Subgroups — #33239028

Order 18688.

658 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618531	C₂	1	2	yes	maximal
2	#22611719	C₂	1	6	yes	maximal, complemented
3	#22611715	C₂	1	3	yes	maximal, complemented
4	C₂×#17988413	C₂	1	3	yes	maximal, complemented
5	C₂×#504×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989070	—	2	2	no	—
7	#17988629	—	2	6	no	—
8	#17988625	C₂²	1	6	yes	—
9	C₂×#14556525	C₂²	1	6	yes	—
10	#17988415	C₂²	1	6	yes	—
11	#17988413	C₂²	1	1	yes	characteristic
12	#17988413	C₂²	1	3	yes	—
13	#17988413	C₂²	1	6	yes	—
14	C₂×#131×C₇₃	C₂²	1	1	yes	characteristic
15	#504×C₇₃	C₄	1	8	yes	complemented
16	C₂×M₃₂×C₇₃	C₂²	1	3	yes	—
17	C₂²×C₁₆×C₇₃	C₂²	1	3	yes	—
18	Q₈×Dic₇₃	—	4	2	no	—
19	D₄×Dic₇₃	—	4	6	no	—
20	#14556550	—	2	6	no	—
21	C₂×C₄⋊Dic₇₃	—	2	6	no	—
22	#14556525	—	2	6	no	—
23	#14556525	C₂³	1	6	yes	—
24	#14556525	C₂³	1	2	yes	—
25	#14556523	—	2	6	no	—
26	C₂×#41×C₇₃	D₄	1	1	yes	characteristic
27	#131×C₇₃	C₂×C₄	1	4	yes	—
28	#131×C₇₃	Q₈	1	4	yes	—
29	C₂×M₁₆×C₇₃	D₄	1	3	yes	—
30	C₂²×C₈×C₇₃	C₂³	1	3	yes	—
31	M₃₂×C₇₃	C₂×C₄	1	12	yes	complemented
32	C₂×C₁₆×C₇₃	C₂×C₄	1	12	yes	complemented
33	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
34	C₂×C₁₆×C₇₃	C₂³	1	3	yes	—
35	C₂²×Dic₇₃	—	4	6	no	—
36	C₄⋊Dic₇₃	—	2	2	no	—
37	C₄⋊Dic₇₃	—	2	6	no	—
38	C₄⋊Dic₇₃	—	4	6	no	—
39	C₂²⋊Dic₇₃	—	4	6	no	—
40	C₄×Dic₇₃	—	4	6	no	—
41	#41×C₇₃	C₄⋊C₄	1	4	yes	—
42	#41×C₇₃	Q₁₆	1	4	yes	—
43	C₂×Q₈×C₇₃	D₈	1	1	yes	characteristic
44	C₂×D₄×C₇₃	D₈	1	3	yes	—
45	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
46	M₁₆×C₇₃	C₄⋊C₄	1	12	yes	—
47	C₂×C₈×C₇₃	C₂²×C₄	1	6	yes	—
48	C₂×C₈×C₇₃	C₂×D₄	1	3	yes	—
49	C₂×C₈×C₇₃	C₂×Q₈	1	6	yes	—
50	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
51	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
52	C₁₆×C₇₃	C₂²×C₄	1	6	yes	—
53	C₂³×C₇₃	C₂×D₈	1	3	yes	—
54	Q₈×C₇₃	#107	1	4	yes	—
55	D₄×C₇₃	#107	1	12	yes	—
56	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	6	yes	—
57	C₂×C₄×C₇₃	C₂×D₈	1	3	yes	—
58	C₂×C₄×C₇₃	C₂×Q₁₆	1	6	yes	—
59	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
60	C₂×Dic₇₃	—	4	2	no	—
61	C₂×Dic₇₃	—	4	6	no	—
62	C₂×Dic₇₃	—	16	6	no	—
63	C₈×C₇₃	C₂×C₄⋊C₄	1	6	yes	—
64	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
65	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
66	C₂²×C₇₃	—	2	3	no	—
67	C₂²×C₇₃	—	2	3	no	—
68	C₂²×C₇₃	C₂×#107	1	6	yes	—
69	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
70	C₄×C₇₃	C₂×#107	1	6	yes	—
71	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
73	Dic₇₃	—	8	6	no	—
74	Dic₇₃	—	8	2	no	—
75	#33410	—	73	1	no	maximal, Sylow
76	C₂×C₇₃	—	2	6	no	—
77	C₂×C₇₃	C₂²×#107	1	1	yes	characteristic
78	C₂×C₇₃	#3415	1	1	yes	characteristic
79	C₂×C₇₃	#3417	1	1	yes	characteristic
80	C₂×#504	D₇₃	1	1	yes	characteristic
81	#2913	—	73	2	no	—
82	#2159	—	73	6	no	—
83	#2158	—	73	3	no	—
84	C₂×#366	—	73	3	no	—
85	C₇₃	#33410	1	1	yes	characteristic, complemented, Sylow
