Subgroups — #34012703

Order 18688.

296 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989509	C₂	1	1	yes	maximal, characteristic
2	#22626152	C₂	1	2	yes	maximal
3	#22626137	C₂	1	1	yes	maximal, characteristic
4	#22624981	C₂	1	2	yes	maximal
5	#22609812	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556665	C₂²	1	2	yes	—
7	C₂×#14556665	C₂²	1	1	yes	characteristic
8	C₂×#14556665	C₂²	1	1	yes	characteristic
9	#17989509	C₄	1	2	yes	complemented
10	#17989509	C₄	1	2	yes	complemented
11	#17989479	—	2	4	no	—
12	#17989476	C₄	1	2	yes	complemented
13	#17989473	C₄	1	2	yes	complemented
14	C₂×#14556527	C₄	1	1	yes	characteristic
15	#17988543	C₂²	1	2	yes	—
16	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
17	#17988346	C₄	1	2	yes	—
18	#427×C₇₃	C₄	1	1	yes	characteristic
19	#14556665	—	2	2	no	—
20	#14556665	—	2	1	no	—
21	#14556665	—	2	1	no	—
22	#14556665	C₂×C₄	1	4	yes	—
23	#14556665	D₄	1	2	yes	—
24	#14556665	Q₈	1	2	yes	—
25	C₂×#11924031	—	2	2	no	—
26	C₂×#11924031	C₂×C₄	1	2	yes	—
27	C₂×#11924031	C₂×C₄	1	2	yes	—
28	C₂×#11924012	C₂×C₄	1	2	yes	—
29	C₂²×#10596831	C₂³	1	1	yes	characteristic
30	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
31	#14556527	—	2	1	no	—
32	#14556527	—	2	1	no	—
33	#14556502	—	2	2	no	—
34	C₄×#10596831	—	2	2	no	—
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	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	#117×C₇₃	C₂×C₄	1	2	yes	—
41	#107×C₇₃	—	2	1	no	—
42	#106×C₇₃	—	2	1	no	—
43	#11924031	—	2	2	no	—
44	#11924031	—	2	4	no	—
45	#11924031	C₄⋊C₄	1	4	yes	—
46	C₂²×Dic₇₃	—	2	1	no	—
47	#11924012	—	2	2	no	—
48	#11924012	—	2	2	no	—
49	C₂×#10596831	—	2	2	no	—
50	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
51	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
52	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
53	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
54	C₂×#10596831	#41	1	1	yes	characteristic
55	C₂×#10596831	#41	1	1	yes	characteristic
56	C₄×Dic₇₃	—	2	1	no	—
57	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
58	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
61	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
62	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
63	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
64	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
65	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
66	C₄⋊C₄×C₇₃	—	2	2	no	—
67	C₂²⋊C₄×C₇₃	—	2	2	no	—
68	C₄²×C₇₃	—	2	2	no	—
69	C₂³×C₇₃	#131	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	4	2	no	—
71	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
77	C₂×Dic₇₃	—	2	1	no	—
78	C₂×Dic₇₃	—	2	1	no	—
79	C₂×Dic₇₃	—	4	2	no	—
80	C₈×C₇₃	#97	1	1	yes	characteristic
81	C₈×C₇₃	#97	1	1	yes	characteristic
82	C₈×C₇₃	#97	1	1	yes	characteristic
83	C₈×C₇₃	#97	1	1	yes	characteristic
84	#10596831	—	4	2	no	—
85	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
86	#10596831	#117	1	1	yes	characteristic
87	#10596831	C₄×D₄	1	1	yes	characteristic
88	#10596831	C₄×Q₈	1	1	yes	characteristic
89	C₂²×C₇₃	—	4	1	no	—
90	C₂²×C₇₃	#405	1	1	yes	characteristic
91	C₂²×C₇₃	#436	1	1	yes	characteristic
92	C₂²×C₇₃	#446	1	1	yes	characteristic
93	C₄×C₇₃	—	8	2	no	—
94	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
96	C₄×C₇₃	#436	1	1	yes	characteristic
97	C₄×C₇₃	#446	1	1	yes	characteristic
98	Dic₇₃	—	2	2	no	—
99	Dic₇₃	—	4	1	no	—
100	#12667	—	73	1	no	maximal, Sylow
101	C₂×C₇₃	—	2	1	no	—
102	C₂×C₇₃	—	2	1	no	—
103	C₂×C₇₃	—	2	1	no	—
