Subgroups — #34001937

Order 18688.

558 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625646	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556646	C₂	1	1	yes	maximal, characteristic
3	C₂×#17989429	C₂	1	2	yes	maximal
4	C₂×#17989428	C₂	1	2	yes	maximal
5	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556646	C₄	1	4	yes	—
7	C₂×C₄×#10596833	C₂²	1	2	yes	—
8	C₂×C₄×#10596833	C₂²	1	2	yes	—
9	#17989453	C₄	1	4	yes	—
10	#17989429	C₄	1	8	yes	complemented
11	#17989428	C₄	1	8	yes	complemented
12	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
17	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
19	C₂²×#10596833	C₂×C₄	1	4	yes	—
20	C₂²×#10596833	C₂×C₄	1	4	yes	—
21	#14556646	—	2	8	no	—
22	C₄×#10596833	C₂×C₄	1	4	yes	—
23	C₄×#10596833	C₂×C₄	1	4	yes	—
24	C₄×#10596833	C₂×C₄	1	4	yes	—
25	C₄×#10596833	C₂×C₄	1	4	yes	—
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
32	C₄×#10596831	—	2	2	no	—
33	C₈×Dic₇₃	—	2	2	no	—
34	C₈×Dic₇₃	—	2	2	no	—
35	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
36	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₇₃	—	2	2	no	—
41	C₂×#10596833	—	2	8	no	—
42	C₂×#10596833	—	2	8	no	—
43	C₂×#10596833	C₄²	1	8	yes	—
44	C₂×#10596833	C₄²	1	8	yes	—
45	C₂²×Dic₇₃	C₄²	1	2	yes	—
46	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×#10596831	—	2	1	no	—
49	C₂×#10596831	—	2	1	no	—
50	C₂×#10596831	C₄²	1	2	yes	—
51	C₂×#10596831	C₄²	1	2	yes	—
52	C₂×#10596831	M₁₆	1	2	yes	—
53	C₂×#10596831	M₁₆	1	2	yes	—
54	C₄×Dic₇₃	—	2	1	no	—
55	C₄×Dic₇₃	—	2	1	no	—
56	C₄×Dic₇₃	—	2	4	no	—
57	C₄×Dic₇₃	M₁₆	1	8	yes	—
58	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
59	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
61	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	—	2	1	no	—
63	C₂×C₈×C₇₃	—	2	1	no	—
64	C₂×C₈×C₇₃	C₄²	1	2	yes	—
65	C₂×C₈×C₇₃	C₄²	1	2	yes	—
66	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
67	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
68	C₄²×C₇₃	—	2	1	no	—
69	C₄²×C₇₃	—	2	1	no	—
70	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	2	2	no	—
73	C₂×C₄×C₇₃	#97	1	4	yes	—
74	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×Dic₇₃	—	2	2	no	—
81	C₂×Dic₇₃	—	2	2	no	—
82	C₂×Dic₇₃	#97	1	4	yes	—
83	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
84	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
85	C₂×Dic₇₃	C₂×M₁₆	1	4	yes	—
86	C₂×Dic₇₃	#131	1	4	yes	—
87	#10596833	—	2	8	no	—
88	#10596833	—	2	8	no	—
89	C₈×C₇₃	—	2	2	no	—
90	C₈×C₇₃	#97	1	4	yes	complemented
91	#10596831	—	2	2	no	—
92	#10596831	#97	1	4	yes	complemented
93	C₂²×C₇₃	C₄³	1	1	yes	characteristic
94	C₂²×C₇₃	C₂×#97	1	2	yes	—
95	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
96	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
97	C₂²×C₇₃	#405	1	2	yes	—
98	C₄×C₇₃	—	2	4	no	—
99	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
100	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
101	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
102	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
103	Dic₇₃	—	2	4	no	—
104	Dic₇₃	C₄×M₁₆	1	4	yes	—
105	Dic₇₃	#405	1	4	yes	—
106	#7842	—	73	1	no	maximal, Sylow
107	C₂×C₇₃	#1451	1	2	yes	—
108	C₂×C₇₃	#1451	1	2	yes	—
109	C₂×C₇₃	C₄×#97	1	1	yes	characteristic
110	C₂×C₇₃	C₄×#97	1	1	yes	characteristic
111	C₂×C₇₃	#1726	1	1	yes	characteristic
112	#1744	—	73	1	no	—
113	C₄×#97	—	73	1	no	—
114	C₄²×C₈	—	73	1	no	—
115	C₂×#322	—	73	2	no	—
116	C₂×#322	—	73	2	no	—
117	C₇₃	#7842	1	1	yes	characteristic, complemented, Sylow
