Subgroups — #33004044

Order 18688.

274 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988417	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988413	C₂	1	1	yes	maximal, characteristic
3	#22610716	C₂	1	2	yes	maximal
4	#22608040	C₂	1	2	yes	maximal
5	C₇₃×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556527	C₂²	1	2	yes	—
7	#17988630	—	2	2	no	—
8	C₂×#14556525	C₂²	1	2	yes	—
9	#17988417	C₄	1	4	yes	complemented
10	#17988413	—	2	1	no	—
11	#17988413	—	2	1	no	—
12	#17988355	—	2	2	no	—
13	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	#405×C₇₃	C₂²	1	1	yes	characteristic
16	#346×C₇₃	—	2	1	no	—
17	C₄×C₁₆×C₇₃	—	2	1	no	—
18	C₂×#11924012	—	2	2	no	—
19	C₂×C₄⋊Dic₇₃	—	2	2	no	—
20	#14556527	—	2	2	no	—
21	#14556527	C₂×C₄	1	4	yes	—
22	#14556525	—	2	2	no	—
23	#14556525	—	2	2	no	—
24	#14556524	—	4	2	no	—
25	#14556522	—	4	2	no	—
26	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
28	#117×C₇₃	D₄	1	1	yes	characteristic
29	M₃₂×C₇₃	—	2	1	no	—
30	M₃₂×C₇₃	Q₈	1	2	yes	—
31	C₂×C₁₆×C₇₃	—	2	1	no	—
32	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₃	Q₈	1	2	yes	—
36	#97×C₇₃	—	2	1	no	—
37	C₄×C₈×C₇₃	—	2	1	no	—
38	C₂²×Dic₇₃	—	4	2	no	—
39	#11924012	—	2	4	no	—
40	#11924012	—	4	2	no	—
41	C₂×#10596831	—	4	2	no	—
42	C₄⋊Dic₇₃	—	4	2	no	—
43	C₄⋊Dic₇₃	—	4	2	no	—
44	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
45	M₁₆×C₇₃	—	2	1	no	—
46	M₁₆×C₇₃	Q₁₆	1	2	yes	—
47	C₂×C₈×C₇₃	—	2	1	no	—
48	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
53	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
55	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
56	C₂²⋊C₄×C₇₃	—	2	1	no	—
57	C₄²×C₇₃	—	2	1	no	—
58	C₁₆×C₇₃	—	2	2	no	—
59	C₁₆×C₇₃	C₄⋊C₄	1	2	yes	—
60	C₁₆×C₇₃	C₄⋊C₄	1	2	yes	—
61	C₂³×C₇₃	#135	1	1	yes	characteristic
62	C₂×C₄×C₇₃	—	2	1	no	—
63	C₂×C₄×C₇₃	—	2	1	no	—
64	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
70	C₂×Dic₇₃	—	8	2	no	—
71	C₂×Dic₇₃	—	16	2	no	—
72	C₈×C₇₃	—	2	2	no	—
73	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
75	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₇₃	#125	1	1	yes	characteristic
77	#10596831	—	4	4	no	—
78	C₂²×C₇₃	—	2	1	no	—
79	C₂²×C₇₃	—	2	1	no	—
80	C₂²×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
81	C₂²×C₇₃	#466	1	1	yes	characteristic
82	C₂²×C₇₃	#499	1	1	yes	characteristic
83	C₄×C₇₃	—	4	1	no	—
84	C₄×C₇₃	—	4	1	no	—
85	C₄×C₇₃	#389	1	1	yes	characteristic
86	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
87	C₄×C₇₃	#467	1	1	yes	characteristic
88	C₄×C₇₃	#500	1	1	yes	characteristic
89	Dic₇₃	—	16	2	no	—
90	#13171	—	73	1	no	maximal, Sylow
91	C₂×C₇₃	—	2	1	no	—
92	C₂×C₇₃	—	2	1	no	—
93	C₂×C₇₃	#1942	1	1	yes	characteristic
94	C₂×C₇₃	#2221	1	1	yes	characteristic
95	C₂×C₇₃	#2222	1	1	yes	characteristic
96	C₂×#368	—	73	1	no	—
97	C₂×#366	—	73	1	no	—
98	#2108	D₇₃	1	1	yes	characteristic
99	#1952	—	73	2	no	—
100	#1387	—	73	2	no	—
101	C₇₃	#13171	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
103	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
104	C₂×#108	—	73	2	no	—
105	#428	—	146	2	no	—
106	C₂×#107	—	73	2	no	—
107	#405	C₂×D₇₃	1	1	yes	characteristic
108	#368	—	73	4	no	—
109	#366	—	146	1	no	—
