Subgroups — #33004136

Order 18688.

280 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22611704	C₂	1	2	yes	maximal
2	#22610103	C₂	1	1	yes	maximal, characteristic
3	#22609810	C₂	1	1	yes	maximal, characteristic
4	#22608043	C₂	1	2	yes	maximal
5	C₂×#364×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556527	—	2	1	no	—
7	C₂×#14556527	C₂²	1	2	yes	—
8	C₂×#14556527	C₂²	1	1	yes	characteristic
9	#17988625	C₂²	1	1	yes	characteristic
10	#17988611	—	2	1	no	—
11	#17988542	—	2	1	no	—
12	#17988410	—	2	4	no	—
13	#17988347	—	2	1	no	—
14	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	#364×C₇₃	C₄	1	2	yes	complemented
17	#364×C₇₃	C₄	1	2	yes	complemented
18	C₂×#11924012	—	2	1	no	—
19	C₂×#11924012	—	2	2	no	—
20	#14556550	—	2	1	no	—
21	#14556527	—	2	2	no	—
22	#14556527	—	2	1	no	—
23	#14556527	—	2	2	no	—
24	#14556527	—	4	1	no	—
25	#14556527	—	4	1	no	—
26	#14556527	Q₈	1	2	yes	—
27	#14556525	—	2	1	no	—
28	#14556523	D₄	1	2	yes	—
29	C₄⋊#10596831	—	4	2	no	—
30	#14556501	—	4	1	no	—
31	C₈×Dic₇₃	—	4	1	no	—
32	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
35	M₃₂×C₇₃	—	2	2	no	—
36	M₃₂×C₇₃	C₂×C₄	1	4	yes	—
37	C₂×C₁₆×C₇₃	—	2	2	no	—
38	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
42	#11924012	—	2	2	no	—
43	#11924012	—	4	1	no	—
44	#11924012	—	4	2	no	—
45	#11924012	—	4	2	no	—
46	C₂×#10596831	—	4	1	no	—
47	C₂×#10596831	—	4	2	no	—
48	C₄⋊Dic₇₃	—	2	2	no	—
49	C₂²⋊Dic₇₃	—	4	1	no	—
50	C₄×Dic₇₃	—	4	1	no	—
51	C₂²×C₄×C₇₃	—	2	1	no	—
52	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
57	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
58	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
59	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
64	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
65	C₄²×C₇₃	—	2	1	no	—
66	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
68	C₁₆×C₇₃	—	2	4	no	—
69	C₂³×C₇₃	#137	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	2	1	no	—
71	C₂×C₄×C₇₃	—	2	1	no	—
72	C₂×C₄×C₇₃	—	4	2	no	—
73	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×Dic₇₃	—	4	2	no	—
80	C₈×C₇₃	#106	1	1	yes	characteristic
81	C₈×C₇₃	#106	1	1	yes	characteristic
82	C₈×C₇₃	#106	1	1	yes	characteristic
83	C₈×C₇₃	#106	1	1	yes	characteristic
84	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
85	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₇₃	#125	1	1	yes	characteristic
87	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
88	#10596831	—	4	2	no	—
89	#10596831	—	8	2	no	—
90	C₂²×C₇₃	—	4	1	no	—
91	C₂²×C₇₃	#428	1	1	yes	characteristic
92	C₂²×C₇₃	#461	1	1	yes	characteristic
93	C₂²×C₇₃	#475	1	1	yes	characteristic
94	C₄×C₇₃	—	2	2	no	—
95	C₄×C₇₃	—	4	1	no	—
96	C₄×C₇₃	#389	1	1	yes	characteristic
97	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
98	C₄×C₇₃	#460	1	1	yes	characteristic
99	C₄×C₇₃	#476	1	1	yes	characteristic
100	Dic₇₃	—	8	2	no	—
101	#13188	—	73	1	no	maximal, Sylow
102	C₂×C₇₃	—	2	1	no	—
103	C₂×C₇₃	—	2	1	no	—
104	C₂×C₇₃	—	2	1	no	—
105	C₂×C₇₃	#1948	1	1	yes	characteristic
106	#2155	—	73	2	no	—
107	C₂×#364	D₇₃	1	1	yes	characteristic
108	#1848	—	73	1	no	—
109	#1772	—	73	1	no	—
110	#1388	—	73	2	no	—
111	C₇₃	#13188	1	1	yes	characteristic, complemented, Sylow
112	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
113	C₂×#108	—	73	1	no	—
114	C₂×#108	—	73	2	no	—
115	C₂×#108	—	146	1	no	—
116	#427	—	73	1	no	—
117	#423	—	146	1	no	—
118	#405	—	146	1	no	—
119	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
120	#365	—	146	4	no	—
121	#364	Dic₇₃	1	2	yes	complemented
