Finite Group Databasebeta

Subgroups — #34001068

Order 18688.

178 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₂×#17989519	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556641	C₂	1	1	yes	maximal, characteristic
3	#22611530	C₂	1	1	yes	maximal, characteristic
4	#17989519	—	2	1	no	—
5	#17989519	C₄	1	2	yes	complemented
6	C₂×#14556641	—	2	1	no	—
7	C₂×#14556641	—	2	1	no	—
8	C₂×#14556641	—	2	1	no	—
9	C₂×#14556641	C₄	1	2	yes	complemented
10	C₂²×#11924003	C₂²	1	1	yes	characteristic
11	#17988542	C₄	1	1	yes	characteristic
12	#17988362	—	2	1	no	—
13	C₁₆×Dic₇₃	—	2	1	no	—
14	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
15	#14556641	—	2	2	no	—
16	#14556641	—	2	2	no	—
17	C₂×#11924012	C₈	1	1	yes	characteristic
18	#14556550	C₈	1	1	yes	characteristic
19	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
20	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
21	C₂×#11924003	D₄	1	1	yes	characteristic
22	C₂×#11924003	D₄	1	1	yes	characteristic
23	C₂×#11924003	D₄	1	1	yes	characteristic
24	C₂×#11924003	Q₈	1	1	yes	characteristic
25	#14556501	—	2	1	no	—
26	C₈×Dic₇₃	—	2	1	no	—
27	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₇₃	—	2	1	no	—
29	M₃₂×C₇₃	C₈	1	2	yes	—
30	C₂×C₁₆×C₇₃	—	2	1	no	—
31	#11924012	—	2	1	no	—
32	#11924012	C₁₆	1	2	yes	—
33	C₂×#10596831	—	2	1	no	—
34	C₄⋊Dic₇₃	C₁₆	1	2	yes	—
35	C₂²⋊Dic₇₃	—	2	1	no	—
36	C₄×Dic₇₃	—	2	1	no	—
37	#11924003	C₂²⋊C₄	1	1	yes	characteristic
38	#11924003	C₂²⋊C₄	1	1	yes	characteristic
39	#11924003	C₄⋊C₄	1	1	yes	characteristic
40	#11924003	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
47	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
48	C₁₆×C₇₃	—	2	2	no	—
49	C₂³×C₇₃	M₃₂	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
56	C₂×Dic₇₃	—	2	2	no	—
57	C₈×C₇₃	#95	1	1	yes	characteristic
58	C₈×C₇₃	#97	1	1	yes	characteristic
59	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
60	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
61	#10596831	—	2	2	no	—
62	C₂²×C₇₃	—	2	2	no	—
63	C₂²×C₇₃	#346	1	1	yes	characteristic
64	C₂²×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
65	C₂²×C₇₃	C₄⋊C₁₆	1	1	yes	characteristic
66	C₄×C₇₃	#336	1	1	yes	characteristic
67	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
68	C₄×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
69	C₄×C₇₃	C₄⋊C₁₆	1	1	yes	characteristic
70	Dic₇₃	—	4	2	no	—
71	#7470	—	73	1	no	maximal, Sylow
72	C₂×C₇₃	—	2	1	no	—
73	C₂×C₇₃	—	2	1	no	—
74	C₂×C₇₃	#1374	1	1	yes	characteristic
75	C₂×C₇₃	#1401	1	2	yes	—
76	C₂×M₆₄	—	73	1	no	—
77	C₂²×C₃₂	—	73	1	no	—
78	#2108	—	73	1	no	—
79	C₇₃	#7470	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	#64408	1	1	yes	characteristic
81	C₂²×C₁₆	—	73	1	no	—
82	#405	—	73	1	no	—
83	M₆₄	—	73	2	no	—
84	M₆₄	—	146	1	no	—
85	C₂×C₃₂	—	73	2	no	—
86	C₂×C₃₂	—	146	1	no	—
87	C₂×C₃₂	—	146	1	no	—
88	C₂×C₃₂	—	146	1	no	—
89	#346	—	146	1	no	—
90	C₄×C₁₆	—	146	1	no	—
91	C₂×M₁₆	—	73	1	no	—
92	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
93	#117	—	73	1	no	—
94	M₃₂	—	2	1	no	—
95	M₃₂	#10596833	1	2	yes	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	—	73	1	no	—
98	C₂×C₁₆	—	73	1	no	—
99	C₂×C₁₆	—	73	1	no	—
100	C₂×C₁₆	—	73	1	no	—
101	C₂×C₁₆	—	73	1	no	—
102	C₂×C₁₆	—	73	1	no	—
103	#97	—	146	1	no	—
104	C₄×C₈	—	146	1	no	—
105	C₃₂	—	146	2	no	—
106	C₃₂	—	146	2	no	—
107	C₂²×C₄	#11924036	1	1	yes	characteristic
108	M₁₆	—	73	2	no	—
109	M₁₆	—	146	1	no	—
110	C₂×C₈	—	146	1	no	—
111	C₂×C₈	C₄×#64408	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
113	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
114	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
115	C₂×C₈	#11924037	1	1	yes	characteristic
116	C₂×C₈	#11924038	1	1	yes	characteristic
117	C₄⋊C₄	—	73	2	no	—
118	C₂²⋊C₄	—	146	1	no	—
119	C₄²	—	146	1	no	—
120	C₁₆	—	2	2	no	—
121	C₁₆	—	73	1	no	—
122	C₁₆	—	73	1	no	—
123	C₁₆	—	73	1	no	—
124	C₁₆	—	73	1	no	—
125	C₂³	#14556666	1	1	yes	characteristic
126	C₂×C₄	—	146	2	no	—
127	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
128	C₂×C₄	#14556648	1	1	yes	characteristic
129	C₂×C₄	#14556658	1	1	yes	characteristic
130	C₂×C₄	#14556664	1	1	yes	characteristic
131	C₂×C₄	#14556664	1	1	yes	characteristic
132	C₂×C₄	#14556666	1	1	yes	characteristic
133	C₈	—	146	2	no	—
134	C₈	(C₂×#64408)⋊C₄	1	1	yes	characteristic
135	C₈	#14556646	1	1	yes	characteristic
136	C₈	C₂²⋊#10596833	1	1	yes	characteristic
137	C₈	C₄⋊#10596833	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	#17989476	1	1	yes	characteristic
140	C₂²	#17989482	1	1	yes	characteristic
141	C₂²	#17989508	1	1	yes	characteristic
142	C₄	—	292	2	no	—
143	C₄	#17989453	1	1	yes	characteristic
144	C₄	#17989473	1	1	yes	characteristic
145	C₄	#17989482	1	1	yes	characteristic
146	C₄	#17989508	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#22624970	1	1	yes	characteristic
150	C₂	#22625049	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊#64408
commutator	C₄×C₇₃	C₄×C₁₆
Frattini	C₂²×C₈	C₂×#64408
Fitting	C₂×M₃₂×C₇₃	C₄
socle	C₂²×C₇₃	C₂²⋊C₁₆
prime radical	C₂³×C₇₃	M₃₂
cosocle	C₂²×#11924003	C₂²
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	#7470
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
5. #17989519	C₄	semidirect
9. C₂×#14556641	C₄	semidirect
79. C₇₃	#7470	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989519	2
C₂²×#14556641	2
#22611530	2
#7470	73