Subgroups — C₂²×#17989676 #34034512

Order 18688.

942 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#10596842	C₂	1	2	yes	maximal
2	C₂×#17989676	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×#64408	C₂	1	1	yes	maximal, characteristic
4	C₂³×#10596842	—	2	2	no	—
5	C₂×C₄⋊#10596842	—	2	6	no	—
6	C₂×C₄⋊#10596842	C₂²	1	12	yes	—
7	#17989676	C₂²	1	16	yes	direct product factor
8	C₂²×#11924036	C₄	1	1	yes	characteristic
9	C₂²×C₄×#64408	C₂²	1	1	yes	characteristic
10	C₂×C₈×#64408	C₄	1	6	yes	—
11	C₂×C₈×#64408	C₂²	1	6	yes	—
12	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
13	C₂²×#10596842	—	2	12	no	—
14	C₂²×#10596842	—	4	8	no	—
15	C₄⋊#10596842	—	2	12	no	—
16	C₄⋊#10596842	C₂³	1	8	yes	—
17	C₂³×#64408	D₄	1	1	yes	characteristic
18	C₂³×#64408	Q₈	1	1	yes	characteristic
19	C₂×#11924036	—	2	3	no	—
20	C₂×#11924036	C₂×C₄	1	6	yes	—
21	C₂²×#10596833	—	2	1	no	—
22	C₂×C₄×#64408	C₂×C₄	1	3	yes	—
23	C₂×C₄×#64408	D₄	1	3	yes	—
24	C₂×C₄×#64408	Q₈	1	3	yes	—
25	C₂×C₄×#64408	C₂³	1	3	yes	—
26	C₈×#64408	C₂×C₄	1	12	yes	—
27	C₈×#64408	C₂³	1	4	yes	—
28	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×#10596831	C₈	1	1	yes	characteristic, complemented
30	C₂×C₈×D₇₃	C₈	1	6	yes	complemented
31	C₂×C₈×D₇₃	C₂×C₄	1	6	yes	—
32	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic, complemented
33	C₂×#10596842	—	2	8	no	—
34	C₂×#10596842	—	4	24	no	—
35	C₂²×#64408	—	2	4	no	—
36	C₂²×#64408	C₄⋊C₄	1	3	yes	—
37	C₂²×#64408	C₄⋊C₄	1	3	yes	—
38	C₂²×#64408	SD₁₆	1	8	yes	—
39	C₂²×#64408	C₂×D₄	1	3	yes	—
40	C₂²×#64408	C₂×Q₈	1	3	yes	—
41	#11924036	—	2	6	no	—
42	#11924036	C₂²×C₄	1	4	yes	—
43	C₂×#10596833	—	2	6	no	—
44	C₄×#64408	C₄⋊C₄	1	6	yes	—
45	C₄×#64408	C₂²×C₄	1	3	yes	—
46	C₄×#64408	C₂×D₄	1	3	yes	—
47	C₄×#64408	C₂×Q₈	1	3	yes	—
48	C₄×#64408	C₂⁴	1	1	yes	characteristic
49	C₂³×D₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×Dic₇₃	M₁₆	1	1	yes	characteristic
51	C₂×C₄×D₇₃	C₄⋊C₄	1	3	yes	—
52	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
53	C₂×C₄×D₇₃	M₁₆	1	3	yes	—
54	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
55	C₂×#10596831	C₂×C₈	1	6	yes	complemented
56	C₈×D₇₃	C₂×C₈	1	12	yes	complemented
57	C₈×D₇₃	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
59	C₂×C₈×C₇₃	C₂×C₈	1	6	yes	complemented
60	C₂³×C₇₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂²×D₇₃	C₄⋊C₈	1	3	yes	—
62	C₂²×D₇₃	#106	1	4	yes	—
63	C₂²×D₇₃	#108	1	4	yes	—
64	C₂²×D₇₃	C₂×C₄⋊C₄	1	3	yes	—
65	C₂×#64408	—	2	6	no	—
66	C₂×#64408	—	2	6	no	—
67	C₂×#64408	#106	1	12	yes	—
68	C₂×#64408	C₂×C₄⋊C₄	1	3	yes	—
69	C₂×#64408	C₂×C₄⋊C₄	1	3	yes	—
