Subgroups — C₂²×C₄×#11924036 #34031060

Order 18688.

4638 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11924036	C₂	1	2	yes	maximal
2	C₂×C₄×#11924036	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄×#10596833	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924036	C₄	1	32	yes	direct product factor
6	C₂²×#11924036	C₂²	1	24	yes	—
7	C₂³×#10596833	C₂²	1	4	yes	—
8	C₄×#11924036	C₂²	1	64	yes	direct product factor
9	C₂×C₄×#10596833	C₂²	1	48	yes	complemented
10	C₂³×C₄×D₇₃	C₄	1	2	yes	—
11	C₂³×C₄×D₇₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×Dic₇₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×Dic₇₃	C₂²	1	2	yes	—
14	C₂×C₄²×D₇₃	C₄	1	12	yes	—
15	C₂×C₄²×D₇₃	C₂²	1	12	yes	—
16	C₂²×C₄²×C₇₃	C₄	1	1	yes	characteristic
17	C₂×#11924036	C₂×C₄	1	192	yes	direct product factor
18	C₂×#11924036	C₂³	1	32	yes	—
19	C₂²×#10596833	C₂×C₄	1	32	yes	complemented
20	C₂²×#10596833	C₂³	1	24	yes	—
21	C₄×#10596833	C₂³	1	64	yes	complemented
22	C₂⁴×D₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×Dic₇₃	C₂×C₄	1	2	yes	—
24	C₂³×Dic₇₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×D₇₃	C₈	1	32	yes	complemented
26	C₂²×C₄×D₇₃	C₂×C₄	1	16	yes	—
27	C₂²×C₄×D₇₃	C₂×C₄	1	6	yes	—
28	C₂²×C₄×D₇₃	C₂×C₄	1	24	yes	—
29	C₂²×C₄×D₇₃	C₂³	1	6	yes	—
30	C₂×C₄×Dic₇₃	C₂×C₄	1	12	yes	—
31	C₂×C₄×Dic₇₃	C₂×C₄	1	12	yes	—
32	C₂×C₄×Dic₇₃	C₂³	1	12	yes	—
33	C₄²×D₇₃	C₂×C₄	1	48	yes	—
34	C₄²×D₇₃	C₂³	1	16	yes	—
35	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₇₃	C₂×C₄	1	12	yes	—
38	#11924036	C₂²×C₄	1	128	yes	direct product factor
39	C₂×#10596833	C₂²×C₄	1	96	yes	complemented
40	C₂×#10596833	C₂⁴	1	16	yes	—
41	C₂³×D₇₃	C₄²	1	8	yes	—
42	C₂³×D₇₃	C₂×C₈	1	16	yes	—
43	C₂³×D₇₃	C₂²×C₄	1	6	yes	—
44	C₂²×Dic₇₃	C₂×C₈	1	16	yes	complemented
45	C₂²×Dic₇₃	C₂²×C₄	1	3	yes	—
46	C₂²×Dic₇₃	C₂²×C₄	1	8	yes	—
47	C₂²×Dic₇₃	C₂²×C₄	1	12	yes	—
48	C₂²×Dic₇₃	C₂⁴	1	3	yes	—
49	C₂×C₄×D₇₃	C₄²	1	48	yes	—
50	C₂×C₄×D₇₃	C₂×C₈	1	192	yes	complemented
51	C₂×C₄×D₇₃	C₂²×C₄	1	12	yes	—
52	C₂×C₄×D₇₃	C₂²×C₄	1	48	yes	—
53	C₂×C₄×D₇₃	C₂²×C₄	1	32	yes	—
54	C₂×C₄×D₇₃	C₂⁴	1	4	yes	—
55	C₄×Dic₇₃	C₂²×C₄	1	24	yes	—
56	C₄×Dic₇₃	C₂²×C₄	1	16	yes	—
57	C₄×Dic₇₃	C₂⁴	1	8	yes	—
58	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	C₄²	1	8	yes	—
60	C₂²×C₄×C₇₃	C₂×C₈	1	16	yes	complemented
61	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₇₃	C₂²×C₄	1	12	yes	—
63	C₄²×C₇₃	C₂²×C₄	1	16	yes	—
64	C₂³×C₇₃	C₂×C₄²	1	4	yes	—
65	C₂³×C₇₃	C₂²×C₈	1	8	yes	—
66	C₂³×C₇₃	C₂³×C₄	1	3	yes	—
67	C₂²×D₇₃	C₄×C₈	1	64	yes	complemented
68	C₂²×D₇₃	C₂×C₄²	1	24	yes	—
69	C₂²×D₇₃	C₂²×C₈	1	48	yes	—
70	C₂²×D₇₃	C₂³×C₄	1	4	yes	—
71	C₂×C₄×C₇₃	C₂×C₄²	1	24	yes	—
72	C₂×C₄×C₇₃	C₂²×C₈	1	48	yes	complemented
73	C₂×C₄×C₇₃	C₂³×C₄	1	8	yes	—