104	C₂×C₇₃	#1770	1	1	yes	characteristic
105	C₂×#364	—	73	1	no	—
106	#2110	—	73	2	no	—
107	C₄×M₃₂	—	73	1	no	—
108	#1772	—	73	1	no	—
109	#1378	—	73	2	no	—
110	C₇₃	#12667	1	1	yes	characteristic, complemented, Sylow
111	C₂×M₃₂	—	73	1	no	—
112	C₂×M₃₂	—	73	1	no	—
113	C₂×M₃₂	—	73	2	no	—
114	C₂×#108	—	73	1	no	—
115	#427	#64408	1	1	yes	characteristic
116	#405	—	73	2	no	—
117	C₂×C₄×C₈	—	73	1	no	—
118	#364	—	73	2	no	—
119	#364	—	73	2	no	—
120	#347	—	146	4	no	—
121	#346	—	73	2	no	—
122	C₄×C₁₆	—	73	2	no	—
123	#339	—	73	2	no	—
124	C₂×M₁₆	—	73	2	no	—
125	C₂²×C₈	—	73	1	no	—
126	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
127	#117	C₂×#64408	1	2	yes	—
128	C₂×C₄²	—	73	1	no	—
129	M₃₂	—	73	2	no	—
130	M₃₂	—	73	2	no	—
131	M₃₂	—	73	4	no	—
132	M₃₂	—	146	2	no	—
133	M₃₂	—	146	1	no	—
134	M₃₂	—	146	1	no	—
135	C₂×C₁₆	—	73	2	no	—
136	C₂×C₁₆	—	73	2	no	—
137	C₂×C₁₆	—	146	2	no	—
138	#108	—	146	1	no	—
139	#108	—	146	1	no	—
140	#107	—	2	1	no	—
141	#106	—	2	1	no	—
142	#97	—	146	2	no	—
143	C₄×C₈	—	73	1	no	—
144	C₄×C₈	—	73	1	no	—
145	C₄×C₈	—	73	1	no	—
146	C₄×C₈	—	73	1	no	—
147	C₄×C₈	—	146	2	no	—
148	C₂²×C₄	—	146	1	no	—
149	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
150	M₁₆	—	146	2	no	—
151	M₁₆	—	146	2	no	—
152	C₂×C₈	—	73	1	no	—
153	C₂×C₈	—	73	1	no	—
154	C₂×C₈	—	73	1	no	—
155	C₂×C₈	—	73	1	no	—
156	C₂×C₈	—	73	1	no	—
157	C₂×C₈	—	73	1	no	—
158	C₂×C₈	—	146	2	no	—
159	C₂×C₈	C₄×#64408	1	1	yes	characteristic
160	C₂×C₈	C₄×#64408	1	1	yes	characteristic
161	C₂×C₈	#11924037	1	1	yes	characteristic
162	C₂×C₈	#11924037	1	1	yes	characteristic
163	C₂×C₈	#11924038	1	1	yes	characteristic
164	C₂×C₈	#11924038	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	73	1	no	—
169	C₄²	—	73	1	no	—
170	C₄²	—	146	1	no	—
171	C₁₆	—	73	4	no	—
172	C₁₆	—	146	2	no	—
173	C₁₆	—	146	4	no	—
174	C₂³	#14556678	1	1	yes	characteristic
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	292	2	no	—
177	C₂×C₄	—	146	1	no	—
178	C₂×C₄	—	146	1	no	—
179	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
180	C₂×C₄	D₄×#64408	1	1	yes	characteristic
181	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
182	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
183	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
184	C₂×C₄	#14556679	1	1	yes	characteristic
185	C₈	—	292	2	no	—
186	C₈	—	73	1	no	—
187	C₈	—	73	1	no	—
188	C₈	—	73	1	no	—
189	C₈	—	73	1	no	—
190	C₈	#14556646	1	1	yes	characteristic
191	C₈	#14556646	1	1	yes	characteristic
192	C₈	#14556646	1	1	yes	characteristic
193	C₈	#14556646	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#17989552	1	1	yes	characteristic
196	C₂²	#17989622	1	1	yes	characteristic
197	C₂²	#17989637	1	1	yes	characteristic
198	C₄	—	8	2	no	—
199	C₄	—	292	1	no	—
200	C₄	—	146	2	no	—
201	C₄	C₄⋊C₄×#64408	1	1	yes	characteristic
202	C₄	C₂×#14556646	1	1	yes	characteristic
203	C₄	#17989621	1	1	yes	characteristic
204	C₄	#17989638	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#22625727	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊C₄×#64408
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#427×C₇₃	C₄
socle	C₂×C₇₃	#1770
prime radical	C₂³×C₇₃	#131
cosocle	C₂²×#10596831	C₂³
hypercenter	#427	#64408
hypocenter	C₇₃	#12667
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
9. #17989509	C₄	semidirect
10. #17989509	C₄	semidirect
12. #17989476	C₄	semidirect
13. #17989473	C₄	semidirect
110. C₇₃	#12667	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989509	2
#22626152	2
#22626137	2
#22624981	2
#22609812	2
#12667	73