	16	no	—
59	C₂²×D₁₇₃	—	4	16	no	—
60	C₂²×D₁₇₃	—	4	8	no	—
61	C₂²×D₁₇₃	—	4	4	no	—
62	C₂²×D₁₇₃	C₂²×C₄	1	2	yes	—
63	C₂²×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×D₁₇₃	C₂²×C₄	1	8	yes	—
65	C₂²×D₁₇₃	C₂²×C₄	1	4	yes	—
66	C₂²×D₁₇₃	C₂²×C₄	1	16	yes	complemented
67	C₂²×D₁₇₃	C₂²×C₄	1	8	yes	complemented
68	C₂²×D₁₇₃	C₂⁴	1	2	yes	—
69	C₂²×D₁₇₃	C₂⁴	1	8	yes	—
70	C₂²×D₁₇₃	C₂⁴	1	1	yes	characteristic
71	D₆₉₂	C₂²×C₄	1	16	yes	complemented
72	C₂²⋊D₁₇₃	C₂²×C₄	1	32	yes	complemented
73	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₇₃	C₂²×C₄	1	4	yes	complemented
75	C₄×D₁₇₃	C₂²×C₄	1	4	yes	—
76	C₄×D₁₇₃	C₂²×C₄	1	8	yes	complemented
77	C₄×D₁₇₃	C₂⁴	1	4	yes	—
78	C₂³×C₁₇₃	—	4	4	no	—
79	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂³×C₁₇₃	C₂²×C₄	1	4	yes	—
81	C₂³×C₁₇₃	C₂²×C₄	1	8	yes	complemented
82	D₄×C₁₇₃	C₂²×C₄	1	16	yes	complemented
83	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₇₃	C₂²×C₄	1	4	yes	complemented
85	C₂²×C₁₇₃	—	2	2	no	—
86	C₂²×C₁₇₃	—	2	8	no	—
87	C₂²×C₁₇₃	—	4	8	no	—
88	C₂²×C₁₇₃	C₂³×C₄	1	2	yes	—
89	C₂²×C₁₇₃	C₂³×C₄	1	8	yes	—
90	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
91	C₂×D₁₇₃	—	2	2	no	—
92	C₂×D₁₇₃	—	2	8	no	—
93	C₂×D₁₇₃	—	2	4	no	—
94	C₂×D₁₇₃	—	2	16	no	—
95	C₂×D₁₇₃	—	2	4	no	—
96	C₂×D₁₇₃	—	2	16	no	—
97	C₂×D₁₇₃	—	4	16	no	—
98	C₂×D₁₇₃	—	4	8	no	—
99	C₂×D₁₇₃	C₂³×C₄	1	2	yes	—
100	C₂×D₁₇₃	C₂³×C₄	1	8	yes	—
101	C₂×D₁₇₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₁₇₃	H(5,𝔽₂)	1	4	yes	—
103	C₂×D₁₇₃	C₂⁵	1	1	yes	characteristic
104	#10621924	—	2	16	no	—
105	C₄×C₁₇₃	C₂³×C₄	1	4	yes	—
106	Dic₁₇₃	C₂³×C₄	1	4	yes	—
107	C₂×C₁₇₃	—	2	2	no	—
108	C₂×C₁₇₃	—	2	8	no	—
109	C₂×C₁₇₃	#518	1	2	yes	complemented
110	C₂×C₁₇₃	C₂⁴×C₄	1	1	yes	characteristic
111	D₁₇₃	—	2	2	no	—
112	D₁₇₃	—	2	8	no	—
113	D₁₇₃	#518	1	2	yes	complemented
114	D₁₇₃	C₂×H(5,𝔽₂)	1	2	yes	—
115	C₁₇₃	C₂×#518	1	1	yes	characteristic, complemented, Sylow
116	C₂×#518	—	173	1	no	maximal, Sylow
117	C₂³×D₄	—	173	1	no	—
118	#518	—	173	16	no	—
119	C₂×C₄×D₄	—	173	8	no	—
120	C₂×#117	—	173	2	no	—
121	C₂²×C₂²⋊C₄	—	173	4	no	—
122	C₂⁵	—	173	2	no	—
123	C₂²×D₄	—	173	2	no	—
124	C₂²×D₄	—	173	4	no	—
125	C₂²×D₄	—	173	8	no	—
126	C₂²×D₄	—	173	4	no	—
127	C₂²×D₄	—	173	8	no	—
128	C₂²×D₄	—	173	1	no	—
129	C₂²×D₄	#10621924	1	1	yes	characteristic, complemented
130	C₂³×C₄	—	173	1	no	—
131	C₂³×C₄	—	173	8	no	—
132	C₄×D₄	—	173	64	no	—
133	#117	—	173	16	no	—
134	C₂×C₄⋊C₄	—	173	4	no	—
135	C₂×C₂²⋊C₄	—	173	4	no	—
136	C₂×C₂²⋊C₄	—	173	8	no	—
