Subgroups — #32997626

Order 18688.

242 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22608690	C₂	1	2	yes	maximal
2	#22608003	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
4	#17988611	—	2	2	no	—
5	#17988340	—	2	2	no	—
6	#17988340	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₇₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
10	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₇₃	C₂²	1	2	yes	—
12	C₂×#11924012	—	4	2	no	—
13	C₄⋊#10596831	—	2	4	no	—
14	C₄⋊#10596831	—	4	2	no	—
15	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₃	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₃	—	2	4	no	—
19	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
23	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
25	#11924012	—	8	4	no	—
26	C₂×#10596831	—	4	4	no	—
27	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
29	C₂²×C₄×C₇₃	Q₁₆	1	1	yes	characteristic
30	C₂×C₈×C₇₃	—	2	2	no	—
31	C₂×C₈×C₇₃	—	2	2	no	—
32	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₇₃	D₈	1	1	yes	characteristic
38	C₄²×C₇₃	Q₁₆	1	1	yes	characteristic
39	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₇₃	—	2	2	no	—
41	C₁₆×C₇₃	M₁₆	1	4	yes	complemented
42	C₂³×C₇₃	#107	1	1	yes	characteristic
43	C₂×C₄×C₇₃	—	2	2	no	—
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
47	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
48	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
49	C₂×C₄×C₇₃	D₁₆	1	2	yes	—
50	C₂×C₄×C₇₃	Q₃₂	1	2	yes	—
51	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
54	C₈×C₇₃	—	2	1	no	—
55	C₈×C₇₃	—	2	1	no	—
56	C₈×C₇₃	#108	1	2	yes	—
57	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
58	#10596831	—	8	4	no	—
59	C₂²×C₇₃	—	2	1	no	—
60	C₂²×C₇₃	—	2	1	no	—
61	C₂²×C₇₃	#366	1	2	yes	—
62	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
63	C₄×C₇₃	—	2	1	no	—
64	C₄×C₇₃	—	2	1	no	—
65	C₄×C₇₃	#366	1	2	yes	—
66	C₄×C₇₃	#368	1	2	yes	—
67	C₄×C₇₃	#423	1	1	yes	characteristic
68	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
69	C₄×C₇₃	C₂×D₁₆	1	1	yes	characteristic
70	C₄×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
71	#12022	—	73	1	no	maximal, Sylow
72	C₂×C₇₃	—	2	2	no	—
73	C₂×C₇₃	#1567	1	1	yes	characteristic
74	C₂×C₇₃	C₂×#366	1	1	yes	characteristic
75	C₂×C₇₃	C₂×#368	1	1	yes	characteristic
76	C₂×C₄×C₁₆	D₇₃	1	1	yes	characteristic
77	#1567	—	73	2	no	—
78	#1371	—	73	4	no	—
79	C₇₃	#12022	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₁₆	Dic₇₃	1	2	yes	—
81	#423	—	146	2	no	—
82	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
83	C₄×C₁₆	Dic₇₃	1	1	yes	characteristic
84	C₄×C₁₆	Dic₇₃	1	1	yes	characteristic
85	C₄×C₁₆	C₂×D₇₃	1	2	yes	—
86	#335	—	73	4	no	—
87	#335	—	146	2	no	—
88	C₂×M₁₆	—	292	2	no	—
89	C₂²×C₈	Dic₁₄₆	1	1	yes	characteristic
90	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
91	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
92	C₂×C₁₆	—	2	4	no	—
93	C₂×C₁₆	C₂×Dic₇₃	1	2	yes	—
94	C₂×C₁₆	C₂×Dic₇₃	1	2	yes	—
95	C₄⋊C₈	—	292	2	no	—
96	C₄⋊C₈	—	146	4	no	—
97	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
98	C₄×C₈	D₂₉₂	1	1	yes	characteristic
99	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
100	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
101	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
102	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
103	C₂²×C₄	Dic₂₉₂	1	1	yes	characteristic
104	M₁₆	—	584	4	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	292	4	no	—
108	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
109	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
110	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
111	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
112	C₄²	C₄⋊Dic₇₃	1	1	yes	characteristic
113	C₄²	D₅₈₄	1	1	yes	characteristic
114	C₄²	Dic₂₉₂	1	1	yes	characteristic
115	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
116	C₁₆	—	2	2	no	—
117	C₁₆	#11924012	1	4	yes	complemented
118	C₂³	#14556525	1	1	yes	characteristic
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	#14556525	1	1	yes	characteristic
123	C₂×C₄	#14556525	1	1	yes	characteristic
124	C₂×C₄	#14556525	1	1	yes	characteristic
125	C₂×C₄	D₁₁₆₈	1	2	yes	—
126	C₂×C₄	Dic₅₈₄	1	2	yes	—
127	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
128	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	584	4	no	—
133	C₈	#14556527	1	2	yes	—
134	C₈	C₂×#11924012	1	2	yes	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	#17988413	1	2	yes	—
138	C₂²	C₂×#14556525	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	#17988413	1	2	yes	—
142	C₄	#17988417	1	2	yes	—
143	C₄	#17988611	1	1	yes	characteristic
144	C₄	C₂×#14556527	1	1	yes	characteristic
145	C₄	C₂×D₁₁₆₈	1	1	yes	characteristic
146	C₄	C₂×Dic₅₈₄	1	1	yes	characteristic
147	C₂	—	2	2	no	—
148	C₂	#22608690	1	1	yes	characteristic
149	C₂	C₂×#17988413	1	1	yes	characteristic
150	C₂	C₂×#17988417	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₈₄
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₂×C₄×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂³×C₇₃	#107
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₃
hypocenter	C₇₃	#12022
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22608003	C₂	semidirect
41. C₁₆×C₇₃	M₁₆	semidirect
79. C₇₃	#12022	semidirect
117. C₁₆	#11924012	semidirect

Maximal subgroups

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

subgroup	index
#22608690	2
#22608003	2
C₂×C₄×C₁₆×C₇₃	2
#12022	73