Subgroups — #34012320

Order 18688.

358 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989476	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989473	C₂	1	1	yes	maximal, characteristic
3	#22624970	C₂	1	2	yes	maximal
4	#22624969	C₂	1	2	yes	maximal
5	#22609619	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924031	C₂²	1	2	yes	—
7	C₂²×#11924031	C₂²	1	2	yes	—
8	#17989476	C₄	1	4	yes	complemented
9	#17989473	C₄	1	4	yes	complemented
10	C₂×#14556501	C₄	1	1	yes	characteristic
11	C₂×#14556501	C₄	1	1	yes	characteristic
12	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	#17988433	C₄	1	1	yes	characteristic
16	C₂×#97×C₇₃	C₄	1	1	yes	characteristic
17	C₂×#11924031	—	2	2	no	—
18	C₂×#11924031	—	2	2	no	—
19	C₂×#11924031	—	2	2	no	—
20	C₂×#11924031	—	2	2	no	—
21	C₂×#11924031	C₂×C₄	1	4	yes	—
22	C₂×#11924031	C₂×C₄	1	4	yes	—
23	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
24	C₂²×#10596831	C₂³	1	1	yes	characteristic
25	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	#14556501	—	2	2	no	—
28	#14556501	C₈	1	4	yes	—
29	C₄×#10596831	—	2	2	no	—
30	C₈×Dic₇₃	—	2	2	no	—
31	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
32	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
33	#14556498	C₈	1	4	yes	—
34	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
37	#97×C₇₃	—	2	2	no	—
38	#11924031	—	2	4	no	—
39	#11924031	—	2	4	no	—
40	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
41	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
42	C₂×#10596831	—	2	1	no	—
43	C₂×#10596831	—	2	1	no	—
44	C₂×#10596831	C₂×C₈	1	2	yes	—
45	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
46	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
47	C₂×#10596831	#41	1	2	yes	—
48	C₂×#10596831	#41	1	2	yes	—
49	C₄×Dic₇₃	—	2	1	no	—
50	C₄×Dic₇₃	—	2	1	no	—
51	C₄×Dic₇₃	C₄²	1	2	yes	—
52	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₃	—	2	1	no	—
56	C₂×C₈×C₇₃	—	2	1	no	—
57	C₂×C₈×C₇₃	C₄²	1	2	yes	—
58	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
60	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
61	C₄²×C₇₃	—	2	1	no	—
62	C₄²×C₇₃	—	2	1	no	—
63	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	—	2	4	no	—
66	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
68	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
72	C₂×Dic₇₃	—	2	4	no	—
73	C₂×Dic₇₃	—	2	2	no	—
74	C₂×Dic₇₃	—	2	4	no	—
75	C₂×Dic₇₃	C₄×C₈	1	4	yes	—
76	C₈×C₇₃	—	2	2	no	—
77	C₈×C₇₃	#97	1	4	yes	—
78	#10596831	—	2	2	no	—
79	#10596831	#117	1	2	yes	—
80	#10596831	#117	1	2	yes	—
81	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
82	C₂²×C₇₃	#431	1	1	yes	characteristic
83	C₂²×C₇₃	#431	1	1	yes	characteristic
84	C₂²×C₇₃	#504	1	4	yes	—
85	C₄×C₇₃	—	4	2	no	—
86	C₄×C₇₃	#376	1	1	yes	characteristic
87	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
88	C₄×C₇₃	#431	1	1	yes	characteristic
89	C₄×C₇₃	#431	1	1	yes	characteristic
90	Dic₇₃	—	2	4	no	—
91	Dic₇₃	—	4	2	no	—
92	#12592	—	73	1	no	maximal, Sylow
93	C₂×C₇₃	#1744	1	1	yes	characteristic
94	C₂×C₇₃	#2108	1	2	yes	—
95	C₂×C₇₃	#2164	1	2	yes	—
96	C₂×C₇₃	#2164	1	2	yes	—
