Subgroups — #33086139

Order 18688.

926 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988611	C₂	1	2	yes	maximal
2	#22610051	C₂	1	8	yes	maximal, complemented
3	#22609625	C₂	1	4	yes	maximal
4	C₂×#376×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924012	C₂²	1	4	yes	—
6	#17988611	—	2	8	no	—
7	C₂×C₄⋊#10596831	C₂²	1	8	yes	—
8	#17988342	C₂²	1	16	yes	—
9	C₂²×C₄²×C₇₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
11	#376×C₇₃	C₄	1	4	yes	—
12	#376×C₇₃	C₂²	1	4	yes	—
13	C₂×#95×C₇₃	C₄	1	4	yes	—
14	C₂×#11924012	—	2	8	no	—
15	C₂×#11924012	—	2	16	no	—
16	C₂²×#10596831	C₂³	1	8	yes	—
17	C₄⋊#10596831	—	2	16	no	—
18	C₂³×C₄×C₇₃	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₇₃	—	2	8	no	—
22	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₇₃	D₄	1	4	yes	—
25	C₂×C₄²×C₇₃	Q₈	1	4	yes	—
26	C₂×C₄²×C₇₃	C₂³	1	4	yes	—
27	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
28	#95×C₇₃	C₂×C₄	1	16	yes	—
29	#11924012	—	4	16	no	—
30	C₂×#10596831	—	2	8	no	—
31	C₂×#10596831	—	2	16	no	—
32	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	—	2	8	no	—
34	C₂²×C₄×C₇₃	—	4	4	no	—
35	C₂²×C₄×C₇₃	—	4	8	no	—
36	C₂²×C₄×C₇₃	C₄⋊C₄	1	8	yes	—
37	C₂²×C₄×C₇₃	C₂²×C₄	1	8	yes	—
38	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
41	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
42	C₂²×C₄×C₇₃	#41	1	8	yes	—
43	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
44	C₄²×C₇₃	—	2	8	no	—
45	C₄²×C₇₃	—	2	8	no	—
46	C₄²×C₇₃	C₄⋊C₄	1	8	yes	—
47	C₄²×C₇₃	C₂×D₄	1	4	yes	—
48	C₄²×C₇₃	C₂×Q₈	1	4	yes	—
49	C₂³×C₇₃	—	2	4	no	—
50	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
51	C₂³×C₇₃	#117	1	4	yes	—
52	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	—	2	8	no	—
54	C₂×C₄×C₇₃	—	2	8	no	—
55	C₂×C₄×C₇₃	—	2	16	no	—
56	C₂×C₄×C₇₃	—	4	8	no	—
57	C₂×C₄×C₇₃	—	4	16	no	—
58	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	8	yes	—
59	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
60	C₂×C₄×C₇₃	#117	1	4	yes	—
61	C₂×C₄×C₇₃	#124	1	8	yes	—
62	C₂×C₄×C₇₃	#125	1	8	yes	—
63	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂×#41	1	4	yes	—
66	#10596831	—	4	8	no	—
67	C₂²×C₇₃	—	2	4	no	—
68	C₂²×C₇₃	—	2	8	no	—
69	C₂²×C₇₃	#382	1	4	yes	—
70	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₇₃	C₂×#117	1	2	yes	—
72	C₂²×C₇₃	#568	1	4	yes	—
73	C₄×C₇₃	—	2	8	no	—
74	C₄×C₇₃	—	4	8	no	—
75	C₄×C₇₃	#382	1	4	yes	—
76	C₄×C₇₃	C₂×#124	1	2	yes	—
77	C₄×C₇₃	C₂×#125	1	2	yes	—
78	#20753	—	73	1	no	maximal, Sylow
79	C₂×C₇₃	—	2	4	no	—
80	C₂×C₇₃	C₂×#382	1	1	yes	characteristic
81	C₂×C₇₃	#2905	1	2	yes	—
82	C₂×C₇₃	#2921	1	4	yes	—
83	C₂×#423	—	73	2	no	—
84	C₂×#376	D₇₃	1	1	yes	characteristic
85	#1832	—	73	8	no	—
86	#1728	—	73	4	no	—
87	C₇₃	#20753	1	1	yes	characteristic, complemented, Sylow
88	C₂²×M₁₆	—	73	4	no	—
89	C₂²×C₄²	C₂×D₇₃	1	2	yes	—
