Finite Group Databasebeta

Subgroups — #33006594

Order 18688.

184 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612238	C₂	1	1	yes	maximal, characteristic
2	#22611812	C₂	1	1	yes	maximal, characteristic, complemented
3	#22611811	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608134	C₂	1	1	yes	maximal, characteristic, complemented
5	#22608116	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608064	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1417	C₂	1	1	yes	maximal, characteristic, complemented
8	#17989019	C₂²	1	1	yes	characteristic
9	#17988704	—	2	1	no	—
10	#17988703	—	2	1	no	—
11	#17988427	C₄	1	1	yes	characteristic, complemented
12	#17988425	C₄	1	1	yes	characteristic, complemented
13	#17988424	C₄	1	1	yes	characteristic, complemented
14	#17988422	C₄	1	1	yes	characteristic, complemented
15	C₂×#14556497	C₂²	1	1	yes	characteristic
16	#17988417	C₂²	1	1	yes	characteristic
17	#17988394	—	2	1	no	—
18	#17988393	—	2	1	no	—
19	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
20	#508×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
22	#368×C₇₃	C₂²	1	1	yes	characteristic
23	#14556613	—	2	1	no	—
24	#14556599	—	4	1	no	—
25	#14556598	—	4	1	no	—
26	Dic₅₈₄	C₂×C₄	1	1	yes	characteristic
27	#14556536	—	2	1	no	—
28	D₁₁₆₈	C₂×C₄	1	1	yes	characteristic
29	C₂×#11924003	D₄	1	1	yes	characteristic
30	#14556527	—	2	1	no	—
31	#14556518	—	4	1	no	—
32	#14556517	—	4	1	no	—
33	C₈×Dic₇₃	D₄	1	1	yes	characteristic
34	#14556497	C₂×C₄	1	1	yes	characteristic
35	#14556497	C₂×C₄	1	1	yes	characteristic
36	#135×C₇₃	—	2	1	no	—
37	Q₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
38	SD₃₂×C₇₃	—	2	1	no	—
39	D₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
41	#108×C₇₃	—	2	1	no	—
42	C₃₂×C₇₃	—	2	1	no	—
43	#11924028	—	4	1	no	—
44	Dic₂₉₂	—	2	1	no	—
45	#11924017	—	4	1	no	—
46	D₅₈₄	—	2	1	no	—
47	#11924012	—	4	1	no	—
48	#11924012	—	8	1	no	—
49	#11924011	—	8	1	no	—
50	C₂×#10596831	—	8	1	no	—
51	C₂×#10596831	D₈	1	1	yes	characteristic
52	C₈×D₇₃	—	8	1	no	—
53	C₄×Dic₇₃	D₈	1	1	yes	characteristic
54	#11924003	—	2	1	no	—
55	#41×C₇₃	—	4	1	no	—
56	Q₁₆×C₇₃	—	2	1	no	—
57	SD₁₆×C₇₃	—	4	1	no	—
58	D₈×C₇₃	—	2	1	no	—
59	M₁₆×C₇₃	—	4	1	no	—
60	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₁₆×C₇₃	#41	1	1	yes	characteristic
63	Q₈×C₇₃	—	4	1	no	—
64	D₄×C₇₃	—	4	1	no	—
65	D₄×C₇₃	—	8	1	no	—
66	C₂×C₄×C₇₃	—	8	1	no	—
67	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
68	C₂²⋊D₇₃	—	8	1	no	—
69	C₂×Dic₇₃	—	2	1	no	—
70	D₂₉₂	—	4	1	no	—
71	C₄×D₇₃	—	8	1	no	—
72	Dic₁₄₆	—	4	1	no	—
73	C₈×C₇₃	—	8	1	no	—
74	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
75	C₈×C₇₃	#135	1	1	yes	characteristic
76	#10596831	—	8	1	no	—
77	#10596831	D₁₆	1	1	yes	characteristic
78	#10596831	D₁₆	1	1	yes	characteristic
79	C₂²×C₇₃	—	8	1	no	—
80	C₂²×C₇₃	#508	1	1	yes	characteristic
81	C₂×D₇₃	—	8	1	no	—
82	C₄×C₇₃	—	8	1	no	—
83	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
84	C₄×C₇₃	C₂×D₁₆	1	1	yes	characteristic
85	Dic₇₃	—	4	1	no	—
86	Dic₇₃	—	8	1	no	—
87	#13656	—	73	1	no	maximal, Sylow
88	C₂×C₇₃	—	2	1	no	—
89	C₂×C₇₃	—	16	1	no	—
90	C₂×C₇₃	C₄×D₁₆	1	1	yes	characteristic
91	D₇₃	—	16	1	no	—
92	#2262	—	73	1	no	—
93	#2178	—	73	1	no	—
94	#2178	—	73	1	no	—
