Subgroups — C₂×#22627162 #34035028

Order 18688.

502 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989696	C₂	1	2	yes	maximal, complemented
2	#22627162	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17989695	C₂	1	2	yes	maximal, complemented
4	C₂×#17989694	C₂	1	2	yes	maximal, complemented
5	C₂²×#14556664	C₂	1	1	yes	maximal, characteristic
6	#17989696	C₂²	1	8	yes	complemented
7	#17989695	C₂²	1	8	yes	complemented
8	#17989694	C₂²	1	8	yes	complemented
9	C₂×#14556681	C₂²	1	4	yes	complemented
10	C₂×#14556664	C₄	1	4	yes	—
11	C₂×#14556664	C₄	1	1	yes	characteristic
12	C₂×#14556664	C₄	1	1	yes	characteristic
13	C₂×#14556664	C₂²	1	1	yes	characteristic
14	C₂×#14556664	C₂²	1	4	yes	—
15	C₂×#14556664	C₂²	1	1	yes	characteristic
16	C₂²×#11924031	C₄	1	1	yes	characteristic
17	C₂²×#11924031	C₂²	1	1	yes	characteristic
18	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
19	#14556681	C₂³	1	8	yes	complemented
20	#14556664	C₂×C₄	1	2	yes	—
21	#14556664	C₂×C₄	1	4	yes	—
22	#14556664	C₂×C₄	1	4	yes	—
23	#14556664	C₂×C₄	1	2	yes	—
24	#14556664	C₂³	1	2	yes	—
25	#14556664	C₂³	1	2	yes	—
26	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
27	C₂×#11924031	C₂×C₄	1	4	yes	—
28	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
29	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
30	C₂×#11924031	C₂×C₄	1	2	yes	—
31	C₂×#11924031	C₂³	1	1	yes	characteristic
32	C₂×#11924031	C₂³	1	2	yes	—
33	C₂²×C₄×D₇₃	C₈	1	1	yes	characteristic
34	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₇₃	C₈	1	4	yes	—
36	C₂×C₈×D₇₃	C₈	1	1	yes	characteristic
37	C₂×C₈×D₇₃	C₈	1	1	yes	characteristic
38	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
39	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic
42	#11924031	C₂²×C₄	1	2	yes	—
43	#11924031	C₂²×C₄	1	2	yes	—
44	#11924031	C₂²×C₄	1	1	yes	characteristic
45	#11924031	C₂²×C₄	1	2	yes	—
46	#11924031	C₂⁴	1	1	yes	characteristic
47	C₂³×D₇₃	C₁₆	1	1	yes	characteristic
48	C₂²×Dic₇₃	C₁₆	1	1	yes	characteristic
49	C₂×C₄×D₇₃	C₁₆	1	1	yes	characteristic
50	C₂×C₄×D₇₃	C₁₆	1	4	yes	—
51	C₂×C₄×D₇₃	C₁₆	1	1	yes	characteristic
52	C₂×C₄×D₇₃	C₂×C₈	1	4	yes	—
53	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
54	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
55	C₂×#10596831	C₂×C₈	1	2	yes	—
56	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
57	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
58	C₂×#10596831	C₂²×C₄	1	2	yes	—
59	C₈×D₇₃	C₂×C₈	1	2	yes	—
60	C₈×D₇₃	C₂×C₈	1	4	yes	—
61	C₈×D₇₃	C₂×C₈	1	2	yes	—
62	C₈×D₇₃	C₂×C₈	1	4	yes	—
63	C₈×D₇₃	C₂²×C₄	1	2	yes	—
64	C₈×D₇₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
66	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