90	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
91	#423	—	146	8	no	—
92	C₂×C₄⋊C₈	—	73	8	no	—
93	#376	Dic₇₃	1	4	yes	—
94	#376	C₂×D₇₃	1	4	yes	—
95	C₂×#95	Dic₇₃	1	4	yes	—
96	#336	—	73	16	no	—
97	C₂³×C₄	C₂×Dic₇₃	1	4	yes	—
98	C₂³×C₄	C₂×Dic₇₃	1	2	yes	—
99	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
100	C₂×M₁₆	—	146	8	no	—
101	C₂×M₁₆	—	146	16	no	—
102	C₂²×C₈	—	73	8	no	—
103	C₂×C₄²	—	2	8	no	—
104	C₂×C₄²	Dic₁₄₆	1	4	yes	—
105	C₂×C₄²	D₂₉₂	1	4	yes	—
106	C₂×C₄²	C₂×Dic₇₃	1	4	yes	—
107	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
108	C₂×C₄²	C₂²×D₇₃	1	4	yes	—
109	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
110	C₄⋊C₈	—	146	16	no	—
111	#95	C₂×Dic₇₃	1	16	yes	—
112	C₂⁴	C₂²×Dic₇₃	1	1	yes	characteristic
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	—	4	4	no	—
115	C₂²×C₄	—	4	8	no	—
116	C₂²×C₄	C₄⋊Dic₇₃	1	8	yes	—
117	C₂²×C₄	C₂²×Dic₇₃	1	8	yes	—
118	C₂²×C₄	C₂²×Dic₇₃	1	4	yes	—
119	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
120	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
121	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
122	C₂²×C₄	#11924028	1	8	yes	—
123	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
124	M₁₆	—	292	16	no	—
125	C₂×C₈	—	146	8	no	—
126	C₂×C₈	—	146	16	no	—
127	C₄²	—	2	8	no	—
128	C₄²	—	2	8	no	—
129	C₄²	C₄⋊Dic₇₃	1	8	yes	—
130	C₄²	C₂×D₂₉₂	1	4	yes	—
131	C₄²	C₂×Dic₁₄₆	1	4	yes	—
132	C₂³	—	2	4	no	—
133	C₂³	C₂×C₄⋊Dic₇₃	1	2	yes	—
134	C₂³	#14556550	1	4	yes	—
135	C₂³	C₂³×Dic₇₃	1	1	yes	characteristic
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	16	no	—
139	C₂×C₄	—	4	8	no	—
140	C₂×C₄	—	4	16	no	—
141	C₂×C₄	C₂×C₄⋊Dic₇₃	1	8	yes	—
142	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
143	C₂×C₄	#14556550	1	4	yes	—
144	C₂×C₄	#14556579	1	8	yes	—
145	C₂×C₄	#14556583	1	8	yes	—
146	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
147	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
148	C₂×C₄	C₂×#11924028	1	4	yes	—
149	C₈	—	292	8	no	—
150	C₂²	—	2	4	no	—
151	C₂²	—	2	8	no	—
152	C₂²	#17988453	1	4	yes	—
153	C₂²	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
154	C₂²	C₂×#14556550	1	2	yes	—
155	C₂²	#17989354	1	4	yes	—
156	C₄	—	2	8	no	—
157	C₄	—	4	8	no	—
158	C₄	#17988453	1	4	yes	—
159	C₄	C₂×#14556579	1	2	yes	—
160	C₄	C₂×#14556583	1	2	yes	—
161	C₂	—	2	4	no	—
162	C₂	C₂×#17988453	1	1	yes	characteristic
163	C₂	#22618495	1	2	yes	—
164	C₂	#22618569	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₃
commutator, socle	C₂³×C₇₃	C₂³×C₄
Fitting	C₂×#376×C₇₃	C₂
prime radical	C₂⁴×C₇₃	C₂²×C₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#376	D₇₃
hypocenter	C₇₃	#20753
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
2. #22610051	C₂	semidirect
87. C₇₃	#20753	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988611	2
#22610051	2
#22609625	2
C₂×#376×C₇₃	2
#20753	73