Subgroups — C₂²×#17988366 #33238648

Order 18688.

1470 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×#11924003	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17988366	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₄×D₇₃	C₄	1	1	yes	characteristic
6	C₂³×#10596831	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₇₃	—	2	4	no	—
8	C₂²×C₈×D₇₃	—	2	4	no	—
9	C₂²×C₈×D₇₃	C₄	1	6	yes	—
10	C₂²×C₈×D₇₃	C₂²	1	6	yes	—
11	C₂²×#11924003	—	2	4	no	—
12	C₂²×#11924003	C₂²	1	6	yes	complemented
13	#17988366	C₂²	1	16	yes	direct product factor
14	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₇₃	—	2	4	no	—
16	C₂²×C₁₆×C₇₃	C₂²	1	6	yes	complemented
17	C₂⁴×D₇₃	C₈	1	1	yes	characteristic
18	C₂³×Dic₇₃	C₈	1	1	yes	characteristic
19	C₂²×C₄×D₇₃	—	2	4	no	—
20	C₂²×C₄×D₇₃	—	2	4	no	—
21	C₂²×C₄×D₇₃	C₈	1	6	yes	—
22	C₂²×C₄×D₇₃	C₂×C₄	1	6	yes	—
23	C₂²×#10596831	—	2	4	no	—
24	C₂²×#10596831	C₂×C₄	1	6	yes	—
25	C₂×C₈×D₇₃	—	2	24	no	—
26	C₂×C₈×D₇₃	—	2	12	no	—
27	C₂×C₈×D₇₃	—	2	12	no	—
28	C₂×C₈×D₇₃	C₂×C₄	1	12	yes	—
29	C₂×C₈×D₇₃	C₂³	1	4	yes	—
30	C₂×#11924003	—	2	12	no	—
31	C₂×#11924003	C₂³	1	4	yes	complemented
32	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂×C₄	1	3	yes	—
34	C₂²×C₈×C₇₃	D₄	1	4	yes	—
35	C₂²×C₈×C₇₃	D₄	1	4	yes	—
36	C₂²×C₈×C₇₃	C₂³	1	3	yes	—
37	C₂×C₁₆×C₇₃	—	2	12	no	—
38	C₂×C₁₆×C₇₃	C₂³	1	4	yes	complemented
39	C₂³×D₇₃	—	2	4	no	—
40	C₂³×D₇₃	—	2	4	no	—
41	C₂³×D₇₃	C₁₆	1	8	yes	complemented
42	C₂³×D₇₃	C₂×C₈	1	6	yes	—
43	C₂²×Dic₇₃	—	2	4	no	—
44	C₂²×Dic₇₃	C₂×C₈	1	6	yes	—
45	C₂×C₄×D₇₃	—	2	24	no	—
46	C₂×C₄×D₇₃	—	2	12	no	—
47	C₂×C₄×D₇₃	—	2	12	no	—
48	C₂×C₄×D₇₃	C₂×C₈	1	12	yes	—
49	C₂×C₄×D₇₃	C₂²×C₄	1	4	yes	—
50	C₂×#10596831	—	2	12	no	—
51	C₂×#10596831	C₂²×C₄	1	4	yes	—
52	C₈×D₇₃	—	2	24	no	—
53	C₈×D₇₃	—	2	4	no	—
54	C₈×D₇₃	—	2	4	no	—
55	#11924003	—	2	4	no	—
56	C₂⁴×C₇₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
58	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
59	C₂²×C₄×C₇₃	M₁₆	1	4	yes	—
60	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
61	C₂×C₈×C₇₃	C₂²⋊C₄	1	12	yes	—
62	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
63	C₂×C₈×C₇₃	C₂×D₄	1	6	yes	—
64	C₂×C₈×C₇₃	C₂×D₄	1	6	yes	—
65	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
66	C₁₆×C₇₃	—	2	4	no	—
67	C₂³×C₇₃	C₂²⋊C₈	1	4	yes	—
68	C₂³×C₇₃	C₂×C₁₆	1	4	yes	—
69	C₂³×C₇₃	M₃₂	1	4	yes	—
70	C₂³×C₇₃	C₂²×C₈	1	3	yes	—
71	C₂²×D₇₃	—	2	32	no	—
72	C₂²×D₇₃	—	2	12	no	—
73	C₂²×D₇₃	—	2	12	no	—
74	C₂²×D₇₃	C₂×C₁₆	1	24	yes	complemented
75	C₂²×D₇₃	C₂²×C₈	1	4	yes	—
76	C₂×C₄×C₇₃	C₂²⋊C₈	1	12	yes	—
