Subgroups — #33239646

Order 18688.

460 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22611840	C₂	1	4	yes	maximal, complemented
2	Q₈×#11924003	C₂	1	4	yes	maximal, complemented
3	#22611738	C₂	1	4	yes	maximal
4	#22611730	C₂	1	2	yes	maximal
5	C₇₃×#2960	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11924003	C₂²	1	2	yes	—
7	C₄⋊#11924003	C₂²	1	2	yes	—
8	C₄⋊#11924003	C₂²	1	8	yes	—
9	C₂²⋊#11924003	C₂²	1	4	yes	—
10	#17988363	C₂²	1	4	yes	—
11	C₄×#11924003	C₂²	1	4	yes	—
12	C₄×#11924003	C₂²	1	4	yes	—
13	#533×C₇₃	C₄	1	1	yes	characteristic
14	#446×C₇₃	C₄	1	4	yes	—
15	C₈×Q₈×C₇₃	C₂²	1	4	yes	—
16	#424×C₇₃	C₄	1	2	yes	—
17	#423×C₇₃	C₄	1	1	yes	characteristic
18	#405×C₇₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×#11924003	C₂³	1	4	yes	—
21	C₂×#11924003	C₂³	1	4	yes	—
22	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
25	C₄⋊Q₈×C₇₃	C₈	1	2	yes	—
26	#125×C₇₃	C₈	1	2	yes	—
27	C₂²⋊Q₈×C₇₃	C₈	1	4	yes	—
28	C₄×Q₈×C₇₃	C₂×C₄	1	4	yes	—
29	#117×C₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
32	C₄⋊C₈×C₇₃	C₂×C₄	1	8	yes	—
33	#97×C₇₃	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₃	C₂³	1	2	yes	—
36	C₄×C₈×C₇₃	C₂³	1	4	yes	—
37	#11924003	—	2	4	no	—
38	#11924003	—	2	4	no	—
39	C₂×Q₈×C₇₃	C₂×C₈	1	4	yes	—
40	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
42	M₁₆×C₇₃	—	4	2	no	—
43	C₂×C₈×C₇₃	—	2	2	no	—
44	C₂×C₈×C₇₃	—	4	1	no	—
45	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
48	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
50	C₄⋊C₄×C₇₃	C₂×C₈	1	8	yes	—
51	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
52	C₂²⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
53	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
56	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
57	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
58	Q₈×C₇₃	—	2	8	no	—
59	C₂×C₄×C₇₃	—	2	1	no	—
60	C₂×C₄×C₇₃	—	2	1	no	—
61	C₂×C₄×C₇₃	—	2	4	no	—
62	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
64	C₂×C₄×C₇₃	C₂²×C₈	1	8	yes	—
65	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
67	C₈×C₇₃	—	2	1	no	—
68	C₈×C₇₃	—	2	2	no	—
69	C₈×C₇₃	—	2	4	no	—
70	C₈×C₇₃	2₋¹⁺⁴	1	2	yes	—
71	C₂²×C₇₃	—	2	2	no	—
72	C₂²×C₇₃	—	2	1	no	—
73	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
74	C₄×C₇₃	—	2	2	no	—
75	C₄×C₇₃	—	2	8	no	—
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	#504	1	4	yes	—
78	C₄×C₇₃	#519	1	1	yes	characteristic
79	C₄×C₇₃	#568	1	1	yes	characteristic
80	#33500	—	73	1	no	maximal, Sylow
81	C₂×C₇₃	—	4	1	no	—
82	C₂×C₇₃	#2982	1	1	yes	characteristic
83	C₂×C₇₃	C₂×#504	1	2	yes	—
84	#2960	D₇₃	1	1	yes	characteristic
85	#2183	—	73	4	no	—
86	Q₈×C₁₆	—	73	4	no	—
87	#2164	—	73	4	no	—
88	#2162	—	73	2	no	—
89	C₇₃	#33500	1	1	yes	characteristic, complemented, Sylow
90	#533	Dic₇₃	1	1	yes	characteristic
91	#446	Dic₇₃	1	4	yes	—
