Finite Group Databasebeta

Subgroups — #34033894

Order 18688.

222 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22627125	C₂	1	2	yes	maximal
2	#22626994	C₂	1	2	yes	maximal, complemented
3	#22626993	C₂	1	2	yes	maximal, complemented
4	C₂×#17989474	C₂	1	1	yes	maximal, characteristic
5	#17989706	—	2	2	no	—
6	#17989692	—	2	2	no	—
7	#17989691	C₂²	1	4	yes	—
8	C₂×#14556664	C₄	1	1	yes	characteristic
9	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
10	#17989474	C₄	1	1	yes	characteristic
11	#17989474	C₄	1	1	yes	characteristic
12	#17989474	C₂²	1	1	yes	characteristic
13	#17989474	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic
15	#14556688	—	2	4	no	—
16	#14556687	—	4	2	no	—
17	#14556686	—	4	2	no	—
18	C₂×#11924036	D₄	1	1	yes	characteristic
19	C₂×C₄×#64408	Q₈	1	1	yes	characteristic
20	#14556664	—	2	1	no	—
21	#14556664	C₂×C₄	1	1	yes	characteristic
22	#14556664	C₂×C₄	1	1	yes	characteristic
23	C₂×#11924031	—	2	1	no	—
24	C₈×#64408	C₂×C₄	1	1	yes	characteristic
25	C₈×#64408	D₄	1	1	yes	characteristic
26	C₈×#64408	Q₈	1	1	yes	characteristic
27	C₈×#64408	C₂³	1	1	yes	characteristic
28	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×#11924003	C₈	1	1	yes	characteristic
30	C₁₆×D₇₃	C₈	1	1	yes	characteristic
31	C₁₆×D₇₃	C₈	1	1	yes	characteristic
32	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
33	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₇₃	C₈	1	1	yes	characteristic
35	#11924040	—	4	4	no	—
36	C₂²×#64408	—	2	1	no	—
37	#11924036	C₄⋊C₄	1	1	yes	characteristic
38	#11924036	SD₁₆	1	2	yes	—
39	#11924036	C₂×D₄	1	1	yes	characteristic
40	C₂×#10596833	SD₁₆	1	2	yes	—
41	C₄×#64408	—	2	1	no	—
42	C₄×#64408	C₄⋊C₄	1	1	yes	characteristic
43	C₄×#64408	C₂×Q₈	1	1	yes	characteristic
44	#11924031	—	2	1	no	—
45	#11924031	—	2	1	no	—
46	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×#10596831	M₁₆	1	1	yes	characteristic
48	C₈×D₇₃	C₄⋊C₄	1	1	yes	characteristic
49	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
50	C₈×D₇₃	M₁₆	1	1	yes	characteristic
51	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
52	#11924003	C₂×C₈	1	1	yes	characteristic
53	#11924003	C₂×C₈	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
55	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
56	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×D₇₃	#108	1	1	yes	characteristic
58	C₂×#64408	—	2	1	no	—
59	C₂×#64408	—	2	1	no	—
60	C₂×#64408	—	4	2	no	—
61	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×Dic₇₃	#106	1	1	yes	characteristic
63	C₄×D₇₃	C₄⋊C₈	1	1	yes	characteristic
64	C₄×D₇₃	#106	1	1	yes	characteristic
65	C₄×D₇₃	#108	1	1	yes	characteristic
66	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	#10596833	#106	1	2	yes	—
68	#10596833	C₂×SD₁₆	1	2	yes	—
69	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
70	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
71	#10596831	C₄⋊C₈	1	1	yes	characteristic
72	#10596831	C₂×M₁₆	1	1	yes	characteristic
73	C₂²×C₇₃	#334	1	1	yes	characteristic
74	C₂×D₇₃	—	2	2	no	—
75	C₂×D₇₃	#334	1	1	yes	characteristic
76	C₂×D₇₃	C₂×#108	1	1	yes	characteristic
77	#64408	—	4	1	no	—
78	#64408	—	4	1	no	—