86	C₂×#131	C₂×D₇₃	1	1	yes	characteristic
87	#520	—	146	2	no	—
88	#504	Dic₇₃	1	8	yes	complemented
89	C₂×M₃₂	C₂×D₇₃	1	3	yes	—
90	C₂²×C₁₆	C₂×D₇₃	1	3	yes	—
91	#428	—	146	6	no	—
92	#427	—	73	6	no	—
93	C₂×#107	—	73	6	no	—
94	#367	—	73	6	no	—
95	#366	—	73	3	no	—
96	#366	—	73	6	no	—
97	#366	—	73	1	no	—
98	C₂×#41	D₂₉₂	1	1	yes	characteristic
99	#131	Dic₁₄₆	1	4	yes	—
100	#131	C₂×Dic₇₃	1	4	yes	—
101	C₂×M₁₆	D₂₉₂	1	3	yes	—
102	C₂²×C₈	C₂²×D₇₃	1	3	yes	—
103	C₄×Q₈	—	292	2	no	—
104	C₄×D₄	—	292	6	no	—
105	#117	—	146	6	no	—
106	C₂×C₄⋊C₄	—	146	6	no	—
107	M₃₂	C₂×Dic₇₃	1	12	yes	complemented
108	C₂×C₁₆	C₂×Dic₇₃	1	12	yes	complemented
109	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
110	C₂×C₁₆	C₂²×D₇₃	1	3	yes	—
111	#107	—	73	6	no	—
112	#107	—	73	2	no	—
113	#107	—	146	6	no	—
114	#106	—	146	6	no	—
115	#41	C₄⋊Dic₇₃	1	4	yes	—
116	#41	Dic₂₉₂	1	4	yes	—
117	C₂×Q₈	D₅₈₄	1	1	yes	characteristic
118	C₂×D₄	D₅₈₄	1	3	yes	—
119	C₂²×C₄	—	292	6	no	—
120	C₂²×C₄	C₂×D₂₉₂	1	3	yes	—
121	M₁₆	C₄⋊Dic₇₃	1	12	yes	—
122	C₂×C₈	C₂²×Dic₇₃	1	6	yes	—
123	C₂×C₈	C₂×D₂₉₂	1	3	yes	—
124	C₂×C₈	C₂×Dic₁₄₆	1	6	yes	—
125	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
126	C₄⋊C₄	—	292	6	no	—
127	C₄⋊C₄	—	146	2	no	—
128	C₄⋊C₄	—	146	6	no	—
129	C₂²⋊C₄	—	292	6	no	—
130	C₄²	—	292	6	no	—
131	C₁₆	C₂²×Dic₇₃	1	2	yes	—
132	C₁₆	C₂²×Dic₇₃	1	6	yes	—
133	C₂³	C₂×D₅₈₄	1	3	yes	—
134	Q₈	#14556525	1	4	yes	—
135	D₄	#14556525	1	12	yes	—
136	C₂×C₄	—	292	2	no	—
137	C₂×C₄	—	292	6	no	—
138	C₂×C₄	—	1168	6	no	—
139	C₂×C₄	C₂×C₄⋊Dic₇₃	1	6	yes	—
140	C₂×C₄	C₂×D₅₈₄	1	3	yes	—
141	C₂×C₄	C₂×Dic₂₉₂	1	6	yes	—
142	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
143	C₈	C₂×C₄⋊Dic₇₃	1	6	yes	—
144	C₈	C₂³×Dic₇₃	1	1	yes	characteristic
145	C₈	C₂²×Dic₁₄₆	1	1	yes	characteristic
146	C₂²	—	2	3	no	—
147	C₂²	—	2	3	no	—
148	C₂²	C₂×#14556525	1	6	yes	—
149	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
150	C₄	—	584	6	no	—
151	C₄	—	584	2	no	—
152	C₄	C₂×#14556525	1	6	yes	—
153	C₄	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
154	C₄	C₂²×Dic₂₉₂	1	1	yes	characteristic
155	C₂	—	2	6	no	—
156	C₂	C₂²×#14556525	1	1	yes	characteristic
157	C₂	#22623072	1	1	yes	characteristic
158	C₂	#22623084	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₈₄
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#504×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×#41×C₇₃	D₄
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#504	D₇₃
hypocenter	C₇₃	#33410
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
2. #22611719	C₂	semidirect
3. #22611715	C₂	semidirect
4. C₂×#17988413	C₂	semidirect
15. #504×C₇₃	C₄	semidirect
31. M₃₂×C₇₃	C₂×C₄	semidirect
32. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
85. C₇₃	#33410	semidirect
88. #504	Dic₇₃	semidirect
107. M₃₂	C₂×Dic₇₃	semidirect
108. C₂×C₁₆	C₂×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22618531	2
#22611719	2
#22611715	2
C₂×#17988413	2
C₂×#504×C₇₃	2
#33410	73