Subgroups — #33001082

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	#22611552	C₂	1	1	yes	maximal, characteristic
3	#22611528	C₂	1	2	yes	maximal, complemented
4	#22608017	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1757	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	2	yes	—
10	C₂×#14556531	C₂²	1	1	yes	characteristic
11	C₂²×#11924003	C₄	1	2	yes	complemented
12	C₂²×#11924003	C₂²	1	1	yes	characteristic
13	#17988369	C₄	1	4	yes	complemented
14	#17988363	—	2	2	no	—
15	C₄×#11924003	—	2	2	no	—
16	C₂×#131×C₇₃	C₂²	1	1	yes	characteristic
17	#413×C₇₃	C₄	1	1	yes	characteristic
18	#410×C₇₃	C₄	1	1	yes	characteristic
19	#405×C₇₃	C₂²	1	2	yes	—
20	#344×C₇₃	C₄	1	2	yes	—
21	#14556531	—	2	4	no	—
22	#14556531	—	2	2	no	—
23	#14556531	—	2	2	no	—
24	#14556531	C₂×C₄	1	4	yes	—
25	C₂×#11924003	—	2	2	no	—
26	C₂×#11924003	—	2	2	no	—
27	C₂×#11924003	—	2	2	no	—
28	C₂×#11924003	C₂×C₄	1	2	yes	—
29	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
30	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
31	C₂×#11924003	D₄	1	2	yes	—
32	C₂×#11924003	D₄	1	2	yes	—
33	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
34	#131×C₇₃	—	2	2	no	—
35	#131×C₇₃	—	2	2	no	—
36	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
41	#117×C₇₃	C₂×C₄	1	2	yes	—
42	#101×C₇₃	C₈	1	2	yes	—
43	#100×C₇₃	C₈	1	2	yes	—
44	#99×C₇₃	C₈	1	4	yes	—
45	#97×C₇₃	—	2	2	no	—
46	C₄×C₈×C₇₃	—	2	2	no	—
47	#11924003	—	2	2	no	—
48	#11924003	—	2	4	no	—
49	#11924003	C₂²⋊C₄	1	1	yes	characteristic
50	#11924003	C₂²⋊C₄	1	1	yes	characteristic
51	#11924003	C₂²⋊C₄	1	2	yes	—
52	#41×C₇₃	—	2	2	no	—
53	#41×C₇₃	—	2	2	no	—
54	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
55	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
56	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
58	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	M₁₆×C₇₃	—	2	2	no	—
60	M₁₆×C₇₃	—	2	2	no	—
61	M₁₆×C₇₃	—	2	4	no	—
62	M₁₆×C₇₃	C₂×C₈	1	4	yes	—
63	C₂×C₈×C₇₃	—	2	2	no	—
64	C₂×C₈×C₇₃	—	2	2	no	—
65	C₂×C₈×C₇₃	—	2	2	no	—
66	C₂×C₈×C₇₃	C₄²	1	2	yes	—
67	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₇₃	#41	1	2	yes	—
71	C₄⋊C₄×C₇₃	—	2	2	no	—
72	C₂²⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
73	C₄²×C₇₃	—	2	2	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	—	2	4	no	—
82	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
83	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₇₃	#131	1	2	yes	—
87	C₈×C₇₃	—	2	2	no	—
88	C₈×C₇₃	—	2	4	no	—
89	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₈×C₇₃	#117	1	1	yes	characteristic
91	C₈×C₇₃	C₄×D₄	1	2	yes	—
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	—	4	4	no	—
99	C₄×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
100	C₄×C₇₃	#405	1	1	yes	characteristic
101	C₄×C₇₃	C₈×D₄	1	2	yes	—
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	—	73	1	no	—
109	#2108	—	73	2	no	—
110	#1757	D₇₃	1	1	yes	characteristic
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	—	73	2	no	—
119	C₂²×C₁₆	—	73	1	no	—
120	C₂²×C₁₆	—	73	2	no	—
121	#413	Dic₇₃	1	1	yes	characteristic
122	#410	Dic₇₃	1	1	yes	characteristic
123	#405	C₂×D₇₃	1	2	yes	—
124	#349	—	73	4	no	—
