Subgroups — #32977241

Order 18688.

398 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556523	C₂	1	1	yes	maximal, characteristic
2	#22609801	C₂	1	2	yes	maximal
3	C₂×#17988308	C₂	1	1	yes	maximal, characteristic
4	#22607770	C₂	1	2	yes	maximal
5	C₂×#335×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556523	C₄	1	2	yes	—
7	C₂×#14556501	—	2	2	no	—
8	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
11	#17988350	C₄	1	2	yes	—
12	#17988341	—	2	2	no	—
13	#17988308	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	#335×C₇₃	C₄	1	4	yes	complemented
17	C₂²×#10596831	—	2	1	no	—
18	C₂²×#10596831	D₄	1	2	yes	—
19	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
20	C₂×C₄×Dic₇₃	—	2	2	no	—
21	#14556523	—	2	4	no	—
22	#14556501	—	4	4	no	—
23	C₄×#10596831	—	2	1	no	—
24	C₄×#10596831	C₂×C₄	1	2	yes	—
25	C₄×#10596831	C₂×C₄	1	2	yes	—
26	C₄×#10596831	Q₈	1	2	yes	—
27	#14556498	—	2	2	no	—
28	C₂²×C₈×C₇₃	—	2	2	no	—
29	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₇₃	—	2	2	no	—
33	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
36	C₂²×Dic₇₃	—	2	4	no	—
37	C₂×#10596831	—	2	2	no	—
38	C₂×#10596831	—	2	2	no	—
39	C₂×#10596831	—	2	2	no	—
40	C₂×#10596831	—	4	2	no	—
41	C₂×#10596831	C₄⋊C₄	1	4	yes	—
42	C₄⋊Dic₇₃	—	2	4	no	—
43	C₄⋊Dic₇₃	—	2	4	no	—
44	C₄×Dic₇₃	—	4	2	no	—
45	C₄×Dic₇₃	—	4	2	no	—
46	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
49	C₂×C₈×C₇₃	—	2	2	no	—
50	C₂×C₈×C₇₃	—	2	4	no	—
51	C₂×C₈×C₇₃	—	4	2	no	—
52	C₂×C₈×C₇₃	—	4	2	no	—
53	C₂×C₈×C₇₃	C₄²	1	2	yes	—
54	C₂×C₈×C₇₃	C₄²	1	2	yes	—
55	C₂×C₈×C₇₃	M₁₆	1	4	yes	—
56	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₇₃	#41	1	1	yes	characteristic
60	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	—	2	2	no	—
62	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₇₃	#131	1	2	yes	—
70	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₇₃	#137	1	2	yes	—
72	C₂×Dic₇₃	—	4	4	no	—
73	C₂×Dic₇₃	—	4	4	no	—
74	C₂×Dic₇₃	—	4	4	no	—
75	C₈×C₇₃	—	2	2	no	—
76	C₈×C₇₃	—	4	4	no	—
77	C₈×C₇₃	#97	1	4	yes	complemented
78	#10596831	—	2	4	no	—
79	#10596831	—	4	2	no	—
80	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₇₃	#436	1	2	yes	—
82	C₂²×C₇₃	#440	1	2	yes	—
83	C₂²×C₇₃	#444	1	2	yes	—
84	C₄×C₇₃	—	2	1	no	—
85	C₄×C₇₃	—	2	1	no	—
86	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
87	C₄×C₇₃	#405	1	1	yes	characteristic
88	C₄×C₇₃	#428	1	2	yes	—
89	C₄×C₇₃	#441	1	1	yes	characteristic
90	C₄×C₇₃	#442	1	1	yes	characteristic
91	C₄×C₇₃	#446	1	2	yes	—
92	Dic₇₃	—	8	4	no	—
93	#8286	—	73	1	no	maximal, Sylow
94	C₂×C₇₃	#1592	1	4	yes	—
95	C₂×C₇₃	#1770	1	1	yes	characteristic
96	C₂×C₇₃	#1773	1	1	yes	characteristic
97	C₂×C₇₃	#1776	1	1	yes	characteristic
98	C₄×#106	—	73	1	no	—
99	#1770	—	73	2	no	—
100	C₂×#335	D₇₃	1	1	yes	characteristic
101	C₂×#322	—	73	1	no	—
102	#1279	—	73	2	no	—
103	C₇₃	#8286	1	1	yes	characteristic, complemented, Sylow
104	C₂×#106	—	73	2	no	—
105	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
106	C₂×#97	—	146	2	no	—
107	C₂×C₄×C₈	—	73	1	no	—
108	C₂×C₄×C₈	—	73	1	no	—
109	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	73	2	no	—
111	#340	—	73	2	no	—
