Finite Group Databasebeta

Subgroups — #34013299

Order 18688.

274 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 100 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	#22625872	C₂	1	1	yes	maximal, characteristic
2	#22625871	C₂	1	1	yes	maximal, characteristic
3	#22624991	C₂	1	4	yes	maximal
4	#346×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556662	—	2	1	no	—
6	C₂×#14556662	C₂²	1	2	yes	—
7	C₂×#14556662	C₂²	1	2	yes	—
8	C₂×#14556661	—	2	1	no	—
9	#17989592	—	2	1	no	—
10	#17989591	—	2	1	no	—
11	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
12	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
13	#17988363	C₄	1	1	yes	characteristic
14	#17988362	C₄	1	2	yes	—
15	#346×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924038	—	2	2	no	—
17	C₂×#11924038	—	2	2	no	—
18	#14556662	—	2	4	no	—
19	#14556662	—	2	4	no	—
20	#14556662	—	4	2	no	—
21	#14556661	—	4	2	no	—
22	#14556658	—	4	2	no	—
23	C₄⋊#10596833	—	4	2	no	—
24	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
25	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
26	C₄²×D₇₃	D₄	1	1	yes	characteristic
27	C₂×#11924003	C₂×C₄	1	2	yes	—
28	C₁₆×D₇₃	—	2	2	no	—
29	C₁₆×D₇₃	—	2	2	no	—
30	C₄×#10596831	Q₈	1	1	yes	characteristic
31	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
33	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	#11924038	—	4	2	no	—
36	#11924038	—	4	2	no	—
37	#11924038	—	4	2	no	—
38	#11924038	—	4	2	no	—
39	C₂×#10596833	—	4	2	no	—
40	C₂×#10596833	—	4	2	no	—
41	C₂×C₄×D₇₃	Q₁₆	1	2	yes	—
42	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
43	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
45	C₈×D₇₃	SD₁₆	1	2	yes	—
46	C₈×D₇₃	SD₁₆	1	2	yes	—
47	C₈×D₇₃	#41	1	2	yes	—
48	C₈×D₇₃	#41	1	2	yes	—
49	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₄×Dic₇₃	D₈	1	2	yes	—
51	#11924003	—	2	2	no	—
52	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
55	C₁₆×C₇₃	—	2	2	no	—
56	C₂²×D₇₃	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#107	1	2	yes	—
58	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×Dic₇₃	#107	1	2	yes	—
60	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
61	C₄×D₇₃	—	4	2	no	—
62	C₄×D₇₃	#124	1	1	yes	characteristic
63	C₄×D₇₃	#124	1	1	yes	characteristic
64	C₄×D₇₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₄×D₇₃	C₂×SD₁₆	1	1	yes	characteristic
66	#10596833	—	8	2	no	—
67	#10596833	—	8	2	no	—
68	C₈×C₇₃	#106	1	2	yes	—
69	C₈×C₇₃	#117	1	2	yes	—
70	#10596831	#106	1	2	yes	—
71	#10596831	#125	1	2	yes	—
72	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
73	C₂×D₇₃	—	2	2	no	—
74	C₂×D₇₃	#487	1	1	yes	characteristic
75	C₂×D₇₃	#487	1	1	yes	characteristic
76	C₄×C₇₃	—	2	2	no	—
77	C₄×C₇₃	#382	1	1	yes	characteristic
78	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
79	Dic₇₃	—	2	2	no	—
80	Dic₇₃	#486	1	1	yes	characteristic
81	Dic₇₃	#486	1	1	yes	characteristic
