Subgroups — #33001083

Order 18688.

410 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989004	C₂	1	1	yes	maximal, characteristic
2	#22611530	C₂	1	2	yes	maximal, complemented
3	#22609731	C₂	1	1	yes	maximal, characteristic
4	#22608018	C₂	1	2	yes	maximal, complemented
5	C₇₃×#2115	C₂	1	1	yes	maximal, characteristic
6	#17989004	—	2	2	no	—
7	#17989004	—	2	2	no	—
8	C₂×#14556531	C₄	1	2	yes	complemented
9	C₂×#14556531	C₂²	1	1	yes	characteristic
10	C₂²×#11924003	C₄	1	2	yes	complemented
11	C₂²×#11924003	C₂²	1	1	yes	characteristic
12	#17988571	C₄	1	1	yes	characteristic
13	#17988561	C₄	1	1	yes	characteristic
14	#17988542	C₂²	1	2	yes	—
15	#17988362	—	2	2	no	—
16	C₁₆×Dic₇₃	—	2	2	no	—
17	#17988358	C₄	1	2	yes	—
18	C₂×#131×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
20	#349×C₇₃	C₄	1	4	yes	complemented
21	C₂×#11924012	C₂×C₄	1	2	yes	—
22	#14556550	C₂×C₄	1	2	yes	—
23	#14556531	—	2	4	no	—
24	#14556531	—	2	2	no	—
25	C₂×#11924003	—	2	2	no	—
26	C₂×#11924003	—	2	2	no	—
27	C₂×#11924003	C₂×C₄	1	2	yes	—
28	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
29	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
30	C₂×#11924003	D₄	1	2	yes	—
31	C₂×#11924003	D₄	1	2	yes	—
32	#14556510	C₈	1	2	yes	—
33	#14556508	C₈	1	2	yes	—
34	#14556506	C₈	1	4	yes	—
35	#14556501	—	2	2	no	—
36	C₈×Dic₇₃	—	2	2	no	—
37	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
38	#131×C₇₃	—	2	2	no	—
39	#131×C₇₃	—	2	2	no	—
40	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
44	M₃₂×C₇₃	—	2	2	no	—
45	M₃₂×C₇₃	C₂×C₄	1	4	yes	—
46	C₂×C₁₆×C₇₃	—	2	2	no	—
47	#11924012	—	2	2	no	—
48	#11924012	C₂×C₈	1	4	yes	—
49	C₂×#10596831	—	2	2	no	—
50	C₄⋊Dic₇₃	—	2	2	no	—
51	C₂²⋊Dic₇₃	C₂×C₈	1	4	yes	—
52	C₄×Dic₇₃	—	2	2	no	—
53	#11924003	—	2	2	no	—
54	#11924003	C₂²⋊C₄	1	1	yes	characteristic
55	#11924003	C₂²⋊C₄	1	1	yes	characteristic
56	#11924003	C₂²⋊C₄	1	2	yes	—
57	#41×C₇₃	—	2	2	no	—
58	#41×C₇₃	—	2	2	no	—
59	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
60	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
61	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
64	M₁₆×C₇₃	—	2	2	no	—
65	M₁₆×C₇₃	—	2	4	no	—
66	C₂×C₈×C₇₃	—	2	2	no	—
67	C₂×C₈×C₇₃	—	2	2	no	—
68	C₂×C₈×C₇₃	C₄²	1	2	yes	—
69	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₇₃	#41	1	2	yes	—
73	C₁₆×C₇₃	—	2	4	no	—
74	C₂³×C₇₃	C₄×C₈	1	2	yes	—
75	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
76	Q₈×C₇₃	—	2	2	no	—
77	D₄×C₇₃	—	2	4	no	—
78	D₄×C₇₃	—	2	2	no	—
79	C₂×C₄×C₇₃	—	2	2	no	—
80	C₂×C₄×C₇₃	—	2	2	no	—
81	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
82	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₇₃	#131	1	2	yes	—
86	C₂×Dic₇₃	—	2	4	no	—
87	C₈×C₇₃	—	2	2	no	—
88	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₈×C₇₃	#117	1	1	yes	characteristic
90	C₈×C₇₃	C₄×D₄	1	2	yes	—
91	#10596831	—	2	4	no	—
92	C₂²×C₇₃	—	2	2	no	—
93	C₂²×C₇₃	—	4	1	no	—
94	C₂²×C₇₃	—	4	2	no	—
95	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
96	C₂²×C₇₃	C₈×D₄	1	2	yes	—
97	C₄×C₇₃	—	2	2	no	—
98	C₄×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
99	C₄×C₇₃	#405	1	1	yes	characteristic
100	C₄×C₇₃	C₈×D₄	1	2	yes	—
101	Dic₇₃	—	4	4	no	—
102	#12613	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	—	2	2	no	—
104	C₂×C₇₃	—	2	1	no	—
105	C₂×C₇₃	—	4	2	no	—
106	C₂×C₇₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×#504	—	73	1	no	—
108	#2115	D₇₃	1	1	yes	characteristic
109	#2108	—	73	2	no	—
110	#1757	—	73	1	no	—
111	#1378	—	73	2	no	—
112	C₇₃	#12613	1	1	yes	characteristic, complemented, Sylow
113	C₂×#131	C₂×D₇₃	1	1	yes	characteristic
