Subgroups — #33238986

Order 18688.

684 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22623034	C₂	1	1	yes	maximal, characteristic
2	#131×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#22611696	C₂	1	6	yes	maximal, complemented
4	C₂×#17988407	C₂	1	6	yes	maximal, complemented
5	C₇₃×#3407	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556598	C₄	1	1	yes	characteristic
7	#41×Dic₇₃	C₄	1	1	yes	characteristic
8	#17989004	—	2	4	no	—
9	C₂×#14556531	C₂²	1	6	yes	—
10	#17988545	C₄	1	3	yes	—
11	#17988542	C₂²	1	3	yes	—
12	M₁₆×Dic₇₃	C₄	1	3	yes	—
13	C₂×C₈×Dic₇₃	C₂²	1	3	yes	—
14	#17988407	C₂²	1	4	yes	—
15	#17988407	C₂²	1	12	yes	—
16	C₂×#131×C₇₃	C₂²	1	1	yes	characteristic
17	#504×C₇₃	—	2	4	no	—
18	C₂×M₃₂×C₇₃	C₂²	1	6	yes	—
19	#14556598	C₈	1	8	yes	—
20	C₂×#11924012	C₂×C₄	1	3	yes	—
21	C₂²×#10596831	C₂×C₄	1	3	yes	—
22	Q₈×Dic₇₃	—	2	1	no	—
23	D₄×Dic₇₃	—	2	3	no	—
24	#14556550	C₂×C₄	1	3	yes	—
25	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
26	#14556531	—	2	6	no	—
27	#14556531	C₂³	1	4	yes	—
28	C₂×#11924003	—	2	6	no	—
29	#14556501	C₂×C₄	1	6	yes	—
30	C₈×Dic₇₃	C₂×C₄	1	3	yes	—
31	C₈×Dic₇₃	C₂×C₄	1	3	yes	—
32	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
33	C₈×Dic₇₃	C₂³	1	3	yes	—
34	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
35	#131×C₇₃	D₄	1	4	yes	—
36	#131×C₇₃	Q₈	1	4	yes	—
37	C₂×M₁₆×C₇₃	C₂×C₄	1	3	yes	—
38	C₂²×C₈×C₇₃	C₂³	1	3	yes	—
39	M₃₂×C₇₃	—	2	6	no	—
40	M₃₂×C₇₃	C₂³	1	4	yes	—
41	C₂×C₁₆×C₇₃	—	2	6	no	—
42	C₂²×Dic₇₃	—	2	3	no	—
43	#11924012	C₂×C₈	1	12	yes	—
44	C₂×#10596831	C₂×C₈	1	12	yes	—
45	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
46	C₂×#10596831	C₂²×C₄	1	3	yes	—
47	C₄⋊Dic₇₃	—	2	3	no	—
48	C₂²⋊Dic₇₃	—	2	3	no	—
49	C₄×Dic₇₃	—	2	3	no	—
50	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₄×Dic₇₃	C₂²×C₄	1	3	yes	—
52	#11924003	—	2	4	no	—
53	#41×C₇₃	C₄⋊C₄	1	4	yes	—
54	#41×C₇₃	C₂×C₈	1	4	yes	—
55	C₂×Q₈×C₇₃	M₁₆	1	1	yes	characteristic
56	C₂×D₄×C₇₃	M₁₆	1	3	yes	—
57	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
58	M₁₆×C₇₃	C₄⋊C₄	1	12	yes	—
59	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
60	C₂×C₈×C₇₃	C₂×D₄	1	6	yes	—
61	C₂×C₈×C₇₃	C₂×Q₈	1	6	yes	—
62	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
63	C₁₆×C₇₃	—	2	4	no	—
64	C₂³×C₇₃	C₂×M₁₆	1	3	yes	—
65	Q₈×C₇₃	C₄⋊C₈	1	4	yes	—
66	D₄×C₇₃	C₄⋊C₈	1	12	yes	—
67	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	6	yes	—
68	C₂×C₄×C₇₃	C₂²×C₈	1	6	yes	—
69	C₂×C₄×C₇₃	C₂×M₁₆	1	3	yes	—
70	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
71	C₂×Dic₇₃	—	2	1	no	—
72	C₂×Dic₇₃	—	2	3	no	—
73	C₂×Dic₇₃	—	4	6	no	—
74	C₈×C₇₃	C₂×C₄⋊C₄	1	6	yes	—
75	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
76	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
77	#10596831	C₂²×C₈	1	2	yes	—
78	#10596831	C₂²×C₈	1	6	yes	—
79	C₂²×C₇₃	—	4	3	no	—
80	C₂²×C₇₃	C₂×C₄⋊C₈	1	6	yes	—
81	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
82	C₄×C₇₃	C₂×C₄⋊C₈	1	6	yes	—
83	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
85	Dic₇₃	—	2	2	no	—
86	Dic₇₃	—	4	3	no	—
87	#33401	—	73	1	no	maximal, Sylow
88	C₂×C₇₃	—	2	1	no	—
89	C₂×C₇₃	—	2	6	no	—
90	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