70	C₂×#64408	C₂×SD₁₆	1	12	yes	—
71	C₂×#64408	C₂²×D₄	1	1	yes	characteristic
72	C₂×#64408	C₂²×Q₈	1	1	yes	characteristic
73	#10596842	—	4	8	no	—
74	C₂×C₄×C₇₃	C₄⋊C₈	1	3	yes	—
75	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
76	C₂×Dic₇₃	C₄⋊C₈	1	3	yes	—
77	C₂×Dic₇₃	C₂×M₁₆	1	3	yes	—
78	C₄×D₇₃	C₄⋊C₈	1	6	yes	—
79	C₄×D₇₃	C₂×C₄⋊C₄	1	3	yes	—
80	C₄×D₇₃	C₂²×C₈	1	3	yes	—
81	C₄×D₇₃	C₂×M₁₆	1	3	yes	—
82	C₄×D₇₃	C₂³×C₄	1	1	yes	characteristic
83	#10596833	—	2	4	no	—
84	C₈×C₇₃	C₂²×C₈	1	4	yes	complemented
85	#10596831	C₂²×C₈	1	4	yes	complemented
86	C₂²×C₇₃	#334	1	4	yes	complemented
87	C₂²×C₇₃	C₂×C₄⋊C₈	1	3	yes	—
88	C₂×D₇₃	#334	1	12	yes	complemented
89	C₂×D₇₃	C₂×C₄⋊C₈	1	3	yes	—
90	C₂×D₇₃	C₂×#106	1	6	yes	—
91	C₂×D₇₃	C₂×#108	1	6	yes	—
92	C₂×D₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	#64408	—	2	3	no	—
94	#64408	—	2	1	no	—
95	#64408	C₂×#106	1	6	yes	—
96	#64408	C₂²×SD₁₆	1	2	yes	—
97	C₄×C₇₃	C₂×C₄⋊C₈	1	3	yes	—
98	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
99	Dic₇₃	C₂×C₄⋊C₈	1	3	yes	—
100	Dic₇₃	C₂²×M₁₆	1	1	yes	characteristic
101	C₂²×#334	—	73	1	no	maximal, Sylow
102	C₂×C₇₃	C₂×#334	1	6	yes	complemented
103	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
104	D₇₃	C₂×#334	1	6	yes	complemented
105	D₇₃	C₂²×#106	1	1	yes	characteristic
106	D₇₃	C₂²×#108	1	1	yes	characteristic
107	C₂²×C₄⋊C₈	—	73	2	no	—
108	C₂²×C₄×C₈	—	73	1	no	—
109	C₂×#334	—	73	12	no	—
110	C₇₃	C₂²×#334	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₈	—	73	1	no	—
112	C₂³×C₈	—	73	1	no	—
113	C₂³×C₈	—	146	2	no	—
114	C₂²×C₄²	—	73	1	no	—
115	C₂×C₄⋊C₈	—	73	12	no	—
116	C₂×C₄⋊C₈	—	146	6	no	—
117	C₂×C₄×C₈	—	73	6	no	—
118	C₂×C₄×C₈	—	73	6	no	—
119	#334	—	73	16	no	—
120	C₂³×C₄	—	73	1	no	—
121	C₂³×C₄	—	73	1	no	—
122	C₂³×C₄	—	73	1	no	—
123	C₂²×C₈	—	292	8	no	—
124	C₂²×C₈	—	73	1	no	—
125	C₂²×C₈	—	73	6	no	—
126	C₂²×C₈	—	73	6	no	—
127	C₂²×C₈	—	73	6	no	—
128	C₂²×C₈	—	146	12	no	—
129	C₂²×C₈	—	146	1	no	—
130	C₂²×C₈	—	146	3	no	—
131	C₂²×C₈	#10596842	1	1	yes	characteristic, complemented
132	C₂×C₄²	—	73	3	no	—
133	C₂×C₄²	—	73	3	no	—
134	C₂×C₄²	—	73	3	no	—
135	C₂×C₄²	—	73	3	no	—
136	C₄⋊C₈	—	73	8	no	—
137	C₄⋊C₈	—	146	12	no	—
138	C₄×C₈	—	73	4	no	—
139	C₄×C₈	—	73	12	no	—
140	C₂⁴	—	73	1	no	—
141	C₂²×C₄	—	73	3	no	—
142	C₂²×C₄	—	73	8	no	—
143	C₂²×C₄	—	73	3	no	—
144	C₂²×C₄	—	73	1	no	—
145	C₂²×C₄	—	73	3	no	—
146	C₂²×C₄	—	73	3	no	—
147	C₂²×C₄	—	73	3	no	—
148	C₂²×C₄	—	73	3	no	—
149	C₂²×C₄	—	73	3	no	—
150	C₂²×C₄	—	73	3	no	—
151	C₂²×C₄	—	146	4	no	—
152	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
153	C₂×C₈	—	292	24	no	—
154	C₂×C₈	—	73	6	no	—
155	C₂×C₈	—	73	12	no	—