77	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	6	yes	—
78	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
79	C₂×C₄×C₇₃	C₂×M₁₆	1	6	yes	—
80	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×Dic₇₃	—	2	12	no	—
82	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
83	C₄×D₇₃	—	2	4	no	—
84	C₄×D₇₃	—	2	4	no	—
85	C₄×D₇₃	—	2	24	no	—
86	C₈×C₇₃	C₂×C₂²⋊C₄	1	6	yes	—
87	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
88	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
89	#10596831	—	2	4	no	—
90	C₂²×C₇₃	C₂²⋊C₁₆	1	16	yes	complemented
91	C₂²×C₇₃	C₂×C₂²⋊C₈	1	6	yes	—
92	C₂²×C₇₃	C₂²×C₁₆	1	6	yes	—
93	C₂²×C₇₃	C₂×M₃₂	1	6	yes	—
94	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
95	C₂×D₇₃	—	2	4	no	—
96	C₂×D₇₃	—	2	48	no	—
97	C₂×D₇₃	—	2	4	no	—
98	C₂×D₇₃	C₂²×C₁₆	1	8	yes	complemented
99	C₄×C₇₃	C₂×C₂²⋊C₈	1	6	yes	—
100	C₄×C₇₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
101	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
102	Dic₇₃	—	2	4	no	—
103	C₂²×C₂²⋊C₁₆	—	73	1	no	maximal, Sylow
104	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	12	yes	complemented
105	C₂×C₇₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
106	C₂×C₇₃	C₂³×C₁₆	1	1	yes	characteristic
107	C₂×C₇₃	C₂²×M₃₂	1	1	yes	characteristic
108	D₇₃	—	2	8	no	—
109	C₂⁴×C₈	—	73	1	no	—
110	C₂³×C₁₆	—	73	1	no	—
111	C₂³×C₁₆	D₇₃	1	1	yes	characteristic, complemented
112	C₂×C₂²⋊C₁₆	—	73	12	no	—
113	C₇₃	C₂²×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
114	C₂⁴×C₄	—	73	1	no	—
115	C₂³×C₈	—	73	1	no	—
116	C₂³×C₈	—	73	6	no	—
117	C₂³×C₈	—	73	6	no	—
118	C₂³×C₈	—	146	4	no	—
119	C₂³×C₈	—	146	4	no	—
120	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
121	C₂²×C₁₆	—	2	4	no	—
122	C₂²×C₁₆	—	73	6	no	—
123	C₂²×C₁₆	—	146	4	no	—
124	C₂²×C₁₆	C₂×D₇₃	1	6	yes	complemented
125	C₂²⋊C₁₆	—	73	16	no	—
126	C₂⁵	—	73	1	no	—
127	C₂³×C₄	—	73	1	no	—
128	C₂³×C₄	—	73	6	no	—
129	C₂³×C₄	—	73	6	no	—
130	C₂³×C₄	—	146	4	no	—
131	C₂³×C₄	—	146	4	no	—
132	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
133	C₂²×C₈	—	73	6	no	—
134	C₂²×C₈	—	73	12	no	—
135	C₂²×C₈	—	73	4	no	—
136	C₂²×C₈	—	146	4	no	—
137	C₂²×C₈	—	146	24	no	—
138	C₂²×C₈	—	146	12	no	—
139	C₂²×C₈	—	146	12	no	—
140	C₂²×C₈	C₄×D₇₃	1	3	yes	—
141	C₂²×C₈	D₂₉₂	1	4	yes	—
142	C₂²×C₈	C₂²⋊D₇₃	1	4	yes	—
143	C₂²×C₈	C₂²×D₇₃	1	3	yes	—
144	C₂×C₁₆	—	2	12	no	—
145	C₂×C₁₆	—	73	4	no	—
146	C₂×C₁₆	—	146	12	no	—
147	C₂×C₁₆	C₂²×D₇₃	1	4	yes	complemented
148	C₂⁴	—	73	8	no	—
149	C₂⁴	—	73	6	no	—
150	C₂⁴	—	146	4	no	—
151	C₂⁴	—	146	4	no	—
152	C₂⁴	C₈×D₇₃	1	1	yes	characteristic
