Subgroups — #33017176

Order 18688.

1710 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988457	C₂	1	2	yes	maximal
2	C₂×#17988453	C₂	1	4	yes	maximal
3	#22608161	C₂	1	8	yes	maximal
4	C₂×C₄³×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₇₃	C₂²	1	8	yes	—
6	C₂²×C₄⋊Dic₇₃	C₂²	1	4	yes	—
7	#17988457	—	2	8	no	—
8	#17988453	—	2	16	no	—
9	C₂×#14556498	C₂²	1	16	yes	—
10	C₂²×C₄²×C₇₃	C₂²	1	4	yes	—
11	C₂²×C₄²×C₇₃	C₂²	1	2	yes	—
12	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
13	C₄³×C₇₃	C₄	1	8	yes	complemented
14	C₂³×Dic₇₃	C₂³	1	8	yes	—
15	C₂×C₄⋊Dic₇₃	—	2	16	no	—
16	C₂×C₄⋊Dic₇₃	—	2	16	no	—
17	C₂×C₄⋊Dic₇₃	—	2	8	no	—
18	C₂×C₄⋊Dic₇₃	—	2	32	no	—
19	#14556498	—	2	32	no	—
20	C₂³×C₄×C₇₃	C₂³	1	4	yes	—
21	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₇₃	C₂³	1	2	yes	—
23	C₂×C₄²×C₇₃	—	2	16	no	—
24	C₂×C₄²×C₇₃	C₂×C₄	1	16	yes	—
25	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₇₃	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₇₃	D₄	1	4	yes	—
28	C₂×C₄²×C₇₃	D₄	1	8	yes	—
29	C₂×C₄²×C₇₃	Q₈	1	4	yes	—
30	C₂×C₄²×C₇₃	Q₈	1	8	yes	—
31	C₂²×Dic₇₃	—	2	16	no	—
32	C₂²×Dic₇₃	—	2	8	no	—
33	C₂²×Dic₇₃	—	2	32	no	—
34	C₄⋊Dic₇₃	—	4	32	no	—
35	C₄⋊Dic₇₃	—	4	16	no	—
36	C₂⁴×C₇₃	C₂⁴	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	—	2	16	no	—
38	C₂²×C₄×C₇₃	C₂²×C₄	1	8	yes	—
39	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
41	C₂²×C₄×C₇₃	C₂×D₄	1	8	yes	—
42	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
43	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
45	C₂²×C₄×C₇₃	C₂×Q₈	1	8	yes	—
46	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
47	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
48	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₇₃	#41	1	16	yes	—
50	C₄²×C₇₃	—	2	32	no	—
51	C₄²×C₇₃	C₄⋊C₄	1	16	yes	—
52	C₄²×C₇₃	C₄⋊C₄	1	32	yes	complemented
53	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
54	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
55	C₂³×C₇₃	C₂²×D₄	1	2	yes	—
56	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
57	C₂³×C₇₃	C₂²×Q₈	1	2	yes	—
58	C₂³×C₇₃	C₂×#41	1	8	yes	—
59	C₂×C₄×C₇₃	—	2	16	no	—
60	C₂×C₄×C₇₃	—	2	16	no	—
61	C₂×C₄×C₇₃	—	2	16	no	—
62	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	16	yes	—
63	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	8	yes	—
64	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
65	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	8	yes	—
66	C₂×C₄×C₇₃	#117	1	16	yes	—
67	C₂×C₄×C₇₃	#124	1	16	yes	—
68	C₂×C₄×C₇₃	#125	1	16	yes	—
69	C₂×C₄×C₇₃	#127	1	4	yes	—
70	C₂×C₄×C₇₃	C₄⋊Q₈	1	8	yes	—
71	C₂×C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
72	C₂×Dic₇₃	—	4	16	no	—
73	C₂×Dic₇₃	—	4	8	no	—
74	C₂×Dic₇₃	—	4	32	no	—
75	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₇₃	C₂²×C₄⋊C₄	1	2	yes	—
77	C₂²×C₇₃	C₂×#117	1	4	yes	—
78	C₂²×C₇₃	C₂×#124	1	4	yes	—
79	C₂²×C₇₃	C₂×#125	1	4	yes	—
80	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
81	C₂²×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
82	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
83	C₂²×C₇₃	#532	1	8	yes	—
84	C₂²×C₇₃	#533	1	8	yes	—
85	C₄×C₇₃	—	2	16	no	—
86	C₄×C₇₃	#382	1	16	yes	complemented
87	C₄×C₇₃	#384	1	8	yes	—
88	Dic₇₃	—	8	8	no	—
89	#14687	—	73	1	no	maximal, Sylow
90	C₂×C₇₃	C₂×#382	1	2	yes	—
91	C₂×C₇₃	C₂×#384	1	1	yes	characteristic
92	C₂×C₇₃	#2285	1	4	yes	—
93	C₂×C₇₃	#2858	1	2	yes	—
94	C₂×C₇₃	#2859	1	4	yes	—
95	C₂×C₇₃	#2862	1	2	yes	—
96	C₂×#384	—	73	2	no	—
97	C₂×#382	—	73	4	no	—
98	C₂×C₄³	D₇₃	1	1	yes	characteristic
