Finite Group Databasebeta

Subgroups — #34014493

Order 18688.

178 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 76 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	D₇₃⋊#368	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989515	C₂	1	1	yes	maximal, characteristic
3	#22625095	C₂	1	2	yes	maximal, complemented
4	#22625094	C₂	1	2	yes	maximal, complemented
5	M₆₄×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556662	—	2	1	no	—
7	D₇₃⋊#108	—	2	1	no	—
8	#17989516	—	2	1	no	—
9	#17989515	—	2	1	no	—
10	#17989515	C₂²	1	1	yes	characteristic
11	#17989515	C₂²	1	2	yes	—
12	#17989515	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
14	#17988421	C₄	1	1	yes	characteristic
15	#17988420	C₄	1	2	yes	—
16	C₃₂×D₇₃	C₂²	1	2	yes	—
17	M₆₄×C₇₃	C₄	1	1	yes	characteristic
18	C₂×#11924038	—	4	1	no	—
19	D₇₃⋊M₁₆	—	4	1	no	—
20	#14556662	—	2	2	no	—
21	#14556662	—	2	1	no	—
22	#14556662	—	2	1	no	—
23	#14556662	—	4	1	no	—
24	#14556661	—	4	1	no	—
25	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
26	C₂×#11924003	Q₈	1	1	yes	characteristic
27	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₁₆×D₇₃	D₄	1	1	yes	characteristic
29	C₁₆×D₇₃	Q₈	1	1	yes	characteristic
30	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
31	#14556497	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₃₂×C₇₃	C₂×C₄	1	2	yes	—
34	#11924038	—	4	1	no	—
35	#11924038	—	4	1	no	—
36	#11924038	—	4	2	no	—
37	#11924038	—	8	1	no	—
38	#11924037	—	8	1	no	—
39	#11924036	—	8	1	no	—
40	C₂×#10596833	—	8	1	no	—
41	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
42	C₂×#10596831	Q₁₆	1	1	yes	characteristic
43	C₈×D₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₈×D₇₃	D₈	1	1	yes	characteristic
45	C₈×D₇₃	Q₁₆	1	1	yes	characteristic
46	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
47	#11924003	C₄⋊C₄	1	1	yes	characteristic
48	#11924003	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₁₆×C₇₃	C₄⋊C₄	1	1	yes	characteristic
51	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×D₇₃	Q₃₂	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
54	C₂×Dic₇₃	D₁₆	1	1	yes	characteristic
55	C₄×D₇₃	#107	1	1	yes	characteristic
56	C₄×D₇₃	D₁₆	1	1	yes	characteristic
57	C₄×D₇₃	Q₃₂	1	1	yes	characteristic
58	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
59	#10596833	—	8	2	no	—
60	#10596833	—	8	1	no	—
61	#10596833	—	8	1	no	—
62	C₈×C₇₃	#107	1	1	yes	characteristic
63	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	#10596831	#107	1	1	yes	characteristic
65	#10596831	C₂×Q₁₆	1	1	yes	characteristic
66	C₂²×C₇₃	#366	1	1	yes	characteristic
67	C₂×D₇₃	—	4	1	no	—
68	C₂×D₇₃	#366	1	1	yes	characteristic
69	C₂×D₇₃	C₂×Q₃₂	1	1	yes	characteristic
70	C₄×C₇₃	#366	1	1	yes	characteristic
71	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
