Subgroups — #34030104

Order 18688.

622 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626593	C₂	1	2	yes	maximal
2	#22626287	C₂	1	6	yes	maximal, complemented
3	C₂×#17989516	C₂	1	3	yes	maximal, complemented
4	D₇₃⋊#368	C₂	1	3	yes	maximal, complemented
5	#504×D₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989666	—	2	2	no	—
7	#17989602	—	2	6	no	—
8	C₂×#14556662	C₂²	1	6	yes	—
9	D₇₃⋊#108	C₂²	1	6	yes	—
10	#17989516	C₂²	1	1	yes	characteristic
11	#17989516	C₂²	1	3	yes	—
12	#17989516	C₂²	1	6	yes	—
13	#17989515	C₂²	1	6	yes	—
14	#131×D₇₃	C₂²	1	1	yes	characteristic
15	#17989005	C₄	1	3	yes	—
16	#17989004	C₄	1	1	yes	characteristic
17	#17989003	C₄	1	3	yes	—
18	M₃₂×D₇₃	C₂²	1	3	yes	—
19	C₂×C₁₆×D₇₃	C₂²	1	3	yes	—
20	#504×C₇₃	C₄	1	1	yes	characteristic
21	#14556679	—	4	2	no	—
22	#14556678	—	4	6	no	—
23	C₂×#11924038	—	2	6	no	—
24	D₇₃⋊M₁₆	—	2	6	no	—
25	#14556662	—	2	6	no	—
26	#14556662	C₂³	1	2	yes	—
27	#14556662	C₂³	1	6	yes	—
28	#14556661	—	2	6	no	—
29	#41×D₇₃	D₄	1	1	yes	characteristic
30	#14556599	Q₈	1	3	yes	—
31	#14556598	Q₈	1	1	yes	characteristic
32	#14556597	C₂×C₄	1	3	yes	—
33	M₁₆×D₇₃	D₄	1	3	yes	—
34	C₂×C₈×D₇₃	C₂³	1	3	yes	—
35	#14556531	C₂×C₄	1	3	yes	—
36	#14556530	C₂×C₄	1	6	yes	—
37	C₂×#11924003	C₂×C₄	1	3	yes	—
38	C₁₆×D₇₃	C₂×C₄	1	3	yes	—
39	C₁₆×D₇₃	C₂×C₄	1	3	yes	—
40	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
41	C₁₆×D₇₃	C₂³	1	3	yes	—
42	#131×C₇₃	C₂×C₄	1	1	yes	characteristic
43	M₃₂×C₇₃	C₂×C₄	1	3	yes	—
44	C₂×C₁₆×C₇₃	C₂×C₄	1	3	yes	—
45	#11924038	—	2	2	no	—
46	#11924038	—	2	6	no	—
47	#11924038	—	4	6	no	—
48	#11924037	—	4	6	no	—
49	#11924036	—	4	6	no	—
50	C₂×#10596833	—	4	6	no	—
51	#11924029	D₈	1	1	yes	characteristic
52	#11924028	C₄⋊C₄	1	3	yes	—
53	#11924027	D₈	1	3	yes	—
54	Q₈×D₇₃	Q₁₆	1	1	yes	characteristic
55	D₄×D₇₃	Q₁₆	1	3	yes	—
56	C₂×C₄×D₇₃	C₂×D₄	1	3	yes	—
57	#11924012	C₄⋊C₄	1	3	yes	—
58	#11924011	C₄⋊C₄	1	6	yes	—
59	C₂×#10596831	C₂×Q₈	1	3	yes	—
60	C₈×D₇₃	C₂²×C₄	1	3	yes	—
61	C₈×D₇₃	C₂×D₄	1	3	yes	—
62	C₈×D₇₃	C₂×Q₈	1	3	yes	—
63	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
64	#11924003	C₂²×C₄	1	1	yes	characteristic
65	#11924003	C₂²×C₄	1	3	yes	—
66	#41×C₇₃	C₄⋊C₄	1	1	yes	characteristic
