Subgroups — #34003960

Order 18688.

390 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625478	C₂	1	2	yes	maximal
2	#22625134	C₂	1	2	yes	maximal
3	#22624759	C₂	1	2	yes	maximal
4	#97×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	#17989603	C₂²	1	4	yes	—
6	#17989450	C₄	1	4	yes	complemented
7	#17989448	C₄	1	4	yes	complemented
8	#17989428	—	2	2	no	—
9	C₈×#10596833	—	2	2	no	—
10	C₂×#14556502	C₄	1	1	yes	characteristic
11	C₂×#14556501	C₄	1	1	yes	characteristic
12	C₂×#14556501	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×#97×C₇₃	C₄	1	1	yes	characteristic
17	C₄⋊#10596833	C₂×C₄	1	8	yes	—
18	C₂²⋊#10596833	—	2	4	no	—
19	C₄×#10596833	—	2	4	no	—
20	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
21	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇₃	—	2	2	no	—
23	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
25	#14556502	—	2	2	no	—
26	#14556501	—	2	2	no	—
27	#14556501	C₈	1	4	yes	complemented
28	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
29	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
30	C₈×Dic₇₃	D₄	1	2	yes	—
31	C₈×Dic₇₃	Q₈	1	2	yes	—
32	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
35	#97×C₇₃	C₈	1	4	yes	complemented
36	C₂×#10596833	—	2	8	no	—
37	C₂²×Dic₇₃	—	2	1	no	—
38	C₂²×Dic₇₃	#41	1	2	yes	—
39	C₂×#10596831	—	2	1	no	—
40	C₂×#10596831	—	2	1	no	—
41	C₂×#10596831	C₄²	1	1	yes	characteristic
42	C₂×#10596831	C₄²	1	1	yes	characteristic
43	C₂×#10596831	C₄⋊C₄	1	2	yes	—
44	C₂×#10596831	C₂×C₈	1	2	yes	—
45	C₂×#10596831	M₁₆	1	2	yes	—
46	C₄×Dic₇₃	—	2	1	no	—
47	C₄×Dic₇₃	—	4	4	no	—
48	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
49	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
50	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
51	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₇₃	—	2	1	no	—
53	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₄²	1	2	yes	—
55	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
60	C₄²×C₇₃	—	2	1	no	—
61	C₄²×C₇₃	C₂×C₈	1	2	yes	—
62	C₂³×C₇₃	#117	1	1	yes	characteristic
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	4	2	no	—
65	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×Dic₇₃	—	2	4	no	—
72	C₂×Dic₇₃	—	2	2	no	—
73	C₂×Dic₇₃	—	4	2	no	—
74	C₂×Dic₇₃	C₄×D₄	1	2	yes	—
75	C₂×Dic₇₃	C₄×Q₈	1	2	yes	—
76	#10596833	—	4	8	no	—
77	C₈×C₇₃	C₄×C₈	1	4	yes	—
78	C₈×C₇₃	C₄⋊C₈	1	4	yes	—
79	#10596831	—	2	1	no	—
80	#10596831	—	2	1	no	—
81	#10596831	C₄⋊C₈	1	4	yes	—
82	C₂²×C₇₃	—	2	2	no	—
83	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₇₃	#431	1	1	yes	characteristic
85	C₂²×C₇₃	#431	1	1	yes	characteristic
86	C₄×C₇₃	—	4	2	no	—
87	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
88	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
89	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
90	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
91	Dic₇₃	—	2	2	no	—
92	Dic₇₃	—	4	2	no	—
93	Dic₇₃	#444	1	4	yes	—
94	#8268	—	73	1	no	maximal, Sylow
95	C₂×C₇₃	—	2	1	no	—
96	C₂×C₇₃	—	2	1	no	—
97	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
98	C₂×C₇₃	#1773	1	2	yes	—
99	C₄×#97	—	73	1	no	—
100	#1564	—	73	2	no	—
101	#1450	—	73	2	no	—
102	#1276	—	73	2	no	—
103	C₇₃	#8268	1	1	yes	characteristic, complemented, Sylow
104	#431	—	73	4	no	—
