Subgroups — #34004072

Order 18688.

334 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625478	C₂	1	1	yes	maximal, characteristic
2	#22625138	C₂	1	2	yes	maximal
3	C₂×#17989428	C₂	1	1	yes	maximal, characteristic
4	#22624764	C₂	1	2	yes	maximal
5	#108×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989603	C₂²	1	2	yes	—
7	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
8	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
9	#17989450	C₄	1	2	yes	—
10	#17989448	C₄	1	2	yes	—
11	#17989429	—	2	4	no	—
12	#17989428	C₄	1	2	yes	—
13	#17989428	C₄	1	2	yes	—
14	C₂×#14556527	C₄	1	1	yes	characteristic
15	M₁₆×Dic₇₃	C₂²	1	2	yes	—
16	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
17	#17988351	C₄	1	2	yes	—
18	C₂×#108×C₇₃	C₄	1	1	yes	characteristic
19	C₂²×#10596833	—	2	1	no	—
20	C₂²×#10596833	Q₈	1	2	yes	—
21	C₄⋊#10596833	C₂×C₄	1	4	yes	—
22	C₂²⋊#10596833	—	2	2	no	—
23	C₄×#10596833	—	2	2	no	—
24	C₄×#10596833	—	2	1	no	—
25	C₄×#10596833	C₂×C₄	1	2	yes	—
26	C₄×#10596833	C₂×C₄	1	2	yes	—
27	C₄×#10596833	D₄	1	2	yes	—
28	C₂×#11924012	C₂×C₄	1	2	yes	—
29	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
31	#14556527	—	2	2	no	—
32	#14556501	—	2	2	no	—
33	C₈×Dic₇₃	—	2	2	no	—
34	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
35	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
36	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	#108×C₇₃	—	2	2	no	—
41	C₂×#10596833	—	2	2	no	—
42	C₂×#10596833	—	2	4	no	—
43	C₂×#10596833	—	4	2	no	—
44	C₂×#10596833	—	4	2	no	—
45	C₂×#10596833	C₄⋊C₄	1	4	yes	—
46	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
47	C₂²×Dic₇₃	#41	1	1	yes	characteristic
48	#11924012	—	2	2	no	—
49	#11924012	—	2	2	no	—
50	C₂×#10596831	—	2	1	no	—
51	C₂×#10596831	—	2	1	no	—
52	C₂×#10596831	—	2	2	no	—
53	C₂×#10596831	C₄²	1	1	yes	characteristic
54	C₂×#10596831	C₄²	1	1	yes	characteristic
55	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
58	C₄×Dic₇₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	M₁₆×C₇₃	—	2	2	no	—
61	M₁₆×C₇₃	—	2	2	no	—
62	C₂×C₈×C₇₃	—	2	2	no	—
63	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
64	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
65	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
66	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
67	C₂³×C₇₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
74	C₂×Dic₇₃	—	2	2	no	—
75	C₂×Dic₇₃	—	2	2	no	—
76	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×Dic₇₃	#117	1	1	yes	characteristic
78	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
79	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
80	#10596833	—	2	4	no	—
81	#10596833	—	4	2	no	—
82	#10596833	—	4	4	no	—
83	C₈×C₇₃	—	4	2	no	—
84	C₈×C₇₃	#97	1	2	yes	—
85	C₈×C₇₃	#97	1	2	yes	—
86	#10596831	—	2	1	no	—
87	#10596831	—	2	1	no	—
88	#10596831	—	4	2	no	—
89	C₂²×C₇₃	—	2	1	no	—
90	C₂²×C₇₃	—	2	1	no	—
91	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₇₃	#446	1	1	yes	characteristic
93	C₂²×C₇₃	#446	1	1	yes	characteristic
94	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
95	C₄×C₇₃	#405	1	1	yes	characteristic
96	C₄×C₇₃	#436	1	1	yes	characteristic
97	C₄×C₇₃	#436	1	1	yes	characteristic
98	Dic₇₃	—	2	1	no	—
99	Dic₇₃	—	2	1	no	—
100	Dic₇₃	#430	1	2	yes	—
101	Dic₇₃	#444	1	2	yes	—
102	#8295	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	—	2	1	no	—
104	C₂×C₇₃	—	2	1	no	—
105	C₂×C₇₃	#1770	1	1	yes	characteristic
106	C₂×C₇₃	#1773	1	1	yes	characteristic