114	#504	—	146	2	no	—
115	#504	—	146	2	no	—
116	C₂×M₃₂	—	73	2	no	—
117	C₂×M₃₂	—	73	1	no	—
118	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
119	C₂²×C₁₆	—	73	1	no	—
120	C₂²×C₁₆	—	73	2	no	—
121	#413	—	73	1	no	—
122	#410	—	73	1	no	—
123	#405	—	73	2	no	—
124	#349	Dic₇₃	1	4	yes	complemented
125	#346	—	146	2	no	—
126	C₄×C₁₆	—	146	2	no	—
127	#344	—	73	2	no	—
128	C₂×#41	C₄×D₇₃	1	1	yes	characteristic
129	#131	—	2	2	no	—
130	#131	—	2	2	no	—
131	C₂×M₁₆	—	73	2	no	—
132	C₂×M₁₆	C₄×D₇₃	1	1	yes	characteristic
133	C₂×M₁₆	C₂×Dic₇₃	1	2	yes	—
134	C₂²×C₈	C₄×D₇₃	1	2	yes	—
135	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
136	#117	—	73	2	no	—
137	M₃₂	—	2	2	no	—
138	M₃₂	—	146	4	no	—
139	M₃₂	—	146	2	no	—
140	M₃₂	C₂×Dic₇₃	1	4	yes	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	73	1	no	—
143	C₂×C₁₆	—	73	1	no	—
144	C₂×C₁₆	—	73	2	no	—
145	C₂×C₁₆	—	73	2	no	—
146	C₂×C₁₆	—	73	2	no	—
147	C₂×C₁₆	—	146	2	no	—
148	C₂×C₁₆	—	146	2	no	—
149	#101	—	73	2	no	—
150	#100	—	73	2	no	—
151	#99	—	73	4	no	—
152	#97	—	146	2	no	—
153	C₄×C₈	—	146	2	no	—
154	#41	—	2	2	no	—
155	#41	—	2	2	no	—
156	C₂×Q₈	C₈×D₇₃	1	1	yes	characteristic
157	C₂×D₄	C₈×D₇₃	1	2	yes	—
158	C₂×D₄	C₈×D₇₃	1	1	yes	characteristic
159	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
160	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	4	no	—
163	M₁₆	—	73	4	no	—
164	M₁₆	—	146	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	146	2	no	—
168	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
169	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
170	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
171	C₂×C₈	D₄×D₇₃	1	2	yes	—
172	C₂×C₈	#11924027	1	2	yes	—
173	C₄⋊C₄	—	146	2	no	—
174	C₂²⋊C₄	—	73	4	no	—
175	C₄²	—	146	2	no	—
176	C₁₆	—	2	4	no	—
177	C₁₆	—	73	1	no	—
178	C₁₆	—	73	1	no	—
179	C₁₆	—	73	2	no	—
180	C₁₆	—	146	2	no	—
181	C₂³	C₈×Dic₇₃	1	2	yes	—
182	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
183	Q₈	—	2	2	no	—
184	D₄	—	2	4	no	—
185	D₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	146	4	no	—
189	C₂×C₄	C₈×Dic₇₃	1	2	yes	—
190	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
191	C₂×C₄	#14556551	1	2	yes	—
192	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
193	C₂×C₄	#14556599	1	2	yes	—
194	C₈	—	2	2	no	—
195	C₈	—	146	4	no	—
196	C₈	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
197	C₈	#14556547	1	1	yes	characteristic
198	C₈	D₄×Dic₇₃	1	2	yes	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	2	no	—
202	C₂²	C₂×C₈×Dic₇₃	1	1	yes	characteristic
203	C₂²	#17988650	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	292	4	no	—
206	C₄	C₂²⋊C₄×Dic₇₃	1	1	yes	characteristic
207	C₄	#17988545	1	1	yes	characteristic
208	C₄	#17988650	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	2	1	no	—
211	C₂	—	4	2	no	—
212	C₂	#22609705	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊C₄×D₇₃
commutator	C₂²×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#2115	C₂
socle	C₂×C₇₃	C₈×C₂²⋊C₄
prime radical	C₂×#41×C₇₃	C₂×C₄
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#2115	D₇₃
hypocenter	C₇₃	#12613
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2115

Complements

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

N	H ≅ G/N	type
2. #22611530	C₂	semidirect
4. #22608018	C₂	semidirect
8. C₂×#14556531	C₄	semidirect
10. C₂²×#11924003	C₄	semidirect
20. #349×C₇₃	C₄	semidirect
112. C₇₃	#12613	semidirect
124. #349	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989004	2
#22611530	2
#22609731	2
#22608018	2
C₇₃×#2115	2
#12613	73