Subgroups — C₂²×#17989429 #34015630

Order 18688.

1182 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#10596833	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#10596833	C₂	1	1	yes	maximal, characteristic
3	C₂×#17989429	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#10596833	C₄	1	2	yes	—
6	C₂³×#10596833	C₄	1	2	yes	—
7	C₂×C₄×#10596833	C₄	1	6	yes	—
8	C₂×C₄×#10596833	C₄	1	6	yes	—
9	C₂×C₄×#10596833	C₂²	1	6	yes	—
10	C₂×C₄×#10596833	C₂²	1	6	yes	—
11	#17989429	C₂²	1	16	yes	direct product factor
12	C₂³×#10596831	C₄	1	1	yes	characteristic
13	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₄	1	6	yes	—
15	C₂×C₈×Dic₇₃	C₂²	1	6	yes	—
16	C₂³×C₈×C₇₃	C₄	1	1	yes	characteristic
17	C₂²×#10596833	—	2	8	no	—
18	C₂²×#10596833	C₈	1	16	yes	complemented
19	C₂²×#10596833	C₂×C₄	1	12	yes	—
20	C₂²×#10596833	C₂×C₄	1	12	yes	—
21	C₄×#10596833	C₂×C₄	1	12	yes	—
22	C₄×#10596833	C₂×C₄	1	12	yes	—
23	C₄×#10596833	C₂³	1	4	yes	—
24	C₄×#10596833	C₂³	1	4	yes	—
25	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×#10596831	—	2	4	no	—
28	C₂²×#10596831	C₂×C₄	1	6	yes	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
30	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
31	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
32	C₂×C₄×Dic₇₃	C₂³	1	3	yes	—
33	C₈×Dic₇₃	C₂×C₄	1	12	yes	—
34	C₈×Dic₇₃	C₂³	1	4	yes	—
35	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₃	—	2	4	no	—
37	C₂²×C₈×C₇₃	C₂×C₄	1	6	yes	—
38	C₂×#10596833	—	2	24	no	—
39	C₂×#10596833	C₂×C₈	1	48	yes	complemented
40	C₂×#10596833	C₂²×C₄	1	8	yes	—
41	C₂×#10596833	C₂²×C₄	1	8	yes	—
42	C₂²×Dic₇₃	C₄²	1	3	yes	—
43	C₂²×Dic₇₃	C₄²	1	3	yes	—
44	C₂²×Dic₇₃	C₂×C₈	1	8	yes	—
45	C₂²×Dic₇₃	M₁₆	1	8	yes	—
46	C₂²×Dic₇₃	C₂²×C₄	1	3	yes	—
47	C₂²×Dic₇₃	C₂²×C₄	1	3	yes	—
48	C₂×#10596831	—	2	12	no	—
49	C₂×#10596831	C₂²×C₄	1	4	yes	—
50	C₄×Dic₇₃	C₄²	1	6	yes	—
51	C₄×Dic₇₃	C₂²×C₄	1	3	yes	—
52	C₄×Dic₇₃	C₂²×C₄	1	3	yes	—
53	C₄×Dic₇₃	C₂²×C₄	1	3	yes	—
54	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
55	C₂⁴×C₇₃	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	C₄²	1	3	yes	—
57	C₂²×C₄×C₇₃	M₁₆	1	4	yes	—
58	C₂²×C₄×C₇₃	M₁₆	1	4	yes	—
59	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
60	C₂×C₈×C₇₃	—	2	12	no	—
61	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
62	C₂³×C₇₃	C₄×C₈	1	4	yes	—
63	C₂³×C₇₃	#97	1	4	yes	—
64	C₂³×C₇₃	#97	1	4	yes	—
65	C₂³×C₇₃	C₂×C₄²	1	3	yes	—
66	C₂×C₄×C₇₃	#97	1	12	yes	—