105	C₂×#97	—	73	1	no	—
106	C₂×#97	—	73	1	no	—
107	C₂×#97	—	73	1	no	—
108	C₂×#97	#64408	1	1	yes	characteristic
109	C₂×C₄×C₈	—	73	1	no	—
110	C₂×C₄×C₈	—	73	1	no	—
111	C₄³	—	73	1	no	—
112	#335	—	73	4	no	—
113	#334	—	73	4	no	—
114	#322	—	146	2	no	—
115	C₈²	—	146	2	no	—
116	C₂²×C₈	—	73	1	no	—
117	C₂²×C₈	—	73	1	no	—
118	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
119	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
120	C₂×C₄²	—	73	1	no	—
121	C₂×C₄²	—	73	1	no	—
122	C₂×C₄²	—	146	2	no	—
123	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
124	C₄⋊C₈	—	73	8	no	—
125	C₂²⋊C₈	—	146	4	no	—
126	#97	—	73	4	no	—
127	#97	—	146	2	no	—
128	#97	—	146	2	no	—
129	#97	#10596833	1	4	yes	complemented
130	C₄×C₈	—	73	2	no	—
131	C₄×C₈	—	73	2	no	—
132	C₄×C₈	—	73	2	no	—
133	C₄×C₈	—	73	2	no	—
134	C₄×C₈	—	146	4	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	73	2	no	—
137	C₂²×C₄	—	146	1	no	—
138	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
139	C₂×C₈	—	73	2	no	—
140	C₂×C₈	—	73	2	no	—
141	C₂×C₈	—	73	2	no	—
142	C₂×C₈	—	73	1	no	—
143	C₂×C₈	—	73	1	no	—
144	C₂×C₈	—	146	8	no	—
145	C₂×C₈	—	146	1	no	—
146	C₂×C₈	—	146	1	no	—
147	C₂×C₈	C₄×#64408	1	2	yes	—
148	C₂×C₈	C₄⋊#64408	1	2	yes	—
149	C₂×C₈	C₂×#10596833	1	2	yes	—
150	C₂×C₈	C₂×#10596833	1	2	yes	—
151	C₂×C₈	#11924036	1	2	yes	—
152	C₂×C₈	#11924038	1	2	yes	—
153	C₄²	—	2	1	no	—
154	C₄²	—	292	4	no	—
155	C₄²	—	73	2	no	—
156	C₄²	—	73	1	no	—
157	C₄²	—	73	1	no	—
158	C₄²	—	73	2	no	—
159	C₄²	—	146	1	no	—
160	C₄²	C₂×#10596833	1	2	yes	—
161	C₂³	#14556670	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	292	2	no	—
165	C₂×C₄	—	73	2	no	—
166	C₂×C₄	—	73	2	no	—
167	C₂×C₄	—	146	2	no	—
168	C₂×C₄	—	146	4	no	—
169	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
171	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
172	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
173	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
174	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
175	C₈	—	292	8	no	—
176	C₈	—	73	4	no	—
177	C₈	—	146	1	no	—
178	C₈	—	146	1	no	—
179	C₈	C₄×#10596833	1	4	yes	—
180	C₈	C₄⋊#10596833	1	4	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	C₄×C₄⋊#64408	1	1	yes	characteristic
183	C₂²	#17989603	1	1	yes	characteristic
184	C₂²	#17989606	1	1	yes	characteristic
185	C₄	—	4	2	no	—
186	C₄	—	292	2	no	—
187	C₄	—	73	4	no	—
188	C₄	—	146	2	no	—
189	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
190	C₄	C₄×#11924038	1	1	yes	characteristic
191	C₄	C₂×C₄⋊#10596833	1	1	yes	characteristic
192	C₄	#17989588	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	C₄×C₄⋊#10596833	1	1	yes	characteristic
196	C₂	#22625737	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#97×C₇₃	C₄
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂³×C₇₃	#117
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#97	#64408
hypocenter	C₇₃	#8268
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
6. #17989450	C₄	semidirect
7. #17989448	C₄	semidirect
27. #14556501	C₈	semidirect
35. #97×C₇₃	C₈	semidirect
103. C₇₃	#8268	semidirect
129. #97	#10596833	semidirect

Maximal subgroups

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

subgroup	index
#22625478	2
#22625134	2
#22624759	2
#97×Dic₇₃	2
#8268	73