79	C₄×C₇₃	#334	1	1	yes	characteristic
80	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
81	Dic₇₃	#334	1	1	yes	characteristic
82	Dic₇₃	C₂×#106	1	1	yes	characteristic
83	C₂×#1369	—	73	1	no	maximal, Sylow
84	C₂×C₇₃	—	2	1	no	—
85	C₂×C₇₃	C₂×#334	1	1	yes	characteristic
86	D₇₃	—	2	1	no	—
87	D₇₃	#1369	1	2	yes	complemented
88	C₂×#364	—	73	2	no	—
89	C₂×#346	—	73	1	no	—
90	#1369	—	73	2	no	—
91	#1369	—	73	2	no	—
92	C₇₃	C₂×#1369	1	1	yes	characteristic, complemented, Sylow
93	C₂×M₃₂	—	146	2	no	—
94	C₂²×C₁₆	—	73	1	no	—
95	C₂²×C₁₆	—	73	1	no	—
96	C₂×C₄×C₈	—	73	1	no	—
97	#364	—	73	4	no	—
98	#364	—	146	2	no	—
99	#346	—	73	1	no	—
100	#346	—	73	1	no	—
101	#346	—	73	1	no	—
102	#346	—	73	1	no	—
103	C₂²×C₈	—	73	1	no	—
104	C₂²×C₈	—	73	1	no	—
105	C₂×C₄²	—	73	1	no	—
106	M₃₂	—	292	2	no	—
107	M₃₂	—	146	4	no	—
108	C₂×C₁₆	—	292	2	no	—
109	C₂×C₁₆	—	73	1	no	—
110	C₂×C₁₆	—	73	1	no	—
111	C₂×C₁₆	—	73	1	no	—
112	C₂×C₁₆	—	73	1	no	—
113	C₂×C₁₆	—	73	1	no	—
114	C₂×C₁₆	—	73	1	no	—
115	C₂×C₁₆	—	73	1	no	—
116	C₂×C₁₆	—	146	1	no	—
117	C₂×C₁₆	—	146	1	no	—
118	C₂×C₁₆	#10596842	1	1	yes	characteristic
119	C₄×C₈	—	73	1	no	—
120	C₄×C₈	—	73	1	no	—
121	C₄×C₈	—	73	1	no	—
122	C₄×C₈	—	73	1	no	—
123	C₂²×C₄	—	73	1	no	—
124	C₂²×C₄	—	146	1	no	—
125	C₂×C₈	—	73	2	no	—
126	C₂×C₈	—	73	1	no	—
127	C₂×C₈	—	73	2	no	—
128	C₂×C₈	—	73	1	no	—
129	C₂×C₈	—	73	1	no	—
130	C₂×C₈	—	73	1	no	—
131	C₂×C₈	—	73	1	no	—
132	C₂×C₈	—	73	1	no	—
133	C₂×C₈	—	73	1	no	—
134	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
135	C₄²	—	73	1	no	—
136	C₄²	—	73	1	no	—
137	C₄²	—	146	1	no	—
138	C₁₆	—	292	4	no	—
139	C₁₆	—	73	1	no	—
140	C₁₆	—	73	1	no	—
141	C₁₆	—	146	1	no	—
142	C₁₆	—	146	1	no	—
143	C₁₆	C₂×#10596842	1	1	yes	characteristic
144	C₁₆	C₂×#10596842	1	1	yes	characteristic
145	C₂³	—	73	1	no	—
146	C₂×C₄	—	292	2	no	—
147	C₂×C₄	—	73	1	no	—
148	C₂×C₄	—	73	1	no	—
149	C₂×C₄	—	73	1	no	—
150	C₂×C₄	—	73	1	no	—
151	C₂×C₄	—	73	1	no	—
152	C₂×C₄	—	146	1	no	—
153	C₂×C₄	—	146	1	no	—
154	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
155	C₈	—	73	2	no	—
156	C₈	—	73	2	no	—
157	C₈	—	73	1	no	—
158	C₈	—	73	1	no	—
159	C₈	C₄⋊#10596842	1	1	yes	characteristic
160	C₈	C₂²×#10596842	1	1	yes	characteristic
161	C₂²	—	73	1	no	—
162	C₂²	—	73	1	no	—
163	C₂²	—	146	2	no	—
164	C₂²	#17989676	1	1	yes	characteristic
165	C₄	—	292	1	no	—
166	C₄	—	292	1	no	—
167	C₄	—	73	1	no	—
168	C₄	—	73	1	no	—
169	C₄	#17989676	1	1	yes	characteristic
170	C₄	C₂×C₄⋊#10596842	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	73	2	no	—
173	C₂	—	146	1	no	—
174	C₂	C₂×#17989676	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17989676
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₈	C₂²×#10596842
Fitting	C₂×C₁₆×C₇₃	C₈
socle	C₂×C₇₃	C₂×#334
prime radical	C₂²×D₇₃	#108
cosocle	C₈×#64408	C₂³
hypercenter	C₂×C₁₆	#10596842
hypocenter	C₇₃	C₂×#1369
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22626994	C₂	semidirect
3. #22626993	C₂	semidirect
87. D₇₃	#1369	semidirect
92. C₇₃	C₂×#1369	semidirect

Maximal subgroups

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

subgroup	index
#22627125	2
#22626994	2
#22626993	2
C₂×#17989474	2
C₂×#1369	73