Subgroups — #34014244

Order 18688.

290 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989519	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17989519	C₂	1	1	yes	maximal, characteristic, complemented
3	#22625048	C₂	1	4	yes	maximal, complemented
4	C₂×#17989004	C₂	1	1	yes	maximal, characteristic
5	#17989519	—	2	1	no	—
6	#17989519	—	2	1	no	—
7	#17989519	—	2	1	no	—
8	#17989519	C₂²	1	2	yes	complemented
9	C₂×#14556641	—	2	1	no	—
10	C₂×#14556641	C₂²	1	2	yes	complemented
11	#17989004	—	2	1	no	—
12	#17989004	—	2	1	no	—
13	#17989004	C₄	1	2	yes	—
14	#17989004	C₂²	1	2	yes	—
15	C₂×#14556531	—	2	1	no	—
16	C₂×#14556531	C₄	1	1	yes	characteristic
17	C₂²×#11924003	—	2	1	no	—
18	C₂²×#11924003	C₂²	1	1	yes	characteristic
19	C₂×#131×C₇₃	C₄	1	1	yes	characteristic
20	#14556641	—	2	2	no	—
21	#14556641	—	2	2	no	—
22	#14556531	—	2	2	no	—
23	#14556531	—	2	2	no	—
24	#14556531	—	2	1	no	—
25	#14556531	C₂×C₄	1	2	yes	—
26	C₂×#11924003	—	2	2	no	—
27	C₂×#11924003	—	2	2	no	—
28	C₂×#11924003	—	2	1	no	—
29	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
30	C₂×#11924003	D₄	1	1	yes	characteristic
31	C₂×#11924003	D₄	1	1	yes	characteristic
32	C₂×#11924003	D₄	1	1	yes	characteristic
33	C₂×#11924003	D₄	1	1	yes	characteristic
34	C₂×#11924003	C₂³	1	1	yes	characteristic
35	C₂×#41×C₇₃	C₈	1	1	yes	characteristic
36	#131×C₇₃	—	2	1	no	—
37	#131×C₇₃	—	2	1	no	—
38	#131×C₇₃	C₈	1	2	yes	—
39	#131×C₇₃	C₂×C₄	1	2	yes	—
40	C₂×M₁₆×C₇₃	—	2	1	no	—
41	C₂×M₁₆×C₇₃	C₈	1	1	yes	characteristic
42	C₂²×C₈×C₇₃	—	2	1	no	—
43	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
44	#11924003	—	2	2	no	—
45	#11924003	C₂²⋊C₄	1	1	yes	characteristic
46	#11924003	C₂²⋊C₄	1	1	yes	characteristic
47	#11924003	C₂×D₄	1	1	yes	characteristic
48	#11924003	C₂×D₄	1	1	yes	characteristic
49	#41×C₇₃	—	2	1	no	—
50	#41×C₇₃	—	2	1	no	—
51	#41×C₇₃	C₁₆	1	2	yes	—
52	#41×C₇₃	C₂×C₈	1	2	yes	—
53	C₂×Q₈×C₇₃	C₁₆	1	1	yes	characteristic
54	C₂×D₄×C₇₃	—	2	1	no	—
55	C₂×D₄×C₇₃	C₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	—	2	1	no	—
57	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
58	M₁₆×C₇₃	—	2	1	no	—
59	M₁₆×C₇₃	—	2	2	no	—
60	M₁₆×C₇₃	—	2	2	no	—
61	M₁₆×C₇₃	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₇₃	—	2	1	no	—
63	C₂×C₈×C₇₃	—	2	2	no	—
64	C₂×C₈×C₇₃	—	2	2	no	—
65	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
68	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
69	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
70	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂³×C₇₃	—	2	1	no	—
72	C₂³×C₇₃	M₃₂	1	1	yes	characteristic
73	Q₈×C₇₃	—	2	1	no	—
74	Q₈×C₇₃	C₂×C₁₆	1	2	yes	—
75	D₄×C₇₃	—	2	2	no	—
76	D₄×C₇₃	—	2	2	no	—
77	D₄×C₇₃	—	2	1	no	—
78	D₄×C₇₃	C₂×C₁₆	1	2	yes	—
79	C₂×C₄×C₇₃	—	2	1	no	—
80	C₂×C₄×C₇₃	—	2	2	no	—
81	C₂×C₄×C₇₃	—	2	2	no	—
82	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
86	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
87	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
88	C₈×C₇₃	—	2	2	no	—
89	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
90	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
91	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
93	C₂²×C₇₃	—	2	2	no	—
94	C₂²×C₇₃	—	4	1	no	—
95	C₂²×C₇₃	—	4	2	no	—
96	C₂²×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
97	C₂²×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
98	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
99	C₄×C₇₃	—	2	2	no	—
100	C₄×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
101	C₄×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
102	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
104	#13558	—	73	1	no	maximal, Sylow
105	C₂×C₇₃	—	2	1	no	—
106	C₂×C₇₃	—	2	1	no	—
