Subgroups — #64408⋊#346 #34034033

Order 18688.

510 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	C₂×#17989682	C₂	1	2	yes	maximal
2	#22627087	C₂	1	2	yes	maximal
3	C₂×#64408⋊C₁₆	C₂	1	2	yes	maximal
4	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556686	C₂²	1	4	yes	—
6	#17989682	C₄	1	8	yes	complemented
7	#17989681	C₄	1	8	yes	complemented
8	#64408⋊C₁₆	C₄	1	8	yes	complemented
9	C₄×#11924036	C₂²	1	1	yes	characteristic
10	C₂×C₈×#64408	C₄	1	1	yes	characteristic
11	C₂×C₈×#64408	C₄	1	1	yes	characteristic
12	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
13	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
14	C₄²×#64408	C₄	1	1	yes	characteristic
15	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
16	#14556686	C₂×C₄	1	8	yes	—
17	#14556686	C₂×C₄	1	8	yes	—
18	C₂×#11924040	C₂×C₄	1	8	yes	—
19	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
20	C₂×#11924036	C₂³	1	1	yes	characteristic
21	C₂×C₄×#64408	C₈	1	4	yes	—
22	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
24	C₄×#10596833	C₂×C₄	1	4	yes	—
25	C₈×#64408	C₈	1	4	yes	—
26	C₈×#64408	C₈	1	4	yes	—
27	C₈×#64408	C₂×C₄	1	4	yes	—
28	C₈×#64408	C₂×C₄	1	4	yes	—
29	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×#10596831	C₈	1	1	yes	characteristic
33	C₈×Dic₇₃	C₈	1	1	yes	characteristic
34	C₈×Dic₇₃	C₈	1	1	yes	characteristic
35	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
36	#11924040	C₄²	1	16	yes	complemented
37	C₂²×#64408	C₂×C₈	1	2	yes	—
38	C₂²×#64408	C₂×C₈	1	2	yes	—
39	#11924036	C₂×C₈	1	2	yes	—
40	#11924036	C₂×C₈	1	2	yes	—
41	#11924036	C₂²×C₄	1	2	yes	—
42	#11924036	C₂²×C₄	1	2	yes	—
43	C₂×#10596833	C₄²	1	2	yes	—
44	C₂×#10596833	C₂²×C₄	1	2	yes	—
45	C₄×#64408	—	2	8	no	—
46	C₄×#64408	C₄²	1	2	yes	—
47	C₄×#64408	C₄²	1	2	yes	—
48	C₄×#64408	C₂×C₈	1	2	yes	—
49	C₄×#64408	C₂×C₈	1	2	yes	—
50	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
51	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
53	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
54	C₈×D₇₃	C₄²	1	2	yes	—
55	C₈×D₇₃	C₄²	1	2	yes	—
56	C₈×D₇₃	C₂×C₈	1	2	yes	—
57	C₈×D₇₃	C₂×C₈	1	2	yes	—
58	C₄×Dic₇₃	C₄²	1	2	yes	—
59	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
61	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×#64408	—	2	4	no	—
65	C₂×#64408	C₄×C₈	1	4	yes	—
66	C₂×#64408	C₄×C₈	1	4	yes	—
67	C₂×#64408	M₃₂	1	8	yes	—
68	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
71	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
72	C₄×D₇₃	—	2	4	no	—
73	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
74	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
75	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
76	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
77	#10596833	C₄×C₈	1	4	yes	—
78	#10596833	C₂×C₄²	1	4	yes	—
79	C₈×C₇₃	C₄×C₈	1	2	yes	—
80	C₈×C₇₃	C₄×C₈	1	2	yes	—
81	#10596831	C₄×C₈	1	2	yes	—
82	#10596831	C₄×C₈	1	2	yes	—
83	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
84	C₂×D₇₃	C₂×C₄×C₈	1	1	yes	characteristic
85	C₂×D₇₃	C₂×C₄×C₈	1	1	yes	characteristic
86	C₂×D₇₃	C₂×M₃₂	1	4	yes	—
87	#64408	—	2	4	no	—
88	#64408	#346	1	8	yes	complemented
89	C₄×C₇₃	—	2	2	no	—
90	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
91	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
92	Dic₇₃	—	2	2	no	—
