Subgroups — #32977288

Order 18688.

358 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22608679	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#10596831	C₂	1	2	yes	maximal
3	C₂×#17988308	C₂	1	1	yes	maximal, characteristic
4	#22607769	C₂	1	2	yes	maximal
5	C₄×#108×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
8	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
9	#17988340	C₄	1	2	yes	—
10	#17988338	C₄	1	2	yes	—
11	#17988308	—	2	2	no	—
12	#17988308	C₄	1	2	yes	—
13	#17988308	C₄	1	2	yes	—
14	C₈×#10596831	—	2	2	no	—
15	C₂×#108×C₇₃	C₄	1	2	yes	—
16	C₄×M₁₆×C₇₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	#341×C₇₃	C₄	1	2	yes	—
19	C₂²×#10596831	—	2	1	no	—
20	C₂²×#10596831	Q₈	1	2	yes	—
21	C₄⋊#10596831	C₂×C₄	1	4	yes	—
22	C₂²⋊#10596831	—	2	2	no	—
23	C₄×#10596831	—	2	2	no	—
24	C₄×#10596831	—	2	1	no	—
25	C₄×#10596831	C₂×C₄	1	2	yes	—
26	C₄×#10596831	C₂×C₄	1	2	yes	—
27	C₄×#10596831	D₄	1	2	yes	—
28	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
32	#108×C₇₃	C₈	1	8	yes	complemented
33	#97×C₇₃	—	2	2	no	—
34	C₄×C₈×C₇₃	—	2	2	no	—
35	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×#10596831	—	2	2	no	—
40	C₂×#10596831	—	2	4	no	—
41	C₂×#10596831	—	4	2	no	—
42	C₂×#10596831	—	4	2	no	—
43	C₂×#10596831	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
47	M₁₆×C₇₃	—	2	4	no	—
48	M₁₆×C₇₃	C₂×C₈	1	8	yes	—
49	C₂×C₈×C₇₃	—	2	4	no	—
50	C₂×C₈×C₇₃	—	2	2	no	—
51	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
56	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₇₃	#41	1	1	yes	characteristic
60	C₂³×C₇₃	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₇₃	—	2	2	no	—
62	C₂×C₄×C₇₃	—	2	2	no	—
63	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₇₃	#131	1	2	yes	—
71	C₈×C₇₃	—	2	1	no	—
72	C₈×C₇₃	—	2	1	no	—
73	C₈×C₇₃	—	2	8	no	—
74	C₈×C₇₃	C₄×C₈	1	2	yes	—
75	C₈×C₇₃	C₄×C₈	1	2	yes	—
76	#10596831	—	2	4	no	—
77	#10596831	—	4	2	no	—
78	#10596831	—	4	4	no	—
79	C₂²×C₇₃	—	2	1	no	—
80	C₂²×C₇₃	—	2	1	no	—
81	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₇₃	C₈×Q₈	1	2	yes	—
83	C₄×C₇₃	—	2	1	no	—
84	C₄×C₇₃	—	2	1	no	—
85	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
86	C₄×C₇₃	#405	1	1	yes	characteristic
87	C₄×C₇₃	#430	1	2	yes	—
88	C₄×C₇₃	C₈×D₄	1	2	yes	—
89	C₄×C₇₃	#444	1	2	yes	—
90	#8294	—	73	1	no	maximal, Sylow
91	C₂×C₇₃	—	2	2	no	—
92	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₇₃	#1773	1	1	yes	characteristic
94	C₂×C₇₃	#1778	1	1	yes	characteristic
95	C₄×#108	D₇₃	1	1	yes	characteristic
96	#1564	—	73	1	no	—
97	C₈×M₁₆	—	73	2	no	—
98	C₂×#322	—	73	1	no	—
99	#1278	—	73	2	no	—
100	C₇₃	#8294	1	1	yes	characteristic, complemented, Sylow
101	#431	—	73	2	no	—
102	C₂×#108	Dic₇₃	1	2	yes	—
103	C₄×M₁₆	C₂×D₇₃	1	2	yes	—
104	C₂×C₄×C₈	—	73	1	no	—
105	C₂×C₄×C₈	—	73	1	no	—
106	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
107	#341	Dic₇₃	1	2	yes	—
108	#335	—	73	2	no	—
109	#334	—	73	2	no	—
110	#322	—	73	2	no	—
111	#322	—	73	2	no	—
112	#322	—	146	2	no	—
113	C₈²	—	146	2	no	—
114	C₂×M₁₆	C₂×Dic₇₃	1	4	yes	—
115	C₂²×C₈	—	73	2	no	—
116	C₂²×C₈	—	146	1	no	—
117	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
119	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
120	#108	#10596831	1	8	yes	complemented
121	C₄⋊C₈	—	73	4	no	—
122	C₂²⋊C₈	—	146	2	no	—
123	#97	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	73	2	no	—
126	C₄×C₈	—	73	2	no	—
127	C₄×C₈	—	73	2	no	—
128	C₄×C₈	—	146	2	no	—
129	C₄×C₈	—	146	1	no	—
130	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
131	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
132	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
133	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
134	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
135	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
136	C₂²×C₄	#11924029	1	1	yes	characteristic
137	M₁₆	—	2	4	no	—
138	M₁₆	C₂×#10596831	1	8	yes	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	292	2	no	—
142	C₂×C₈	—	292	2	no	—
143	C₂×C₈	—	73	4	no	—
144	C₂×C₈	—	146	2	no	—
145	C₂×C₈	—	146	4	no	—
146	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
147	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
148	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
149	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
150	C₂×C₈	C₂×#10596831	1	4	yes	—
151	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
152	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
153	C₄²	D₄×D₇₃	1	1	yes	characteristic
154	C₄²	#11924027	1	1	yes	characteristic
155	C₂³	Q₈×Dic₇₃	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
159	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
160	C₂×C₄	#14556549	1	1	yes	characteristic
161	C₂×C₄	#14556553	1	1	yes	characteristic
162	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
163	C₂×C₄	#14556556	1	1	yes	characteristic
164	C₂×C₄	C₂²×#10596831	1	2	yes	—
165	C₂×C₄	#14556598	1	2	yes	—
166	C₈	—	2	8	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	292	2	no	—
170	C₈	—	292	4	no	—
171	C₈	—	146	4	no	—
172	C₈	C₄×#10596831	1	2	yes	—
173	C₈	C₄×#10596831	1	2	yes	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄⋊C₄×Dic₇₃	1	1	yes	characteristic
177	C₂²	Q₈×#10596831	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₂×C₄×#10596831	1	1	yes	characteristic
181	C₄	#17988543	1	1	yes	characteristic
182	C₄	#17988634	1	2	yes	—
183	C₄	D₄×#10596831	1	2	yes	—
184	C₄	#17988708	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	C₄⋊C₄×#10596831	1	1	yes	characteristic
187	C₂	#22609818	1	1	yes	characteristic
188	C₂	#22609839	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₇₃
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₄×#108×C₇₃	C₂
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂³×C₇₃	C₄×Q₈
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₄×#108	D₇₃
hypocenter	C₇₃	#8294
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
32. #108×C₇₃	C₈	semidirect
100. C₇₃	#8294	semidirect
120. #108	#10596831	semidirect

Maximal subgroups

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

subgroup	index
#22608679	2
M₁₆×#10596831	2
C₂×#17988308	2
#22607769	2
C₄×#108×C₇₃	2
#8294	73