Subgroups — C₂×C₈×#11924036 #34015627

Order 18688.

1342 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11924036	C₂	1	2	yes	maximal
2	C₈×#11924036	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₈×#10596833	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924036	C₄	1	4	yes	—
6	C₄×#11924036	C₄	1	8	yes	—
7	C₄×#11924036	C₂²	1	8	yes	—
8	C₂×C₄×#10596833	C₄	1	4	yes	—
9	C₂×C₄×#10596833	C₂²	1	4	yes	—
10	C₈×#10596833	C₂²	1	16	yes	complemented
11	C₂²×C₈×D₇₃	C₄	1	2	yes	—
12	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
15	C₄×C₈×D₇₃	C₄	1	4	yes	—
16	C₄×C₈×D₇₃	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
18	C₂×#11924036	C₈	1	32	yes	direct product factor
19	C₂×#11924036	C₂×C₄	1	16	yes	—
20	C₂²×#10596833	C₂×C₄	1	8	yes	—
21	C₄×#10596833	C₂×C₄	1	16	yes	—
22	C₄×#10596833	C₂×C₄	1	8	yes	—
23	C₄×#10596833	C₂³	1	8	yes	—
24	C₂²×C₄×D₇₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10596831	C₂×C₄	1	2	yes	—
27	C₂×C₈×D₇₃	C₈	1	16	yes	complemented
28	C₂×C₈×D₇₃	C₂×C₄	1	8	yes	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
31	C₄²×D₇₃	C₂×C₄	1	4	yes	—
32	C₄²×D₇₃	C₂×C₄	1	2	yes	—
33	C₄²×D₇₃	C₂³	1	2	yes	—
34	C₄×#10596831	C₂×C₄	1	4	yes	—
35	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
36	C₈×Dic₇₃	C₂³	1	4	yes	—
37	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
40	#11924036	C₂×C₈	1	64	yes	direct product factor
41	C₂×#10596833	C₂×C₈	1	32	yes	complemented
42	C₂×#10596833	C₂²×C₄	1	16	yes	—
43	C₂³×D₇₃	C₄²	1	1	yes	characteristic
44	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
45	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
46	C₂×C₄×D₇₃	C₄²	1	4	yes	—
47	C₂×C₄×D₇₃	C₄²	1	2	yes	—
48	C₂×C₄×D₇₃	C₂×C₈	1	8	yes	—
49	C₂×C₄×D₇₃	C₂×C₈	1	16	yes	—
50	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×D₇₃	C₂²×C₄	1	4	yes	—
52	C₂×#10596831	C₂×C₈	1	8	yes	complemented
53	C₂×#10596831	C₂²×C₄	1	4	yes	—
54	C₈×D₇₃	C₂×C₈	1	32	yes	complemented
55	C₄×Dic₇₃	C₄²	1	4	yes	—
56	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
58	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
59	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
61	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	C₂×C₈	1	8	yes	complemented
63	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
64	C₄²×C₇₃	C₄²	1	2	yes	—
65	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
66	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
67	C₂²×D₇₃	C₄×C₈	1	8	yes	—
68	C₂²×D₇₃	C₄×C₈	1	4	yes	—
69	C₂²×D₇₃	C₂×C₄²	1	2	yes	—
70	C₂×C₄×C₇₃	C₄×C₈	1	8	yes	—
71	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