137	C₂×C₂²⋊C₄	—	173	32	no	—
138	C₂×C₄²	—	173	4	no	—
139	C₂⁴	—	173	1	no	—
140	C₂⁴	—	173	4	no	—
141	C₂⁴	—	173	4	no	—
142	C₂⁴	—	173	2	no	—
143	C₂⁴	—	173	8	no	—
144	C₂⁴	—	173	8	no	—
145	C₂⁴	—	692	8	no	—
146	C₂⁴	C₂×#10621924	1	2	yes	complemented
147	C₂×D₄	—	173	8	no	—
148	C₂×D₄	—	173	16	no	—
149	C₂×D₄	—	173	16	no	—
150	C₂×D₄	—	173	32	no	—
151	C₂×D₄	—	173	16	no	—
152	C₂×D₄	—	173	4	no	—
153	C₂×D₄	—	173	8	no	—
154	C₂×D₄	C₂×#10621924	1	4	yes	complemented
155	C₂×D₄	C₂×#10621924	1	8	yes	complemented
156	C₂²×C₄	—	173	8	no	—
157	C₂²×C₄	—	173	1	no	—
158	C₂²×C₄	—	173	2	no	—
159	C₂²×C₄	—	173	2	no	—
160	C₂²×C₄	—	173	4	no	—
161	C₂²×C₄	—	173	4	no	—
162	C₂²×C₄	—	173	32	no	—
163	C₂²×C₄	—	692	16	no	—
164	C₂²×C₄	C₂×#10621924	1	1	yes	characteristic, complemented
165	C₄⋊C₄	—	173	16	no	—
166	C₂²⋊C₄	—	173	16	no	—
167	C₂²⋊C₄	—	173	32	no	—
168	C₄²	—	173	16	no	—
169	C₂³	—	4	4	no	—
170	C₂³	—	346	16	no	—
171	C₂³	—	346	4	no	—
172	C₂³	—	173	2	no	—
173	C₂³	—	173	8	no	—
174	C₂³	—	173	2	no	—
175	C₂³	—	173	8	no	—
176	C₂³	—	173	1	no	—
177	C₂³	—	173	4	no	—
178	C₂³	—	173	1	no	—
179	C₂³	—	173	16	no	—
180	C₂³	—	173	8	no	—
181	C₂³	—	692	4	no	—
182	C₂³	—	692	8	no	—
183	C₂³	—	692	16	no	—
184	C₂³	—	692	16	no	—
185	C₂³	C₂²×#10621924	1	1	yes	characteristic
186	C₂³	C₂²×#10621924	1	4	yes	—
187	C₂³	C₂²×#10621924	1	8	yes	complemented
188	D₄	—	173	32	no	—
189	D₄	—	173	16	no	—
190	D₄	C₂²×#10621924	1	16	yes	complemented
191	C₂×C₄	—	346	16	no	—
192	C₂×C₄	—	173	16	no	—
193	C₂×C₄	—	173	4	no	—
194	C₂×C₄	—	173	2	no	—
195	C₂×C₄	—	173	4	no	—
196	C₂×C₄	—	173	8	no	—
197	C₂×C₄	—	173	4	no	—
198	C₂×C₄	—	692	32	no	—
199	C₂×C₄	C₂²×#10621924	1	2	yes	—
200	C₂×C₄	C₂²×#10621924	1	4	yes	complemented
201	C₂²	—	2	2	no	—
202	C₂²	—	2	8	no	—
203	C₂²	—	4	8	no	—
204	C₂²	—	346	16	no	—
205	C₂²	—	346	4	no	—
206	C₂²	—	346	4	no	—
207	C₂²	—	346	16	no	—
208	C₂²	—	346	2	no	—
209	C₂²	—	346	8	no	—
210	C₂²	—	173	2	no	—
211	C₂²	—	173	8	no	—
212	C₂²	—	173	4	no	—
213	C₂²	—	173	1	no	—
214	C₂²	—	173	1	no	—
215	C₂²	—	692	8	no	—
216	C₂²	—	692	16	no	—
217	C₂²	C₂³×#10621924	1	2	yes	—
218	C₂²	C₂³×#10621924	1	8	yes	—
219	C₂²	C₂³×#10621924	1	1	yes	characteristic
220	C₄	—	346	16	no	—
221	C₄	—	173	4	no	—
222	C₄	C₂³×#10621924	1	4	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	2	8	no	—
225	C₂	—	346	8	no	—
226	C₂	—	346	2	no	—
227	C₂	—	173	2	no	—
228	C₂	—	173	2	no	—
229	C₂	#26012558	1	2	yes	direct product factor