82	#12892	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	#1834	1	1	yes	characteristic
84	C₂×C₇₃	#2161	1	2	yes	—
85	D₇₃	—	2	1	no	—
86	D₇₃	—	2	1	no	—
87	C₂×#346	—	73	1	no	—
88	#1848	—	73	1	no	—
89	#1848	—	73	1	no	—
90	#1381	—	73	4	no	—
91	C₇₃	#12892	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₁₆	—	73	2	no	—
93	C₂×#108	—	73	2	no	—
94	C₂×#108	—	73	2	no	—
95	C₂×#108	—	146	1	no	—
96	C₂×#108	—	146	1	no	—
97	#423	—	146	1	no	—
98	#423	—	146	1	no	—
99	C₂×C₄×C₈	—	73	1	no	—
100	#346	—	73	2	no	—
101	#346	—	73	1	no	—
102	#346	#64408	1	1	yes	characteristic
103	C₂×M₁₆	—	146	2	no	—
104	C₂×M₁₆	—	146	2	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂²×C₈	—	73	1	no	—
107	C₂×C₄²	—	73	1	no	—
108	C₂×C₁₆	—	73	2	no	—
109	C₂×C₁₆	—	146	2	no	—
110	C₂×C₁₆	—	146	2	no	—
111	C₂×C₁₆	C₂×#64408	1	2	yes	—
112	#108	—	292	2	no	—
113	#108	—	292	2	no	—
114	#108	—	146	4	no	—
115	#108	—	146	4	no	—
116	C₄⋊C₈	—	292	2	no	—
117	C₄⋊C₈	—	292	2	no	—
118	C₄×C₈	—	73	1	no	—
119	C₄×C₈	—	73	1	no	—
120	C₄×C₈	—	73	1	no	—
121	C₄×C₈	C₂×#64408	1	1	yes	characteristic
122	C₂²×C₄	—	73	2	no	—
123	C₂²×C₄	—	73	1	no	—
124	M₁₆	—	292	2	no	—
125	M₁₆	—	292	2	no	—
126	M₁₆	—	292	2	no	—
127	M₁₆	—	292	2	no	—
128	C₂×C₈	—	292	2	no	—
129	C₂×C₈	—	292	2	no	—
130	C₂×C₈	—	73	1	no	—
131	C₂×C₈	—	73	1	no	—
132	C₂×C₈	—	73	2	no	—
133	C₂×C₈	—	73	2	no	—
134	C₂×C₈	—	73	2	no	—
135	C₂×C₈	—	73	2	no	—
136	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
137	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
138	C₄²	—	73	2	no	—
139	C₄²	—	73	1	no	—
140	C₄²	C₄⋊#64408	1	1	yes	characteristic
141	C₁₆	—	2	2	no	—
142	C₁₆	—	146	2	no	—
143	C₂³	—	73	1	no	—
144	C₂×C₄	—	292	2	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	73	1	no	—
148	C₂×C₄	—	73	1	no	—
149	C₂×C₄	—	73	1	no	—
150	C₂×C₄	—	73	2	no	—
151	C₂×C₄	#14556660	1	2	yes	—
152	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
153	C₈	—	584	2	no	—
154	C₈	—	584	2	no	—
155	C₈	—	73	2	no	—
156	C₈	—	73	2	no	—
157	C₈	#14556659	1	2	yes	—
158	C₈	#14556670	1	2	yes	—
159	C₂²	—	73	1	no	—
160	C₂²	—	73	1	no	—
161	C₂²	—	146	2	no	—
162	C₂²	C₂×#14556660	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	73	1	no	—
165	C₄	—	73	1	no	—
166	C₄	—	146	2	no	—
167	C₄	#17989531	1	1	yes	characteristic
168	C₄	C₂×#14556659	1	1	yes	characteristic
169	C₂	—	146	1	no	—
170	C₂	—	146	1	no	—
171	C₂	#22625852	1	1	yes	characteristic
172	C₂	#22626287	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14556660
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#64408
Fitting	#346×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂²×D₇₃	C₂×Q₁₆
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	#346	#64408
hypocenter	C₇₃	#12892
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
91. C₇₃	#12892	semidirect

Maximal subgroups

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

subgroup	index
#22625872	2
#22625871	2
#22624991	2
#346×D₇₃	2
#12892	73