Subgroups — C₂²×#17989451 #34017969

Order 18688.

942 conjugacy classes of subgroups.

The subgroup lattice has 184 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	2	yes	maximal
2	C₂×#17989451	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#10596833	—	2	2	no	—
5	C₂×C₄⋊#10596833	—	2	6	no	—
6	C₂×C₄⋊#10596833	C₂²	1	12	yes	—
7	#17989451	C₂²	1	16	yes	direct product factor
8	C₂³×#10596831	C₄	1	1	yes	characteristic
9	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₇₃	C₄	1	6	yes	—
11	C₂×C₈×Dic₇₃	C₂²	1	6	yes	—
12	C₂³×C₈×C₇₃	C₄	1	1	yes	characteristic
13	C₂²×#10596833	—	2	12	no	—
14	C₂²×#10596833	—	4	8	no	—
15	C₄⋊#10596833	—	2	12	no	—
16	C₄⋊#10596833	C₂³	1	8	yes	—
17	C₂³×Dic₇₃	D₄	1	1	yes	characteristic
18	C₂³×Dic₇₃	Q₈	1	1	yes	characteristic
19	C₂²×#10596831	C₈	1	8	yes	complemented
20	C₂²×#10596831	C₂×C₄	1	6	yes	—
21	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
22	C₂×C₄×Dic₇₃	D₄	1	3	yes	—
23	C₂×C₄×Dic₇₃	Q₈	1	3	yes	—
24	C₂×C₄×Dic₇₃	C₂³	1	3	yes	—
25	C₈×Dic₇₃	C₂×C₄	1	12	yes	—
26	C₈×Dic₇₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₃	—	2	4	no	—
29	C₂²×C₈×C₇₃	C₂×C₄	1	6	yes	—
30	C₂×#10596833	—	2	8	no	—
31	C₂×#10596833	—	4	24	no	—
32	C₂²×Dic₇₃	—	2	4	no	—
33	C₂²×Dic₇₃	C₄⋊C₄	1	3	yes	—
34	C₂²×Dic₇₃	C₄⋊C₄	1	3	yes	—
35	C₂²×Dic₇₃	D₈	1	4	yes	—
36	C₂²×Dic₇₃	Q₁₆	1	4	yes	—
37	C₂²×Dic₇₃	C₂×D₄	1	3	yes	—
38	C₂²×Dic₇₃	C₂×Q₈	1	3	yes	—
39	C₂×#10596831	C₂×C₈	1	24	yes	complemented
40	C₂×#10596831	C₂²×C₄	1	4	yes	—
41	C₄×Dic₇₃	C₄⋊C₄	1	6	yes	—
42	C₄×Dic₇₃	C₂²×C₄	1	3	yes	—
43	C₄×Dic₇₃	C₂×D₄	1	3	yes	—
44	C₄×Dic₇₃	C₂×Q₈	1	3	yes	—
45	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
46	C₂⁴×C₇₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	C₄⋊C₄	1	3	yes	—
48	C₂²×C₄×C₇₃	C₂×C₈	1	4	yes	—
49	C₂²×C₄×C₇₃	M₁₆	1	4	yes	—
50	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
51	C₂×C₈×C₇₃	—	2	12	no	—
52	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
53	C₂³×C₇₃	C₄⋊C₈	1	4	yes	—
54	C₂³×C₇₃	#107	1	4	yes	—
55	C₂³×C₇₃	#108	1	4	yes	—
56	C₂³×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
57	C₂×C₄×C₇₃	C₄⋊C₈	1	12	yes	—
58	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
59	C₂×C₄×C₇₃	C₂²×C₈	1	6	yes	—
60	C₂×C₄×C₇₃	C₂×M₁₆	1	6	yes	—
61	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
62	C₂×Dic₇₃	—	2	6	no	—
63	C₂×Dic₇₃	—	2	6	no	—
64	C₂×Dic₇₃	#107	1	12	yes	—
65	C₂×Dic₇₃	C₂×C₄⋊C₄	1	3	yes	—
66	C₂×Dic₇₃	C₂×C₄⋊C₄	1	3	yes	—
67	C₂×Dic₇₃	C₂×D₈	1	6	yes	—
68	C₂×Dic₇₃	C₂×Q₁₆	1	6	yes	—
69	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
70	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
71	#10596833	—	4	8	no	—
72	C₈×C₇₃	—	2	4	no	—
73	#10596831	C₂²×C₈	1	8	yes	complemented
74	C₂²×C₇₃	#335	1	16	yes	complemented
75	C₂²×C₇₃	C₂×C₄⋊C₈	1	6	yes	—
76	C₂²×C₇₃	C₂×#107	1	6	yes	—
77	C₂²×C₇₃	C₂×#108	1	6	yes	—
78	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
79	C₄×C₇₃	C₂×C₄⋊C₈	1	6	yes	—
80	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
81	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
82	Dic₇₃	—	2	3	no	—
83	Dic₇₃	—	2	1	no	—
84	Dic₇₃	C₂×#107	1	6	yes	—
85	Dic₇₃	C₂²×D₈	1	1	yes	characteristic
86	Dic₇₃	C₂²×Q₁₆	1	1	yes	characteristic
87	C₂²×#335	—	73	1	no	maximal, Sylow
88	C₂×C₇₃	C₂×#335	1	12	yes	complemented
89	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
90	C₂×C₇₃	C₂²×#107	1	1	yes	characteristic
91	C₂×C₇₃	C₂²×#108	1	1	yes	characteristic
