Subgroups — #97×#10596842 #34033219

Order 18688.

606 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627065	C₂	1	1	yes	maximal, characteristic
2	C₄²×#10596842	C₂	1	1	yes	maximal, characteristic
3	C₂×#17989671	C₂	1	2	yes	maximal
4	C₂×C₈×#10596842	C₂	1	2	yes	maximal
5	#97×#64408	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#10596842	C₄	1	4	yes	—
7	C₂×C₄×#10596842	C₂²	1	1	yes	characteristic
8	C₂×C₄×#10596842	C₂²	1	2	yes	—
9	C₂×C₄×#10596842	C₂²	1	1	yes	characteristic
10	(C₂×#10596842)⋊C₄	C₄	1	4	yes	—
11	#17989671	C₄	1	8	yes	complemented
12	C₈×#10596842	C₄	1	8	yes	complemented
13	#17989546	C₄	1	1	yes	characteristic
14	C₂×#14556645	C₄	1	2	yes	—
15	C₂×C₈×#64408	C₂²	1	2	yes	—
16	C₄²×#64408	C₂²	1	1	yes	characteristic
17	#97×D₇₃	C₄	1	1	yes	characteristic
18	C₂²×#10596842	C₂×C₄	1	4	yes	—
19	C₂²×#10596842	C₂×C₄	1	2	yes	—
20	C₂²×#10596842	C₂×C₄	1	2	yes	—
21	C₄×#10596842	—	2	8	no	—
22	C₄×#10596842	C₂×C₄	1	4	yes	—
23	C₄×#10596842	C₂×C₄	1	4	yes	—
24	C₄×#10596842	C₂×C₄	1	4	yes	—
25	C₄×#10596842	C₂×C₄	1	4	yes	—
26	C₂×#11924036	C₂×C₄	1	2	yes	—
27	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
28	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
29	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×#64408	C₂³	1	1	yes	characteristic
31	#14556646	C₈	1	4	yes	complemented
32	#14556645	C₈	1	8	yes	complemented
33	C₈×#64408	C₂×C₄	1	4	yes	—
34	C₈×#64408	C₂×C₄	1	4	yes	—
35	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
36	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
37	#14556502	C₈	1	1	yes	characteristic, complemented
38	#14556501	C₈	1	2	yes	complemented
39	#97×C₇₃	C₈	1	1	yes	characteristic, complemented
40	C₂×#10596842	—	2	4	no	—
41	C₂×#10596842	—	2	8	no	—
42	C₂×#10596842	C₄²	1	4	yes	—
43	C₂×#10596842	C₄²	1	8	yes	—
44	C₂×#10596842	C₄²	1	4	yes	—
45	C₂×#10596842	M₁₆	1	8	yes	—
46	C₂²×#64408	C₄²	1	2	yes	—
47	C₂²×#64408	C₂²×C₄	1	1	yes	characteristic
48	C₂²×#64408	C₂²×C₄	1	1	yes	characteristic
49	#11924036	C₄²	1	4	yes	—
50	#11924036	C₂×C₈	1	4	yes	—
51	C₂×#10596833	C₂×C₈	1	4	yes	—
52	C₄×#64408	—	2	4	no	—
53	C₄×#64408	—	2	4	no	—
54	C₄×#64408	C₂×C₈	1	4	yes	—
55	C₄×#64408	C₂²×C₄	1	2	yes	—
56	C₄×#64408	C₂²×C₄	1	2	yes	—
57	C₂×C₄×D₇₃	C₄²	1	2	yes	—
58	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×#10596831	C₂×C₈	1	2	yes	—
60	C₈×D₇₃	C₄²	1	4	yes	—
61	C₈×D₇₃	C₂×C₈	1	4	yes	—
62	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
63	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
