Subgroups — #32985026

Order 18688.

570 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618329	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988332	C₂	1	4	yes	maximal
4	C₇₃×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556591	—	2	1	no	—
6	C₂×#14556591	—	2	1	no	—
7	C₂×#14556591	C₂²	1	2	yes	—
8	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
9	C₂×C₄×#10596831	—	2	1	no	—
10	#17988457	—	2	2	no	—
11	#17988457	C₄	1	4	yes	—
12	#17988342	—	2	1	no	—
13	#17988332	—	2	8	no	—
14	C₂×C₄⋊C₈×C₇₃	—	2	2	no	—
15	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
16	C₄³×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₄₆	—	2	2	no	—
18	C₂²×#10596831	—	2	2	no	—
19	#14556591	—	2	4	no	—
20	#14556591	—	2	4	no	—
21	#14556591	—	4	4	no	—
22	#14556591	—	4	2	no	—
23	#14556591	—	4	2	no	—
24	C₂×C₄⋊Dic₇₃	—	2	4	no	—
25	C₂×C₄⋊Dic₇₃	—	2	4	no	—
26	C₂×C₄⋊Dic₇₃	—	2	2	no	—
27	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
28	C₄⋊#10596831	—	2	2	no	—
29	C₄⋊#10596831	D₄	1	4	yes	—
30	C₄×#10596831	—	4	2	no	—
31	C₂²×C₈×C₇₃	—	2	2	no	—
32	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₃	D₄	1	2	yes	—
36	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₇₃	—	2	4	no	—
38	C₄⋊C₈×C₇₃	—	4	4	no	—
39	C₂×Dic₁₄₆	—	4	4	no	—
40	C₂×Dic₁₄₆	—	4	2	no	—
41	C₂×Dic₁₄₆	—	4	2	no	—
42	C₂×Dic₁₄₆	—	4	4	no	—
43	C₂²×Dic₇₃	—	2	4	no	—
44	C₂×#10596831	—	2	4	no	—
45	C₂×#10596831	—	4	2	no	—
46	C₂×#10596831	—	4	2	no	—
47	C₄⋊Dic₇₃	—	2	8	no	—
48	C₄⋊Dic₇₃	—	4	8	no	—
49	C₄⋊Dic₇₃	—	4	8	no	—
50	C₄⋊Dic₇₃	—	4	2	no	—
51	C₄⋊Dic₇₃	—	4	2	no	—
52	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₇₃	—	4	2	no	—
58	C₂×C₈×C₇₃	—	4	4	no	—
59	C₄²×C₇₃	—	2	4	no	—
60	C₄²×C₇₃	—	2	2	no	—
61	C₄²×C₇₃	D₈	1	2	yes	—
62	C₄²×C₇₃	SD₁₆	1	4	yes	—
63	C₄²×C₇₃	SD₁₆	1	2	yes	—
64	C₄²×C₇₃	Q₁₆	1	4	yes	—
65	C₄²×C₇₃	C₂×D₄	1	2	yes	—
66	C₄²×C₇₃	#41	1	2	yes	—
67	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	—	2	2	no	—
69	C₂×C₄×C₇₃	—	2	2	no	—
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	#101	1	4	yes	—
72	C₂×C₄×C₇₃	#102	1	4	yes	—
73	C₂×C₄×C₇₃	#103	1	4	yes	—
74	C₂×C₄×C₇₃	#103	1	4	yes	—
75	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₇₃	#121	1	2	yes	—
77	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
84	C₂×Dic₇₃	—	4	4	no	—
85	C₂×Dic₇₃	—	4	4	no	—
86	C₂×Dic₇₃	—	4	4	no	—
87	Dic₁₄₆	—	8	4	no	—
88	Dic₁₄₆	—	8	4	no	—
89	C₈×C₇₃	—	8	2	no	—
90	#10596831	—	4	4	no	—
91	C₂²×C₇₃	#390	1	1	yes	characteristic
92	C₂²×C₇₃	C₂×#101	1	2	yes	—
93	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
94	C₂²×C₇₃	C₂×#103	1	2	yes	—
95	C₂²×C₇₃	#418	1	1	yes	characteristic
96	C₄×C₇₃	—	2	4	no	—
97	C₄×C₇₃	—	2	2	no	—
98	C₄×C₇₃	#332	1	8	yes	complemented
99	C₄×C₇₃	#459	1	2	yes	—
100	C₄×C₇₃	#460	1	2	yes	—
101	C₄×C₇₃	#487	1	2	yes	—
102	C₄×C₇₃	#492	1	1	yes	characteristic
103	C₄×C₇₃	#494	1	1	yes	characteristic
104	Dic₇₃	—	8	4	no	—
105	#9694	—	73	1	no	maximal, Sylow
106	C₂×C₇₃	C₂×#332	1	2	yes	—
107	C₂×C₇₃	#1549	1	2	yes	—
108	C₂×C₇₃	#1956	1	1	yes	characteristic
109	C₂×C₇₃	#1966	1	1	yes	characteristic
110	C₂×C₇₃	#1982	1	1	yes	characteristic
111	#2868	—	73	1	no	—
112	#1842	D₇₃	1	1	yes	characteristic
113	C₄×C₄⋊C₈	—	73	1	no	—
114	C₂×#332	—	73	4	no	—
115	C₇₃	#9694	1	1	yes	characteristic, complemented, Sylow
116	C₂×C₄⋊Q₈	—	73	2	no	—
117	C₂×C₄⋊Q₈	—	146	1	no	—
118	C₂×C₄⋊Q₈	—	146	1	no	—
119	C₂×C₄⋊C₈	—	2	2	no	—
