Subgroups — #33004014

Order 18688.

326 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×#17988413	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988413	C₂	1	1	yes	maximal, characteristic
3	#22610664	C₂	1	2	yes	maximal
4	#22608023	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556525	C₂²	1	2	yes	—
7	C₂×#14556525	C₂²	1	2	yes	—
8	#17988485	—	2	2	no	—
9	#17988413	—	2	2	no	—
10	#17988413	C₄	1	4	yes	complemented
11	#17988350	—	2	2	no	—
12	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₇₃	—	2	2	no	—
16	C₂×C₄⋊Dic₇₃	—	2	2	no	—
17	C₂×C₄⋊Dic₇₃	—	2	2	no	—
18	C₂×#11924007	—	4	2	no	—
19	#14556525	—	2	2	no	—
20	#14556525	—	2	2	no	—
21	#14556525	—	2	2	no	—
22	#14556525	C₂×C₄	1	4	yes	—
23	#14556498	—	4	2	no	—
24	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₃	—	2	1	no	—
28	C₂×C₁₆×C₇₃	—	2	1	no	—
29	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
31	C₂×C₁₆×C₇₃	Q₈	1	2	yes	—
32	C₂×C₁₆×C₇₃	Q₈	1	2	yes	—
33	C₄×C₈×C₇₃	—	2	1	no	—
34	C₄×C₈×C₇₃	—	2	1	no	—
35	C₂²×Dic₇₃	—	4	2	no	—
36	C₂²×Dic₇₃	—	4	2	no	—
37	C₄⋊Dic₇₃	—	2	4	no	—
38	C₄⋊Dic₇₃	—	4	2	no	—
39	C₄⋊Dic₇₃	—	4	2	no	—
40	C₄⋊Dic₇₃	—	4	2	no	—
41	#11924007	—	8	4	no	—
42	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₇₃	—	2	1	no	—
46	C₂×C₈×C₇₃	—	2	1	no	—
47	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
48	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
53	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
54	C₄²×C₇₃	—	2	1	no	—
55	C₄²×C₇₃	—	2	1	no	—
56	C₁₆×C₇₃	—	2	2	no	—
57	C₁₆×C₇₃	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	—	2	1	no	—
60	C₂×C₄×C₇₃	—	2	1	no	—
61	C₂×C₄×C₇₃	—	2	1	no	—
62	C₂×C₄×C₇₃	—	2	1	no	—
63	C₂×C₄×C₇₃	D₁₆	1	2	yes	—
64	C₂×C₄×C₇₃	Q₃₂	1	2	yes	—
65	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
71	C₂×Dic₇₃	—	4	4	no	—
72	C₂×Dic₇₃	—	8	2	no	—
73	C₂×Dic₇₃	—	8	2	no	—
74	C₂×Dic₇₃	—	8	2	no	—
75	C₂×Dic₇₃	—	8	2	no	—
76	C₂×Dic₇₃	—	8	2	no	—
77	C₈×C₇₃	—	2	2	no	—
78	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
80	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₈×C₇₃	#125	1	1	yes	characteristic
82	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
83	C₂²×C₇₃	#466	1	1	yes	characteristic
84	C₂²×C₇₃	#500	1	1	yes	characteristic
85	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
86	C₂²×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
87	C₂²×C₇₃	#508	1	1	yes	characteristic
88	C₂²×C₇₃	#508	1	1	yes	characteristic
89	C₄×C₇₃	—	2	2	no	—
90	C₄×C₇₃	—	2	2	no	—
91	C₄×C₇₃	#389	1	1	yes	characteristic
92	C₄×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
93	C₄×C₇₃	#467	1	1	yes	characteristic
94	C₄×C₇₃	#499	1	1	yes	characteristic
95	Dic₇₃	—	8	4	no	—
96	Dic₇₃	—	16	2	no	—
97	#13165	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	#1942	1	1	yes	characteristic
99	C₂×C₇₃	C₄×D₁₆	1	1	yes	characteristic
100	C₂×C₇₃	C₄×Q₃₂	1	1	yes	characteristic
101	C₂×C₇₃	#2215	1	1	yes	characteristic
102	C₂×C₇₃	#2216	1	1	yes	characteristic
103	C₂×C₇₃	#2252	1	1	yes	characteristic
104	C₂×C₇₃	#2253	1	1	yes	characteristic
105	C₂×#366	—	73	1	no	—
106	C₂×#366	—	73	1	no	—
107	C₂×C₄×C₁₆	D₇₃	1	1	yes	characteristic
108	#1942	—	73	2	no	—
109	#1380	—	73	2	no	—
110	C₇₃	#13165	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
112	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
113	C₂×#107	—	73	2	no	—
114	C₂×#107	—	73	2	no	—
115	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