110	#366	—	146	1	no	—
111	#346	—	2	1	no	—
112	C₄×C₁₆	—	2	1	no	—
113	#343	—	146	2	no	—
114	C₂×M₁₆	—	146	2	no	—
115	C₂×M₁₆	D₂₉₂	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
117	#117	D₂₉₂	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	146	2	no	—
119	M₃₂	—	2	1	no	—
120	M₃₂	Dic₁₄₆	1	2	yes	—
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	Dic₁₄₆	1	2	yes	—
123	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
124	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
125	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
126	#108	—	73	4	no	—
127	#108	—	146	2	no	—
128	#107	—	292	2	no	—
129	#107	—	146	2	no	—
130	#107	—	146	2	no	—
131	#106	—	292	2	no	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₂²×C₄	—	292	2	no	—
135	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
136	M₁₆	—	2	1	no	—
137	M₁₆	—	292	2	no	—
138	M₁₆	—	146	4	no	—
139	M₁₆	Dic₂₉₂	1	2	yes	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	292	2	no	—
142	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
143	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
144	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
146	C₂×C₈	#11924028	1	1	yes	characteristic
147	C₂×C₈	#11924028	1	1	yes	characteristic
148	C₄⋊C₄	—	292	2	no	—
149	C₄⋊C₄	—	292	2	no	—
150	C₄⋊C₄	D₅₈₄	1	1	yes	characteristic
151	C₄⋊C₄	D₅₈₄	1	1	yes	characteristic
152	C₂²⋊C₄	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₁₆	—	2	2	no	—
155	C₁₆	#11924007	1	2	yes	—
156	C₁₆	#11924007	1	2	yes	—
157	C₂³	#14556613	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	584	2	no	—
161	C₂×C₄	—	1168	2	no	—
162	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
163	C₂×C₄	#14556565	1	1	yes	characteristic
164	C₂×C₄	#14556591	1	1	yes	characteristic
165	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
166	C₂×C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
167	C₂×C₄	#14556613	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	292	4	no	—
170	C₈	C₂×#11924007	1	1	yes	characteristic
171	C₈	C₄×Dic₁₄₆	1	1	yes	characteristic
172	C₈	#14556570	1	1	yes	characteristic
173	C₈	#14556583	1	1	yes	characteristic
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄×Dic₂₉₂	1	1	yes	characteristic
177	C₂²	#17988823	1	1	yes	characteristic
178	C₂²	#17988989	1	1	yes	characteristic
179	C₄	—	4	1	no	—
180	C₄	—	4	1	no	—
181	C₄	—	1168	2	no	—
182	C₄	#17988485	1	1	yes	characteristic
183	C₄	C₄×D₅₈₄	1	1	yes	characteristic
184	C₄	#17988828	1	1	yes	characteristic
185	C₄	#17988992	1	1	yes	characteristic
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	#22610664	1	1	yes	characteristic
189	C₂	#22612051	1	1	yes	characteristic
190	C₂	#22612056	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17988823
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#2108	C₂
socle	C₂²×C₇₃	#466
prime radical	C₂³×C₇₃	#135
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#2108	D₇₃
hypocenter	C₇₃	#13171
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
9. #17988417	C₄	semidirect
101. C₇₃	#13171	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988417	2
C₂×#17988413	2
#22610716	2
#22608040	2
C₇₃×#2108	2
#13171	73