122	#364	Dic₇₃	1	2	yes	complemented
123	#339	—	146	1	no	—
124	C₂×M₁₆	—	146	2	no	—
125	C₂×M₁₆	—	146	1	no	—
126	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
128	#117	—	146	1	no	—
129	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
130	M₃₂	—	2	2	no	—
131	M₃₂	C₂×Dic₇₃	1	4	yes	—
132	C₂×C₁₆	—	2	2	no	—
133	#108	—	292	1	no	—
134	#108	—	292	1	no	—
135	#108	—	73	2	no	—
136	#108	—	146	2	no	—
137	#108	—	146	1	no	—
138	#108	—	146	2	no	—
139	#107	—	146	1	no	—
140	#106	—	73	2	no	—
141	C₄⋊C₈	—	292	2	no	—
142	#97	—	292	1	no	—
143	C₄×C₈	—	292	1	no	—
144	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
145	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
146	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
147	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
150	M₁₆	—	292	1	no	—
151	M₁₆	—	292	2	no	—
152	M₁₆	—	292	2	no	—
153	M₁₆	—	146	2	no	—
154	C₂×C₈	—	292	1	no	—
155	C₂×C₈	—	292	2	no	—
156	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
157	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
158	C₂×C₈	#11924017	1	1	yes	characteristic
159	C₂×C₈	#11924017	1	1	yes	characteristic
160	C₂×C₈	#11924017	1	1	yes	characteristic
161	C₂×C₈	#11924017	1	1	yes	characteristic
162	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
163	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
164	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
165	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
166	C₂×C₈	#11924027	1	1	yes	characteristic
167	C₂×C₈	#11924029	1	1	yes	characteristic
168	C₄⋊C₄	—	146	2	no	—
169	C₂²⋊C₄	—	292	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	292	1	no	—
172	C₄²	C₄⋊Dic₇₃	1	1	yes	characteristic
173	C₄²	C₄⋊Dic₇₃	1	1	yes	characteristic
174	C₁₆	—	2	4	no	—
175	C₂³	#14556624	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	292	2	no	—
180	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
181	C₂×C₄	#14556566	1	1	yes	characteristic
182	C₂×C₄	#14556571	1	1	yes	characteristic
183	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
184	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
185	C₂×C₄	#14556619	1	1	yes	characteristic
186	C₈	—	292	2	no	—
187	C₈	—	584	2	no	—
188	C₈	#14556523	1	1	yes	characteristic
189	C₈	#14556523	1	1	yes	characteristic
190	C₈	#14556523	1	1	yes	characteristic
191	C₈	#14556523	1	1	yes	characteristic
192	C₈	D₄×Dic₇₃	1	1	yes	characteristic
193	C₈	Q₈×Dic₇₃	1	1	yes	characteristic
194	C₈	#14556582	1	1	yes	characteristic
195	C₈	#14556590	1	1	yes	characteristic
196	C₂²	—	4	1	no	—
197	C₂²	#17988629	1	1	yes	characteristic
198	C₂²	#17988793	1	1	yes	characteristic
199	C₂²	#17988873	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	584	2	no	—
203	C₄	#17988486	1	1	yes	characteristic
204	C₄	C₂×#14556523	1	1	yes	characteristic
205	C₄	#17988788	1	1	yes	characteristic
206	C₄	#17988878	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#22610695	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17988878
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×#364×C₇₃	C₂
socle	C₂×C₇₃	#1948
prime radical	C₂³×C₇₃	#137
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×#364	D₇₃
hypocenter	C₇₃	#13188
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
16. #364×C₇₃	C₄	semidirect
17. #364×C₇₃	C₄	semidirect
111. C₇₃	#13188	semidirect
121. #364	Dic₇₃	semidirect
122. #364	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22611704	2
#22610103	2
#22609810	2
#22608043	2
C₂×#364×C₇₃	2
#13188	73