118	C₂×#97	—	73	4	no	—
119	C₂×C₄×C₈	—	73	2	no	—
120	C₂×C₄×C₈	—	73	2	no	—
121	C₂×C₄×C₈	—	73	1	no	—
122	C₂×C₄×C₈	—	73	1	no	—
123	C₂×C₄×C₈	—	73	1	no	—
124	C₂×C₄×C₈	—	73	1	no	—
125	C₂×C₄×C₈	—	73	1	no	—
126	C₂×C₄×C₈	#64408	1	1	yes	characteristic
127	C₄³	—	73	1	no	—
128	#336	—	73	4	no	—
129	#322	—	73	8	no	—
130	#322	—	73	8	no	—
131	C₂²×C₈	—	73	1	no	—
132	C₂²×C₈	—	73	4	no	—
133	C₂²×C₈	—	73	4	no	—
134	C₂²×C₈	—	73	1	no	—
135	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
136	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
137	C₂×C₄²	—	73	2	no	—
138	C₂×C₄²	—	73	1	no	—
139	C₂×C₄²	—	73	2	no	—
140	C₂×C₄²	—	73	1	no	—
141	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
142	#97	—	146	8	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	—	73	4	no	—
145	C₄×C₈	—	73	4	no	—
146	C₄×C₈	—	73	4	no	—
147	C₄×C₈	—	73	4	no	—
148	C₄×C₈	—	73	4	no	—
149	C₄×C₈	—	73	4	no	—
150	C₄×C₈	—	146	2	no	—
151	C₄×C₈	—	146	2	no	—
152	C₄×C₈	—	146	2	no	—
153	C₂²×C₄	—	73	1	no	—
154	C₂²×C₄	—	73	1	no	—
155	C₂²×C₄	—	73	2	no	—
156	C₂²×C₄	C₄×#64408	1	2	yes	—
157	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	73	2	no	—
161	C₂×C₈	—	73	2	no	—
162	C₂×C₈	—	73	2	no	—
163	C₂×C₈	—	73	8	no	—
164	C₂×C₈	—	73	8	no	—
165	C₂×C₈	—	73	2	no	—
166	C₂×C₈	—	146	8	no	—
167	C₂×C₈	—	146	8	no	—
168	C₂×C₈	—	146	1	no	—
169	C₂×C₈	—	146	1	no	—
170	C₂×C₈	C₄×#64408	1	2	yes	—
171	C₂×C₈	C₄×#64408	1	2	yes	—
172	C₂×C₈	#11924037	1	2	yes	—
173	C₂×C₈	#11924038	1	2	yes	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	73	8	no	—
177	C₄²	—	73	2	no	—
178	C₄²	—	73	2	no	—
179	C₄²	—	146	4	no	—
180	C₄²	—	146	1	no	—
181	C₄²	—	146	1	no	—
182	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	73	2	no	—
186	C₂×C₄	—	73	2	no	—
187	C₂×C₄	—	73	4	no	—
188	C₂×C₄	—	73	4	no	—
189	C₂×C₄	—	73	4	no	—
190	C₂×C₄	—	146	2	no	—
191	C₂×C₄	—	146	2	no	—
192	C₂×C₄	#14556645	1	4	yes	—
193	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
195	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
196	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
197	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
198	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	73	4	no	—
201	C₈	—	146	8	no	—
202	C₈	—	146	8	no	—
203	C₈	—	146	2	no	—
204	C₈	#14556646	1	4	yes	complemented
205	C₂²	C₄²×#64408	1	1	yes	characteristic
206	C₂²	C₂×#14556645	1	2	yes	—
207	C₂²	C₄×#11924037	1	1	yes	characteristic
208	C₂²	C₄×#11924038	1	1	yes	characteristic
209	C₂²	#17989550	1	2	yes	—
210	C₄	—	2	4	no	—
211	C₄	—	73	4	no	—
212	C₄	—	73	4	no	—
213	C₄	—	146	4	no	—
214	C₄	C₂×#14556646	1	1	yes	characteristic
215	C₄	#17989546	1	1	yes	characteristic
216	C₄	C₄×#11924037	1	1	yes	characteristic
217	C₄	C₄×#11924038	1	1	yes	characteristic
218	C₂	#22625140	1	2	yes	—
219	C₂	#22625141	1	2	yes	—
220	C₂	C₄×#14556645	1	1	yes	characteristic
221	C₂	C₄×#14556646	1	1	yes	characteristic
222	C₂	#22625587	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₄³
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄²
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#7842
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
10. #17989429	C₄	semidirect
11. #17989428	C₄	semidirect
90. C₈×C₇₃	#97	semidirect
92. #10596831	#97	semidirect
117. C₇₃	#7842	semidirect
204. C₈	#14556646	semidirect

Maximal subgroups

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

subgroup	index
#22625646	2
C₄×#14556646	2
C₂×#17989429	2
C₂×#17989428	2
C₄×C₈×Dic₇₃	2
#7842	73