Subgroups — #32979734

Order 18688.

418 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22610083	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556502	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988339	C₂	1	2	yes	maximal
4	C₂×#17988337	C₂	1	2	yes	maximal
5	C₇₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596831	—	2	2	no	—
7	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
8	C₂×#14556502	C₄	1	2	yes	—
9	C₂×#14556502	C₄	1	1	yes	characteristic
10	C₂×#14556502	C₄	1	1	yes	characteristic
11	C₂×C₄×#10596831	C₂²	1	2	yes	—
12	#17988339	—	2	4	no	—
13	#17988337	—	2	4	no	—
14	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₄³×C₇₃	C₂²	1	1	yes	characteristic
17	#336×C₇₃	—	2	2	no	—
18	C₂²×#10596831	—	2	2	no	—
19	C₂²×#10596831	C₂×C₄	1	2	yes	—
20	C₂²×#10596831	C₂×C₄	1	2	yes	—
21	C₄⋊#10596831	—	2	2	no	—
22	C₄⋊#10596831	—	2	2	no	—
23	C₄⋊#10596831	—	4	4	no	—
24	#14556502	—	2	4	no	—
25	#14556502	—	2	2	no	—
26	#14556502	C₈	1	4	yes	complemented
27	C₄×#10596831	D₄	1	4	yes	—
28	C₄×#10596831	Q₈	1	4	yes	—
29	C₂²×C₈×C₇₃	—	2	1	no	—
30	C₂²×C₈×C₇₃	—	2	1	no	—
31	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₃	D₄	1	2	yes	—
34	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
35	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₇₃	—	2	2	no	—
37	C₄⋊C₈×C₇₃	—	2	2	no	—
38	C₂×#10596831	—	2	2	no	—
39	C₂×#10596831	—	2	2	no	—
40	C₂×#10596831	—	4	2	no	—
41	C₂×#10596831	—	4	4	no	—
42	C₂×#10596831	C₄⋊C₄	1	4	yes	—
43	C₂×#10596831	C₄⋊C₄	1	4	yes	—
44	C₂×#10596831	C₂×C₈	1	4	yes	—
45	C₂×#10596831	M₁₆	1	4	yes	—
46	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₇₃	—	4	1	no	—
52	C₂×C₈×C₇₃	—	4	1	no	—
53	C₂×C₈×C₇₃	—	4	2	no	—
54	C₂×C₈×C₇₃	—	4	2	no	—
55	C₄²×C₇₃	—	2	2	no	—
56	C₄²×C₇₃	—	2	2	no	—
57	C₄²×C₇₃	—	2	2	no	—
58	C₄²×C₇₃	—	4	2	no	—
59	C₄²×C₇₃	C₂×C₈	1	2	yes	—
60	C₄²×C₇₃	M₁₆	1	2	yes	—
61	C₄²×C₇₃	C₂×D₄	1	2	yes	—
62	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
63	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	—	2	2	no	—
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	—	2	2	no	—
69	C₂×C₄×C₇₃	—	2	2	no	—
70	C₂×C₄×C₇₃	C₄⋊C₈	1	4	yes	—
71	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
75	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₇₃	#137	1	2	yes	—
80	C₈×C₇₃	—	8	1	no	—
81	C₈×C₇₃	—	8	1	no	—
82	#10596831	—	2	4	no	—
83	#10596831	—	8	2	no	—
84	#10596831	C₄⋊C₈	1	8	yes	complemented
85	C₂²×C₇₃	#384	1	1	yes	characteristic
86	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₇₃	#423	1	1	yes	characteristic
88	C₂²×C₇₃	#428	1	2	yes	—
89	C₂²×C₇₃	#430	1	2	yes	—
90	C₄×C₇₃	—	2	4	no	—
91	C₄×C₇₃	—	2	2	no	—
92	C₄×C₇₃	—	4	2	no	—
93	C₄×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
94	C₄×C₇₃	#423	1	2	yes	—
95	C₄×C₇₃	#496	1	1	yes	characteristic
96	C₄×C₇₃	#497	1	1	yes	characteristic
97	C₄×C₇₃	#501	1	2	yes	—
98	#8808	—	73	1	no	maximal, Sylow
99	C₂×C₇₃	#1565	1	2	yes	—
100	C₂×C₇₃	#1569	1	2	yes	—
101	C₂×C₇₃	#1842	1	1	yes	characteristic
102	C₂×C₇₃	#1847	1	1	yes	characteristic
103	C₂×C₇₃	#1849	1	1	yes	characteristic
104	#1842	—	73	1	no	—
105	#1831	D₇₃	1	1	yes	characteristic
