Subgroups — #33004126

Order 18688.

310 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988410	C₂	1	2	yes	maximal
2	#22610088	C₂	1	1	yes	maximal, characteristic
3	#22609810	C₂	1	1	yes	maximal, characteristic
4	#22608037	C₂	1	2	yes	maximal
5	C₂×#364×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556527	C₂²	1	1	yes	characteristic
7	#17988625	C₂²	1	1	yes	characteristic
8	C₂×#14556523	—	2	1	no	—
9	C₂×#14556523	C₂²	1	2	yes	—
10	#17988542	—	2	1	no	—
11	#17988457	—	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	4	yes	complemented
17	C₂×#11924012	—	2	1	no	—
18	#14556550	—	2	1	no	—
19	C₂×C₄⋊Dic₇₃	—	2	2	no	—
20	C₂×C₄⋊Dic₇₃	—	4	2	no	—
21	#14556527	—	2	1	no	—
22	#14556527	D₄	1	2	yes	—
23	#14556525	Q₈	1	2	yes	—
24	#14556523	—	2	2	no	—
25	#14556523	—	2	1	no	—
26	#14556523	—	2	2	no	—
27	#14556523	—	4	2	no	—
28	#14556501	—	4	1	no	—
29	C₈×Dic₇₃	—	4	1	no	—
30	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
33	M₃₂×C₇₃	—	2	2	no	—
34	M₃₂×C₇₃	C₂×C₄	1	4	yes	—
35	C₂×C₁₆×C₇₃	—	2	2	no	—
36	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₇₃	Q₈	1	2	yes	—
38	C₂²×Dic₇₃	—	4	2	no	—
39	#11924012	—	2	2	no	—
40	#11924012	—	4	1	no	—
41	C₂×#10596831	—	4	1	no	—
42	C₄⋊Dic₇₃	—	2	2	no	—
43	C₄⋊Dic₇₃	—	4	2	no	—
44	C₄⋊Dic₇₃	—	4	2	no	—
45	C₄⋊Dic₇₃	—	8	4	no	—
46	C₂²⋊Dic₇₃	—	4	1	no	—
47	C₄×Dic₇₃	—	4	1	no	—
48	C₂²×C₄×C₇₃	—	2	1	no	—
49	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
54	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
59	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
60	C₄²×C₇₃	—	2	1	no	—
61	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
62	C₁₆×C₇₃	—	2	4	no	—
63	C₂³×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×C₄×C₇₃	—	2	4	no	—
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
72	C₂×Dic₇₃	—	4	2	no	—
73	C₂×Dic₇₃	—	8	4	no	—
74	C₂×Dic₇₃	—	8	2	no	—
75	C₈×C₇₃	#106	1	2	yes	—
76	C₈×C₇₃	#106	1	2	yes	—
77	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
78	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
79	C₈×C₇₃	#125	1	1	yes	characteristic
80	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
81	#10596831	—	4	2	no	—
82	C₂²×C₇₃	—	2	2	no	—
83	C₂²×C₇₃	#428	1	1	yes	characteristic
84	C₂²×C₇₃	#460	1	1	yes	characteristic
85	C₂²×C₇₃	#476	1	1	yes	characteristic
86	C₄×C₇₃	—	2	4	no	—
87	C₄×C₇₃	#389	1	1	yes	characteristic
88	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
89	C₄×C₇₃	#461	1	1	yes	characteristic
90	C₄×C₇₃	#475	1	1	yes	characteristic
91	Dic₇₃	—	8	2	no	—
92	Dic₇₃	—	16	2	no	—
93	#13186	—	73	1	no	maximal, Sylow
94	C₂×C₇₃	—	2	1	no	—
95	C₂×C₇₃	—	2	1	no	—
96	C₂×C₇₃	#1948	1	1	yes	characteristic
97	C₂×C₇₃	#2230	1	2	yes	—
98	C₂×#365	—	73	2	no	—
99	C₂×#364	D₇₃	1	1	yes	characteristic
100	#1844	—	73	1	no	—
101	#1772	—	73	1	no	—
102	#1386	—	73	2	no	—
103	C₇₃	#13186	1	1	yes	characteristic, complemented, Sylow
104	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
105	C₂×#108	—	73	1	no	—
106	#427	—	73	1	no	—
107	C₂×#106	—	73	2	no	—
108	C₂×#106	—	146	1	no	—
109	#405	—	146	1	no	—
110	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
111	#384	—	146	1	no	—
112	#365	—	146	4	no	—