67	M₁₆×C₇₃	C₄⋊C₄	1	3	yes	—
68	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
69	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
70	C₁₆×C₇₃	C₂²×C₄	1	3	yes	—
71	C₂²×D₇₃	C₂×Q₁₆	1	3	yes	—
72	Q₈×C₇₃	#107	1	1	yes	characteristic
73	D₄×C₇₃	#107	1	3	yes	—
74	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
75	C₂²⋊D₇₃	#107	1	6	yes	—
76	C₂×Dic₇₃	C₂×D₈	1	3	yes	—
77	D₂₉₂	#107	1	3	yes	—
78	C₄×D₇₃	C₂×C₄⋊C₄	1	3	yes	—
79	C₄×D₇₃	C₂×D₈	1	3	yes	—
80	C₄×D₇₃	C₂×Q₁₆	1	3	yes	—
81	C₄×D₇₃	C₂²×D₄	1	1	yes	characteristic
82	Dic₁₄₆	#107	1	3	yes	—
83	#10596833	—	4	6	no	—
84	#10596833	—	4	2	no	—
85	C₈×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
86	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
87	#10596831	C₂×C₄⋊C₄	1	3	yes	—
88	#10596831	C₂²×Q₈	1	1	yes	characteristic
89	C₂²×C₇₃	C₂×#107	1	3	yes	—
90	C₂×D₇₃	—	4	3	no	—
91	C₂×D₇₃	C₂×#107	1	3	yes	—
92	C₂×D₇₃	C₂²×Q₁₆	1	1	yes	characteristic
93	C₄×C₇₃	C₂×#107	1	3	yes	—
94	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	Dic₇₃	C₂×#107	1	3	yes	—
96	Dic₇₃	C₂²×D₈	1	1	yes	characteristic
97	#33414	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	—	2	3	no	—
99	C₂×C₇₃	C₂²×#107	1	1	yes	characteristic
100	D₇₃	—	2	1	no	—
101	D₇₃	—	2	3	no	—
102	C₂×#504	—	73	1	no	—
103	#2916	—	73	2	no	—
104	#2161	—	73	6	no	—
105	C₂×#368	—	73	3	no	—
106	C₂×#368	—	73	3	no	—
107	C₇₃	#33414	1	1	yes	characteristic, complemented, Sylow
108	#568	—	146	2	no	—
109	C₂×#131	—	73	1	no	—
110	#504	—	73	1	no	—
111	#504	—	73	3	no	—
112	#504	—	73	3	no	—
113	#504	#64408	1	1	yes	characteristic
114	C₂×M₃₂	—	73	3	no	—
115	C₂²×C₁₆	—	73	3	no	—
116	#430	—	146	6	no	—
117	C₂×#108	—	73	6	no	—
118	C₂×#108	—	73	6	no	—
119	#368	—	73	1	no	—
120	#368	—	73	3	no	—
121	#368	—	73	6	no	—
122	#368	—	73	6	no	—
123	C₂×#41	—	73	1	no	—
124	#131	—	292	6	no	—
125	#131	—	292	2	no	—
126	#131	—	73	1	no	—
127	#131	—	73	3	no	—
128	#131	—	73	3	no	—
129	#131	C₂×#64408	1	1	yes	characteristic
130	C₂×M₁₆	—	73	3	no	—
131	C₂×M₁₆	—	146	6	no	—
132	C₂×M₁₆	—	146	6	no	—
133	C₂²×C₈	—	73	3	no	—
134	M₃₂	—	73	3	no	—
135	M₃₂	—	73	6	no	—
136	M₃₂	C₂×#64408	1	3	yes	—
137	C₂×C₁₆	—	73	3	no	—
138	C₂×C₁₆	—	73	1	no	—
139	C₂×C₁₆	—	73	3	no	—
140	C₂×C₁₆	—	73	3	no	—