65	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×D₇₃	C₂×C₄²	1	1	yes	characteristic
67	C₂×#64408	—	2	2	no	—
68	C₂×#64408	—	2	2	no	—
69	C₂×#64408	C₄×C₈	1	4	yes	—
70	C₂×#64408	C₂×C₄²	1	1	yes	characteristic
71	C₂×#64408	C₂×C₄²	1	1	yes	characteristic
72	C₂×#64408	C₂×C₄²	1	2	yes	—
73	C₂×#64408	C₂×M₁₆	1	4	yes	—
74	C₂×#64408	#131	1	4	yes	—
75	#10596842	—	2	4	no	—
76	#10596842	—	2	8	no	—
77	#10596842	#97	1	8	yes	direct product factor
78	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
79	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
81	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
82	C₄×D₇₃	—	2	2	no	—
83	C₄×D₇₃	—	2	2	no	—
84	C₄×D₇₃	C₂×C₄²	1	2	yes	—
85	C₄×D₇₃	C₂²×C₈	1	2	yes	—
86	#10596833	C₄×C₈	1	8	yes	complemented
87	C₈×C₇₃	C₄×C₈	1	4	yes	complemented
88	#10596831	C₄×C₈	1	4	yes	complemented
89	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
90	C₂×D₇₃	C₄³	1	1	yes	characteristic
91	C₂×D₇₃	C₂×C₄×C₈	1	1	yes	characteristic
92	C₂×D₇₃	C₄×M₁₆	1	2	yes	—
93	C₂×D₇₃	#405	1	2	yes	—
94	#64408	—	2	4	no	—
95	#64408	C₂×#97	1	2	yes	—
96	#64408	C₄×M₁₆	1	2	yes	—
97	#64408	#405	1	4	yes	—
98	C₄×C₇₃	—	2	2	no	—
99	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
100	Dic₇₃	—	2	2	no	—
101	Dic₇₃	C₂×C₄×C₈	1	2	yes	—
102	C₈×#97	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	C₈×M₁₆	1	2	yes	—
104	C₂×C₇₃	C₄²×C₈	1	1	yes	characteristic
105	D₇₃	C₈×M₁₆	1	2	yes	—
106	D₇₃	C₄×#97	1	1	yes	characteristic
107	D₇₃	#1726	1	1	yes	characteristic
108	#1744	—	73	1	no	—
109	C₄×#97	—	73	1	no	—
110	C₄²×C₈	—	73	1	no	—
111	C₂×#322	—	73	2	no	—
112	C₂×C₈²	—	73	2	no	—
113	C₇₃	C₈×#97	1	1	yes	characteristic, complemented, Sylow
114	C₂×#97	—	73	2	no	—
115	C₂×#97	—	73	1	no	—
116	C₂×#97	—	73	1	no	—
117	C₂×C₄×C₈	—	73	1	no	—
118	C₂×C₄×C₈	—	73	2	no	—
119	C₂×C₄×C₈	—	73	1	no	—
120	C₂×C₄×C₈	—	73	2	no	—
121	C₂×C₄×C₈	—	73	4	no	—
122	C₄³	—	73	1	no	—
123	#336	—	73	4	no	—
124	#322	—	73	8	no	—
125	C₈²	—	73	8	no	—
126	C₂²×C₈	—	73	2	no	—
127	C₂²×C₈	—	73	2	no	—
128	C₂²×C₈	—	73	2	no	—
129	C₂²×C₈	—	73	4	no	—
130	C₂²×C₈	—	73	2	no	—
131	C₂×C₄²	—	73	1	no	—
132	C₂×C₄²	—	73	1	no	—
133	C₂×C₄²	—	73	1	no	—
134	C₂×C₄²	—	73	2	no	—
135	C₂×C₄²	—	73	2	no	—
136	#97	—	73	8	no	—
137	#97	—	73	4	no	—
138	#97	—	73	2	no	—
139	#97	—	73	1	no	—
140	#97	#10596842	1	1	yes	characteristic, direct product factor
141	C₄×C₈	—	73	4	no	—
142	C₄×C₈	—	73	4	no	—
143	C₄×C₈	—	73	4	no	—
144	C₄×C₈	—	73	4	no	—
145	C₄×C₈	—	73	4	no	—
146	C₄×C₈	—	73	4	no	—