113	#364	Dic₇₃	1	4	yes	complemented
114	#339	—	146	1	no	—
115	C₂×M₁₆	—	146	1	no	—
116	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
118	#117	—	146	1	no	—
119	C₂×C₄⋊C₄	—	292	2	no	—
120	C₂×C₄⋊C₄	—	146	2	no	—
121	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	C₂×Dic₇₃	1	4	yes	—
124	C₂×C₁₆	—	2	2	no	—
125	#108	—	73	2	no	—
126	#108	—	146	1	no	—
127	#107	—	73	2	no	—
128	#106	—	292	2	no	—
129	#106	—	146	2	no	—
130	#106	—	146	2	no	—
131	#106	—	146	1	no	—
132	#97	—	292	1	no	—
133	C₄×C₈	—	292	1	no	—
134	C₄×C₈	Dic₁₄₆	1	2	yes	—
135	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	292	2	no	—
138	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
139	M₁₆	—	292	1	no	—
140	M₁₆	—	146	2	no	—
141	C₂×C₈	—	292	1	no	—
142	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
143	C₂×C₈	#11924017	1	1	yes	characteristic
144	C₂×C₈	#11924017	1	1	yes	characteristic
145	C₂×C₈	#11924017	1	2	yes	—
146	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
147	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
148	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
149	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
150	C₂×C₈	#11924027	1	1	yes	characteristic
151	C₂×C₈	#11924029	1	1	yes	characteristic
152	C₄⋊C₄	—	292	2	no	—
153	C₄⋊C₄	—	292	2	no	—
154	C₄⋊C₄	—	584	4	no	—
155	C₄⋊C₄	—	146	2	no	—
156	C₂²⋊C₄	—	292	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	292	1	no	—
159	C₄²	C₄⋊Dic₇₃	1	2	yes	—
160	C₁₆	—	2	4	no	—
161	C₂³	#14556619	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	292	2	no	—
165	C₂×C₄	—	584	4	no	—
166	C₂×C₄	—	584	2	no	—
167	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
168	C₂×C₄	#14556566	1	1	yes	characteristic
169	C₂×C₄	#14556571	1	1	yes	characteristic
170	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
171	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
172	C₂×C₄	#14556624	1	1	yes	characteristic
173	C₈	—	292	2	no	—
174	C₈	#14556523	1	2	yes	—
175	C₈	#14556523	1	2	yes	—
176	C₈	D₄×Dic₇₃	1	1	yes	characteristic
177	C₈	Q₈×Dic₇₃	1	1	yes	characteristic
178	C₈	#14556582	1	1	yes	characteristic
179	C₈	#14556590	1	1	yes	characteristic
180	C₂²	—	2	2	no	—
181	C₂²	#17988629	1	1	yes	characteristic
182	C₂²	#17988788	1	1	yes	characteristic
183	C₂²	#17988878	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	—	584	2	no	—
186	C₄	—	1168	2	no	—
187	C₄	#17988486	1	1	yes	characteristic
188	C₄	C₂×#14556523	1	1	yes	characteristic
189	C₄	#17988793	1	1	yes	characteristic
190	C₄	#17988873	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#22610695	1	1	yes	characteristic
194	C₂	#22612103	1	2	yes	—

	subgroup	G/H
center	C₂²	#17988878
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×#364×C₇₃	C₂
socle	C₂²×C₇₃	#476
prime radical	C₂³×C₇₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×#364	D₇₃
hypocenter	C₇₃	#13186
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ 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
103. C₇₃	#13186	semidirect
113. #364	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988410	2
#22610088	2
#22609810	2
#22608037	2
C₂×#364×C₇₃	2
#13186	73