Subgroups — #33003994

Order 18688.

298 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988408	C₂	1	1	yes	maximal, characteristic
2	#22611689	C₂	1	2	yes	maximal
3	#22609810	C₂	1	1	yes	maximal, characteristic
4	#22608019	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	C₂²	1	2	yes	—
7	C₂×#14556527	C₄	1	1	yes	characteristic
8	#17988625	C₄	1	1	yes	characteristic
9	#17988542	C₂²	1	2	yes	—
10	#17988408	—	2	2	no	—
11	#17988405	—	2	4	no	—
12	#17988347	C₄	1	2	yes	—
13	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₇₃	—	2	2	no	—
17	C₂×#11924012	C₂×C₄	1	2	yes	—
18	#14556550	C₂×C₄	1	2	yes	—
19	#14556531	—	2	2	no	—
20	#14556531	—	2	2	no	—
21	C₂×#11924003	—	2	2	no	—
22	#14556527	—	2	2	no	—
23	#14556525	C₈	1	2	yes	—
24	#14556523	C₈	1	2	yes	—
25	#14556501	—	2	2	no	—
26	C₈×Dic₇₃	—	2	2	no	—
27	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₃	—	2	1	no	—
31	C₂×C₁₆×C₇₃	—	2	1	no	—
32	C₂×C₁₆×C₇₃	—	2	1	no	—
33	C₂×C₁₆×C₇₃	—	2	1	no	—
34	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
35	C₂×C₁₆×C₇₃	Q₈	1	2	yes	—
36	C₄×C₈×C₇₃	D₄	1	2	yes	—
37	C₄×C₈×C₇₃	Q₈	1	2	yes	—
38	#11924012	—	2	2	no	—
39	#11924012	—	2	2	no	—
40	C₂×#10596831	—	2	2	no	—
41	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
42	C₂²⋊Dic₇₃	—	2	2	no	—
43	C₄×Dic₇₃	—	2	2	no	—
44	#11924003	—	4	2	no	—
45	C₂²×C₄×C₇₃	—	2	1	no	—
46	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
51	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
57	C₄²×C₇₃	—	2	1	no	—
58	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
59	C₁₆×C₇₃	—	2	2	no	—
60	C₁₆×C₇₃	—	2	2	no	—
61	C₂³×C₇₃	#131	1	1	yes	characteristic
62	C₂×C₄×C₇₃	—	2	4	no	—
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
70	C₂×Dic₇₃	—	2	4	no	—
71	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
72	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
73	C₈×C₇₃	#121	1	1	yes	characteristic
74	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₈×C₇₃	#125	1	1	yes	characteristic
76	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
77	#10596831	—	4	2	no	—
78	C₂²×C₇₃	—	2	2	no	—
79	C₂²×C₇₃	#424	1	1	yes	characteristic
80	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
81	C₂²×C₇₃	#446	1	1	yes	characteristic
82	C₄×C₇₃	—	2	4	no	—
83	C₄×C₇₃	#389	1	1	yes	characteristic
84	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₄×C₇₃	#436	1	1	yes	characteristic
86	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
87	Dic₇₃	—	4	4	no	—
88	#13161	—	73	1	no	maximal, Sylow
89	C₂×C₇₃	—	2	1	no	—
90	C₂×C₇₃	—	2	1	no	—
91	C₂×C₇₃	#1939	1	1	yes	characteristic
92	C₂×C₇₃	#2170	1	2	yes	—
93	C₂×#364	—	73	1	no	—
94	#2151	—	73	2	no	—
95	C₂×C₄×C₁₆	D₇₃	1	1	yes	characteristic
96	#1772	—	73	1	no	—
97	#1379	—	73	2	no	—
98	C₇₃	#13161	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₃₂	—	73	2	no	—
100	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
101	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
102	C₂×#108	—	73	1	no	—
103	#427	—	73	1	no	—
104	#405	—	73	2	no	—
105	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
106	#364	—	146	2	no	—
107	C₄⋊C₁₆	—	146	4	no	—
108	C₄×C₁₆	—	2	2	no	—
109	#339	—	73	2	no	—
110	C₂×M₁₆	—	73	2	no	—
111	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
113	#117	—	73	2	no	—
114	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
115	M₃₂	—	146	2	no	—
116	M₃₂	—	146	2	no	—
117	C₂×C₁₆	—	2	1	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	2	1	no	—
121	C₂×C₁₆	—	146	2	no	—
122	C₂×C₁₆	Dic₁₄₆	1	2	yes	—
123	C₂×C₁₆	D₂₉₂	1	2	yes	—
124	#108	—	146	2	no	—
125	#107	—	73	2	no	—
126	#106	—	73	2	no	—
127	#97	—	146	2	no	—
128	C₄×C₈	—	146	2	no	—
129	C₄×C₈	Dic₁₄₆	1	2	yes	—
130	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
133	M₁₆	—	146	2	no	—
134	M₁₆	—	146	2	no	—
135	C₂×C₈	—	146	2	no	—
136	C₂×C₈	#11924007	1	2	yes	—
137	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
138	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
139	C₂×C₈	#11924011	1	2	yes	—
140	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
141	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
143	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
144	C₂×C₈	#11924028	1	1	yes	characteristic
145	C₂×C₈	#11924028	1	1	yes	characteristic
146	C₄⋊C₄	—	73	4	no	—
147	C₂²⋊C₄	—	146	2	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	146	2	no	—
150	C₄²	#11924007	1	2	yes	—
151	C₁₆	—	2	2	no	—
152	C₁₆	—	2	2	no	—
153	C₁₆	—	292	2	no	—
154	C₂³	#14556597	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	146	4	no	—
158	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
159	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
160	C₂×C₄	C₄×Dic₁₄₆	1	1	yes	characteristic
161	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
162	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
163	C₂×C₄	#14556597	1	1	yes	characteristic
164	C₈	—	292	2	no	—
165	C₈	#14556520	1	2	yes	—
166	C₈	#14556520	1	2	yes	—
167	C₈	#14556565	1	1	yes	characteristic
168	C₈	#14556570	1	1	yes	characteristic
169	C₈	#14556583	1	1	yes	characteristic
170	C₈	#14556591	1	1	yes	characteristic
171	C₂²	—	2	2	no	—
172	C₂²	#17988613	1	1	yes	characteristic
173	C₂²	C₈×D₂₉₂	1	1	yes	characteristic
174	C₂²	#17988717	1	1	yes	characteristic
175	C₄	—	2	4	no	—
176	C₄	—	292	4	no	—
177	C₄	#17988485	1	1	yes	characteristic
178	C₄	C₂×#14556520	1	1	yes	characteristic
179	C₄	#17988666	1	1	yes	characteristic
180	C₄	C₈×Dic₁₄₆	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#22610649	1	1	yes	characteristic
184	C₂	#22611769	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×C₄×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₈×D₄
prime radical	C₂³×C₇₃	#131
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₃
hypocenter	C₇₃	#13161
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
98. C₇₃	#13161	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988408	2
#22611689	2
#22609810	2
#22608019	2
C₂×C₄×C₁₆×C₇₃	2
#13161	73