95	#1425	—	73	1	no	—
96	#1417	—	73	1	no	—
97	#1417	D₇₃	1	1	yes	characteristic, complemented
98	#1397	—	73	1	no	—
99	C₇₃	#13656	1	1	yes	characteristic, complemented, Sylow
100	#508	—	73	1	no	—
101	#508	C₂×D₇₃	1	1	yes	characteristic
102	#443	—	146	1	no	—
103	#443	—	146	1	no	—
104	Q₆₄	—	73	1	no	—
105	SD₆₄	—	73	1	no	—
106	SD₆₄	—	73	1	no	—
107	D₃₂	—	73	1	no	—
108	C₂×C₃₂	—	73	1	no	—
109	C₂×C₃₂	C₂×D₇₃	1	1	yes	characteristic
110	#368	—	73	1	no	—
111	#368	C₂×D₇₃	1	1	yes	characteristic
112	#359	—	146	1	no	—
113	#359	—	146	1	no	—
114	C₄×C₁₆	—	73	1	no	—
115	#135	—	2	1	no	—
116	#135	—	146	1	no	—
117	#131	—	292	1	no	—
118	#131	—	292	1	no	—
119	Q₃₂	—	73	1	no	—
120	Q₃₂	C₄×D₇₃	1	1	yes	characteristic
121	SD₃₂	—	2	1	no	—
122	SD₃₂	—	146	1	no	—
123	D₁₆	—	73	1	no	—
124	D₁₆	C₄×D₇₃	1	1	yes	characteristic
125	C₂×C₁₆	—	73	1	no	—
126	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
127	#108	—	2	1	no	—
128	#108	—	146	1	no	—
129	#104	—	292	1	no	—
130	#104	—	292	1	no	—
131	C₄×C₈	—	73	1	no	—
132	C₃₂	—	2	1	no	—
133	C₃₂	—	73	1	no	—
134	C₃₂	—	73	1	no	—
135	#41	—	4	1	no	—
136	#41	—	292	1	no	—
137	Q₁₆	—	2	1	no	—
138	Q₁₆	—	146	1	no	—
139	SD₁₆	—	4	1	no	—
140	SD₁₆	—	292	1	no	—
141	D₈	—	2	1	no	—
142	D₈	—	146	1	no	—
143	M₁₆	—	4	1	no	—
144	M₁₆	—	292	1	no	—
145	M₁₆	—	584	1	no	—
146	M₁₆	—	584	1	no	—
147	C₂×C₈	—	584	1	no	—
148	C₂×C₈	—	584	1	no	—
149	C₂×C₈	—	73	1	no	—
150	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
151	C₄²	—	73	1	no	—
152	C₁₆	—	146	1	no	—
153	C₁₆	C₂×C₄×D₇₃	1	1	yes	characteristic
154	C₁₆	#11924027	1	1	yes	characteristic
155	Q₈	—	4	1	no	—
156	Q₈	—	292	1	no	—
157	D₄	—	4	1	no	—
158	D₄	—	8	1	no	—
159	D₄	—	292	1	no	—
160	D₄	—	584	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	584	1	no	—
163	C₂×C₄	—	146	1	no	—
164	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
165	C₈	—	8	1	no	—
166	C₈	—	584	1	no	—
167	C₈	—	73	1	no	—
168	C₈	—	73	1	no	—
169	C₈	#14556551	1	1	yes	characteristic
170	C₈	#14556612	1	1	yes	characteristic
171	C₂²	—	8	1	no	—
172	C₂²	—	584	1	no	—
173	C₂²	#17989018	1	1	yes	characteristic
174	C₄	—	8	1	no	—
175	C₄	—	292	1	no	—
176	C₄	—	584	1	no	—
177	C₄	#17988667	1	1	yes	characteristic
178	C₄	D₁₆×D₇₃	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	16	1	no	—
181	C₂	—	1168	1	no	—
182	C₂	#22611775	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₆×D₇₃
commutator	C₁₆×C₇₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#1417	C₂
socle	C₂×C₇₃	C₄×D₁₆
prime radical	#22612238	C₂
cosocle	C₂×C₁₆×C₇₃	C₂³
hypercenter	#1417	D₇₃
hypocenter	C₇₃	#13656
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1417

Complements

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

N	H ≅ G/N	type
2. #22611812	C₂	semidirect
3. #22611811	C₂	semidirect
4. #22608134	C₂	semidirect
5. #22608116	C₂	semidirect
6. #22608064	C₂	semidirect
7. C₇₃×#1417	C₂	semidirect
11. #17988427	C₄	semidirect
12. #17988425	C₄	semidirect
13. #17988424	C₄	semidirect
14. #17988422	C₄	semidirect
97. #1417	D₇₃	semidirect
99. C₇₃	#13656	semidirect

Maximal subgroups

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

subgroup	index
#22612238	2
#22611812	2
#22611811	2
#22608134	2
#22608116	2
#22608064	2
C₇₃×#1417	2
#13656	73