74	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
75	C₂×Dic₇₃	C₂×C₄²	1	12	yes	—
76	C₂×Dic₇₃	C₂²×C₈	1	48	yes	complemented
77	C₂×Dic₇₃	C₂³×C₄	1	8	yes	—
78	C₂×Dic₇₃	C₂³×C₄	1	12	yes	—
79	C₂×Dic₇₃	C₂³×C₄	1	3	yes	—
80	C₂×Dic₇₃	C₂⁵	1	1	yes	characteristic
81	C₄×D₇₃	C₂×C₄²	1	48	yes	—
82	C₄×D₇₃	C₂²×C₈	1	128	yes	complemented
83	C₄×D₇₃	C₂³×C₄	1	16	yes	—
84	#10596833	C₂³×C₄	1	32	yes	complemented
85	C₂²×C₇₃	C₂×C₄×C₈	1	16	yes	complemented
86	C₂²×C₇₃	C₂²×C₄²	1	6	yes	—
87	C₂²×C₇₃	C₂³×C₈	1	12	yes	—
88	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
89	C₂×D₇₃	C₂×C₄×C₈	1	96	yes	complemented
90	C₂×D₇₃	C₂²×C₄²	1	8	yes	—
91	C₂×D₇₃	C₂³×C₈	1	16	yes	—
92	C₄×C₇₃	C₂²×C₄²	1	8	yes	—
93	C₄×C₇₃	C₂³×C₈	1	16	yes	complemented
94	Dic₇₃	C₂²×C₄²	1	6	yes	—
95	Dic₇₃	C₂³×C₈	1	16	yes	complemented
96	Dic₇₃	C₂⁴×C₄	1	2	yes	—
97	C₂³×C₄×C₈	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	C₂²×C₄×C₈	1	12	yes	complemented
99	C₂×C₇₃	C₂³×C₄²	1	1	yes	characteristic
100	C₂×C₇₃	C₂⁴×C₈	1	2	yes	—
101	D₇₃	C₂²×C₄×C₈	1	16	yes	complemented
102	C₂⁴×C₈	—	73	2	no	—
103	C₂³×C₄²	—	73	1	no	—
104	C₂²×C₄×C₈	—	73	4	no	—
105	C₂²×C₄×C₈	—	73	24	no	—
106	C₇₃	C₂³×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
107	C₂⁴×C₄	—	73	2	no	—
108	C₂⁴×C₄	—	73	1	no	—
109	C₂³×C₈	—	73	4	no	—
110	C₂³×C₈	—	73	32	no	—
111	C₂³×C₈	—	73	24	no	—
112	C₂²×C₄²	—	73	2	no	—
113	C₂²×C₄²	—	73	1	no	—
114	C₂²×C₄²	—	73	12	no	—
115	C₂²×C₄²	—	73	12	no	—
116	C₂²×C₄²	#64408	1	1	yes	characteristic
117	C₂×C₄×C₈	—	73	48	no	—
118	C₂×C₄×C₈	—	73	64	no	—
119	C₂⁵	—	73	1	no	—
120	C₂³×C₄	—	73	1	no	—
121	C₂³×C₄	—	73	2	no	—
122	C₂³×C₄	—	73	16	no	—
123	C₂³×C₄	—	73	32	no	—
124	C₂³×C₄	—	73	6	no	—
125	C₂³×C₄	—	73	24	no	—
126	C₂³×C₄	—	73	6	no	—
127	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
128	C₂³×C₄	C₂×#64408	1	2	yes	—
129	C₂²×C₈	—	73	32	no	—
130	C₂²×C₈	—	73	24	no	—
131	C₂²×C₈	—	73	192	no	—
132	C₂²×C₈	—	73	32	no	—
133	C₂×C₄²	—	73	12	no	—
134	C₂×C₄²	—	73	12	no	—
135	C₂×C₄²	—	73	12	no	—
136	C₂×C₄²	—	73	48	no	—
137	C₂×C₄²	—	73	16	no	—
138	C₂×C₄²	C₂×#64408	1	12	yes	—
139	C₄×C₈	—	73	64	no	—
140	C₂⁴	—	73	16	no	—
141	C₂⁴	—	73	6	no	—
142	C₂⁴	—	73	8	no	—
143	C₂⁴	C₂²×#64408	1	1	yes	characteristic
144	C₂²×C₄	—	73	16	no	—
145	C₂²×C₄	—	73	3	no	—
146	C₂²×C₄	—	73	12	no	—
147	C₂²×C₄	—	73	3	no	—
148	C₂²×C₄	—	73	8	no	—
149	C₂²×C₄	—	73	192	no	—
150	C₂²×C₄	—	73	12	no	—
151	C₂²×C₄	—	73	32	no	—
152	C₂²×C₄	—	73	4	no	—
153	C₂²×C₄	—	73	48	no	—
154	C₂²×C₄	—	73	48	no	—
155	C₂²×C₄	C₄×#64408	1	8	yes	—
156	C₂²×C₄	#11924036	1	16	yes	direct product factor
157	C₂²×C₄	C₂²×#64408	1	12	yes	—
158	C₂²×C₄	C₂²×#64408	1	6	yes	—
159	C₂×C₈	—	73	96	no	—
160	C₂×C₈	—	73	16	no	—
161	C₂×C₈	—	73	128	no	—
162	C₄²	—	73	16	no	—