106	C₄×#97	—	73	1	no	—
107	C₂×#335	—	73	2	no	—
108	C₂×#334	—	73	2	no	—
109	C₇₃	#8808	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊C₈	—	73	2	no	—
111	C₂×C₄⋊C₈	—	146	2	no	—
112	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
113	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
114	C₂×#97	—	73	1	no	—
115	C₂×#97	—	73	2	no	—
116	C₂×#97	—	73	1	no	—
117	C₂×C₄×C₈	—	73	2	no	—
118	C₄³	C₂×D₇₃	1	1	yes	characteristic
119	#336	—	2	2	no	—
120	#335	—	146	4	no	—
121	#334	—	146	4	no	—
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	—	2	1	no	—
124	C₂²×C₈	—	73	2	no	—
125	C₂²×C₈	—	73	2	no	—
126	C₂²×C₈	—	146	2	no	—
127	C₂×C₄²	Dic₁₄₆	1	2	yes	—
128	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
129	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
130	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
131	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	292	4	no	—
135	C₄⋊C₈	—	146	2	no	—
136	C₄⋊C₈	—	146	2	no	—
137	#97	—	73	4	no	—
138	#97	—	146	4	no	—
139	#97	—	146	2	no	—
140	C₄×C₈	—	73	4	no	—
141	C₄×C₈	—	73	4	no	—
142	C₂²×C₄	#11924007	1	2	yes	—
143	C₂²×C₄	#11924007	1	2	yes	—
144	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	292	2	no	—
152	C₂×C₈	—	292	4	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	—	146	2	no	—
158	C₂×C₈	—	146	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	C₈×D₇₃	1	2	yes	—
164	C₄²	#11924011	1	2	yes	—
165	C₄²	D₄×D₇₃	1	2	yes	—
166	C₄²	Q₈×D₇₃	1	2	yes	—
167	C₂³	C₂×#11924007	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	#14556520	1	4	yes	—
175	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
176	C₂×C₄	#14556587	1	1	yes	characteristic
177	C₂×C₄	#14556589	1	1	yes	characteristic
178	C₂×C₄	#14556590	1	2	yes	—
179	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
180	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
181	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
182	C₂×C₄	#14556616	1	2	yes	—
183	C₂×C₄	#14556621	1	2	yes	—
184	C₈	—	8	1	no	—
185	C₈	—	8	1	no	—
186	C₈	—	584	2	no	—
187	C₈	—	73	8	no	—
188	C₈	—	146	4	no	—
189	C₂²	#17988456	1	1	yes	characteristic
190	C₂²	C₂×#14556520	1	1	yes	characteristic
191	C₂²	#17988610	1	1	yes	characteristic
192	C₂²	#17988630	1	2	yes	—
193	C₂²	#17988635	1	2	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	2	2	no	—
196	C₄	—	4	2	no	—
197	C₄	C₄⋊C₈×D₇₃	1	2	yes	—
198	C₄	#17988609	1	2	yes	—
199	C₄	#17988977	1	1	yes	characteristic
200	C₄	#17988982	1	1	yes	characteristic
201	C₄	#17988997	1	2	yes	—
202	C₂	#22608684	1	2	yes	—
203	C₂	#22608699	1	2	yes	—
204	C₂	#22610082	1	1	yes	characteristic
205	C₂	#22610100	1	1	yes	characteristic
206	C₂	#22610105	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1831	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1831	D₇₃
hypocenter	C₇₃	#8808
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
26. #14556502	C₈	semidirect
84. #10596831	C₄⋊C₈	semidirect
109. C₇₃	#8808	semidirect

Maximal subgroups

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

subgroup	index
#22610083	2
C₄×#14556502	2
C₂×#17988339	2
C₂×#17988337	2
C₇₃×#1831	2
#8808	73