Subgroups — #33004135

Order 18688.

280 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22611703	C₂	1	2	yes	maximal
2	C₂×#17988408	C₂	1	1	yes	maximal, characteristic
3	#22610103	C₂	1	1	yes	maximal, characteristic
4	#22608042	C₂	1	2	yes	maximal
5	C₇₃×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	C₂²	1	2	yes	—
7	C₂×#14556527	—	2	1	no	—
8	C₂×#14556527	C₂²	1	2	yes	—
9	#17988611	—	2	1	no	—
10	#17988409	—	2	4	no	—
11	#17988408	C₄	1	2	yes	complemented
12	#17988408	C₄	1	2	yes	complemented
13	C₂×#108×C₇₃	C₂²	1	1	yes	characteristic
14	#427×C₇₃	C₂²	1	1	yes	characteristic
15	#405×C₇₃	—	2	1	no	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	#339×C₇₃	—	2	1	no	—
18	C₂×#11924012	—	2	2	no	—
19	#14556531	—	2	2	no	—
20	#14556531	C₂×C₄	1	4	yes	—
21	C₂×#11924003	—	2	2	no	—
22	#14556527	—	2	2	no	—
23	#14556527	—	2	2	no	—
24	#14556527	—	4	1	no	—
25	#14556527	—	4	1	no	—
26	C₄⋊#10596831	—	4	2	no	—
27	C₂×M₁₆×C₇₃	—	2	1	no	—
28	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
30	#117×C₇₃	—	2	1	no	—
31	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
32	#108×C₇₃	—	2	1	no	—
33	#108×C₇₃	Q₈	1	2	yes	—
34	#107×C₇₃	—	2	1	no	—
35	#106×C₇₃	D₄	1	2	yes	—
36	#97×C₇₃	—	4	1	no	—
37	C₄×C₈×C₇₃	—	4	1	no	—
38	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
42	#11924012	—	4	2	no	—
43	#11924012	—	4	2	no	—
44	C₂×#10596831	—	4	2	no	—
45	#11924003	—	2	4	no	—
46	C₂²×C₄×C₇₃	—	2	1	no	—
47	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	M₁₆×C₇₃	—	2	2	no	—
49	M₁₆×C₇₃	—	4	1	no	—
50	C₂×C₈×C₇₃	—	4	1	no	—
51	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
62	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
63	C₄⋊C₄×C₇₃	—	2	2	no	—
64	C₂²⋊C₄×C₇₃	—	4	1	no	—
65	C₄²×C₇₃	—	2	1	no	—
66	C₄²×C₇₃	—	4	1	no	—
67	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂³×C₇₃	#137	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	2	1	no	—
71	C₂×C₄×C₇₃	—	2	1	no	—
72	C₂×C₄×C₇₃	—	4	2	no	—
73	C₂×C₄×C₇₃	—	4	2	no	—
74	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₈×C₇₃	—	4	2	no	—
81	C₈×C₇₃	#106	1	1	yes	characteristic
82	C₈×C₇₃	#106	1	1	yes	characteristic
83	C₈×C₇₃	#106	1	1	yes	characteristic
84	C₈×C₇₃	#106	1	1	yes	characteristic
85	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
86	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
87	C₈×C₇₃	#125	1	1	yes	characteristic
88	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
89	#10596831	—	8	2	no	—
90	C₂²×C₇₃	—	4	1	no	—
91	C₂²×C₇₃	#428	1	1	yes	characteristic
92	C₂²×C₇₃	#461	1	1	yes	characteristic
93	C₂²×C₇₃	#475	1	1	yes	characteristic
94	C₄×C₇₃	—	2	2	no	—
95	C₄×C₇₃	—	4	1	no	—
96	C₄×C₇₃	—	8	2	no	—
97	C₄×C₇₃	#389	1	1	yes	characteristic
98	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
99	C₄×C₇₃	#460	1	1	yes	characteristic
100	C₄×C₇₃	#476	1	1	yes	characteristic
101	#13188	—	73	1	no	maximal, Sylow
102	C₂×C₇₃	—	2	1	no	—
103	C₂×C₇₃	—	2	1	no	—
104	C₂×C₇₃	—	2	1	no	—
105	C₂×C₇₃	#1948	1	1	yes	characteristic
106	#2155	—	73	2	no	—
107	C₂×#364	—	73	1	no	—
108	#1848	—	73	1	no	—
109	#1772	D₇₃	1	1	yes	characteristic
110	#1388	—	73	2	no	—
111	C₇₃	#13188	1	1	yes	characteristic, complemented, Sylow
112	C₂×M₃₂	—	73	2	no	—
113	C₂×#108	—	73	2	no	—
114	C₂×#108	—	146	1	no	—
115	C₂×#108	C₂×D₇₃	1	1	yes	characteristic
116	#427	C₂×D₇₃	1	1	yes	characteristic
117	#423	—	146	1	no	—
118	#405	—	2	1	no	—
