Finite Group Databasebeta

Subgroups — #33006603

Order 18688.

182 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22611817	C₂	1	1	yes	maximal, characteristic
2	#22611802	C₂	1	1	yes	maximal, characteristic, complemented
3	#22608137	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608124	C₂	1	1	yes	maximal, characteristic, complemented
5	#22608118	C₂	1	1	yes	maximal, characteristic
6	#22608066	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#2264	C₂	1	1	yes	maximal, characteristic
8	#17988704	—	2	1	no	—
9	Q₁₆×Dic₇₃	—	2	1	no	—
10	#17988421	C₂²	1	1	yes	characteristic
11	#17988417	C₂²	1	1	yes	characteristic
12	#17988415	C₂²	1	1	yes	characteristic
13	#17988394	—	2	1	no	—
14	#17988391	—	2	1	no	—
15	#17988362	C₂²	1	1	yes	characteristic
16	#508×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
18	Q₆₄×C₇₃	C₄	1	2	yes	complemented
19	SD₆₄×C₇₃	C₄	1	2	yes	complemented
20	M₆₄×C₇₃	C₂²	1	1	yes	characteristic
21	#14556598	—	4	1	no	—
22	Q₈×Dic₇₃	—	4	1	no	—
23	C₂×#11924003	D₄	1	1	yes	characteristic
24	#14556527	—	2	1	no	—
25	#14556525	—	2	1	no	—
26	#14556518	—	4	1	no	—
27	#14556515	—	4	1	no	—
28	C₈×Dic₇₃	D₄	1	1	yes	characteristic
29	#14556497	—	2	1	no	—
30	#135×C₇₃	—	2	1	no	—
31	C₂×Q₁₆×C₇₃	—	2	1	no	—
32	Q₃₂×C₇₃	—	2	1	no	—
33	Q₃₂×C₇₃	C₂×C₄	1	2	yes	—
34	Q₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
35	SD₃₂×C₇₃	—	2	1	no	—
36	D₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
38	C₃₂×C₇₃	C₂×C₄	1	2	yes	—
39	#11924012	—	4	1	no	—
40	#11924012	—	8	1	no	—
41	C₂×#10596831	—	8	1	no	—
42	C₂×#10596831	D₈	1	1	yes	characteristic
43	C₄⋊Dic₇₃	—	4	1	no	—
44	C₄⋊Dic₇₃	—	8	1	no	—
45	C₄×Dic₇₃	—	8	1	no	—
46	C₄×Dic₇₃	D₈	1	1	yes	characteristic
47	#11924003	—	2	1	no	—
48	#41×C₇₃	—	4	1	no	—
49	C₂×Q₈×C₇₃	—	4	1	no	—
50	Q₁₆×C₇₃	—	2	1	no	—
51	Q₁₆×C₇₃	—	2	2	no	—
52	Q₁₆×C₇₃	—	4	1	no	—
53	SD₁₆×C₇₃	—	4	1	no	—
54	D₈×C₇₃	—	2	1	no	—
55	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₁₆×C₇₃	#41	1	1	yes	characteristic
58	Q₈×C₇₃	—	4	1	no	—
59	Q₈×C₇₃	—	4	2	no	—
60	Q₈×C₇₃	—	8	1	no	—
61	D₄×C₇₃	—	4	1	no	—
62	D₄×C₇₃	—	8	1	no	—
63	C₂×C₄×C₇₃	—	8	1	no	—
64	C₂×C₄×C₇₃	—	8	1	no	—
65	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
66	C₂×Dic₇₃	—	2	1	no	—
67	C₂×Dic₇₃	—	8	1	no	—
68	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
69	C₈×C₇₃	#135	1	1	yes	characteristic
70	#10596831	—	2	1	no	—
71	#10596831	—	8	1	no	—
72	C₂²×C₇₃	—	8	1	no	—
73	C₂²×C₇₃	#509	1	1	yes	characteristic
74	C₄×C₇₃	—	8	1	no	—
75	C₄×C₇₃	—	8	2	no	—
76	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
77	C₄×C₇₃	#509	1	1	yes	characteristic
78	Dic₇₃	—	4	1	no	—
79	Dic₇₃	—	16	1	no	—
80	#13657	—	73	1	no	maximal, Sylow
81	C₂×C₇₃	—	2	1	no	—
82	C₂×C₇₃	—	16	1	no	—
83	C₂×C₇₃	#2177	1	1	yes	characteristic
84	#2264	D₇₃	1	1	yes	characteristic
85	#2179	—	73	1	no	—
86	#2176	—	73	1	no	—
87	#1426	—	73	1	no	—
88	#1420	—	73	1	no	—
