Subgroups — #33002413

Order 18688.

318 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988409	C₂	1	4	yes	maximal
2	C₄×#14556531	C₂	1	1	yes	maximal, characteristic
3	#22610103	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556531	C₂²	1	2	yes	—
6	C₂×#14556527	—	2	1	no	—
7	C₂×#14556527	C₂²	1	2	yes	—
8	#17988611	—	2	1	no	—
9	#17988409	—	2	8	no	—
10	#17988363	C₄	1	2	yes	complemented
11	C₄×#11924003	C₄	1	2	yes	complemented
12	C₂×#107×C₇₃	—	2	1	no	—
13	C₂×#106×C₇₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	#382×C₇₃	—	2	1	no	—
16	C₂×#11924012	—	2	2	no	—
17	#14556531	D₄	1	4	yes	—
18	#14556531	Q₈	1	4	yes	—
19	C₂×#11924003	C₂×C₄	1	4	yes	—
20	#14556527	—	2	4	no	—
21	#14556527	—	4	2	no	—
22	C₄⋊#10596831	—	4	2	no	—
23	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
26	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
27	#107×C₇₃	—	4	2	no	—
28	#106×C₇₃	—	2	4	no	—
29	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
33	#95×C₇₃	—	4	2	no	—
34	#11924012	—	4	2	no	—
35	#11924012	—	4	2	no	—
36	C₂×#10596831	—	4	2	no	—
37	#11924003	C₄⋊C₄	1	8	yes	—
38	C₂²×C₄×C₇₃	—	4	2	no	—
39	C₂²×C₄×C₇₃	SD₁₆	1	2	yes	—
40	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
44	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
45	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
49	C₄⋊C₄×C₇₃	—	4	2	no	—
50	C₄⋊C₄×C₇₃	—	4	2	no	—
51	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₇₃	SD₁₆	1	2	yes	—
54	C₂³×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₄×C₇₃	—	4	2	no	—
56	C₂×C₄×C₇₃	—	8	4	no	—
57	C₂×C₄×C₇₃	—	8	2	no	—
58	C₂×C₄×C₇₃	#106	1	2	yes	—
59	C₂×C₄×C₇₃	#106	1	2	yes	—
60	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₈×C₇₃	#106	1	2	yes	—
67	C₈×C₇₃	#106	1	2	yes	—
68	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
69	C₈×C₇₃	C₄⋊Q₈	1	2	yes	—
70	#10596831	—	8	2	no	—
71	C₂²×C₇₃	—	2	2	no	—
72	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
73	C₂²×C₇₃	#492	1	1	yes	characteristic
74	C₂²×C₇₃	#498	1	1	yes	characteristic
75	C₄×C₇₃	—	2	2	no	—
76	C₄×C₇₃	—	2	2	no	—
77	C₄×C₇₃	—	16	2	no	—
78	C₄×C₇₃	#384	1	1	yes	characteristic
79	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
80	C₄×C₇₃	#492	1	1	yes	characteristic
81	C₄×C₇₃	#498	1	1	yes	characteristic
82	#12904	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	—	2	1	no	—
84	C₂×C₇₃	—	2	1	no	—
85	C₂×C₇₃	#1844	1	1	yes	characteristic
86	C₂×C₇₃	#2155	1	2	yes	—
87	C₂×#365	—	73	4	no	—
88	C₄×M₃₂	—	73	1	no	—
89	#1848	—	73	1	no	—
90	#1846	D₇₃	1	1	yes	characteristic
91	C₇₃	#12904	1	1	yes	characteristic, complemented, Sylow
92	C₂×M₃₂	—	73	2	no	—
93	C₂×#108	—	73	2	no	—
94	C₂×#108	—	146	1	no	—
95	C₂×#107	—	2	1	no	—
96	C₂×#106	C₂×D₇₃	1	2	yes	—
97	#423	—	146	1	no	—
98	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
99	#382	—	2	1	no	—
100	#365	—	146	8	no	—
101	#346	—	73	2	no	—
102	C₄×C₁₆	—	73	2	no	—
103	C₂×M₁₆	—	146	2	no	—
104	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
105	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
108	M₃₂	—	73	4	no	—
109	M₃₂	—	73	4	no	—
110	C₂×C₁₆	—	73	4	no	—
111	#108	—	292	2	no	—
112	#108	—	146	4	no	—
113	#107	—	4	2	no	—
114	#106	—	2	4	no	—
115	C₄⋊C₈	—	292	2	no	—
116	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
117	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
118	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
119	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
120	#95	—	4	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	#11924016	1	2	yes	—
123	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
124	M₁₆	—	292	2	no	—
125	M₁₆	—	292	2	no	—
126	C₂×C₈	—	292	2	no	—
127	C₂×C₈	#11924007	1	1	yes	characteristic
128	C₂×C₈	#11924007	1	1	yes	characteristic
129	C₂×C₈	#11924015	1	2	yes	—
130	C₂×C₈	#11924016	1	2	yes	—
131	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
132	C₂×C₈	D₄×D₇₃	1	2	yes	—
133	C₂×C₈	Q₈×D₇₃	1	2	yes	—
134	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄²	#11924007	1	1	yes	characteristic
138	C₄²	#11924007	1	1	yes	characteristic
139	C₄²	#11924015	1	2	yes	—
140	C₁₆	—	73	8	no	—
141	C₂³	C₂×#11924016	1	1	yes	characteristic
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	#14556522	1	2	yes	—
146	C₂×C₄	#14556522	1	2	yes	—
147	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
148	C₂×C₄	#14556587	1	1	yes	characteristic
149	C₂×C₄	#14556589	1	1	yes	characteristic
150	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
151	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
152	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
153	C₈	—	584	2	no	—
154	C₈	#14556522	1	2	yes	—
155	C₈	#14556522	1	2	yes	—
156	C₈	C₄⋊C₄×D₇₃	1	2	yes	—
157	C₈	#14556590	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	C₂×#14556522	1	1	yes	characteristic
160	C₂²	#17988959	1	1	yes	characteristic
161	C₂²	#17988983	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	16	2	no	—
165	C₄	#17988456	1	1	yes	characteristic
166	C₄	C₂×#14556522	1	1	yes	characteristic
167	C₄	#17988961	1	1	yes	characteristic
168	C₄	#17988983	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#22610087	1	1	yes	characteristic
172	C₂	#22611703	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14556522
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1846	C₂
socle	C₂²×C₇₃	C₂×#106
prime radical	C₂³×C₇₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1846	D₇₃
hypocenter	C₇₃	#12904
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
10. #17988363	C₄	semidirect
11. C₄×#11924003	C₄	semidirect
91. C₇₃	#12904	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988409	2
C₄×#14556531	2
#22610103	2
C₇₃×#1846	2
#12904	73