72	Dic₇₃	#366	1	1	yes	characteristic
73	Dic₇₃	C₂×D₁₆	1	1	yes	characteristic
74	#13636	—	73	1	no	maximal, Sylow
75	C₂×C₇₃	—	2	1	no	—
76	C₂×C₇₃	C₂×#366	1	1	yes	characteristic
77	D₇₃	—	2	1	no	—
78	D₇₃	—	2	1	no	—
79	C₂×M₆₄	—	73	1	no	—
80	C₂×#368	—	73	1	no	—
81	C₂×#368	—	73	1	no	—
82	#1425	—	73	2	no	—
83	#1425	—	73	2	no	—
84	C₇₃	#13636	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₁₆	—	73	1	no	—
86	C₂×#108	—	146	1	no	—
87	C₂×#108	—	146	1	no	—
88	M₆₄	—	73	1	no	—
89	M₆₄	—	73	2	no	—
90	M₆₄	#64408	1	1	yes	characteristic
91	C₂×C₃₂	—	73	2	no	—
92	#368	—	73	2	no	—
93	#368	—	73	1	no	—
94	#368	—	73	1	no	—
95	#368	—	146	1	no	—
96	#368	—	146	1	no	—
97	C₂×M₁₆	—	292	1	no	—
98	C₂×M₁₆	—	292	1	no	—
99	C₂²×C₈	—	73	1	no	—
100	C₂×C₁₆	—	73	1	no	—
101	C₂×C₁₆	—	73	1	no	—
102	C₂×C₁₆	—	73	1	no	—
103	C₂×C₁₆	—	73	1	no	—
104	C₂×C₁₆	—	73	1	no	—
105	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
106	#108	—	292	1	no	—
107	#108	—	292	1	no	—
108	#108	—	146	2	no	—
109	#108	—	146	1	no	—
110	#108	—	146	1	no	—
111	C₃₂	—	73	2	no	—
112	C₃₂	C₂×#64408	1	2	yes	—
113	C₂²×C₄	—	73	1	no	—
114	M₁₆	—	292	1	no	—
115	M₁₆	—	292	1	no	—
116	M₁₆	—	292	2	no	—
117	M₁₆	—	584	1	no	—
118	M₁₆	—	584	1	no	—
119	C₂×C₈	—	584	1	no	—
120	C₂×C₈	—	584	1	no	—
121	C₂×C₈	—	73	1	no	—
122	C₂×C₈	—	73	1	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	—	73	1	no	—
126	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
127	C₁₆	—	73	1	no	—
128	C₁₆	—	73	1	no	—
129	C₁₆	C₄⋊#64408	1	1	yes	characteristic
130	C₁₆	C₂²×#64408	1	1	yes	characteristic
131	C₂³	—	73	1	no	—
132	C₂×C₄	—	73	1	no	—
133	C₂×C₄	—	73	1	no	—
134	C₂×C₄	—	73	1	no	—
135	C₂×C₄	—	73	1	no	—
136	C₂×C₄	—	73	1	no	—
137	C₂×C₄	#14556660	1	1	yes	characteristic
138	C₈	—	584	2	no	—
139	C₈	—	584	1	no	—
140	C₈	—	584	1	no	—
141	C₈	—	73	1	no	—
142	C₈	—	73	1	no	—
143	C₈	#14556660	1	1	yes	characteristic
144	C₈	C₂×C₄⋊#64408	1	1	yes	characteristic
145	C₂²	—	292	1	no	—
146	C₂²	—	73	1	no	—
147	C₂²	—	73	1	no	—
148	C₂²	#17989513	1	1	yes	characteristic
149	C₄	—	73	1	no	—
150	C₄	—	73	1	no	—
151	C₄	#17989513	1	1	yes	characteristic
152	C₄	C₂×#14556660	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	146	1	no	—
155	C₂	—	146	1	no	—
156	C₂	C₂×#17989513	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#17989513
commutator	C₁₆×C₇₃	C₂²×C₄
Frattini	C₁₆	C₂²×#64408
Fitting	M₆₄×C₇₃	C₄
socle	C₂×C₇₃	C₂×#366
prime radical	C₂²×D₇₃	Q₃₂
cosocle	C₁₆×D₇₃	C₂³
hypercenter	M₆₄	#64408
hypocenter	C₇₃	#13636
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
3. #22625095	C₂	semidirect
4. #22625094	C₂	semidirect
84. C₇₃	#13636	semidirect

Maximal subgroups

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

subgroup	index
D₇₃⋊#368	2
C₂×#17989515	2
#22625095	2
#22625094	2
M₆₄×D₇₃	2
#13636	73