Subgroups — #33001070

Order 18688.

486 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11924003	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#11924003	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988363	C₂	1	2	yes	maximal, complemented
4	#22608009	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊C₄×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924003	—	2	4	no	—
7	C₂²×#11924003	C₄	1	4	yes	complemented
8	C₂²×#11924003	C₂²	1	1	yes	characteristic
9	C₂²×#11924003	C₂²	1	2	yes	—
10	C₂²×#11924003	C₂²	1	1	yes	characteristic
11	C₂²⋊#11924003	C₄	1	4	yes	complemented
12	#17988363	—	2	4	no	—
13	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₈×C₇₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
16	C₄×C₂²⋊C₄×C₇₃	C₄	1	1	yes	characteristic
17	#336×C₇₃	C₄	1	2	yes	—
18	C₂×#11924003	—	2	8	no	—
19	C₂×#11924003	—	2	2	no	—
20	C₂×#11924003	—	2	2	no	—
21	C₂×#11924003	—	2	2	no	—
22	C₂×#11924003	C₂×C₄	1	4	yes	—
23	C₂×#11924003	C₂×C₄	1	2	yes	—
24	C₂×#11924003	C₂×C₄	1	2	yes	—
25	C₂×#11924003	D₄	1	4	yes	—
26	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₃	—	2	2	no	—
28	C₂²×C₈×C₇₃	—	2	2	no	—
29	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₇₃	C₈	1	2	yes	—
35	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
36	C₂²⋊C₈×C₇₃	C₈	1	4	yes	—
37	C₄×C₈×C₇₃	—	2	2	no	—
38	C₄×C₈×C₇₃	—	2	2	no	—
39	#95×C₇₃	C₈	1	2	yes	—
40	#11924003	—	2	2	no	—
41	#11924003	—	2	4	no	—
42	#11924003	C₂²⋊C₄	1	4	yes	complemented
43	C₂⁴×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	—	2	2	no	—
45	C₂²×C₄×C₇₃	—	2	2	no	—
46	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
47	C₂²×C₄×C₇₃	C₂×C₈	1	4	yes	—
48	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₃	—	2	2	no	—
52	C₂×C₈×C₇₃	—	2	2	no	—
53	C₂×C₈×C₇₃	—	2	2	no	—
54	C₂×C₈×C₇₃	—	2	2	no	—
55	C₂×C₈×C₇₃	—	2	2	no	—
56	C₂×C₈×C₇₃	—	2	4	no	—
57	C₂×C₈×C₇₃	C₄²	1	2	yes	—
58	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
59	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₇₃	#41	1	2	yes	—
63	C₂²⋊C₄×C₇₃	—	2	4	no	—
64	C₄²×C₇₃	—	2	2	no	—
65	C₄²×C₇₃	—	2	2	no	—
66	C₂³×C₇₃	—	2	2	no	—
67	C₂³×C₇₃	—	2	2	no	—
68	C₂³×C₇₃	C₄×C₈	1	2	yes	—
69	C₂³×C₇₃	M₃₂	1	4	yes	—
70	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	2	2	no	—
73	C₂×C₄×C₇₃	—	2	2	no	—
74	C₂×C₄×C₇₃	—	2	4	no	—
75	C₂×C₄×C₇₃	—	2	4	no	—
76	C₂×C₄×C₇₃	—	2	4	no	—
77	C₂×C₄×C₇₃	—	2	4	no	—
78	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
79	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₇₃	#131	1	2	yes	—
83	C₈×C₇₃	—	2	2	no	—
84	C₈×C₇₃	—	2	4	no	—
85	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₈×C₇₃	#117	1	1	yes	characteristic
87	C₈×C₇₃	C₄×D₄	1	2	yes	—
88	C₂²×C₇₃	—	2	2	no	—
89	C₂²×C₇₃	—	2	8	no	—
90	C₂²×C₇₃	—	2	2	no	—
91	C₂²×C₇₃	#346	1	4	yes	complemented
92	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
93	C₂²×C₇₃	C₈×D₄	1	2	yes	—
94	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
95	C₂²×C₇₃	#504	1	2	yes	—
96	C₄×C₇₃	—	2	2	no	—
97	C₄×C₇₃	—	4	4	no	—
98	C₄×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
99	C₄×C₇₃	#405	1	1	yes	characteristic
100	C₄×C₇₃	C₈×D₄	1	2	yes	—
101	#12611	—	73	1	no	maximal, Sylow
102	C₂×C₇₃	—	2	4	no	—
103	C₂×C₇₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
105	C₂×C₇₃	#2108	1	1	yes	characteristic
106	C₂×C₇₃	#2168	1	4	yes	—
107	C₂³×C₁₆	—	73	1	no	—
108	C₂×C₂²⋊C₁₆	—	73	1	no	—
109	C₂×#346	—	73	2	no	—
110	C₈×C₂²⋊C₄	D₇₃	1	1	yes	characteristic
111	#1374	—	73	2	no	—
