Subgroups — #33086623

Order 18688.

770 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618564	C₂	1	2	yes	maximal
2	#22610106	C₂	1	2	yes	maximal, complemented
3	#22610072	C₂	1	2	yes	maximal, complemented
4	#22609969	C₂	1	4	yes	maximal
5	#22609837	C₂	1	4	yes	maximal, complemented
6	C₂²×#97×C₇₃	C₂	1	1	yes	maximal, characteristic
7	#17988644	—	2	4	no	—
8	#17988640	—	2	4	no	—
9	C₂×#14556527	C₂²	1	8	yes	—
10	#17988611	C₂²	1	4	yes	—
11	#17988551	C₂²	1	4	yes	—
12	C₄×#11924012	C₂²	1	4	yes	—
13	#17988352	C₂²	1	8	yes	—
14	C₂³×C₈×C₇₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#97×C₇₃	C₄	1	8	yes	—
17	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₇₃	C₂²	1	2	yes	—
19	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×#11924012	C₂³	1	8	yes	—
21	#14556527	—	2	16	no	—
22	C₄⋊#10596831	—	2	8	no	—
23	C₂²⋊#10596831	—	2	8	no	—
24	#14556502	—	2	4	no	—
25	C₄×#10596831	—	2	4	no	—
26	C₂³×C₄×C₇₃	Q₈	1	2	yes	—
27	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₇₃	—	2	8	no	—
29	C₂²×C₈×C₇₃	C₂×C₄	1	8	yes	—
30	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
31	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
32	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₇₃	D₄	1	4	yes	—
34	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
35	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
37	#97×C₇₃	C₂×C₄	1	4	yes	—
38	#97×C₇₃	C₂×C₄	1	8	yes	—
39	#97×C₇₃	C₂×C₄	1	4	yes	—
40	#11924012	—	2	16	no	—
41	C₂×#10596831	—	2	8	no	—
42	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	—	2	4	no	—
44	C₂²×C₄×C₇₃	—	4	4	no	—
45	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
52	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
53	C₂×C₈×C₇₃	—	2	4	no	—
54	C₂×C₈×C₇₃	—	2	16	no	—
55	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
56	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₇₃	#41	1	8	yes	—
58	C₄²×C₇₃	C₄⋊C₄	1	8	yes	—
59	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
60	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
61	C₄²×C₇₃	C₂×D₄	1	2	yes	—
62	C₄²×C₇₃	C₂×D₄	1	2	yes	—
63	C₂³×C₇₃	—	2	2	no	—
64	C₂³×C₇₃	—	2	4	no	—
65	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
66	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	—	2	8	no	—
69	C₂×C₄×C₇₃	—	2	4	no	—
70	C₂×C₄×C₇₃	—	2	4	no	—
71	C₂×C₄×C₇₃	—	4	4	no	—
72	C₂×C₄×C₇₃	—	4	2	no	—
73	C₂×C₄×C₇₃	—	4	2	no	—
74	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
76	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
77	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂×#41	1	4	yes	—
80	C₈×C₇₃	—	2	4	no	—
81	C₈×C₇₃	#117	1	8	yes	—
82	#10596831	—	4	8	no	—
83	C₂²×C₇₃	—	2	8	no	—
84	C₂²×C₇₃	—	2	2	no	—
85	C₂²×C₇₃	—	2	2	no	—
86	C₂²×C₇₃	—	2	2	no	—
87	C₂²×C₇₃	#430	1	4	yes	—
88	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₇₃	#533	1	2	yes	—
90	C₄×C₇₃	—	2	4	no	—
91	C₄×C₇₃	—	2	4	no	—
92	C₄×C₇₃	—	2	2	no	—
93	C₄×C₇₃	C₂×#117	1	2	yes	—
94	C₄×C₇₃	#532	1	2	yes	—
95	#20830	—	73	1	no	maximal, Sylow
96	C₂×C₇₃	—	2	2	no	—
97	C₂×C₇₃	—	2	4	no	—
98	C₂×C₇₃	#2285	1	1	yes	characteristic
99	C₂×C₇₃	C₂×#430	1	2	yes	—
100	#2920	—	73	2	no	—
101	C₂²×#97	D₇₃	1	1	yes	characteristic
102	#1849	—	73	2	no	—
103	#1838	—	73	2	no	—
104	#1810	—	73	4	no	—
105	#1778	—	73	4	no	—
106	C₇₃	#20830	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₈	C₂×D₇₃	1	2	yes	—