120	C₂×C₄⋊C₈	—	73	2	no	—
121	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
122	C₂×C₄×C₈	—	146	1	no	—
123	#384	—	73	4	no	—
124	#384	—	146	2	no	—
125	C₄³	C₂×D₇₃	1	1	yes	characteristic
126	#336	—	146	1	no	—
127	#332	—	146	8	no	—
128	C₂²×Q₈	—	146	2	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	146	2	no	—
131	C₄⋊Q₈	—	292	4	no	—
132	C₄⋊Q₈	—	292	2	no	—
133	C₄⋊Q₈	—	292	2	no	—
134	C₄⋊Q₈	—	146	4	no	—
135	C₄⋊Q₈	—	146	4	no	—
136	C₂×C₄⋊C₄	—	73	4	no	—
137	C₂×C₄⋊C₄	—	146	4	no	—
138	C₂×C₄⋊C₄	—	146	4	no	—
139	C₂×C₄⋊C₄	—	146	2	no	—
140	C₂×C₄²	C₄×D₇₃	1	2	yes	—
141	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
142	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
144	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	—	4	4	no	—
147	C₄⋊C₈	—	73	4	no	—
148	C₄⋊C₈	—	146	2	no	—
149	C₄×C₈	—	292	2	no	—
150	C₂×Q₈	—	292	4	no	—
151	C₂×Q₈	—	292	2	no	—
152	C₂×Q₈	—	292	2	no	—
153	C₂×Q₈	—	292	4	no	—
154	C₂²×C₄	—	146	4	no	—
155	C₂²×C₄	#11924005	1	2	yes	—
156	C₂²×C₄	#11924005	1	2	yes	—
157	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
159	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	4	4	no	—
162	C₂×C₈	—	292	2	no	—
163	C₂×C₈	—	292	2	no	—
164	C₂×C₈	—	146	4	no	—
165	C₄⋊C₄	—	292	8	no	—
166	C₄⋊C₄	—	292	8	no	—
167	C₄⋊C₄	—	292	2	no	—
168	C₄⋊C₄	—	292	2	no	—
169	C₄⋊C₄	—	146	8	no	—
170	C₄²	—	2	4	no	—
171	C₄²	—	2	2	no	—
172	C₄²	D₅₈₄	1	2	yes	—
173	C₄²	#11924016	1	4	yes	—
174	C₄²	#11924017	1	2	yes	—
175	C₄²	#11924018	1	4	yes	—
176	C₄²	D₄×D₇₃	1	2	yes	—
177	C₄²	#11924029	1	2	yes	—
178	C₂³	C₂×#11924005	1	1	yes	characteristic
179	Q₈	—	584	4	no	—
180	Q₈	—	584	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	292	4	no	—
185	C₂×C₄	—	292	4	no	—
186	C₂×C₄	—	292	4	no	—
187	C₂×C₄	#14556509	1	4	yes	—
188	C₂×C₄	#14556513	1	4	yes	—
189	C₂×C₄	#14556514	1	4	yes	—
190	C₂×C₄	#14556514	1	4	yes	—
191	C₂×C₄	#14556553	1	2	yes	—
192	C₂×C₄	#14556566	1	2	yes	—
193	C₂×C₄	#14556580	1	1	yes	characteristic
194	C₂×C₄	#14556587	1	1	yes	characteristic
195	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
196	C₂×C₄	C₂×#11924016	1	2	yes	—
197	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
198	C₂×C₄	C₂×#11924018	1	2	yes	—
199	C₂×C₄	#14556619	1	2	yes	—
200	C₈	—	8	2	no	—
201	C₈	—	292	4	no	—
202	C₂²	#17988488	1	1	yes	characteristic
203	C₂²	C₂×#14556509	1	2	yes	—
204	C₂²	C₂×#14556513	1	1	yes	characteristic
205	C₂²	C₂×#14556514	1	2	yes	—
206	C₂²	#17988593	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	584	4	no	—
210	C₄	#17988332	1	8	yes	complemented
211	C₄	#17988783	1	2	yes	—
212	C₄	#17988788	1	2	yes	—
213	C₄	#17988939	1	2	yes	—
214	C₄	#17988959	1	1	yes	characteristic
215	C₄	#17988966	1	1	yes	characteristic
216	C₂	C₂×#17988332	1	2	yes	—
217	C₂	#22608617	1	2	yes	—
218	C₂	#22610732	1	1	yes	characteristic
219	C₂	#22610774	1	1	yes	characteristic
220	C₂	#22610857	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11924005
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1842	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1842	D₇₃
hypocenter	C₇₃	#9694
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
98. C₄×C₇₃	#332	semidirect
115. C₇₃	#9694	semidirect
210. C₄	#17988332	semidirect

Maximal subgroups

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

subgroup	index
#22618329	2
C₄×C₄⋊#10596831	2
C₂×#17988332	2
C₇₃×#1842	2
#9694	73