107	C₂×C₇₃	#1778	1	1	yes	characteristic
108	C₄×#108	—	73	1	no	—
109	#1564	—	73	1	no	—
110	#1451	—	73	2	no	—
111	C₂×#322	—	73	1	no	—
112	#1278	—	73	2	no	—
113	C₇₃	#8295	1	1	yes	characteristic, complemented, Sylow
114	#431	—	73	2	no	—
115	C₂×#108	—	73	1	no	—
116	C₂×#108	#64408	1	1	yes	characteristic
117	C₄×M₁₆	—	73	2	no	—
118	C₂×C₄×C₈	—	73	1	no	—
119	C₂×C₄×C₈	—	73	1	no	—
120	C₂×C₄×C₈	—	73	1	no	—
121	#341	—	73	2	no	—
122	#335	—	73	2	no	—
123	#334	—	73	2	no	—
124	#322	—	73	2	no	—
125	#322	—	73	2	no	—
126	#322	—	146	4	no	—
127	C₂×M₁₆	—	73	2	no	—
128	C₂×M₁₆	C₂×#64408	1	2	yes	—
129	C₂²×C₈	—	73	2	no	—
130	C₂²×C₈	—	73	1	no	—
131	C₂²×C₈	—	146	1	no	—
132	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
133	C₂×C₄²	—	73	1	no	—
134	#108	—	2	2	no	—
135	#108	—	146	2	no	—
136	C₄⋊C₈	—	73	4	no	—
137	C₂²⋊C₈	—	146	2	no	—
138	#97	—	146	2	no	—
139	C₄×C₈	—	73	2	no	—
140	C₄×C₈	—	73	2	no	—
141	C₄×C₈	—	73	1	no	—
142	C₄×C₈	—	73	1	no	—
143	C₄×C₈	—	73	1	no	—
144	C₄×C₈	—	73	1	no	—
145	C₄×C₈	—	73	2	no	—
146	C₄×C₈	—	146	2	no	—
147	C₄×C₈	—	146	2	no	—
148	C₄×C₈	—	146	1	no	—
149	C₂²×C₄	—	73	1	no	—
150	C₂²×C₄	—	73	1	no	—
151	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	2	no	—
154	M₁₆	—	146	2	no	—
155	M₁₆	—	146	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	292	2	no	—
158	C₂×C₈	—	292	2	no	—
159	C₂×C₈	—	73	4	no	—
160	C₂×C₈	—	73	1	no	—
161	C₂×C₈	—	73	1	no	—
162	C₂×C₈	—	146	2	no	—
163	C₂×C₈	—	146	4	no	—
164	C₂×C₈	—	146	2	no	—
165	C₂×C₈	—	146	1	no	—
166	C₂×C₈	—	146	1	no	—
167	C₂×C₈	C₄×#64408	1	1	yes	characteristic
168	C₂×C₈	C₄×#64408	1	1	yes	characteristic
169	C₂×C₈	#11924037	1	2	yes	—
170	C₂×C₈	#11924038	1	2	yes	—
171	C₄²	—	73	1	no	—
172	C₄²	—	73	1	no	—
173	C₄²	—	73	1	no	—
174	C₄²	—	73	1	no	—
175	C₂³	Q₈×#64408	1	1	yes	characteristic
176	C₂×C₄	—	73	1	no	—
177	C₂×C₄	—	73	1	no	—
178	C₂×C₄	—	73	1	no	—
179	C₂×C₄	—	73	1	no	—
180	C₂×C₄	—	146	2	no	—
181	C₂×C₄	—	146	2	no	—
182	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
183	C₂×C₄	D₄×#64408	1	1	yes	characteristic
184	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
185	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
186	C₂×C₄	#14556678	1	1	yes	characteristic
187	C₂×C₄	#14556679	1	1	yes	characteristic
188	C₈	—	4	2	no	—
189	C₈	—	292	4	no	—
190	C₈	—	292	2	no	—
191	C₈	—	292	2	no	—
192	C₈	—	146	4	no	—
193	C₈	—	146	1	no	—
194	C₈	—	146	1	no	—
195	C₈	#14556646	1	2	yes	—
196	C₈	#14556646	1	2	yes	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	C₄⋊C₄×#64408	1	1	yes	characteristic
200	C₂²	#17989637	1	1	yes	characteristic
201	C₂²	#17989638	1	1	yes	characteristic
202	C₄	—	73	2	no	—
203	C₄	—	73	2	no	—
204	C₄	—	146	1	no	—
205	C₄	—	146	1	no	—
206	C₄	C₂×#14556646	1	1	yes	characteristic
207	C₄	#17989552	1	1	yes	characteristic
208	C₄	#17989621	1	1	yes	characteristic
209	C₄	#17989622	1	1	yes	characteristic
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#22625727	1	1	yes	characteristic
213	C₂	#22625740	1	1	yes	characteristic
214	C₂	#22625752	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#108×C₇₃	C₄
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂³×C₇₃	C₄×Q₈
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#108	#64408
hypocenter	C₇₃	#8295
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
113. C₇₃	#8295	semidirect

Maximal subgroups

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

subgroup	index
#22625478	2
#22625138	2
C₂×#17989428	2
#22624764	2
#108×Dic₇₃	2
#8295	73