91	#3407	—	73	1	no	—
92	#3407	D₇₃	1	1	yes	characteristic
93	C₄×#131	—	73	1	no	—
94	#2153	—	73	6	no	—
95	C₂×#364	—	73	6	no	—
96	C₇₃	#33401	1	1	yes	characteristic, complemented, Sylow
97	C₂×#131	—	73	1	no	—
98	C₂×#131	C₂×D₇₃	1	1	yes	characteristic
99	C₄×#41	—	73	1	no	—
100	#504	—	2	4	no	—
101	#504	—	146	4	no	—
102	C₂×M₃₂	—	73	6	no	—
103	C₂×M₃₂	C₂×D₇₃	1	6	yes	—
104	#405	—	73	3	no	—
105	#405	—	73	3	no	—
106	C₄×M₁₆	—	73	3	no	—
107	C₂×C₄×C₈	—	73	3	no	—
108	#364	—	73	4	no	—
109	#364	—	73	12	no	—
110	C₂×#41	C₄×D₇₃	1	1	yes	characteristic
111	#131	—	73	8	no	—
112	#131	Dic₁₄₆	1	4	yes	—
113	#131	C₂²⋊D₇₃	1	4	yes	—
114	C₂×M₁₆	—	73	3	no	—
115	C₂×M₁₆	C₄×D₇₃	1	3	yes	—
116	C₂²×C₈	—	73	3	no	—
117	C₂²×C₈	C₂²×D₇₃	1	3	yes	—
118	C₄×Q₈	—	146	1	no	—
119	C₄×D₄	—	146	3	no	—
120	#117	—	73	3	no	—
121	C₂×C₄²	—	73	3	no	—
122	M₃₂	—	2	6	no	—
123	M₃₂	—	73	4	no	—
124	M₃₂	—	146	6	no	—
125	M₃₂	C₂²×D₇₃	1	4	yes	—
126	C₂×C₁₆	—	2	6	no	—
127	C₂×C₁₆	—	146	6	no	—
128	#97	—	73	6	no	—
129	C₄×C₈	—	73	3	no	—
130	C₄×C₈	—	73	3	no	—
131	C₄×C₈	—	73	1	no	—
132	C₄×C₈	—	73	3	no	—
133	#41	#11924007	1	4	yes	—
134	#41	C₈×D₇₃	1	4	yes	—
135	C₂×Q₈	#11924011	1	1	yes	characteristic
136	C₂×D₄	#11924011	1	3	yes	—
137	C₂²×C₄	—	146	3	no	—
138	C₂²×C₄	C₂×C₄×D₇₃	1	3	yes	—
139	M₁₆	—	73	12	no	—
140	M₁₆	#11924007	1	12	yes	—
141	C₂×C₈	—	73	12	no	—
142	C₂×C₈	—	73	1	no	—
143	C₂×C₈	—	73	3	no	—
144	C₂×C₈	C₂×C₄×D₇₃	1	3	yes	—
145	C₂×C₈	C₂×C₂²⋊D₇₃	1	6	yes	—
146	C₂×C₈	C₂×Dic₁₄₆	1	6	yes	—
147	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
148	C₄⋊C₄	—	146	3	no	—
149	C₂²⋊C₄	—	146	3	no	—
150	C₄²	—	73	1	no	—
151	C₄²	—	73	3	no	—
152	C₄²	—	146	3	no	—
153	C₁₆	—	2	4	no	—
154	C₁₆	—	146	4	no	—
155	C₂³	C₂×#11924011	1	3	yes	—
156	Q₈	#14556520	1	4	yes	—
157	D₄	#14556520	1	12	yes	—
158	C₂×C₄	—	292	6	no	—
159	C₂×C₄	—	146	3	no	—
160	C₂×C₄	—	146	1	no	—
161	C₂×C₄	C₂×#11924007	1	6	yes	—
162	C₂×C₄	C₂×C₈×D₇₃	1	6	yes	—
163	C₂×C₄	C₂×#11924011	1	3	yes	—
164	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
165	C₈	—	73	2	no	—
166	C₈	—	73	6	no	—
167	C₈	C₂×#11924007	1	6	yes	—
168	C₈	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
169	C₈	C₂²×Dic₁₄₆	1	1	yes	characteristic
170	C₂²	—	4	3	no	—
171	C₂²	C₂×#14556520	1	6	yes	—
172	C₂²	C₂²×#11924011	1	1	yes	characteristic
173	C₄	—	292	3	no	—
174	C₄	—	146	2	no	—
175	C₄	C₂×#14556520	1	6	yes	—
176	C₄	C₂²×#11924007	1	1	yes	characteristic
177	C₄	C₂²×C₈×D₇₃	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	6	no	—
180	C₂	C₂²×#14556520	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×C₂²⋊D₇₃
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#3407	C₂
socle	C₂×C₇₃	C₂²×C₄⋊C₈
prime radical	C₂×#41×C₇₃	C₂×C₄
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#3407	D₇₃
hypocenter	C₇₃	#33401
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×#131 ◁ #3407

Complements

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

N	H ≅ G/N	type
3. #22611696	C₂	semidirect
4. C₂×#17988407	C₂	semidirect
96. C₇₃	#33401	semidirect

Maximal subgroups

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

subgroup	index
#22623034	2
#131×Dic₇₃	2
#22611696	2
C₂×#17988407	2
C₇₃×#3407	2
#33401	73