Finite Group Databasebeta

Subgroups — #34000584

Order 18688.

208 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 72 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	#22626270	C₂	1	1	yes	maximal, characteristic
2	#22625870	C₂	1	1	yes	maximal, characteristic
3	#364×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#17989643	—	2	1	no	—
5	C₂×#14556661	—	2	1	no	—
6	C₂×#14556661	—	2	1	no	—
7	#17989591	C₄	1	2	yes	—
8	#17989510	C₄	1	2	yes	—
9	#17989509	—	2	1	no	—
10	M₃₂×D₇₃	—	2	1	no	—
11	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	#17988408	C₄	1	1	yes	characteristic
13	#17988407	—	2	1	no	—
14	#364×C₇₃	C₄	1	1	yes	characteristic
15	C₂×#11924038	—	2	2	no	—
16	#14556666	—	4	1	no	—
17	#14556665	—	2	2	no	—
18	#14556664	—	4	1	no	—
19	#14556661	—	4	1	no	—
20	#14556661	—	4	1	no	—
21	#14556661	—	4	1	no	—
22	#14556661	—	4	1	no	—
23	C₄⋊#10596833	—	4	2	no	—
24	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
25	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
26	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
27	#14556531	—	2	1	no	—
28	#14556530	—	4	1	no	—
29	C₁₆×D₇₃	—	4	1	no	—
30	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₇₃	D₄	1	1	yes	characteristic
32	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
33	M₃₂×C₇₃	—	2	1	no	—
34	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	#11924038	—	4	2	no	—
36	#11924038	—	4	2	no	—
37	C₂×#10596833	—	4	2	no	—
38	#11924031	—	4	2	no	—
39	C₂×C₄×D₇₃	—	2	1	no	—
40	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
43	C₈×D₇₃	D₈	1	1	yes	characteristic
44	C₈×D₇₃	D₈	1	1	yes	characteristic
45	C₈×D₇₃	SD₁₆	1	1	yes	characteristic
46	C₈×D₇₃	SD₁₆	1	1	yes	characteristic
47	C₈×D₇₃	SD₁₆	1	1	yes	characteristic
48	C₈×D₇₃	SD₁₆	1	1	yes	characteristic
49	C₈×D₇₃	Q₁₆	1	1	yes	characteristic
50	C₈×D₇₃	Q₁₆	1	1	yes	characteristic
51	C₄×Dic₇₃	—	2	1	no	—
52	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
53	#11924003	—	4	1	no	—
54	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
56	C₄²×C₇₃	C₄²	1	1	yes	characteristic
57	C₁₆×C₇₃	—	4	1	no	—
58	C₂²×D₇₃	#101	1	1	yes	characteristic
59	C₂×C₄×C₇₃	—	2	1	no	—
60	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
61	C₂×Dic₇₃	—	2	1	no	—
62	C₂×Dic₇₃	#100	1	1	yes	characteristic
63	C₄×D₇₃	—	4	2	no	—
64	C₄×D₇₃	#102	1	1	yes	characteristic
65	C₄×D₇₃	#102	1	1	yes	characteristic
66	C₄×D₇₃	#103	1	1	yes	characteristic
67	C₄×D₇₃	#103	1	1	yes	characteristic
68	#10596833	—	8	2	no	—
69	C₈×C₇₃	#102	1	1	yes	characteristic
70	C₈×C₇₃	#103	1	1	yes	characteristic
71	C₈×C₇₃	#106	1	1	yes	characteristic
72	C₈×C₇₃	#107	1	1	yes	characteristic
73	#10596831	#102	1	1	yes	characteristic
74	#10596831	#103	1	1	yes	characteristic
75	#10596831	#106	1	1	yes	characteristic
76	#10596831	#107	1	1	yes	characteristic
77	C₂²×C₇₃	#343	1	1	yes	characteristic
78	C₂×D₇₃	—	4	1	no	—
79	C₂×D₇₃	#332	1	1	yes	characteristic
80	C₂×D₇₃	#332	1	1	yes	characteristic
81	C₄×C₇₃	—	2	2	no	—
82	C₄×C₇₃	#340	1	1	yes	characteristic
83	C₄×C₇₃	#340	1	1	yes	characteristic
84	Dic₇₃	—	4	1	no	—