73	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
74	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
75	C₂×Dic₇₃	C₄×C₈	1	4	yes	—
76	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
77	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
78	C₂×Dic₇₃	C₂²×C₈	1	8	yes	—
79	C₂×Dic₇₃	C₂³×C₄	1	2	yes	—
80	C₄×D₇₃	C₄×C₈	1	16	yes	—
81	C₄×D₇₃	C₄×C₈	1	8	yes	—
82	C₄×D₇₃	C₂²×C₈	1	16	yes	—
83	C₄×D₇₃	C₂²×C₈	1	8	yes	—
84	#10596833	C₂²×C₈	1	32	yes	complemented
85	C₈×C₇₃	C₂²×C₈	1	8	yes	complemented
86	#10596831	C₂²×C₈	1	8	yes	complemented
87	C₂²×C₇₃	C₂×C₄×C₈	1	4	yes	—
88	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
89	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
90	C₂×D₇₃	C₈²	1	16	yes	complemented
91	C₂×D₇₃	C₂×C₄×C₈	1	8	yes	—
92	C₂×D₇₃	C₂×C₄×C₈	1	4	yes	—
93	C₄×C₇₃	C₂×C₄×C₈	1	8	yes	—
94	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
95	C₄×C₇₃	C₂³×C₈	1	2	yes	—
96	Dic₇₃	C₂×C₄×C₈	1	4	yes	—
97	Dic₇₃	C₂×C₄×C₈	1	4	yes	—
98	Dic₇₃	C₂³×C₈	1	4	yes	—
99	C₂²×C₈²	—	73	1	no	maximal, Sylow
100	C₂×C₇₃	C₂×C₈²	1	4	yes	complemented
101	C₂×C₇₃	C₂²×C₄×C₈	1	2	yes	—
102	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
103	D₇₃	C₂×C₈²	1	8	yes	complemented
104	C₂²×C₄×C₈	—	73	2	no	—
105	C₂²×C₄×C₈	—	73	1	no	—
106	C₂×C₈²	—	73	4	no	—
107	C₂×C₈²	—	73	8	no	—
108	C₇₃	C₂²×C₈²	1	1	yes	characteristic, complemented, Sylow
109	C₂³×C₈	—	73	4	no	—
110	C₂³×C₈	—	73	2	no	—
111	C₂²×C₄²	—	73	1	no	—
112	C₂×C₄×C₈	—	73	4	no	—
113	C₂×C₄×C₈	—	73	4	no	—
114	C₂×C₄×C₈	—	73	8	no	—
115	C₂×C₄×C₈	—	73	8	no	—
116	C₂×C₄×C₈	—	73	2	no	—
117	C₂×C₄×C₈	—	73	1	no	—
118	C₂×C₄×C₈	—	73	4	no	—
119	C₂×C₄×C₈	—	73	4	no	—
120	C₂×C₄×C₈	#64408	1	1	yes	characteristic
121	C₈²	—	73	16	no	—
122	C₂³×C₄	—	73	2	no	—
123	C₂³×C₄	—	73	1	no	—
124	C₂²×C₈	—	73	8	no	—
125	C₂²×C₈	—	73	32	no	—
126	C₂²×C₈	—	73	16	no	—
127	C₂²×C₈	—	73	2	no	—
128	C₂²×C₈	—	73	16	no	—
129	C₂²×C₈	—	73	8	no	—
130	C₂²×C₈	C₂×#64408	1	2	yes	—
131	C₂×C₄²	—	73	2	no	—
132	C₂×C₄²	—	73	1	no	—
133	C₂×C₄²	—	73	4	no	—
134	C₂×C₄²	—	73	2	no	—
135	C₂×C₄²	—	73	2	no	—
136	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
137	C₄×C₈	—	73	8	no	—
138	C₄×C₈	—	73	16	no	—
139	C₄×C₈	—	73	8	no	—
140	C₄×C₈	—	73	4	no	—
141	C₄×C₈	—	73	4	no	—
142	C₄×C₈	—	73	4	no	—
143	C₄×C₈	C₂×#64408	1	4	yes	—
144	C₂⁴	—	73	1	no	—
145	C₂²×C₄	—	73	1	no	—
146	C₂²×C₄	—	73	2	no	—
147	C₂²×C₄	—	73	8	no	—
148	C₂²×C₄	—	73	2	no	—
149	C₂²×C₄	—	73	16	no	—
150	C₂²×C₄	—	73	4	no	—
151	C₂²×C₄	—	73	2	no	—
152	C₂²×C₄	—	73	4	no	—
153	C₂²×C₄	C₄×#64408	1	2	yes	—
154	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
155	C₂×C₈	—	73	32	no	—
