Subgroups — #32995028

Order 18688.

250 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22611734	C₂	1	1	yes	maximal, characteristic
2	#22608220	C₂	1	1	yes	maximal, characteristic
3	#22607853	C₂	1	2	yes	maximal, complemented
4	#22607851	C₂	1	2	yes	maximal, complemented
5	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₄	1	1	yes	characteristic
7	C₄×#11924012	C₄	1	1	yes	characteristic
8	C₄⋊#11924003	C₄	1	2	yes	—
9	C₂²⋊#11924003	C₄	1	2	yes	—
10	#17988363	C₂²	1	2	yes	—
11	#17988309	C₄	1	2	yes	—
12	#17988308	C₂²	1	2	yes	—
13	C₂²×C₁₆×C₇₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	#346×C₇₃	C₄	1	2	yes	—
16	#346×C₇₃	C₂²	1	1	yes	characteristic
17	#346×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×#11924012	—	2	1	no	—
19	C₂×#11924003	C₂×C₄	1	4	yes	—
20	C₄⋊#10596831	—	2	1	no	—
21	C₂²⋊#10596831	—	2	1	no	—
22	#14556502	—	2	1	no	—
23	C₄×#10596831	C₂×C₄	1	2	yes	—
24	C₄×#10596831	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₃	—	2	4	no	—
29	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	#11924012	—	4	2	no	—
36	C₂×#10596831	—	2	2	no	—
37	C₂×#10596831	—	2	2	no	—
38	#11924003	—	2	4	no	—
39	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
42	C₂×C₈×C₇₃	—	2	2	no	—
43	C₂×C₈×C₇₃	—	2	2	no	—
44	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
46	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₇₃	C₄²	1	1	yes	characteristic
49	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
50	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
51	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₁₆×C₇₃	—	2	2	no	—
53	C₁₆×C₇₃	M₁₆	1	4	yes	complemented
54	C₂³×C₇₃	#97	1	1	yes	characteristic
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	2	1	no	—
57	C₂×C₄×C₇₃	—	2	1	no	—
58	C₂×C₄×C₇₃	—	4	1	no	—
59	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
65	C₈×C₇₃	—	2	1	no	—
66	C₈×C₇₃	—	2	1	no	—
67	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
68	C₈×C₇₃	#131	1	2	yes	—
69	#10596831	—	4	2	no	—
70	#10596831	—	4	2	no	—
71	C₂²×C₇₃	—	2	1	no	—
72	C₂²×C₇₃	—	2	1	no	—
73	C₂²×C₇₃	#347	1	2	yes	—
74	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	—	2	1	no	—
78	C₄×C₇₃	—	2	1	no	—
79	C₄×C₇₃	—	2	1	no	—
80	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
81	C₄×C₇₃	#405	1	1	yes	characteristic
82	#11413	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	—	2	2	no	—
84	C₂×C₇₃	#1451	1	1	yes	characteristic
85	C₂×C₇₃	C₂×#347	1	1	yes	characteristic
86	C₂×C₇₃	#2110	1	1	yes	characteristic
87	#2163	—	73	1	no	—
88	C₂×#346	D₇₃	1	1	yes	characteristic
89	#1451	—	73	1	no	—
90	#1313	—	73	2	no	—
91	#1313	—	73	2	no	—
92	C₇₃	#11413	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₁₆	Dic₇₃	1	2	yes	—
94	#431	—	73	1	no	—
95	C₄×M₁₆	—	73	1	no	—
96	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
97	C₄⋊C₁₆	—	73	2	no	—
98	C₂²⋊C₁₆	—	73	2	no	—
99	#346	—	73	2	no	—
100	#346	Dic₇₃	1	2	yes	—
101	#346	C₂×D₇₃	1	1	yes	characteristic
102	#346	C₂×D₇₃	1	1	yes	characteristic
103	#322	—	73	2	no	—
104	#322	—	73	2	no	—
105	C₂×M₁₆	—	146	1	no	—
106	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
108	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
109	C₂×C₁₆	—	2	4	no	—
110	C₂×C₁₆	—	73	4	no	—
111	C₂×C₁₆	C₂×Dic₇₃	1	2	yes	—
112	C₂×C₁₆	C₂×Dic₇₃	1	2	yes	—
113	C₄⋊C₈	—	146	1	no	—
114	C₂²⋊C₈	—	146	1	no	—
115	#97	—	146	1	no	—
116	C₄×C₈	—	73	2	no	—
117	C₄×C₈	—	73	2	no	—
118	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
119	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
122	C₂²×C₄	C₄×Dic₇₃	1	1	yes	characteristic
123	C₂²×C₄	#11924011	1	1	yes	characteristic
124	C₂²×C₄	#11924011	1	1	yes	characteristic
125	M₁₆	—	292	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	146	2	no	—
129	C₂×C₈	—	146	2	no	—
130	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
131	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
133	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
134	C₄²	C₄×Dic₇₃	1	1	yes	characteristic
135	C₄²	#11924011	1	1	yes	characteristic
136	C₄²	#11924011	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
138	C₁₆	—	2	2	no	—
139	C₁₆	—	146	4	no	—
140	C₁₆	#11924012	1	4	yes	complemented
141	C₂³	#14556501	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	#14556501	1	1	yes	characteristic
147	C₂×C₄	#14556501	1	1	yes	characteristic
148	C₂×C₄	#14556501	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
150	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
151	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	292	2	no	—
155	C₈	—	292	2	no	—
156	C₈	C₂×#11924012	1	2	yes	—
157	C₈	#14556597	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#17988364	1	2	yes	—
161	C₂²	C₂×#14556501	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	C₄×#11924012	1	1	yes	characteristic
168	C₄	#17988542	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	#22608220	1	1	yes	characteristic
171	C₂	C₂×#17988364	1	1	yes	characteristic
172	C₂	#22611535	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₂×#346×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#97
prime radical	C₂³×C₇₃	#97
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	#11413
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #22607853	C₂	semidirect
4. #22607851	C₂	semidirect
53. C₁₆×C₇₃	M₁₆	semidirect
92. C₇₃	#11413	semidirect
140. C₁₆	#11924012	semidirect

Maximal subgroups

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

subgroup	index
#22611734	2
#22608220	2
#22607853	2
#22607851	2
C₂×#346×C₇₃	2
#11413	73