119	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
120	#365	—	146	4	no	—
121	#364	—	73	2	no	—
122	#364	—	73	2	no	—
123	#339	—	2	1	no	—
124	C₂×M₁₆	—	2	1	no	—
125	C₂×M₁₆	—	146	2	no	—
126	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
128	#117	—	2	1	no	—
129	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
130	M₃₂	—	73	4	no	—
131	M₃₂	—	146	2	no	—
132	C₂×C₁₆	—	146	2	no	—
133	#108	—	2	1	no	—
134	#108	—	292	1	no	—
135	#108	—	292	1	no	—
136	#108	—	146	2	no	—
137	#108	—	146	2	no	—
138	#108	Dic₁₄₆	1	2	yes	—
139	#107	—	2	1	no	—
140	#106	D₂₉₂	1	2	yes	—
141	C₄⋊C₈	—	292	2	no	—
142	#97	—	4	1	no	—
143	C₄×C₈	—	4	1	no	—
144	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
145	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
146	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
147	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	4	1	no	—
152	M₁₆	—	292	2	no	—
153	M₁₆	—	292	2	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	292	2	no	—
156	C₂×C₈	#11924007	1	1	yes	characteristic
157	C₂×C₈	#11924007	1	1	yes	characteristic
158	C₂×C₈	#11924015	1	1	yes	characteristic
159	C₂×C₈	#11924015	1	1	yes	characteristic
160	C₂×C₈	#11924016	1	1	yes	characteristic
161	C₂×C₈	#11924016	1	1	yes	characteristic
162	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
163	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
164	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
165	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
166	C₂×C₈	#11924028	1	1	yes	characteristic
167	C₂×C₈	#11924028	1	1	yes	characteristic
168	C₄⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	#11924007	1	1	yes	characteristic
173	C₄²	#11924007	1	1	yes	characteristic
174	C₁₆	—	146	4	no	—
175	C₂³	#14556621	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
181	C₂×C₄	#14556565	1	1	yes	characteristic
182	C₂×C₄	#14556570	1	1	yes	characteristic
183	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
184	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
185	C₂×C₄	#14556616	1	1	yes	characteristic
186	C₈	—	4	2	no	—
187	C₈	—	584	2	no	—
188	C₈	#14556522	1	1	yes	characteristic
189	C₈	#14556522	1	1	yes	characteristic
190	C₈	#14556522	1	1	yes	characteristic
191	C₈	#14556522	1	1	yes	characteristic
192	C₈	C₄×D₂₉₂	1	1	yes	characteristic
193	C₈	C₄×Dic₁₄₆	1	1	yes	characteristic
194	C₈	#14556583	1	1	yes	characteristic
195	C₈	#14556591	1	1	yes	characteristic
196	C₂²	—	4	1	no	—
197	C₂²	#17988630	1	1	yes	characteristic
198	C₂²	#17988792	1	1	yes	characteristic
199	C₂²	#17988872	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	8	2	no	—
203	C₄	#17988485	1	1	yes	characteristic
204	C₄	C₂×#14556522	1	1	yes	characteristic
205	C₄	#17988787	1	1	yes	characteristic
206	C₄	#17988877	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#22610694	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17988877
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1772	C₂
socle	C₂×C₇₃	#1948
prime radical	C₂³×C₇₃	#137
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1772	D₇₃
hypocenter	C₇₃	#13188
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
11. #17988408	C₄	semidirect
12. #17988408	C₄	semidirect
111. C₇₃	#13188	semidirect

Maximal subgroups

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

subgroup	index
#22611703	2
C₂×#17988408	2
#22610103	2
#22608042	2
C₇₃×#1772	2
#13188	73