93	Dic₇₃	C₄³	1	1	yes	characteristic
94	Dic₇₃	C₂×C₄×C₈	1	1	yes	characteristic
95	C₄×#346	—	73	1	no	maximal, Sylow
96	C₂×C₇₃	C₄²×C₈	1	1	yes	characteristic
97	C₂×C₇₃	C₄×M₃₂	1	2	yes	—
98	D₇₃	C₂×#346	1	2	yes	—
99	D₇₃	C₄×M₃₂	1	2	yes	—
100	C₂×#346	—	73	2	no	—
101	C₂×#346	—	73	2	no	—
102	C₂×C₄×C₁₆	—	73	2	no	—
103	C₄²×C₈	—	73	1	no	—
104	C₇₃	C₄×#346	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₁₆	—	73	4	no	—
106	C₂×C₄×C₈	—	73	1	no	—
107	C₂×C₄×C₈	—	73	1	no	—
108	C₂×C₄×C₈	—	73	1	no	—
109	C₂×C₄×C₈	—	73	1	no	—
110	C₂×C₄×C₈	—	73	1	no	—
111	C₂×C₄×C₈	—	73	1	no	—
112	C₄³	—	73	1	no	—
113	#346	—	73	8	no	—
114	#346	—	73	8	no	—
115	C₄×C₁₆	—	73	8	no	—
116	C₂²×C₈	—	73	1	no	—
117	C₂²×C₈	—	73	1	no	—
118	C₂²×C₈	—	73	1	no	—
119	C₂²×C₈	—	73	1	no	—
120	C₂×C₄²	—	73	1	no	—
121	C₂×C₄²	—	73	1	no	—
122	C₂×C₄²	—	73	1	no	—
123	C₂×C₄²	—	73	4	no	—
124	C₂×C₁₆	—	73	8	no	—
125	C₂×C₁₆	—	73	8	no	—
126	C₂×C₁₆	—	73	8	no	—
127	C₄×C₈	—	73	4	no	—
128	C₄×C₈	—	73	4	no	—
129	C₄×C₈	—	73	4	no	—
130	C₄×C₈	—	73	4	no	—
131	C₄×C₈	—	73	4	no	—
132	C₄×C₈	—	73	1	no	—
133	C₄×C₈	—	73	1	no	—
134	C₄×C₈	—	73	1	no	—
135	C₄×C₈	#10596842	1	1	yes	characteristic
136	C₂²×C₄	—	73	2	no	—
137	C₂²×C₄	—	73	1	no	—
138	C₂²×C₄	—	73	2	no	—
139	C₂²×C₄	—	73	2	no	—
140	C₂×C₈	—	73	2	no	—
141	C₂×C₈	—	73	2	no	—
142	C₂×C₈	—	73	2	no	—
143	C₂×C₈	—	73	2	no	—
144	C₂×C₈	—	73	2	no	—
145	C₂×C₈	—	73	2	no	—
146	C₂×C₈	—	73	1	no	—
147	C₂×C₈	—	73	1	no	—
148	C₂×C₈	—	73	2	no	—
149	C₂×C₈	—	73	2	no	—
150	C₂×C₈	—	73	2	no	—
151	C₂×C₈	—	73	2	no	—
152	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
153	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
154	C₄²	—	73	2	no	—
155	C₄²	—	73	2	no	—
156	C₄²	—	73	2	no	—
157	C₄²	—	73	2	no	—
158	C₄²	—	73	2	no	—
159	C₄²	—	73	1	no	—
160	C₄²	—	146	8	no	—
161	C₄²	C₂×#10596842	1	1	yes	characteristic
162	C₁₆	—	73	16	no	—
163	C₂³	—	73	1	no	—
164	C₂×C₄	—	73	4	no	—
165	C₂×C₄	—	73	8	no	—
166	C₂×C₄	—	73	1	no	—
167	C₂×C₄	—	73	1	no	—
168	C₂×C₄	—	73	1	no	—
169	C₂×C₄	—	73	1	no	—
170	C₂×C₄	—	73	1	no	—
171	C₂×C₄	—	73	4	no	—
172	C₂×C₄	—	73	2	no	—
173	C₂×C₄	—	146	4	no	—
174	C₂×C₄	—	146	4	no	—
175	C₂×C₄	C₄×#10596842	1	2	yes	—
176	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
177	C₈	—	73	4	no	—
178	C₈	—	73	4	no	—
179	C₈	—	73	2	no	—
180	C₈	—	73	2	no	—
181	C₈	C₄×#10596842	1	2	yes	—
182	C₈	C₄×#10596842	1	2	yes	—
183	C₂²	—	73	4	no	—
184	C₂²	—	73	1	no	—
185	C₂²	—	73	1	no	—
186	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	73	8	no	—
189	C₄	—	73	1	no	—
190	C₄	—	73	1	no	—
191	C₄	—	146	4	no	—
192	C₄	—	146	2	no	—
193	C₄	C₂×C₄×#10596842	1	1	yes	characteristic
194	C₄	C₂×C₄×#10596842	1	1	yes	characteristic
195	C₂	—	73	2	no	—
196	C₂	—	73	2	no	—
197	C₂	C₄²×#10596842	1	1	yes	characteristic
198	C₂	#64408⋊M₃₂	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×#10596842
commutator	C₂×C₇₃	C₄²×C₈
Frattini	C₂×C₄	C₂²×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#11924036	C₂³
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	C₄×#346
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #17989682	C₄	semidirect
7. #17989681	C₄	semidirect
8. #64408⋊C₁₆	C₄	semidirect
36. #11924040	C₄²	semidirect
88. #64408	#346	semidirect
104. C₇₃	C₄×#346	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989682	2
#22627087	2
C₂×#64408⋊C₁₆	2
C₄×C₈×#64408	2
C₄×#346	73