67	C₂×C₄×C₇₃	C₂×C₄²	1	3	yes	—
68	C₂×C₄×C₇₃	C₂×M₁₆	1	6	yes	—
69	C₂×C₄×C₇₃	C₂×M₁₆	1	6	yes	—
70	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
71	C₂×Dic₇₃	C₄×C₈	1	12	yes	—
72	C₂×Dic₇₃	#97	1	12	yes	—
73	C₂×Dic₇₃	C₂×C₄²	1	3	yes	—
74	C₂×Dic₇₃	C₂×C₄²	1	3	yes	—
75	C₂×Dic₇₃	C₂²×C₈	1	12	yes	—
76	C₂×Dic₇₃	C₂×M₁₆	1	12	yes	—
77	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
78	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
79	#10596833	—	2	8	no	—
80	#10596833	C₂²×C₈	1	16	yes	complemented
81	C₈×C₇₃	—	2	4	no	—
82	#10596831	—	2	4	no	—
83	C₂²×C₇₃	#322	1	16	yes	complemented
84	C₂²×C₇₃	C₂×C₄×C₈	1	6	yes	—
85	C₂²×C₇₃	C₂×#97	1	6	yes	—
86	C₂²×C₇₃	C₂×#97	1	6	yes	—
87	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
88	C₄×C₇₃	C₂×#97	1	6	yes	—
89	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
90	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
91	Dic₇₃	C₂×C₄×C₈	1	6	yes	—
92	Dic₇₃	C₂×#97	1	6	yes	—
93	Dic₇₃	C₂³×C₈	1	2	yes	—
94	Dic₇₃	C₂²×M₁₆	1	2	yes	—
95	C₂²×#322	—	73	1	no	maximal, Sylow
96	C₂×C₇₃	C₂×#322	1	12	yes	complemented
97	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
98	C₂×C₇₃	C₂²×#97	1	1	yes	characteristic
99	C₂×C₇₃	C₂²×#97	1	1	yes	characteristic
100	C₂²×C₄×C₈	—	73	1	no	—
101	C₂²×C₄×C₈	—	73	1	no	—
102	C₂²×C₄×C₈	—	73	1	no	—
103	C₂×#322	—	73	12	no	—
104	C₇₃	C₂²×#322	1	1	yes	characteristic, complemented, Sylow
105	C₂³×C₈	—	73	2	no	—
106	C₂³×C₈	—	73	2	no	—
107	C₂³×C₈	—	73	1	no	—
108	C₂³×C₈	#64408	1	1	yes	characteristic
109	C₂²×C₄²	—	73	1	no	—
110	C₂×C₄×C₈	—	73	6	no	—
111	C₂×C₄×C₈	—	73	6	no	—
112	C₂×C₄×C₈	—	73	6	no	—
113	C₂×C₄×C₈	—	73	6	no	—
114	C₂×C₄×C₈	—	73	6	no	—
115	C₂×C₄×C₈	—	73	6	no	—
116	#322	—	73	16	no	—
117	C₂³×C₄	—	73	1	no	—
118	C₂³×C₄	—	73	1	no	—
119	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
120	C₂²×C₈	—	2	4	no	—
121	C₂²×C₈	—	73	16	no	—
122	C₂²×C₈	—	73	12	no	—
123	C₂²×C₈	—	73	12	no	—
124	C₂²×C₈	—	73	6	no	—
125	C₂²×C₈	—	146	8	no	—
126	C₂²×C₈	—	146	4	no	—
127	C₂²×C₈	C₂×#64408	1	6	yes	—
128	C₂×C₄²	—	73	3	no	—
129	C₂×C₄²	—	73	3	no	—
130	C₂×C₄²	—	73	3	no	—
131	C₂×C₄²	—	73	3	no	—
132	C₄×C₈	—	73	4	no	—
133	C₄×C₈	—	73	4	no	—
134	C₄×C₈	—	73	12	no	—
135	C₄×C₈	—	73	12	no	—
136	C₄×C₈	—	73	4	no	—
137	C₄×C₈	—	73	12	no	—
138	C₂⁴	C₄×#64408	1	1	yes	characteristic
139	C₂²×C₄	—	73	8	no	—
140	C₂²×C₄	—	73	8	no	—
141	C₂²×C₄	—	73	3	no	—
142	C₂²×C₄	—	73	3	no	—
143	C₂²×C₄	—	73	3	no	—
144	C₂²×C₄	—	73	3	no	—