92	C₂²×C₄⋊C₈	—	73	2	no	—
93	C₂²×C₄×C₈	—	73	1	no	—
94	C₂×#335	—	73	12	no	—
95	C₇₃	C₂²×#335	1	1	yes	characteristic, complemented, Sylow
96	C₂³×C₈	—	73	1	no	—
97	C₂³×C₈	—	146	2	no	—
98	C₂³×C₈	#64408	1	1	yes	characteristic
99	C₂²×C₄²	—	73	1	no	—
100	C₂×C₄⋊C₈	—	73	12	no	—
101	C₂×C₄⋊C₈	—	146	6	no	—
102	C₂×C₄×C₈	—	73	6	no	—
103	C₂×C₄×C₈	—	73	6	no	—
104	#335	—	73	16	no	—
105	C₂³×C₄	—	73	1	no	—
106	C₂³×C₄	—	73	1	no	—
107	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
108	C₂²×C₈	—	2	4	no	—
109	C₂²×C₈	—	292	8	no	—
110	C₂²×C₈	—	73	8	no	—
111	C₂²×C₈	—	73	6	no	—
112	C₂²×C₈	—	146	12	no	—
113	C₂²×C₈	C₂×#64408	1	6	yes	—
114	C₂×C₄²	—	73	3	no	—
115	C₂×C₄²	—	73	3	no	—
116	C₂×C₄²	—	73	3	no	—
117	C₂×C₄²	—	73	3	no	—
118	C₄⋊C₈	—	73	8	no	—
119	C₄⋊C₈	—	146	12	no	—
120	C₄×C₈	—	73	4	no	—
121	C₄×C₈	—	73	12	no	—
122	C₂⁴	C₄⋊#64408	1	1	yes	characteristic
123	C₂²×C₄	—	73	4	no	—
124	C₂²×C₄	—	73	4	no	—
125	C₂²×C₄	—	73	3	no	—
126	C₂²×C₄	—	73	3	no	—
127	C₂²×C₄	—	73	3	no	—
128	C₂²×C₄	—	73	3	no	—
129	C₂²×C₄	—	146	4	no	—
130	C₂²×C₄	C₄⋊#64408	1	3	yes	—
131	C₂²×C₄	#11924036	1	4	yes	—
132	C₂²×C₄	#11924037	1	4	yes	—
133	C₂²×C₄	C₂²×#64408	1	3	yes	—
134	C₂×C₈	—	2	12	no	—
135	C₂×C₈	—	292	24	no	—
136	C₂×C₈	—	73	24	no	—
137	C₂×C₈	—	73	4	no	—
138	C₂×C₈	—	146	8	no	—
139	C₂×C₈	C₂²×#64408	1	4	yes	—
140	C₄²	—	73	3	no	—
141	C₄²	—	73	1	no	—
142	C₄²	—	73	6	no	—
143	C₄²	—	73	3	no	—
144	C₄²	—	73	3	no	—
145	C₂³	#14556658	1	4	yes	—
146	C₂³	#14556660	1	4	yes	—
147	C₂³	#14556661	1	4	yes	—
148	C₂³	C₂×C₄⋊#64408	1	3	yes	—
149	C₂×C₄	—	73	1	no	—
150	C₂×C₄	—	73	3	no	—
151	C₂×C₄	—	73	6	no	—
152	C₂×C₄	—	73	12	no	—
153	C₂×C₄	—	73	6	no	—
154	C₂×C₄	—	73	3	no	—
155	C₂×C₄	—	73	1	no	—
156	C₂×C₄	—	146	6	no	—
157	C₂×C₄	—	146	6	no	—
158	C₂×C₄	#14556658	1	12	yes	—
159	C₂×C₄	C₂×C₄⋊#64408	1	3	yes	—
160	C₂×C₄	C₂×#11924036	1	6	yes	—
161	C₂×C₄	C₂×#11924037	1	6	yes	—
162	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₈	—	292	8	no	—
165	C₈	—	73	8	no	—
166	C₂²	#17989451	1	16	yes	direct product factor
167	C₂²	C₂×#14556658	1	6	yes	—
168	C₂²	C₂×#14556660	1	6	yes	—
169	C₂²	C₂×#14556661	1	6	yes	—
170	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
171	C₄	—	73	1	no	—
172	C₄	—	73	1	no	—
173	C₄	—	73	6	no	—
174	C₄	—	146	3	no	—
175	C₄	—	146	1	no	—
176	C₄	C₂×#14556658	1	6	yes	—
177	C₄	C₂²×#11924036	1	1	yes	characteristic
178	C₄	C₂²×#11924037	1	1	yes	characteristic
179	C₂	C₂×#17989451	1	12	yes	direct product factor
180	C₂	C₂²×#14556658	1	1	yes	characteristic
181	C₂	C₂²×#14556660	1	1	yes	characteristic
182	C₂	C₂²×#14556661	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₄⋊#64408
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂³×C₈×C₇₃	C₄
socle, prime radical	C₂⁴×C₇₃	C₄⋊C₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₂³×C₈	#64408
hypocenter	C₇₃	C₂²×#335
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#17989451	C₂	direct (N × H)
7. #17989451	C₂²	direct (N × H)
19. C₂²×#10596831	C₈	semidirect
39. C₂×#10596831	C₂×C₈	semidirect
73. #10596831	C₂²×C₈	semidirect
74. C₂²×C₇₃	#335	semidirect
88. C₂×C₇₃	C₂×#335	semidirect
95. C₇₃	C₂²×#335	semidirect
166. C₂²	#17989451	direct (N × H)
179. C₂	C₂×#17989451	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×#17989451	2
C₂²×C₈×Dic₇₃	2
C₂²×#335	73

Subgroups — C22×#17989451 #34017969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17989451 #34017969