112	#336	—	146	2	no	—
113	#335	Dic₇₃	1	4	yes	complemented
114	#322	—	73	4	no	—
115	C₂²×C₈	—	2	2	no	—
116	C₂²×C₈	—	73	2	no	—
117	C₂²×C₈	—	146	1	no	—
118	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
119	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	73	4	no	—
121	C₂×C₄²	—	146	2	no	—
122	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
123	#106	—	146	4	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
126	#97	—	292	4	no	—
127	C₄×C₈	—	73	2	no	—
128	C₄×C₈	—	73	2	no	—
129	C₄×C₈	—	73	2	no	—
130	C₄×C₈	—	146	1	no	—
131	C₄×C₈	C₄×D₇₃	1	2	yes	—
132	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
133	#95	—	146	2	no	—
134	C₂²×C₄	—	146	4	no	—
135	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
136	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
137	C₂²×C₄	#11924029	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	292	2	no	—
143	C₂×C₈	—	73	4	no	—
144	C₂×C₈	—	146	2	no	—
145	C₂×C₈	—	146	2	no	—
146	C₂×C₈	—	146	2	no	—
147	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
148	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
149	C₂×C₈	#11924011	1	4	yes	—
150	C₄⋊C₄	—	146	4	no	—
151	C₄⋊C₄	—	146	4	no	—
152	C₄²	—	292	2	no	—
153	C₄²	—	292	2	no	—
154	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
155	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
156	C₄²	Q₈×D₇₃	1	1	yes	characteristic
157	C₄²	#11924027	1	1	yes	characteristic
158	C₂³	#14556553	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	292	4	no	—
161	C₂×C₄	—	292	4	no	—
162	C₂×C₄	—	292	4	no	—
163	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
164	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
165	C₂×C₄	#14556549	1	1	yes	characteristic
166	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
167	C₂×C₄	#14556556	1	1	yes	characteristic
168	C₂×C₄	Q₈×Dic₇₃	1	1	yes	characteristic
169	C₂×C₄	C₂×#11924011	1	2	yes	—
170	C₂×C₄	#14556599	1	2	yes	—
171	C₂×C₄	#14556615	1	2	yes	—
172	C₂×C₄	#14556623	1	2	yes	—
173	C₈	—	2	2	no	—
174	C₈	—	4	4	no	—
175	C₈	—	292	2	no	—
176	C₈	—	146	4	no	—
177	C₈	#14556501	1	4	yes	complemented
178	C₂²	C₄⋊C₄×Dic₇₃	1	1	yes	characteristic
179	C₂²	#17988664	1	2	yes	—
180	C₂²	#17988686	1	2	yes	—
181	C₂²	#17988708	1	2	yes	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	584	4	no	—
185	C₄	C₂×#14556501	1	1	yes	characteristic
186	C₄	#17988545	1	1	yes	characteristic
187	C₄	#17988628	1	2	yes	—
188	C₄	#17988694	1	1	yes	characteristic
189	C₄	#17988699	1	1	yes	characteristic
190	C₄	#17988714	1	2	yes	—
191	C₂	#22608832	1	4	yes	—
192	C₂	#22609801	1	1	yes	characteristic
193	C₂	#22609818	1	1	yes	characteristic
194	C₂	#22609830	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#335×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#335	D₇₃
hypocenter	C₇₃	#8286
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
13. #17988308	C₄	semidirect
16. #335×C₇₃	C₄	semidirect
77. C₈×C₇₃	#97	semidirect
103. C₇₃	#8286	semidirect
113. #335	Dic₇₃	semidirect
177. C₈	#14556501	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556523	2
#22609801	2
C₂×#17988308	2
#22607770	2
C₂×#335×C₇₃	2
#8286	73