145	C₂²×C₄	C₄×#64408	1	3	yes	—
146	C₂²×C₄	#11924037	1	4	yes	—
147	C₂²×C₄	#11924038	1	4	yes	—
148	C₂²×C₄	C₂²×#64408	1	3	yes	—
149	C₂×C₈	—	2	12	no	—
150	C₂×C₈	—	73	48	no	—
151	C₂×C₈	—	73	8	no	—
152	C₂×C₈	—	73	8	no	—
153	C₂×C₈	—	73	4	no	—
154	C₂×C₈	—	146	24	no	—
155	C₂×C₈	—	146	12	no	—
156	C₂×C₈	C₂²×#64408	1	4	yes	—
157	C₄²	—	73	3	no	—
158	C₄²	—	73	1	no	—
159	C₄²	—	73	6	no	—
160	C₄²	—	73	3	no	—
161	C₄²	—	73	3	no	—
162	C₂³	C₈×#64408	1	4	yes	—
163	C₂³	#14556645	1	4	yes	—
164	C₂³	#14556646	1	4	yes	—
165	C₂³	C₂×C₄×#64408	1	3	yes	—
166	C₂×C₄	—	73	1	no	—
167	C₂×C₄	—	73	1	no	—
168	C₂×C₄	—	73	3	no	—
169	C₂×C₄	—	73	3	no	—
170	C₂×C₄	—	73	12	no	—
171	C₂×C₄	—	73	12	no	—
172	C₂×C₄	—	73	12	no	—
173	C₂×C₄	—	73	12	no	—
174	C₂×C₄	#14556646	1	12	yes	—
175	C₂×C₄	C₂×C₄×#64408	1	3	yes	—
176	C₂×C₄	C₂×#11924037	1	6	yes	—
177	C₂×C₄	C₂×#11924038	1	6	yes	—
178	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
179	C₈	—	2	4	no	—
180	C₈	—	73	16	no	—
181	C₈	—	146	8	no	—
182	C₈	—	146	4	no	—
183	C₂²	#17989429	1	16	yes	direct product factor
184	C₂²	C₂×C₈×#64408	1	6	yes	—
185	C₂²	C₂×#14556645	1	6	yes	—
186	C₂²	C₂×#14556646	1	6	yes	—
187	C₂²	C₂²×C₄×#64408	1	1	yes	characteristic
188	C₄	—	73	2	no	—
189	C₄	—	73	6	no	—
190	C₄	—	73	2	no	—
191	C₄	—	73	6	no	—
192	C₄	C₂×#14556646	1	6	yes	—
193	C₄	C₂²×#11924037	1	1	yes	characteristic
194	C₄	C₂²×#11924038	1	1	yes	characteristic
195	C₂	C₂×#17989429	1	12	yes	direct product factor
196	C₂	C₂²×C₈×#64408	1	1	yes	characteristic
197	C₂	C₂²×#14556645	1	1	yes	characteristic
198	C₂	C₂²×#14556646	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#64408
commutator	C₂×C₇₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂³×C₈×C₇₃	C₄
socle, prime radical	C₂⁴×C₇₃	C₄²
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₂³×C₈	#64408
hypocenter	C₇₃	C₂²×#322
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#17989429	C₂	direct (N × H)
11. #17989429	C₂²	direct (N × H)
18. C₂²×#10596833	C₈	semidirect
39. C₂×#10596833	C₂×C₈	semidirect
80. #10596833	C₂²×C₈	semidirect
83. C₂²×C₇₃	#322	semidirect
96. C₂×C₇₃	C₂×#322	semidirect
104. C₇₃	C₂²×#322	semidirect
183. C₂²	#17989429	direct (N × H)
195. C₂	C₂×#17989429	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×#10596833	2
C₂×#17989429	2
C₂²×C₈×Dic₇₃	2
C₂²×#322	73

Subgroups — C22×#17989429 #34015630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17989429 #34015630