112	C₇₃	#12611	1	1	yes	characteristic, complemented, Sylow
113	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
114	C₂²×C₁₆	—	73	1	no	—
115	C₂²×C₁₆	—	73	4	no	—
116	C₂²×C₁₆	—	73	2	no	—
117	C₂²×C₁₆	—	73	1	no	—
118	C₂²×C₁₆	—	146	4	no	—
119	C₂×C₂²⋊C₈	Dic₇₃	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
121	C₄×C₂²⋊C₄	Dic₇₃	1	1	yes	characteristic
122	C₂²⋊C₁₆	—	73	4	no	—
123	#346	—	146	4	no	—
124	#336	Dic₇₃	1	2	yes	—
125	C₂³×C₄	C₂×Dic₇₃	1	1	yes	characteristic
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	C₄×D₇₃	1	2	yes	—
129	C₂²×C₈	C₄×D₇₃	1	2	yes	—
130	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
131	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
132	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	#10596831	1	2	yes	—
134	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
135	C₂×C₁₆	—	73	2	no	—
136	C₂×C₁₆	—	73	4	no	—
137	C₂×C₁₆	—	73	4	no	—
138	C₂×C₁₆	—	73	2	no	—
139	C₂×C₁₆	—	146	8	no	—
140	C₂×C₁₆	—	146	2	no	—
141	C₂×C₁₆	—	146	2	no	—
142	C₂×C₁₆	—	146	2	no	—
143	C₂²⋊C₈	#10596831	1	4	yes	—
144	C₄×C₈	—	2	2	no	—
145	C₄×C₈	—	2	2	no	—
146	#95	#10596831	1	2	yes	—
147	C₂⁴	C₂×#10596831	1	1	yes	characteristic
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
151	C₂²×C₄	C₂×#10596831	1	4	yes	—
152	C₂²×C₄	C₂×#10596831	1	1	yes	characteristic
153	C₂²×C₄	C₂×#10596831	1	2	yes	—
154	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
162	C₂×C₈	C₂×#10596831	1	4	yes	—
163	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
164	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
165	C₂×C₈	D₄×D₇₃	1	2	yes	—
166	C₂×C₈	#11924027	1	2	yes	—
167	C₂²⋊C₄	—	2	4	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	2	no	—
170	C₁₆	—	73	4	no	—
171	C₁₆	—	146	2	no	—
172	C₁₆	—	146	4	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	C₄×#10596831	1	2	yes	—
176	C₂³	#14556531	1	4	yes	—
177	C₂³	C₂²×#10596831	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	C₄×#10596831	1	2	yes	—
186	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
188	C₂×C₄	C₂²×#10596831	1	1	yes	characteristic
189	C₂×C₄	#14556598	1	2	yes	—
190	C₈	—	2	4	no	—
191	C₈	—	2	2	no	—
192	C₈	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
193	C₈	#14556547	1	1	yes	characteristic
194	C₈	#14556551	1	2	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	8	no	—
197	C₂²	—	2	2	no	—
198	C₂²	#17988363	1	4	yes	complemented
199	C₂²	C₂×C₄×#10596831	1	1	yes	characteristic
200	C₂²	D₄×#10596831	1	2	yes	—
201	C₂²	C₂×#14556531	1	2	yes	—
202	C₂²	#17989004	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	4	4	no	—
205	C₄	C₂²⋊C₄×Dic₇₃	1	1	yes	characteristic
206	C₄	#17988543	1	1	yes	characteristic
207	C₄	D₄×#10596831	1	2	yes	—
208	C₂	—	2	4	no	—
209	C₂	C₂²⋊C₄×#10596831	1	1	yes	characteristic
210	C₂	C₂×#17988363	1	1	yes	characteristic
211	C₂	#22611528	1	1	yes	characteristic
212	C₂	#22611757	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₈×C₂²⋊C₄×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂⁴×C₇₃	C₂×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₈×C₂²⋊C₄	D₇₃
hypocenter	C₇₃	#12611
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂×#17988363	C₂	semidirect
4. #22608009	C₂	semidirect
7. C₂²×#11924003	C₄	semidirect
11. C₂²⋊#11924003	C₄	semidirect
42. #11924003	C₂²⋊C₄	semidirect
91. C₂²×C₇₃	#346	semidirect
112. C₇₃	#12611	semidirect
198. C₂²	#17988363	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#11924003	2
C₂×C₂²⋊#11924003	2
C₂×#17988363	2
#22608009	2
C₈×C₂²⋊C₄×C₇₃	2
#12611	73