89	#1418	—	73	1	no	—
90	#1398	—	73	1	no	—
91	C₇₃	#13657	1	1	yes	characteristic, complemented, Sylow
92	#508	C₂×D₇₃	1	1	yes	characteristic
93	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
94	#443	—	146	1	no	—
95	C₄×Q₁₆	—	146	1	no	—
96	Q₆₄	Dic₇₃	1	2	yes	complemented
97	SD₆₄	Dic₇₃	1	2	yes	complemented
98	M₆₄	—	73	1	no	—
99	M₆₄	C₂×D₇₃	1	1	yes	characteristic
100	#368	—	73	1	no	—
101	#367	—	73	1	no	—
102	#359	—	146	1	no	—
103	#358	—	146	1	no	—
104	#346	—	73	1	no	—
105	#135	—	2	1	no	—
106	C₂×Q₁₆	—	2	1	no	—
107	#131	—	292	1	no	—
108	C₄×Q₈	—	292	1	no	—
109	Q₃₂	—	2	1	no	—
110	Q₃₂	C₂×Dic₇₃	1	2	yes	—
111	Q₃₂	C₂×Dic₇₃	1	1	yes	characteristic
112	SD₃₂	—	2	1	no	—
113	D₁₆	C₂×Dic₇₃	1	1	yes	characteristic
114	C₂×C₁₆	—	73	1	no	—
115	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
116	#108	—	146	1	no	—
117	#107	—	146	1	no	—
118	#104	—	292	1	no	—
119	#103	—	292	1	no	—
120	C₄×C₈	—	73	1	no	—
121	C₃₂	—	146	1	no	—
122	C₃₂	C₂×Dic₇₃	1	2	yes	—
123	#41	—	4	1	no	—
124	C₂×Q₈	—	4	1	no	—
125	Q₁₆	—	2	1	no	—
126	Q₁₆	—	2	2	no	—
127	Q₁₆	—	4	1	no	—
128	SD₁₆	—	4	1	no	—
129	D₈	—	2	1	no	—
130	M₁₆	—	292	1	no	—
131	M₁₆	—	584	1	no	—
132	C₂×C₈	—	584	1	no	—
133	C₂×C₈	—	73	1	no	—
134	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
135	C₄⋊C₄	—	292	1	no	—
136	C₄⋊C₄	—	584	1	no	—
137	C₄²	—	584	1	no	—
138	C₄²	—	73	1	no	—
139	C₁₆	—	146	1	no	—
140	C₁₆	C₂²×Dic₇₃	1	1	yes	characteristic
141	C₁₆	#11924027	1	1	yes	characteristic
142	Q₈	—	4	1	no	—
143	Q₈	—	4	2	no	—
144	Q₈	—	8	1	no	—
145	D₄	—	4	1	no	—
146	D₄	—	8	1	no	—
147	C₂×C₄	—	8	1	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	584	1	no	—
150	C₂×C₄	—	146	1	no	—
151	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
152	C₈	—	584	1	no	—
153	C₈	—	146	1	no	—
154	C₈	D₄×Dic₇₃	1	1	yes	characteristic
155	C₈	#14556610	1	1	yes	characteristic
156	C₂²	—	8	1	no	—
157	C₂²	#17989021	1	1	yes	characteristic
158	C₄	—	8	1	no	—
159	C₄	—	8	2	no	—
160	C₄	—	292	1	no	—
161	C₄	—	1168	1	no	—
162	C₄	D₈×Dic₇₃	1	1	yes	characteristic
163	C₄	#17989021	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	16	1	no	—
166	C₂	#22611807	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22611807
commutator	C₁₆×C₇₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#2264	C₂
socle	C₂×C₇₃	#2177
prime radical	#508×C₇₃	C₂²
cosocle	C₂×C₁₆×C₇₃	C₂³
hypercenter	#2264	D₇₃
hypocenter	C₇₃	#13657
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2264

Complements

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

N	H ≅ G/N	type
2. #22611802	C₂	semidirect
3. #22608137	C₂	semidirect
4. #22608124	C₂	semidirect
6. #22608066	C₂	semidirect
18. Q₆₄×C₇₃	C₄	semidirect
19. SD₆₄×C₇₃	C₄	semidirect
91. C₇₃	#13657	semidirect
96. Q₆₄	Dic₇₃	semidirect
97. SD₆₄	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22611817	2
#22611802	2
#22608137	2
#22608124	2
#22608118	2
#22608066	2
C₇₃×#2264	2
#13657	73