92	C₈×Q₈	C₂×D₇₃	1	4	yes	—
93	#424	Dic₇₃	1	2	yes	—
94	#423	Dic₇₃	1	1	yes	characteristic
95	#405	C₂×D₇₃	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
97	C₄⋊C₁₆	—	73	2	no	—
98	C₄⋊C₁₆	—	73	2	no	—
99	C₄⋊C₁₆	—	73	8	no	—
100	C₂²⋊C₁₆	—	73	4	no	—
101	#346	—	73	4	no	—
102	C₄×C₁₆	—	73	4	no	—
103	C₄×C₁₆	—	73	4	no	—
104	C₂×M₁₆	C₂×Dic₇₃	1	2	yes	—
105	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
106	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
107	C₄⋊Q₈	#10596831	1	2	yes	—
108	#125	#10596831	1	2	yes	—
109	C₂²⋊Q₈	#10596831	1	4	yes	—
110	C₄×Q₈	C₂×Dic₇₃	1	4	yes	—
111	#117	C₂×Dic₇₃	1	2	yes	—
112	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
113	C₂×C₁₆	—	73	4	no	—
114	C₂×C₁₆	—	73	4	no	—
115	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
116	C₄⋊C₈	C₂×Dic₇₃	1	8	yes	—
117	#97	C₂×Dic₇₃	1	4	yes	—
118	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
119	C₄×C₈	C₂²×D₇₃	1	2	yes	—
120	C₄×C₈	C₂²×D₇₃	1	4	yes	—
121	C₂×Q₈	C₂×#10596831	1	4	yes	—
122	C₂²×C₄	C₂×#10596831	1	2	yes	—
123	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
128	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
129	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
130	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
131	C₄⋊C₄	C₂×#10596831	1	4	yes	—
132	C₄⋊C₄	C₂×#10596831	1	8	yes	—
133	C₄⋊C₄	C₂×#10596831	1	4	yes	—
134	C₂²⋊C₄	C₂×#10596831	1	4	yes	—
135	C₄²	C₂×#10596831	1	1	yes	characteristic
136	C₄²	C₂×#10596831	1	1	yes	characteristic
137	C₄²	C₂²×Dic₇₃	1	2	yes	—
138	C₄²	C₂²×Dic₇₃	1	4	yes	—
139	C₁₆	—	146	4	no	—
140	C₁₆	—	146	4	no	—
141	C₂³	C₂²×#10596831	1	1	yes	characteristic
142	Q₈	—	2	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	C₂²×#10596831	1	1	yes	characteristic
147	C₂×C₄	C₂²×#10596831	1	2	yes	—
148	C₂×C₄	C₂²×#10596831	1	8	yes	—
149	C₂×C₄	C₂²×#10596831	1	2	yes	—
150	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	2	no	—
153	C₈	—	2	4	no	—
154	C₈	#14556639	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	1	no	—
157	C₂²	C₂³×#10596831	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	8	no	—
160	C₄	—	2	1	no	—
161	C₄	#17989004	1	4	yes	—
162	C₄	#17989065	1	1	yes	characteristic
163	C₄	#17989354	1	1	yes	characteristic
164	C₂	—	4	1	no	—
165	C₂	#22619012	1	1	yes	characteristic
166	C₂	C₂×#17989004	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₃
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₇₃×#2960	C₂
prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2960	D₇₃
hypocenter	C₇₃	#33500
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #22611840	C₂	semidirect
2. Q₈×#11924003	C₂	semidirect
89. C₇₃	#33500	semidirect

Maximal subgroups

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

subgroup	index
#22611840	2
Q₈×#11924003	2
#22611738	2
#22611730	2
C₇₃×#2960	2
#33500	73