97	C₂×#346	—	73	1	no	—
98	C₂×C₄×C₁₆	—	73	1	no	—
99	#1726	—	73	1	no	—
100	#1374	—	73	2	no	—
101	#1374	—	73	2	no	—
102	C₇₃	#12592	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₁₆	—	73	2	no	—
104	C₂²×C₁₆	—	73	2	no	—
105	C₂×#97	—	73	1	no	—
106	C₂×#97	—	73	1	no	—
107	C₂×#97	#64408	1	1	yes	characteristic
108	C₂×C₄×C₈	—	73	1	no	—
109	C₂×C₄×C₈	—	73	1	no	—
110	C₂×C₄×C₈	—	73	1	no	—
111	#376	—	73	1	no	—
112	#346	—	73	4	no	—
113	C₄×C₁₆	—	73	4	no	—
114	C₂²×C₈	—	73	1	no	—
115	C₂²×C₈	—	73	1	no	—
116	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
117	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
118	C₂×C₄²	—	73	1	no	—
119	C₂×C₄²	—	73	1	no	—
120	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
121	C₂×C₁₆	—	73	4	no	—
122	C₂×C₁₆	—	73	4	no	—
123	C₂×C₁₆	—	146	2	no	—
124	C₂×C₁₆	—	146	2	no	—
125	C₂×C₁₆	—	146	2	no	—
126	C₂×C₁₆	—	146	2	no	—
127	#97	—	2	2	no	—
128	#97	—	73	4	no	—
129	#97	—	146	2	no	—
130	C₄×C₈	—	73	2	no	—
131	C₄×C₈	—	73	2	no	—
132	C₄×C₈	—	146	2	no	—
133	C₄×C₈	—	146	2	no	—
134	#95	—	73	4	no	—
135	C₂²×C₄	—	73	2	no	—
136	C₂²×C₄	—	73	2	no	—
137	C₂²×C₄	#11924036	1	2	yes	—
138	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	73	1	no	—
142	C₂×C₈	—	73	1	no	—
143	C₂×C₈	—	73	2	no	—
144	C₂×C₈	—	73	2	no	—
145	C₂×C₈	—	73	2	no	—
146	C₂×C₈	—	146	1	no	—
147	C₂×C₈	—	146	1	no	—
148	C₂×C₈	C₄×#64408	1	2	yes	—
149	C₂×C₈	#11924036	1	2	yes	—
150	C₂×C₈	#11924037	1	2	yes	—
151	C₂×C₈	#11924038	1	2	yes	—
152	C₄²	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	73	2	no	—
155	C₄²	—	73	2	no	—
156	C₄²	—	146	1	no	—
157	C₄²	—	146	1	no	—
158	C₁₆	—	146	4	no	—
159	C₁₆	—	146	4	no	—
160	C₂³	C₂×#11924036	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	73	4	no	—
164	C₂×C₄	—	146	4	no	—
165	C₂×C₄	—	146	2	no	—
166	C₂×C₄	—	146	4	no	—
167	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
168	C₂×C₄	#14556670	1	1	yes	characteristic
169	C₂×C₄	#14556670	1	1	yes	characteristic
170	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
171	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
172	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	73	2	no	—
175	C₈	—	73	2	no	—
176	C₈	—	146	2	no	—
177	C₈	#14556646	1	4	yes	—
178	C₂²	C₄×#11924036	1	1	yes	characteristic
179	C₂²	#17989604	1	1	yes	characteristic
180	C₂²	#17989606	1	1	yes	characteristic
181	C₂²	#17989646	1	4	yes	—
182	C₄	—	4	2	no	—
183	C₄	—	292	2	no	—
184	C₄	—	146	4	no	—
185	C₄	#17989523	1	1	yes	characteristic
186	C₄	C₂×#14556646	1	1	yes	characteristic
187	C₄	#17989604	1	1	yes	characteristic
188	C₄	#17989606	1	1	yes	characteristic
189	C₂	#22625646	1	1	yes	characteristic
190	C₂	#22626141	1	2	yes	—
191	C₂	#22626297	1	2	yes	—
192	C₂	#22626299	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₂×#97×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×#97	#64408
hypocenter	C₇₃	#12592
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
8. #17989476	C₄	semidirect
9. #17989473	C₄	semidirect
102. C₇₃	#12592	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989476	2
C₂×#17989473	2
#22624970	2
#22624969	2
#22609619	2
#12592	73