85	Dic₇₃	#331	1	1	yes	characteristic
86	Dic₇₃	#333	1	1	yes	characteristic
87	#7270	—	73	1	no	maximal, Sylow
88	C₂×C₇₃	—	2	1	no	—
89	C₂×C₇₃	#1296	1	1	yes	characteristic
90	D₇₃	—	2	1	no	—
91	D₇₃	—	2	1	no	—
92	C₂×#364	—	73	1	no	—
93	C₂×#364	—	73	1	no	—
94	#1848	—	73	1	no	—
95	C₇₃	#7270	1	1	yes	characteristic, complemented, Sylow
96	C₂×M₃₂	—	146	1	no	—
97	C₂×M₃₂	—	146	1	no	—
98	C₂×#108	—	146	1	no	—
99	C₂×#108	—	146	1	no	—
100	#423	—	73	2	no	—
101	C₂×C₄×C₈	—	73	1	no	—
102	#364	—	73	2	no	—
103	#364	—	73	1	no	—
104	#364	—	146	1	no	—
105	#364	—	146	1	no	—
106	#364	#64408	1	1	yes	characteristic
107	C₂×M₁₆	—	146	2	no	—
108	C₂²×C₈	—	73	1	no	—
109	C₂²×C₈	—	73	1	no	—
110	C₂×C₄²	—	73	1	no	—
111	M₃₂	—	2	1	no	—
112	M₃₂	—	292	1	no	—
113	M₃₂	—	292	1	no	—
114	M₃₂	—	146	2	no	—
115	M₃₂	—	146	1	no	—
116	C₂×C₁₆	—	292	1	no	—
117	C₂×C₁₆	—	292	1	no	—
118	#108	—	292	1	no	—
119	#108	—	292	1	no	—
120	#108	—	292	1	no	—
121	#108	—	292	1	no	—
122	C₄⋊C₈	—	292	2	no	—
123	C₄×C₈	—	73	1	no	—
124	C₄×C₈	—	73	1	no	—
125	C₄×C₈	—	73	1	no	—
126	C₄×C₈	C₂×#64408	1	1	yes	characteristic
127	C₂²×C₄	—	73	1	no	—
128	C₂²×C₄	—	146	1	no	—
129	M₁₆	—	292	2	no	—
130	M₁₆	—	292	2	no	—
131	C₂×C₈	—	292	2	no	—
132	C₂×C₈	—	73	1	no	—
133	C₂×C₈	—	73	1	no	—
134	C₂×C₈	—	73	1	no	—
135	C₂×C₈	—	73	1	no	—
136	C₂×C₈	—	73	1	no	—
137	C₂×C₈	—	73	1	no	—
138	C₂×C₈	—	73	1	no	—
139	C₂×C₈	—	73	1	no	—
140	C₂×C₈	—	73	1	no	—
141	C₂×C₈	—	73	1	no	—
142	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
143	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
144	C₄²	—	73	1	no	—
145	C₄²	—	146	1	no	—
146	C₄²	C₄×#64408	1	1	yes	characteristic
147	C₁₆	—	4	1	no	—
148	C₁₆	—	292	2	no	—
149	C₁₆	—	292	1	no	—
150	C₂³	—	73	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	292	2	no	—
153	C₂×C₄	—	73	1	no	—
154	C₂×C₄	—	73	1	no	—
155	C₂×C₄	—	73	1	no	—
156	C₂×C₄	—	73	1	no	—
157	C₂×C₄	—	73	1	no	—
158	C₂×C₄	—	146	1	no	—
159	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
160	C₈	—	584	2	no	—
161	C₈	—	73	1	no	—
162	C₈	—	73	1	no	—
163	C₈	—	73	1	no	—
164	C₈	—	73	1	no	—
165	C₈	#14556653	1	1	yes	characteristic
166	C₈	#14556654	1	1	yes	characteristic
167	C₈	#14556659	1	1	yes	characteristic
168	C₈	#14556660	1	1	yes	characteristic
169	C₂²	—	292	1	no	—
170	C₂²	—	73	1	no	—
171	C₂²	—	73	1	no	—
172	C₂²	#17989466	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	292	1	no	—
175	C₄	—	73	1	no	—
176	C₄	—	73	1	no	—
177	C₄	#17989460	1	1	yes	characteristic
178	C₄	#17989461	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	146	1	no	—
181	C₂	—	146	1	no	—
182	C₂	#22624810	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22624810
commutator	C₄²×C₇₃	C₄²
Frattini	C₄×C₈	C₂×#64408
Fitting	#364×C₇₃	C₄
socle	C₂×C₇₃	#1296
prime radical	C₂²×D₇₃	#101
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	#364	#64408
hypocenter	C₇₃	#7270
LCS	G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
95. C₇₃	#7270	semidirect

Maximal subgroups

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

subgroup	index
#22626270	2
#22625870	2
#364×D₇₃	2
#7270	73