163	C₄²	—	73	24	no	—
164	C₄²	—	73	8	no	—
165	C₄²	C₂²×#64408	1	16	yes	—
166	C₂³	—	73	24	no	—
167	C₂³	—	73	48	no	—
168	C₂³	—	73	4	no	—
169	C₂³	—	73	64	no	—
170	C₂³	C₂×C₄×#64408	1	4	yes	—
171	C₂³	C₂×#11924036	1	8	yes	—
172	C₂³	C₂³×#64408	1	3	yes	—
173	C₂×C₄	—	73	12	no	—
174	C₂×C₄	—	73	12	no	—
175	C₂×C₄	—	73	48	no	—
176	C₂×C₄	—	73	1	no	—
177	C₂×C₄	—	73	3	no	—
178	C₂×C₄	—	73	8	no	—
179	C₂×C₄	—	73	16	no	—
180	C₂×C₄	—	73	48	no	—
181	C₂×C₄	—	73	128	no	—
182	C₂×C₄	C₂×C₄×#64408	1	24	yes	—
183	C₂×C₄	C₂×#11924036	1	48	yes	direct product factor
184	C₂×C₄	C₂³×#64408	1	4	yes	—
185	C₂×C₄	C₂³×#64408	1	8	yes	—
186	C₈	—	73	32	no	—
187	C₂²	—	73	8	no	—
188	C₂²	—	73	16	no	—
189	C₂²	—	73	96	no	—
190	C₂²	C₄×#11924036	1	16	yes	direct product factor
191	C₂²	C₂²×C₄×#64408	1	6	yes	—
192	C₂²	C₂²×#11924036	1	12	yes	—
193	C₂²	C₂⁴×#64408	1	1	yes	characteristic
194	C₄	—	73	2	no	—
195	C₄	—	73	6	no	—
196	C₄	—	73	16	no	—
197	C₄	C₂²×C₄×#64408	1	8	yes	—
198	C₄	C₂²×#11924036	1	16	yes	direct product factor
199	C₂	—	73	16	no	—
200	C₂	C₂×C₄×#11924036	1	12	yes	direct product factor
201	C₂	C₂³×C₄×#64408	1	1	yes	characteristic
202	C₂	C₂³×#11924036	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	#64408
commutator, hypocenter	C₇₃	C₂³×C₄×C₈
Frattini	C₂²	C₂⁴×#64408
Fitting	C₂²×C₄²×C₇₃	C₄
socle	C₂⁴×C₇₃	C₂²×C₄
prime radical	C₂⁴×D₇₃	C₂×C₄
cosocle	C₂×Dic₇₃	C₂⁵
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#11924036	C₂	direct (N × H)
3. C₂²×C₄×#10596833	C₂	semidirect
5. C₂²×#11924036	C₄	direct (N × H)
8. C₄×#11924036	C₂²	direct (N × H)
9. C₂×C₄×#10596833	C₂²	semidirect
17. C₂×#11924036	C₂×C₄	direct (N × H)
19. C₂²×#10596833	C₂×C₄	semidirect
21. C₄×#10596833	C₂³	semidirect
25. C₂²×C₄×D₇₃	C₈	semidirect
38. #11924036	C₂²×C₄	direct (N × H)
39. C₂×#10596833	C₂²×C₄	semidirect
44. C₂²×Dic₇₃	C₂×C₈	semidirect
50. C₂×C₄×D₇₃	C₂×C₈	semidirect
60. C₂²×C₄×C₇₃	C₂×C₈	semidirect
67. C₂²×D₇₃	C₄×C₈	semidirect
72. C₂×C₄×C₇₃	C₂²×C₈	semidirect
76. C₂×Dic₇₃	C₂²×C₈	semidirect
82. C₄×D₇₃	C₂²×C₈	semidirect
84. #10596833	C₂³×C₄	semidirect
85. C₂²×C₇₃	C₂×C₄×C₈	semidirect
89. C₂×D₇₃	C₂×C₄×C₈	semidirect
93. C₄×C₇₃	C₂³×C₈	semidirect
95. Dic₇₃	C₂³×C₈	semidirect
98. C₂×C₇₃	C₂²×C₄×C₈	semidirect
101. D₇₃	C₂²×C₄×C₈	semidirect
106. C₇₃	C₂³×C₄×C₈	semidirect
156. C₂²×C₄	#11924036	direct (N × H)
183. C₂×C₄	C₂×#11924036	direct (N × H)
190. C₂²	C₄×#11924036	direct (N × H)
198. C₄	C₂²×#11924036	direct (N × H)
200. C₂	C₂×C₄×#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
C₂³×#11924036	2
C₂×C₄×#11924036	2
C₂²×C₄×#10596833	2
C₂²×C₄²×D₇₃	2
C₂³×C₄×C₈	73

Subgroups — C22×C4×#11924036 #34031060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#11924036 #34031060