156	C₂×C₈	—	73	16	no	—
157	C₂×C₈	—	73	64	no	—
158	C₂×C₈	—	73	8	no	—
159	C₂×C₈	—	73	4	no	—
160	C₂×C₈	—	73	32	no	—
161	C₂×C₈	#11924036	1	8	yes	direct product factor
162	C₂×C₈	C₂²×#64408	1	4	yes	—
163	C₄²	—	73	1	no	—
164	C₄²	—	73	1	no	—
165	C₄²	—	73	4	no	—
166	C₄²	—	73	4	no	—
167	C₄²	—	73	2	no	—
168	C₄²	C₄×#64408	1	2	yes	—
169	C₄²	C₂²×#64408	1	2	yes	—
170	C₂³	—	73	2	no	—
171	C₂³	—	73	8	no	—
172	C₂³	—	73	4	no	—
173	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
174	C₂×C₄	—	73	2	no	—
175	C₂×C₄	—	73	2	no	—
176	C₂×C₄	—	73	2	no	—
177	C₂×C₄	—	73	8	no	—
178	C₂×C₄	—	73	4	no	—
179	C₂×C₄	—	73	8	no	—
180	C₂×C₄	—	73	16	no	—
181	C₂×C₄	—	73	16	no	—
182	C₂×C₄	—	73	8	no	—
183	C₂×C₄	C₈×#64408	1	8	yes	—
184	C₂×C₄	C₂×C₄×#64408	1	4	yes	—
185	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
186	C₂×C₄	C₂×#11924036	1	4	yes	—
187	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
188	C₈	—	73	32	no	—
189	C₈	—	73	8	no	—
190	C₈	C₂×#11924036	1	8	yes	direct product factor
191	C₂²	—	73	16	no	—
192	C₂²	—	73	8	no	—
193	C₂²	—	73	4	no	—
194	C₂²	C₂×C₈×#64408	1	4	yes	—
195	C₂²	C₄×#11924036	1	2	yes	—
196	C₂²	C₂²×C₄×#64408	1	1	yes	characteristic
197	C₄	—	73	4	no	—
198	C₄	—	73	4	no	—
199	C₄	—	73	4	no	—
200	C₄	C₂×C₈×#64408	1	8	yes	—
201	C₄	C₄×#11924036	1	2	yes	—
202	C₄	C₂²×#11924036	1	2	yes	—
203	C₂	—	73	8	no	—
204	C₂	C₈×#11924036	1	4	yes	direct product factor
205	C₂	C₂²×C₈×#64408	1	2	yes	—
206	C₂	C₂×C₄×#11924036	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₈	#64408
commutator, hypocenter	C₇₃	C₂²×C₈²
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂×C₄²
prime radical	C₂³×D₇₃	C₄²
cosocle	C₄×Dic₇₃	C₂⁴
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₈×#11924036	C₂	direct (N × H)
3. C₂×C₈×#10596833	C₂	semidirect
10. C₈×#10596833	C₂²	semidirect
18. C₂×#11924036	C₈	direct (N × H)
27. C₂×C₈×D₇₃	C₈	semidirect
40. #11924036	C₂×C₈	direct (N × H)
41. C₂×#10596833	C₂×C₈	semidirect
52. C₂×#10596831	C₂×C₈	semidirect
54. C₈×D₇₃	C₂×C₈	semidirect
62. C₂×C₈×C₇₃	C₂×C₈	semidirect
84. #10596833	C₂²×C₈	semidirect
85. C₈×C₇₃	C₂²×C₈	semidirect
86. #10596831	C₂²×C₈	semidirect
90. C₂×D₇₃	C₈²	semidirect
100. C₂×C₇₃	C₂×C₈²	semidirect
103. D₇₃	C₂×C₈²	semidirect
108. C₇₃	C₂²×C₈²	semidirect
161. C₂×C₈	#11924036	direct (N × H)
190. C₈	C₂×#11924036	direct (N × H)
204. C₂	C₈×#11924036	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×C₄×#11924036	2
C₈×#11924036	2
C₂×C₈×#10596833	2
C₂×C₄×C₈×D₇₃	2
C₂²×C₈²	73

Subgroups — C2×C8×#11924036 #34015627

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×#11924036 #34015627