67	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
68	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
69	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
70	C₂²×D₇₃	—	2	4	no	—
71	C₂²×D₇₃	C₂×C₁₆	1	1	yes	characteristic
72	C₂²×D₇₃	C₂×C₁₆	1	4	yes	—
73	C₂²×D₇₃	C₂×C₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
75	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×Dic₇₃	C₂×C₁₆	1	4	yes	—
79	C₂×Dic₇₃	C₂×C₁₆	1	1	yes	characteristic
80	C₂×Dic₇₃	C₂×C₁₆	1	1	yes	characteristic
81	C₄×D₇₃	C₂×C₁₆	1	4	yes	—
82	C₄×D₇₃	C₂×C₁₆	1	2	yes	—
83	C₄×D₇₃	C₂×C₁₆	1	2	yes	—
84	C₄×D₇₃	C₂×C₁₆	1	4	yes	—
85	C₄×D₇₃	C₂²×C₈	1	2	yes	—
86	C₄×D₇₃	C₂²×C₈	1	2	yes	—
87	C₈×C₇₃	C₂²×C₈	1	2	yes	—
88	C₈×C₇₃	C₂²×C₈	1	2	yes	—
89	#10596831	C₂²×C₈	1	1	yes	characteristic
90	#10596831	C₂²×C₈	1	2	yes	—
91	#10596831	C₂³×C₄	1	1	yes	characteristic
92	C₂²×C₇₃	—	2	2	no	—
93	C₂²×C₇₃	C₂²×C₁₆	1	2	yes	—
94	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
95	C₂×D₇₃	—	2	2	no	—
96	C₂×D₇₃	—	2	8	no	—
97	C₂×D₇₃	M₆₄	1	4	yes	complemented
98	C₂×D₇₃	C₂²×C₁₆	1	2	yes	—
99	C₂×D₇₃	C₂²×C₁₆	1	2	yes	—
100	C₄×C₇₃	C₂²×C₁₆	1	2	yes	—
101	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
102	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
103	Dic₇₃	C₂²×C₁₆	1	2	yes	—
104	Dic₇₃	C₂²×C₁₆	1	2	yes	—
105	C₂²×M₆₄	—	73	1	no	maximal, Sylow
106	C₂×C₇₃	—	2	2	no	—
107	C₂×C₇₃	C₂×M₆₄	1	2	yes	complemented
108	C₂×C₇₃	C₂³×C₁₆	1	1	yes	characteristic
109	D₇₃	—	2	2	no	—
110	D₇₃	C₂×M₆₄	1	4	yes	complemented
111	C₂³×C₁₆	—	73	1	no	—
112	C₂×M₆₄	—	73	2	no	—
113	C₂×M₆₄	—	73	8	no	—
114	C₂×M₆₄	—	73	2	no	—
115	C₂²×C₃₂	—	73	2	no	—
116	C₇₃	C₂²×M₆₄	1	1	yes	characteristic, complemented, Sylow
117	C₂³×C₈	—	73	1	no	—
118	C₂²×C₁₆	—	73	1	no	—
119	C₂²×C₁₆	—	73	1	no	—
120	C₂²×C₁₆	—	73	1	no	—
121	C₂²×C₁₆	—	73	4	no	—
122	C₂²×C₁₆	—	73	1	no	—
123	C₂²×C₁₆	—	73	1	no	—
124	C₂²×C₁₆	—	73	1	no	—
125	C₂²×C₁₆	—	73	4	no	—
126	M₆₄	—	73	8	no	—
127	M₆₄	—	73	8	no	—
128	C₂×C₃₂	—	73	4	no	—
129	C₂×C₃₂	—	73	8	no	—
130	C₂³×C₄	—	73	1	no	—
131	C₂²×C₈	—	73	1	no	—
132	C₂²×C₈	—	73	4	no	—
133	C₂²×C₈	—	73	1	no	—
134	C₂²×C₈	—	73	4	no	—
135	C₂²×C₈	—	73	1	no	—
136	C₂²×C₈	—	73	1	no	—
137	C₂²×C₈	—	73	1	no	—
138	C₂²×C₈	#10596842	1	1	yes	characteristic
139	C₂×C₁₆	—	73	1	no	—
140	C₂×C₁₆	—	73	2	no	—
141	C₂×C₁₆	—	73	2	no	—
142	C₂×C₁₆	—	73	2	no	—
143	C₂×C₁₆	—	73	1	no	—
144	C₂×C₁₆	—	73	4	no	—
145	C₂×C₁₆	—	73	1	no	—
146	C₂×C₁₆	—	73	2	no	—
147	C₂×C₁₆	—	73	1	no	—
148	C₂×C₁₆	—	73	2	no	—
149	C₂×C₁₆	—	73	4	no	—
150	C₂×C₁₆	—	73	2	no	—
151	C₂×C₁₆	—	73	4	no	—
152	C₃₂	—	73	8	no	—
153	C₂⁴	—	73	1	no	—
154	C₂²×C₄	—	73	1	no	—
155	C₂²×C₄	—	73	1	no	—
156	C₂²×C₄	—	73	4	no	—
157	C₂²×C₄	—	73	1	no	—
158	C₂²×C₄	—	73	4	no	—