141	C₂×C₁₆	—	73	3	no	—
142	C₂×C₁₆	C₂×#64408	1	3	yes	—
143	#108	—	73	2	no	—
144	#108	—	73	6	no	—
145	#108	—	146	6	no	—
146	#108	—	146	6	no	—
147	#41	—	73	1	no	—
148	#41	—	73	3	no	—
149	#41	—	73	3	no	—
150	#41	C₄⋊#64408	1	1	yes	characteristic
151	C₂×Q₈	—	73	1	no	—
152	C₂×D₄	—	73	3	no	—
153	C₂²×C₄	—	73	3	no	—
154	M₁₆	—	292	6	no	—
155	M₁₆	—	292	6	no	—
156	M₁₆	—	73	3	no	—
157	M₁₆	—	73	6	no	—
158	M₁₆	—	146	2	no	—
159	M₁₆	—	146	6	no	—
160	M₁₆	C₄⋊#64408	1	3	yes	—
161	C₂×C₈	—	292	6	no	—
162	C₂×C₈	—	292	6	no	—
163	C₂×C₈	—	73	3	no	—
164	C₂×C₈	—	73	3	no	—
165	C₂×C₈	—	73	1	no	—
166	C₂×C₈	—	73	3	no	—
167	C₂×C₈	—	73	3	no	—
168	C₂×C₈	C₂²×#64408	1	3	yes	—
169	C₁₆	—	73	1	no	—
170	C₁₆	—	73	3	no	—
171	C₁₆	C₂²×#64408	1	1	yes	characteristic
172	C₁₆	C₂²×#64408	1	3	yes	—
173	C₂³	—	73	3	no	—
174	Q₈	—	73	3	no	—
175	Q₈	#14556660	1	1	yes	characteristic
176	D₄	—	73	6	no	—
177	D₄	—	73	3	no	—
178	D₄	#14556660	1	3	yes	—
179	C₂×C₄	—	73	3	no	—
180	C₂×C₄	—	73	1	no	—
181	C₂×C₄	—	73	3	no	—
182	C₂×C₄	—	73	3	no	—
183	C₂×C₄	—	73	3	no	—
184	C₂×C₄	C₂×C₄⋊#64408	1	3	yes	—
185	C₈	—	292	6	no	—
186	C₈	—	292	2	no	—
187	C₈	—	73	1	no	—
188	C₈	—	73	3	no	—
189	C₈	C₂×C₄⋊#64408	1	3	yes	—
190	C₈	C₂³×#64408	1	1	yes	characteristic
191	C₂²	—	292	3	no	—
192	C₂²	—	73	1	no	—
193	C₂²	—	73	3	no	—
194	C₂²	C₂×#14556660	1	3	yes	—
195	C₄	—	73	1	no	—
196	C₄	—	73	3	no	—
197	C₄	C₂×#14556660	1	3	yes	—
198	C₄	C₂²×C₄⋊#64408	1	1	yes	characteristic
199	C₂	—	2	3	no	—
200	C₂	—	146	1	no	—
201	C₂	—	146	3	no	—
202	C₂	C₂²×#14556660	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×#14556660
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₈	C₂³×#64408
Fitting	#504×C₇₃	C₄
socle	C₂×C₇₃	C₂²×#107
prime radical	#41×D₇₃	D₄
cosocle	C₈×D₇₃	C₂⁴
hypercenter	#504	#64408
hypocenter	C₇₃	#33414
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

Complements

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

N	H ≅ G/N	type
2. #22626287	C₂	semidirect
3. C₂×#17989516	C₂	semidirect
4. D₇₃⋊#368	C₂	semidirect
107. C₇₃	#33414	semidirect

Maximal subgroups

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

subgroup	index
#22626593	2
#22626287	2
C₂×#17989516	2
D₇₃⋊#368	2
#504×D₇₃	2
#33414	73