107	C₂×C₇₃	—	2	1	no	—
108	C₂×C₇₃	—	4	2	no	—
109	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
110	C₂×#504	—	73	1	no	—
111	C₂×M₆₄	—	73	1	no	—
112	C₂×M₆₄	—	73	1	no	—
113	#1401	—	73	4	no	—
114	C₇₃	#13558	1	1	yes	characteristic, complemented, Sylow
115	C₂×#131	#64408	1	1	yes	characteristic
116	#504	—	73	2	no	—
117	#504	—	73	2	no	—
118	#504	—	146	1	no	—
119	#504	—	146	1	no	—
120	C₂×M₃₂	—	73	1	no	—
121	C₂×M₃₂	—	146	1	no	—
122	C₂²×C₁₆	—	73	1	no	—
123	C₂²×C₁₆	—	146	1	no	—
124	M₆₄	—	73	2	no	—
125	M₆₄	—	146	1	no	—
126	M₆₄	—	146	1	no	—
127	M₆₄	—	146	1	no	—
128	C₂×C₃₂	—	73	2	no	—
129	C₂×C₃₂	—	146	1	no	—
130	C₂×#41	#10596833	1	1	yes	characteristic
131	#131	—	2	1	no	—
132	#131	—	2	1	no	—
133	#131	#10596833	1	2	yes	—
134	#131	C₂×#64408	1	2	yes	—
135	C₂×M₁₆	—	2	1	no	—
136	C₂×M₁₆	#10596833	1	1	yes	characteristic
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
139	M₃₂	—	73	2	no	—
140	M₃₂	—	146	2	no	—
141	M₃₂	—	146	2	no	—
142	M₃₂	—	146	1	no	—
143	C₂×C₁₆	—	73	1	no	—
144	C₂×C₁₆	—	73	1	no	—
145	C₂×C₁₆	—	73	1	no	—
146	C₂×C₁₆	—	73	1	no	—
147	C₂×C₁₆	—	73	1	no	—
148	C₂×C₁₆	—	73	1	no	—
149	C₂×C₁₆	—	146	2	no	—
150	C₂×C₁₆	—	146	2	no	—
151	C₂×C₁₆	—	146	1	no	—
152	C₃₂	—	146	2	no	—
153	C₃₂	—	146	2	no	—
154	#41	—	2	1	no	—
155	#41	—	2	1	no	—
156	#41	#11924031	1	2	yes	—
157	#41	C₂×#10596833	1	2	yes	—
158	C₂×Q₈	#11924031	1	1	yes	characteristic
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	#11924031	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
163	M₁₆	—	2	1	no	—
164	M₁₆	—	2	2	no	—
165	M₁₆	—	2	2	no	—
166	M₁₆	C₂×#10596833	1	2	yes	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
172	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
173	C₂×C₈	#11924037	1	1	yes	characteristic
174	C₂×C₈	#11924037	1	1	yes	characteristic
175	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
176	C₁₆	—	73	1	no	—
177	C₁₆	—	73	1	no	—
178	C₁₆	—	73	1	no	—
179	C₁₆	—	73	1	no	—
180	C₁₆	—	146	2	no	—
181	C₂³	—	2	1	no	—
182	C₂³	#14556665	1	1	yes	characteristic
183	Q₈	—	2	1	no	—
184	Q₈	C₂×#11924031	1	2	yes	—
185	D₄	—	2	2	no	—
186	D₄	—	2	2	no	—
187	D₄	—	2	1	no	—
188	D₄	C₂×#11924031	1	2	yes	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
193	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
194	C₂×C₄	C₂×#11924031	1	1	yes	characteristic
195	C₂×C₄	C₂×#11924031	1	1	yes	characteristic
196	C₂×C₄	#14556665	1	1	yes	characteristic
197	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	C₂²⋊#10596833	1	1	yes	characteristic
200	C₈	C₂²⋊#10596833	1	1	yes	characteristic
201	C₈	C₂×C₂²⋊#64408	1	1	yes	characteristic
202	C₈	C₂×#11924037	1	1	yes	characteristic
203	C₂²	—	2	2	no	—
204	C₂²	—	4	1	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²⋊#11924031	1	1	yes	characteristic
207	C₂²	C₂²⋊#11924031	1	1	yes	characteristic
208	C₂²	C₂×#14556665	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	C₂²⋊#11924031	1	1	yes	characteristic
211	C₄	C₂²⋊#11924031	1	1	yes	characteristic
212	C₄	C₂×C₂²⋊#10596833	1	1	yes	characteristic
213	C₄	C₂²×#11924031	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	—	4	2	no	—
218	C₂	C₂×C₂²⋊#11924031	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊#64408
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂×#131×C₇₃	C₄
socle	C₂×C₇₃	C₂×C₂²⋊C₁₆
prime radical	C₂×#41×C₇₃	C₈
cosocle	C₂×#11924003	C₂³
hypercenter	C₂×#131	#64408
hypocenter	C₇₃	#13558
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#17989519	C₂	semidirect
2. C₂×#17989519	C₂	semidirect
3. #22625048	C₂	semidirect
8. #17989519	C₂²	semidirect
10. C₂×#14556641	C₂²	semidirect
114. C₇₃	#13558	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989519	2
#22625048	2
C₂×#17989004	2
#13558	73