159	C₂²×C₄	—	73	1	no	—
160	C₂²×C₄	—	73	1	no	—
161	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
162	C₂×C₈	—	73	2	no	—
163	C₂×C₈	—	73	1	no	—
164	C₂×C₈	—	73	2	no	—
165	C₂×C₈	—	73	1	no	—
166	C₂×C₈	—	73	4	no	—
167	C₂×C₈	—	73	2	no	—
168	C₂×C₈	—	73	2	no	—
169	C₂×C₈	—	73	4	no	—
170	C₂×C₈	—	73	2	no	—
171	C₂×C₈	—	73	2	no	—
172	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
173	C₂×C₈	C₂×#10596842	1	4	yes	—
174	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
175	C₁₆	—	73	2	no	—
176	C₁₆	—	73	2	no	—
177	C₁₆	—	73	1	no	—
178	C₁₆	—	73	2	no	—
179	C₁₆	—	73	1	no	—
180	C₂³	—	73	1	no	—
181	C₂³	—	73	4	no	—
182	C₂³	—	73	1	no	—
183	C₂³	—	146	4	no	—
184	C₂³	#14556686	1	1	yes	characteristic
185	C₂×C₄	—	73	4	no	—
186	C₂×C₄	—	73	1	no	—
187	C₂×C₄	—	73	1	no	—
188	C₂×C₄	—	73	4	no	—
189	C₂×C₄	—	73	2	no	—
190	C₂×C₄	—	73	4	no	—
191	C₂×C₄	—	73	2	no	—
192	C₂×C₄	—	73	2	no	—
193	C₂×C₄	—	73	2	no	—
194	C₂×C₄	#14556686	1	2	yes	—
195	C₂×C₄	#14556686	1	1	yes	characteristic
196	C₂×C₄	C₂²×#10596842	1	2	yes	—
197	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
198	C₈	—	73	1	no	—
199	C₈	—	73	1	no	—
200	C₈	—	73	2	no	—
201	C₈	C₂²×#10596842	1	2	yes	—
202	C₈	C₂²×#10596842	1	2	yes	—
203	C₂²	—	2	2	no	—
204	C₂²	—	73	4	no	—
205	C₂²	—	73	2	no	—
206	C₂²	—	73	2	no	—
207	C₂²	—	146	8	no	—
208	C₂²	—	146	2	no	—
209	C₂²	C₂×#14556686	1	2	yes	—
210	C₂²	C₂×#14556686	1	1	yes	characteristic
211	C₄	—	73	2	no	—
212	C₄	—	73	2	no	—
213	C₄	C₂×#14556686	1	2	yes	—
214	C₄	C₂×#14556686	1	1	yes	characteristic
215	C₄	C₂³×#10596842	1	1	yes	characteristic
216	C₂	—	2	2	no	—
217	C₂	—	73	4	no	—
218	C₂	—	146	2	no	—
219	C₂	#22627162	1	2	yes	direct product factor
220	C₂	C₂²×#14556686	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×#10596842
commutator	C₂×C₇₃	C₂³×C₁₆
Frattini	C₄	C₂³×#10596842
Fitting	C₂²×C₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂²×C₁₆
prime radical	C₂³×D₇₃	C₁₆
cosocle	#11924031	C₂⁴
hypercenter	C₂²×C₈	#10596842
hypocenter	C₇₃	C₂²×M₆₄
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#17989696	C₂	semidirect
2. #22627162	C₂	direct (N × H)
3. C₂×#17989695	C₂	semidirect
4. C₂×#17989694	C₂	semidirect
6. #17989696	C₂²	semidirect
7. #17989695	C₂²	semidirect
8. #17989694	C₂²	semidirect
9. C₂×#14556681	C₂²	semidirect
19. #14556681	C₂³	semidirect
97. C₂×D₇₃	M₆₄	semidirect
107. C₂×C₇₃	C₂×M₆₄	semidirect
110. D₇₃	C₂×M₆₄	semidirect
116. C₇₃	C₂²×M₆₄	semidirect
219. C₂	#22627162	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×#17989696	2
#22627162	2
C₂×#17989695	2
C₂×#17989694	2
C₂²×#14556664	2
C₂²×M₆₄	73

Subgroups — C2×#22627162 #34035028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22627162 #34035028