125	#346	—	146	2	no	—
126	C₄×C₁₆	—	146	2	no	—
127	#344	Dic₇₃	1	2	yes	—
128	C₂×#41	C₂×Dic₇₃	1	1	yes	characteristic
129	#131	—	2	2	no	—
130	#131	—	2	2	no	—
131	C₂×M₁₆	C₄×D₇₃	1	2	yes	—
132	C₂×M₁₆	C₂×Dic₇₃	1	2	yes	—
133	C₂×M₁₆	C₂×Dic₇₃	1	1	yes	characteristic
134	C₂²×C₈	C₄×D₇₃	1	2	yes	—
135	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
136	#117	C₂×Dic₇₃	1	2	yes	—
137	M₃₂	—	73	4	no	—
138	M₃₂	—	146	4	no	—
139	M₃₂	—	146	2	no	—
140	M₃₂	—	146	2	no	—
141	C₂×C₁₆	—	73	1	no	—
142	C₂×C₁₆	—	73	1	no	—
143	C₂×C₁₆	—	73	2	no	—
144	C₂×C₁₆	—	73	2	no	—
145	C₂×C₁₆	—	73	2	no	—
146	C₂×C₁₆	—	146	2	no	—
147	C₂×C₁₆	—	146	2	no	—
148	C₂×C₁₆	—	146	2	no	—
149	#101	#10596831	1	2	yes	—
150	#100	#10596831	1	2	yes	—
151	#99	#10596831	1	4	yes	—
152	#97	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	#41	—	2	2	no	—
155	#41	—	2	2	no	—
156	C₂×Q₈	C₂×#10596831	1	1	yes	characteristic
157	C₂×D₄	C₂×#10596831	1	2	yes	—
158	C₂×D₄	C₂×#10596831	1	1	yes	characteristic
159	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
160	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	2	no	—
163	M₁₆	—	2	4	no	—
164	M₁₆	C₂×#10596831	1	4	yes	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
169	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
170	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
171	C₂×C₈	D₄×D₇₃	1	2	yes	—
172	C₂×C₈	#11924027	1	2	yes	—
173	C₄⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	C₂×#10596831	1	4	yes	—
175	C₄²	—	2	2	no	—
176	C₁₆	—	73	1	no	—
177	C₁₆	—	73	1	no	—
178	C₁₆	—	73	2	no	—
179	C₁₆	—	146	2	no	—
180	C₁₆	—	146	4	no	—
181	C₂³	C₄×#10596831	1	2	yes	—
182	C₂³	C₂²×#10596831	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	—	2	4	no	—
189	C₂×C₄	C₄×#10596831	1	2	yes	—
190	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
191	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
192	C₂×C₄	C₂²×#10596831	1	1	yes	characteristic
193	C₂×C₄	#14556598	1	2	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	2	2	no	—
196	C₈	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
197	C₈	#14556547	1	1	yes	characteristic
198	C₈	#14556551	1	2	yes	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	2	no	—
202	C₂²	C₂×C₄×#10596831	1	1	yes	characteristic
203	C₂²	D₄×#10596831	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	4	4	no	—
206	C₄	C₂²⋊C₄×Dic₇₃	1	1	yes	characteristic
207	C₄	#17988543	1	1	yes	characteristic
208	C₄	D₄×#10596831	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	2	1	no	—
211	C₂	—	4	2	no	—
212	C₂	C₂²⋊C₄×#10596831	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×#1757	C₂
socle	C₂×C₇₃	C₈×C₂²⋊C₄
prime radical	C₂×#41×C₇₃	C₂×C₄
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1757	D₇₃
hypocenter	C₇₃	#12613
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1757

Complements

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

N	H ≅ G/N	type
3. #22611528	C₂	semidirect
4. #22608017	C₂	semidirect
8. C₂×#14556531	C₄	semidirect
11. C₂²×#11924003	C₄	semidirect
13. #17988369	C₄	semidirect
112. C₇₃	#12613	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
#22611552	2
#22611528	2
#22608017	2
C₇₃×#1757	2
#12613	73