116	#389	—	146	2	no	—
117	#366	—	73	4	no	—
118	#366	—	146	2	no	—
119	C₄×C₁₆	—	2	2	no	—
120	#340	—	146	2	no	—
121	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	292	2	no	—
124	C₂×C₄⋊C₄	—	146	2	no	—
125	C₂×C₄⋊C₄	—	146	2	no	—
126	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
127	C₂×C₁₆	—	2	1	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	Dic₁₄₆	1	2	yes	—
130	C₂×C₁₆	Dic₁₄₆	1	2	yes	—
131	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
132	C₂×C₁₆	C₂²⋊D₇₃	1	2	yes	—
133	#107	—	73	4	no	—
134	#107	—	146	2	no	—
135	#107	—	146	2	no	—
136	#107	—	146	2	no	—
137	C₄×C₈	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	#95	—	292	2	no	—
140	C₂²×C₄	—	292	2	no	—
141	C₂²×C₄	—	292	2	no	—
142	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
143	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	Dic₂₉₂	1	2	yes	—
148	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
149	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
150	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
151	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
152	C₂×C₈	#11924028	1	1	yes	characteristic
153	C₂×C₈	#11924028	1	1	yes	characteristic
154	C₄⋊C₄	—	292	2	no	—
155	C₄⋊C₄	—	292	2	no	—
156	C₄⋊C₄	—	292	2	no	—
157	C₄⋊C₄	—	584	4	no	—
158	C₄⋊C₄	—	146	4	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₁₆	—	2	2	no	—
162	C₁₆	#11924007	1	4	yes	complemented
163	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	292	4	no	—
169	C₂×C₄	—	584	2	no	—
170	C₂×C₄	—	584	2	no	—
171	C₂×C₄	—	584	2	no	—
172	C₂×C₄	—	584	2	no	—
173	C₂×C₄	—	584	2	no	—
174	C₂×C₄	D₁₁₆₈	1	2	yes	—
175	C₂×C₄	Dic₅₈₄	1	2	yes	—
176	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
177	C₂×C₄	#14556565	1	1	yes	characteristic
178	C₂×C₄	#14556591	1	1	yes	characteristic
179	C₂×C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
180	C₂×C₄	#14556613	1	1	yes	characteristic
181	C₂×C₄	#14556613	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	C₂×#11924007	1	1	yes	characteristic
184	C₈	C₄×Dic₁₄₆	1	1	yes	characteristic
185	C₈	#14556570	1	1	yes	characteristic
186	C₈	#14556583	1	1	yes	characteristic
187	C₂²	C₄×D₅₈₄	1	1	yes	characteristic
188	C₂²	#17988823	1	1	yes	characteristic
189	C₂²	#17988992	1	1	yes	characteristic
190	C₂²	C₂×D₁₁₆₈	1	1	yes	characteristic
191	C₂²	C₂×Dic₅₈₄	1	1	yes	characteristic
192	C₂²	#17989019	1	1	yes	characteristic
193	C₂²	#17989019	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	584	4	no	—
197	C₄	—	1168	2	no	—
198	C₄	#17988485	1	1	yes	characteristic
199	C₄	C₄×Dic₂₉₂	1	1	yes	characteristic
200	C₄	#17988828	1	1	yes	characteristic
201	C₄	#17988989	1	1	yes	characteristic
202	C₂	#22610664	1	1	yes	characteristic
203	C₂	C₄×D₁₁₆₈	1	1	yes	characteristic
204	C₂	C₄×Dic₅₈₄	1	1	yes	characteristic
205	C₂	#22612017	1	1	yes	characteristic
206	C₂	#22612022	1	1	yes	characteristic
207	C₂	#22612206	1	1	yes	characteristic
208	C₂	#22612210	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₈₄
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×C₄×C₁₆×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₃
hypocenter	C₇₃	#13165
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. #17988413	C₄	semidirect
57. C₁₆×C₇₃	C₄⋊C₄	semidirect
110. C₇₃	#13165	semidirect
162. C₁₆	#11924007	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988413	2
#22610664	2
#22608023	2
C₂×C₄×C₁₆×C₇₃	2
#13165	73