147	C₄×C₈	—	146	8	no	—
148	C₂²×C₄	—	73	1	no	—
149	C₂²×C₄	—	73	1	no	—
150	C₂²×C₄	—	73	1	no	—
151	C₂²×C₄	—	73	2	no	—
152	C₂²×C₄	—	73	2	no	—
153	C₂×C₈	—	73	4	no	—
154	C₂×C₈	—	73	4	no	—
155	C₂×C₈	—	73	4	no	—
156	C₂×C₈	—	73	4	no	—
157	C₂×C₈	—	73	8	no	—
158	C₂×C₈	—	73	2	no	—
159	C₂×C₈	—	73	4	no	—
160	C₂×C₈	—	73	4	no	—
161	C₂×C₈	—	73	8	no	—
162	C₂×C₈	—	73	4	no	—
163	C₂×C₈	—	146	4	no	—
164	C₂×C₈	—	146	8	no	—
165	C₂×C₈	C₂×#10596842	1	2	yes	—
166	C₄²	—	73	4	no	—
167	C₄²	—	73	2	no	—
168	C₄²	—	73	2	no	—
169	C₄²	—	73	2	no	—
170	C₄²	—	73	1	no	—
171	C₄²	—	146	4	no	—
172	C₄²	—	146	4	no	—
173	C₄²	C₂×#10596842	1	1	yes	characteristic
174	C₂³	—	73	1	no	—
175	C₂×C₄	—	73	1	no	—
176	C₂×C₄	—	73	1	no	—
177	C₂×C₄	—	73	2	no	—
178	C₂×C₄	—	73	4	no	—
179	C₂×C₄	—	73	4	no	—
180	C₂×C₄	—	73	1	no	—
181	C₂×C₄	—	73	2	no	—
182	C₂×C₄	—	73	2	no	—
183	C₂×C₄	—	73	4	no	—
184	C₂×C₄	—	73	2	no	—
185	C₂×C₄	—	146	2	no	—
186	C₂×C₄	—	146	2	no	—
187	C₂×C₄	—	146	2	no	—
188	C₂×C₄	—	146	2	no	—
189	C₂×C₄	C₄×#10596842	1	2	yes	—
190	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
191	C₈	—	73	8	no	—
192	C₈	—	73	8	no	—
193	C₈	—	73	4	no	—
194	C₈	—	146	4	no	—
195	C₈	—	146	8	no	—
196	C₈	C₄×#10596842	1	4	yes	complemented
197	C₂²	—	73	2	no	—
198	C₂²	—	73	2	no	—
199	C₂²	—	73	1	no	—
200	C₂²	—	73	1	no	—
201	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	73	2	no	—
204	C₄	—	73	2	no	—
205	C₄	—	73	4	no	—
206	C₄	—	73	2	no	—
207	C₄	—	146	4	no	—
208	C₄	—	146	2	no	—
209	C₄	C₂×C₄×#10596842	1	2	yes	—
210	C₂	—	73	1	no	—
211	C₂	—	73	1	no	—
212	C₂	—	73	2	no	—
213	C₂	M₁₆×#10596842	1	2	yes	—
214	C₂	C₄²×#10596842	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator	C₂×C₇₃	C₄²×C₈
Fitting	#97×C₇₃	C₈
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂²×D₇₃	C₂×C₄²
cosocle	C₂×C₄×#64408	C₂³
hypercenter	#97	#10596842
hypocenter	C₇₃	C₈×#97
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
11. #17989671	C₄	semidirect
12. C₈×#10596842	C₄	semidirect
31. #14556646	C₈	semidirect
32. #14556645	C₈	semidirect
37. #14556502	C₈	semidirect
38. #14556501	C₈	semidirect
39. #97×C₇₃	C₈	semidirect
77. #10596842	#97	direct (N × H)
86. #10596833	C₄×C₈	semidirect
87. C₈×C₇₃	C₄×C₈	semidirect
88. #10596831	C₄×C₈	semidirect
113. C₇₃	C₈×#97	semidirect
140. #97	#10596842	direct (N × H)
196. C₈	C₄×#10596842	semidirect

Maximal subgroups

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

subgroup	index
#22627065	2
C₄²×#10596842	2
C₂×#17989671	2
C₂×C₈×#10596842	2
#97×#64408	2
C₈×#97	73