156	C₂×C₈	—	73	4	no	—
157	C₂×C₈	—	73	4	no	—
158	C₂×C₈	—	146	8	no	—
159	C₂×C₈	—	146	6	no	—
160	C₂×C₈	—	146	6	no	—
161	C₂×C₈	C₂×#10596842	1	6	yes	complemented
162	C₄²	—	73	6	no	—
163	C₄²	—	73	3	no	—
164	C₄²	—	73	3	no	—
165	C₄²	—	73	3	no	—
166	C₄²	—	73	1	no	—
167	C₂³	—	73	4	no	—
168	C₂³	—	73	4	no	—
169	C₂³	—	73	3	no	—
170	C₂³	—	73	3	no	—
171	C₂³	C₄⋊#10596842	1	1	yes	characteristic
172	C₂×C₄	—	73	1	no	—
173	C₂×C₄	—	73	3	no	—
174	C₂×C₄	—	73	12	no	—
175	C₂×C₄	—	73	12	no	—
176	C₂×C₄	—	73	3	no	—
177	C₂×C₄	—	73	3	no	—
178	C₂×C₄	—	73	3	no	—
179	C₂×C₄	—	73	3	no	—
180	C₂×C₄	—	73	6	no	—
181	C₂×C₄	—	73	1	no	—
182	C₂×C₄	—	73	3	no	—
183	C₂×C₄	—	73	3	no	—
184	C₂×C₄	—	73	1	no	—
185	C₂×C₄	—	146	6	no	—
186	C₂×C₄	—	146	6	no	—
187	C₂×C₄	C₄⋊#10596842	1	3	yes	—
188	C₂×C₄	C₂²×#10596842	1	3	yes	—
189	C₈	—	292	8	no	—
190	C₈	—	73	4	no	—
191	C₈	—	146	4	no	—
192	C₈	C₂²×#10596842	1	4	yes	complemented
193	C₂²	—	73	1	no	—
194	C₂²	—	73	3	no	—
195	C₂²	—	73	6	no	—
196	C₂²	—	73	6	no	—
197	C₂²	—	73	12	no	—
198	C₂²	#17989676	1	4	yes	direct product factor
199	C₂²	C₂×C₄⋊#10596842	1	3	yes	—
200	C₄	—	73	2	no	—
201	C₄	—	73	6	no	—
202	C₄	—	73	1	no	—
203	C₄	—	73	3	no	—
204	C₄	—	146	3	no	—
205	C₄	—	146	1	no	—
206	C₄	C₂×C₄⋊#10596842	1	3	yes	—
207	C₄	C₂³×#10596842	1	1	yes	characteristic
208	C₂	—	73	1	no	—
209	C₂	—	73	1	no	—
210	C₂	—	73	6	no	—
211	C₂	C₂×#17989676	1	6	yes	direct product factor
212	C₂	C₂²×C₄⋊#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₄⋊#10596842
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₄	C₂³×#10596842
Fitting	C₂²×C₈×C₇₃	C₈
socle	C₂³×C₇₃	C₄⋊C₈
prime radical	C₂³×D₇₃	C₄⋊C₄
cosocle	C₄×#64408	C₂⁴
hypercenter	C₂²×C₈	#10596842
hypocenter	C₇₃	C₂²×#334
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#17989676	C₂	direct (N × H)
7. #17989676	C₂²	direct (N × H)
29. C₂²×#10596831	C₈	semidirect
30. C₂×C₈×D₇₃	C₈	semidirect
32. C₂²×C₈×C₇₃	C₈	semidirect
55. C₂×#10596831	C₂×C₈	semidirect
56. C₈×D₇₃	C₂×C₈	semidirect
59. C₂×C₈×C₇₃	C₂×C₈	semidirect
84. C₈×C₇₃	C₂²×C₈	semidirect
85. #10596831	C₂²×C₈	semidirect
86. C₂²×C₇₃	#334	semidirect
88. C₂×D₇₃	#334	semidirect
102. C₂×C₇₃	C₂×#334	semidirect
104. D₇₃	C₂×#334	semidirect
110. C₇₃	C₂²×#334	semidirect
131. C₂²×C₈	#10596842	semidirect
161. C₂×C₈	C₂×#10596842	semidirect
192. C₈	C₂²×#10596842	semidirect
198. C₂²	#17989676	direct (N × H)
211. C₂	C₂×#17989676	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⋊#10596842	2
C₂×#17989676	2
C₂²×C₈×#64408	2
C₂²×#334	73

Subgroups — C22×#17989676 #34034512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17989676 #34034512