Finite Group Databasebeta

Subgroups — #33000732

Order 18688.

246 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22612213	C₂	1	2	yes	maximal
2	#22612206	C₂	1	2	yes	maximal
3	#22609549	C₂	1	1	yes	maximal, characteristic
4	#22609541	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1311	C₂	1	1	yes	maximal, characteristic
6	#17988992	—	2	1	no	—
7	#17988992	C₂²	1	1	yes	characteristic
8	#17988992	C₂²	1	1	yes	characteristic
9	#17988992	C₂²	1	2	yes	—
10	#17988967	—	2	1	no	—
11	#17988938	—	2	1	no	—
12	#17988938	—	2	1	no	—
13	#17988415	—	2	4	no	—
14	#17988413	—	2	4	no	—
15	C₄×C₁₆×C₇₃	C₂²	1	2	yes	—
16	C₈²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₉₂	—	4	2	no	—
18	#14556591	—	2	1	no	—
19	#14556591	—	2	1	no	—
20	#14556591	—	2	2	no	—
21	#14556525	—	2	2	no	—
22	#14556525	—	2	2	no	—
23	#14556525	—	2	2	no	—
24	#14556525	—	2	2	no	—
25	#14556525	—	4	2	no	—
26	#14556516	—	4	2	no	—
27	#14556516	—	4	2	no	—
28	C₂×C₁₆×C₇₃	D₄	1	4	yes	—
29	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₄₆	—	4	1	no	—
33	C₂×Dic₁₄₆	—	4	1	no	—
34	C₂×Dic₁₄₆	—	4	2	no	—
35	Dic₂₉₂	—	8	4	no	—
36	C₄⋊Dic₇₃	—	4	1	no	—
37	C₄⋊Dic₇₃	—	4	2	no	—
38	C₄⋊Dic₇₃	—	4	1	no	—
39	C₄⋊Dic₇₃	—	4	1	no	—
40	C₄⋊Dic₇₃	—	4	1	no	—
41	C₄⋊Dic₇₃	—	4	2	no	—
42	C₂×C₈×C₇₃	D₈	1	2	yes	—
43	C₂×C₈×C₇₃	D₈	1	2	yes	—
44	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
45	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₁₆×C₇₃	Q₁₆	1	8	yes	complemented
47	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₇₃	—	8	1	no	—
51	C₂×Dic₇₃	—	8	2	no	—
52	C₂×Dic₇₃	—	8	1	no	—
53	Dic₁₄₆	—	8	4	no	—
54	Dic₁₄₆	—	8	2	no	—
55	Dic₁₄₆	—	8	2	no	—
56	C₈×C₇₃	—	2	2	no	—
57	C₈×C₇₃	—	2	2	no	—
58	C₈×C₇₃	C₂×Q₁₆	1	4	yes	—
59	C₂²×C₇₃	#493	1	1	yes	characteristic
60	C₄×C₇₃	#494	1	2	yes	—
61	C₄×C₇₃	#509	1	2	yes	—
62	C₄×C₇₃	#510	1	2	yes	—
63	Dic₇₃	—	16	2	no	—
64	Dic₇₃	—	16	1	no	—
65	Dic₇₃	—	16	1	no	—
66	#12524	—	73	1	no	maximal, Sylow
67	C₂×C₇₃	#1713	1	1	yes	characteristic
68	C₂×C₇₃	#2250	1	1	yes	characteristic
69	C₂×C₇₃	#2251	1	1	yes	characteristic
70	#2254	—	73	2	no	—
71	#2252	—	73	2	no	—
72	#1713	—	73	1	no	—
73	#1710	—	73	1	no	—
74	#1311	D₇₃	1	1	yes	characteristic
75	C₇₃	#12524	1	1	yes	characteristic, complemented, Sylow
76	#500	—	73	1	no	—
77	#500	—	73	1	no	—
78	#500	—	73	2	no	—
79	#500	—	146	1	no	—
80	#494	—	146	1	no	—
81	#487	—	146	1	no	—
82	#487	—	146	1	no	—
83	#367	—	146	4	no	—
84	#366	—	146	4	no	—
85	C₄×C₁₆	C₂×D₇₃	1	2	yes	—
86	C₈²	C₂×D₇₃	1	1	yes	characteristic
87	C₂×Q₁₆	—	292	2	no	—
88	C₄⋊Q₈	—	146	1	no	—
89	C₄⋊Q₈	—	146	1	no	—
90	C₄⋊Q₈	—	146	2	no	—
91	C₂×C₁₆	D₂₉₂	1	4	yes	—
92	#107	—	292	2	no	—
93	#107	—	146	2	no	—
94	#107	—	146	2	no	—
95	#107	—	146	2	no	—
96	#107	—	146	2	no	—
97	#103	—	292	2	no	—
98	#103	—	292	2	no	—
99	C₄×C₈	D₂₉₂	1	1	yes	characteristic
100	C₄×C₈	D₂₉₂	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
102	C₂×Q₈	—	292	1	no	—
103	C₂×Q₈	—	292	1	no	—
104	C₂×Q₈	—	292	2	no	—
105	Q₁₆	—	584	4	no	—
106	C₂×C₈	D₅₈₄	1	2	yes	—
107	C₂×C₈	D₅₈₄	1	2	yes	—
108	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
109	C₄⋊C₄	—	292	1	no	—
110	C₄⋊C₄	—	292	2	no	—
111	C₄⋊C₄	—	292	1	no	—
112	C₄⋊C₄	—	292	1	no	—
113	C₄⋊C₄	—	292	1	no	—
114	C₄⋊C₄	—	292	2	no	—
115	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
116	C₁₆	Dic₂₉₂	1	8	yes	complemented
117	Q₈	—	584	4	no	—
118	Q₈	—	584	2	no	—
119	Q₈	—	584	2	no	—
120	C₂×C₄	—	584	1	no	—
121	C₂×C₄	—	584	2	no	—
122	C₂×C₄	—	584	1	no	—
123	C₂×C₄	#14556588	1	1	yes	characteristic
124	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
125	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₈	C₂×Dic₂₉₂	1	4	yes	—
129	C₂²	#17988964	1	1	yes	characteristic
130	C₄	—	1168	2	no	—
131	C₄	—	1168	1	no	—
132	C₄	—	1168	1	no	—
133	C₄	#17988967	1	2	yes	—
134	C₄	#17989025	1	2	yes	—
135	C₄	#17989030	1	2	yes	—
136	C₂	#22609549	1	1	yes	characteristic
137	C₂	#22612197	1	1	yes	characteristic
138	C₂	#22612202	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17988964
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1311	C₂
socle, prime radical	C₂²×C₇₃	#493
hypercenter	#1311	D₇₃
hypocenter	C₇₃	#12524
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

Complements

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

N	H ≅ G/N	type
46. C₁₆×C₇₃	Q₁₆	semidirect
75. C₇₃	#12524	semidirect
116. C₁₆	Dic₂₉₂	semidirect

Maximal subgroups

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

subgroup	index
#22612213	2
#22612206	2
#22609549	2
#22609541	2
C₇₃×#1311	2
#12524	73