99	C₄⋊#95	—	73	8	no	—
100	C₇₃	#14687	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₄⋊C₄	—	73	8	no	—
102	C₂²×C₄⋊C₄	—	73	4	no	—
103	C₂²×C₄²	C₂×D₇₃	1	4	yes	—
104	C₂²×C₄²	C₂×D₇₃	1	2	yes	—
105	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
106	#384	—	146	8	no	—
107	#382	—	146	16	no	—
108	C₂×#95	—	73	16	no	—
109	C₄³	Dic₇₃	1	8	yes	complemented
110	C₂³×C₄	—	73	8	no	—
111	C₂³×C₄	C₂²×D₇₃	1	4	yes	—
112	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
113	C₂³×C₄	C₂²×D₇₃	1	2	yes	—
114	C₂×C₄⋊C₄	—	146	16	no	—
115	C₂×C₄⋊C₄	—	146	16	no	—
116	C₂×C₄⋊C₄	—	146	8	no	—
117	C₂×C₄⋊C₄	—	146	32	no	—
118	C₂×C₄²	—	2	16	no	—
119	C₂×C₄²	Dic₁₄₆	1	4	yes	—
120	C₂×C₄²	Dic₁₄₆	1	8	yes	—
121	C₂×C₄²	D₂₉₂	1	4	yes	—
122	C₂×C₄²	D₂₉₂	1	8	yes	—
123	C₂×C₄²	C₂×Dic₇₃	1	16	yes	—
124	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
125	C₂×C₄²	C₂×Dic₇₃	1	8	yes	—
126	#95	—	146	32	no	—
127	C₂⁴	C₂³×D₇₃	1	1	yes	characteristic
128	C₂²×C₄	—	2	16	no	—
129	C₂²×C₄	—	146	16	no	—
130	C₂²×C₄	—	146	8	no	—
131	C₂²×C₄	—	146	32	no	—
132	C₂²×C₄	C₂²×Dic₇₃	1	8	yes	—
133	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
134	C₂²×C₄	C₂²×Dic₇₃	1	4	yes	—
135	C₂²×C₄	C₂×D₂₉₂	1	8	yes	—
136	C₂²×C₄	C₂×D₂₉₂	1	4	yes	—
137	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
138	C₂²×C₄	C₂×D₂₉₂	1	4	yes	—
139	C₂²×C₄	C₂×Dic₁₄₆	1	8	yes	—
140	C₂²×C₄	C₂×Dic₁₄₆	1	4	yes	—
141	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
142	C₂²×C₄	C₂×Dic₁₄₆	1	4	yes	—
143	C₂²×C₄	#11924028	1	16	yes	—
144	C₄⋊C₄	—	292	32	no	—
145	C₄⋊C₄	—	292	16	no	—
146	C₄²	—	2	32	no	—
147	C₄²	C₄⋊Dic₇₃	1	16	yes	—
148	C₄²	C₄⋊Dic₇₃	1	32	yes	complemented
149	C₂³	C₂³×Dic₇₃	1	1	yes	characteristic
150	C₂³	C₂²×D₂₉₂	1	2	yes	—
151	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
152	C₂³	C₂²×Dic₁₄₆	1	2	yes	—
153	C₂³	C₂²×Dic₁₄₆	1	1	yes	characteristic
154	C₂³	C₂×#11924028	1	8	yes	—
155	C₂×C₄	—	2	16	no	—
156	C₂×C₄	—	2	16	no	—
157	C₂×C₄	—	2	16	no	—
158	C₂×C₄	—	292	8	no	—
159	C₂×C₄	—	292	32	no	—
160	C₂×C₄	—	292	16	no	—
161	C₂×C₄	C₂×C₄⋊Dic₇₃	1	16	yes	—
162	C₂×C₄	C₂×C₄⋊Dic₇₃	1	8	yes	—
163	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₇₃	1	8	yes	—
165	C₂×C₄	#14556550	1	16	yes	—
166	C₂×C₄	#14556579	1	16	yes	—
167	C₂×C₄	#14556583	1	16	yes	—
168	C₂×C₄	#14556588	1	4	yes	—
169	C₂×C₄	#14556591	1	8	yes	—
170	C₂×C₄	#14556591	1	4	yes	—
171	C₂²	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
172	C₂²	C₂²×C₄⋊Dic₇₃	1	2	yes	—
173	C₂²	C₂×#14556550	1	4	yes	—
174	C₂²	C₂×#14556579	1	4	yes	—
175	C₂²	C₂×#14556583	1	4	yes	—
176	C₂²	C₂×#14556588	1	1	yes	characteristic
177	C₂²	C₂×#14556591	1	2	yes	—
178	C₂²	C₂×#14556591	1	1	yes	characteristic
179	C₂²	#17989137	1	8	yes	—
180	C₂²	#17989144	1	8	yes	—
181	C₄	—	2	16	no	—
182	C₄	—	584	8	no	—
183	C₄	#17988453	1	16	yes	complemented
184	C₄	#17988457	1	8	yes	—
185	C₂	C₂×#17988453	1	2	yes	—
186	C₂	C₂×#17988457	1	1	yes	characteristic
187	C₂	#22612338	1	4	yes	—
188	C₂	#22618288	1	2	yes	—
189	C₂	#22618294	1	4	yes	—
190	C₂	#22618309	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₇₃
commutator	C₂³×C₇₃	C₂³×C₄
Fitting	C₂×C₄³×C₇₃	C₂
socle, prime radical, cosocle	C₂⁴×C₇₃	C₂⁴
hypercenter	C₂×C₄³	D₇₃
hypocenter	C₇₃	#14687
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
13. C₄³×C₇₃	C₄	semidirect
52. C₄²×C₇₃	C₄⋊C₄	semidirect
86. C₄×C₇₃	#382	semidirect
100. C₇₃	#14687	semidirect
109. C₄³	Dic₇₃	semidirect
148. C₄²	C₄⋊Dic₇₃	semidirect
183. C₄	#17988453	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988457	2
C₂×#17988453	2
#22608161	2
C₂×C₄³×C₇₃	2
#14687	73