230	C₂	C₂⁴×#10621924	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×#10621924
commutator	C₂×C₁₇₃	C₂⁴×C₄
Frattini	C₂	C₂⁴×#10621924
Fitting	C₂²×D₄×C₁₇₃	C₄
socle	C₂²×C₁₇₃	C₂³×C₄
prime radical	C₂²×D₄×D₁₇₃	C₂
cosocle	C₂×D₁₇₃	C₂⁵
hypercenter	C₂²×D₄	#10621924
hypocenter	C₁₇₃	C₂×#518
LCS	G ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. #26012558	C₂	direct (N × H)
2. C₂×D₄×#10621924	C₂	semidirect
3. C₂×#19149758	C₂	semidirect
4. C₂²×C₂²⋊#10621924	C₂	semidirect
6. C₂³×#10621924	C₂²	semidirect
7. D₄×#10621924	C₂²	semidirect
8. #19149758	C₂²	semidirect
9. C₂×C₄⋊#10621924	C₂²	semidirect
11. C₂×C₂²⋊#10621924	C₂²	semidirect
12. C₂×C₂²⋊#10621924	C₂²	semidirect
13. C₂×C₄×#10621924	C₂²	semidirect
15. C₂²×D₆₉₂	C₄	semidirect
16. C₂²×C₂²⋊D₁₇₃	C₄	semidirect
17. C₂×D₄×D₁₇₃	C₄	semidirect
18. C₂×D₄×D₁₇₃	C₄	semidirect
22. C₂²×D₄×C₁₇₃	C₄	semidirect
25. C₂²×#10621924	C₂³	semidirect
26. C₄⋊#10621924	C₂³	semidirect
28. C₂²⋊#10621924	C₂³	semidirect
29. C₄×#10621924	C₂³	semidirect
31. C₂³×D₁₇₃	C₂×C₄	semidirect
32. C₂³×D₁₇₃	C₂×C₄	semidirect
33. C₂³×D₁₇₃	C₂×C₄	semidirect
37. C₂×D₆₉₂	C₂×C₄	semidirect
38. C₂×D₆₉₂	C₂×C₄	semidirect
39. C₂×C₂²⋊D₁₇₃	C₂×C₄	semidirect
40. C₂×C₂²⋊D₁₇₃	C₂×C₄	semidirect
41. D₄×D₁₇₃	C₂×C₄	semidirect
42. D₄×D₁₇₃	C₂×C₄	semidirect
44. C₂²×Dic₁₇₃	C₂×C₄	semidirect
45. C₂×C₄×D₁₇₃	C₂×C₄	semidirect
46. C₂×C₄×D₁₇₃	C₂×C₄	semidirect
49. C₂⁴×C₁₇₃	C₂×C₄	semidirect
50. C₂×D₄×C₁₇₃	C₂×C₄	semidirect
51. C₂×D₄×C₁₇₃	C₂×C₄	semidirect
52. C₂²×C₄×C₁₇₃	C₂×C₄	semidirect
66. C₂²×D₁₇₃	C₂²×C₄	semidirect
67. C₂²×D₁₇₃	C₂²×C₄	semidirect
71. D₆₉₂	C₂²×C₄	semidirect
72. C₂²⋊D₁₇₃	C₂²×C₄	semidirect
74. C₂×Dic₁₇₃	C₂²×C₄	semidirect
76. C₄×D₁₇₃	C₂²×C₄	semidirect
81. C₂³×C₁₇₃	C₂²×C₄	semidirect
82. D₄×C₁₇₃	C₂²×C₄	semidirect
84. C₂×C₄×C₁₇₃	C₂²×C₄	semidirect
109. C₂×C₁₇₃	#518	semidirect
113. D₁₇₃	#518	semidirect
115. C₁₇₃	C₂×#518	semidirect
129. C₂²×D₄	#10621924	semidirect
146. C₂⁴	C₂×#10621924	semidirect
154. C₂×D₄	C₂×#10621924	semidirect
155. C₂×D₄	C₂×#10621924	semidirect
164. C₂²×C₄	C₂×#10621924	semidirect
187. C₂³	C₂²×#10621924	semidirect
190. D₄	C₂²×#10621924	semidirect
200. C₂×C₄	C₂²×#10621924	semidirect
229. C₂	#26012558	direct (N × H)

Maximal subgroups

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

subgroup	index
#26012558	2
C₂×D₄×#10621924	2
C₂×#19149758	2
C₂²×C₂²⋊#10621924	2
C₂²×D₄×D₁₇₃	2
C₂×#518	173

Subgroups — C2×#26012558 #36370540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26012558 #36370540