108	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
109	#432	—	146	4	no	—
110	#431	—	146	4	no	—
111	C₂×#108	—	73	8	no	—
112	#423	—	73	4	no	—
113	#407	—	73	4	no	—
114	C₄×M₁₆	—	73	4	no	—
115	C₂×#97	Dic₇₃	1	8	yes	—
116	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
117	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
118	C₂×#97	C₂×D₇₃	1	2	yes	—
119	#341	—	73	8	no	—
120	C₂³×C₄	Dic₁₄₆	1	2	yes	—
121	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
122	C₂×M₁₆	—	73	8	no	—
123	C₂²×C₈	—	2	8	no	—
124	C₂²×C₈	C₂×Dic₇₃	1	8	yes	—
125	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
126	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
127	C₂×C₄²	Dic₁₄₆	1	2	yes	—
128	C₂×C₄²	D₂₉₂	1	4	yes	—
129	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
130	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
131	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
132	#108	—	146	16	no	—
133	C₄⋊C₈	—	146	8	no	—
134	C₂²⋊C₈	—	146	8	no	—
135	#97	—	146	4	no	—
136	#97	C₂×Dic₇₃	1	4	yes	—
137	#97	C₂×Dic₇₃	1	8	yes	—
138	#97	C₂×Dic₇₃	1	4	yes	—
139	C₄×C₈	—	146	4	no	—
140	C₂⁴	C₂×Dic₁₄₆	1	1	yes	characteristic
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	C₄⋊Dic₇₃	1	4	yes	—
144	C₂²×C₄	C₄⋊Dic₇₃	1	4	yes	—
145	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
146	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
147	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
148	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
149	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
150	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
151	M₁₆	—	146	16	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	16	no	—
154	C₂×C₈	—	146	8	no	—
155	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
156	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
157	C₂×C₈	#11924028	1	8	yes	—
158	C₄²	C₄⋊Dic₇₃	1	8	yes	—
159	C₄²	C₂²×Dic₇₃	1	2	yes	—
160	C₄²	C₂²×Dic₇₃	1	2	yes	—
161	C₄²	C₂×D₂₉₂	1	2	yes	—
162	C₄²	C₂×D₂₉₂	1	2	yes	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	4	no	—
165	C₂³	C₂×C₄⋊Dic₇₃	1	2	yes	—
166	C₂³	C₂²×Dic₁₄₆	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
175	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
177	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
179	C₂×C₄	C₂×#11924028	1	4	yes	—
180	C₈	—	2	4	no	—
181	C₈	—	292	8	no	—
182	C₈	#14556550	1	8	yes	—
183	C₂²	—	2	8	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	#17988636	1	4	yes	—
188	C₂²	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
189	C₂²	#17989144	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	C₂×#14556550	1	2	yes	—
194	C₄	#17989137	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	#22612338	1	1	yes	characteristic
198	C₂	C₂×#17988636	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂²×#97×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇₃	C₂×Q₈
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂²×#97	D₇₃
hypocenter	C₇₃	#20830
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #22610106	C₂	semidirect
3. #22610072	C₂	semidirect
5. #22609837	C₂	semidirect
106. C₇₃	#20830	semidirect

Maximal subgroups

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

subgroup	index
#22618564	2
#22610106	2
#22610072	2
#22609969	2
#22609837	2
C₂²×#97×C₇₃	2
#20830	73