153	C₂²×C₄	—	73	6	no	—
154	C₂²×C₄	—	73	12	no	—
155	C₂²×C₄	—	73	4	no	—
156	C₂²×C₄	—	146	4	no	—
157	C₂²×C₄	—	146	24	no	—
158	C₂²×C₄	—	146	12	no	—
159	C₂²×C₄	—	146	12	no	—
160	C₂²×C₄	#11924005	1	4	yes	—
161	C₂²×C₄	C₈×D₇₃	1	3	yes	—
162	C₂²×C₄	#11924011	1	4	yes	—
163	C₂²×C₄	C₂×C₄×D₇₃	1	3	yes	—
164	C₂×C₈	—	73	4	no	—
165	C₂×C₈	—	146	12	no	—
166	C₂×C₈	—	146	24	no	—
167	C₂×C₈	—	146	4	no	—
168	C₂×C₈	—	146	4	no	—
169	C₂×C₈	#11924005	1	12	yes	—
170	C₂×C₈	C₂×C₄×D₇₃	1	3	yes	—
171	C₂×C₈	C₂×C₂²⋊D₇₃	1	6	yes	—
172	C₂×C₈	C₂×D₂₉₂	1	6	yes	—
173	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
174	C₁₆	—	2	4	no	—
175	C₁₆	—	146	4	no	—
176	C₂³	—	73	24	no	—
177	C₂³	—	73	4	no	—
178	C₂³	—	146	32	no	—
179	C₂³	—	146	12	no	—
180	C₂³	—	146	12	no	—
181	C₂³	#14556503	1	4	yes	—
182	C₂³	C₁₆×D₇₃	1	4	yes	—
183	C₂³	#14556530	1	4	yes	—
184	C₂³	C₂×C₈×D₇₃	1	3	yes	—
185	C₂×C₄	—	73	4	no	—
186	C₂×C₄	—	146	12	no	—
187	C₂×C₄	—	146	4	no	—
188	C₂×C₄	—	146	24	no	—
189	C₂×C₄	—	146	4	no	—
190	C₂×C₄	#14556503	1	12	yes	—
191	C₂×C₄	C₂×#11924005	1	6	yes	—
192	C₂×C₄	C₂×C₈×D₇₃	1	3	yes	—
193	C₂×C₄	C₂×#11924011	1	6	yes	—
194	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
195	C₈	—	146	4	no	—
196	C₈	C₂×#11924005	1	6	yes	—
197	C₈	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
198	C₈	C₂²×D₂₉₂	1	1	yes	characteristic
199	C₂²	—	73	8	no	—
200	C₂²	—	146	4	no	—
201	C₂²	—	146	48	no	—
202	C₂²	—	146	4	no	—
203	C₂²	#17988366	1	16	yes	direct product factor
204	C₂²	C₂×#14556503	1	6	yes	—
205	C₂²	C₂×C₁₆×D₇₃	1	6	yes	—
206	C₂²	C₂×#14556530	1	6	yes	—
207	C₂²	C₂²×C₈×D₇₃	1	1	yes	characteristic
208	C₄	—	146	4	no	—
209	C₄	C₂×#14556503	1	6	yes	—
210	C₄	C₂²×#11924005	1	1	yes	characteristic
211	C₄	C₂²×#11924011	1	1	yes	characteristic
212	C₂	—	146	8	no	—
213	C₂	C₂×#17988366	1	12	yes	direct product factor
214	C₂	C₂²×#14556503	1	1	yes	characteristic
215	C₂	C₂²×C₁₆×D₇₃	1	1	yes	characteristic
216	C₂	C₂²×#14556530	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₇₃
commutator	C₂×C₇₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂³×C₁₆×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂×C₈
prime radical	C₂⁴×D₇₃	C₈
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂³×C₁₆	D₇₃
hypocenter	C₇₃	C₂²×C₂²⋊C₁₆
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂³×#11924003	C₂	semidirect
3. C₂×#17988366	C₂	direct (N × H)
4. C₂³×C₁₆×C₇₃	C₂	semidirect
12. C₂²×#11924003	C₂²	semidirect
13. #17988366	C₂²	direct (N × H)
16. C₂²×C₁₆×C₇₃	C₂²	semidirect
31. C₂×#11924003	C₂³	semidirect
38. C₂×C₁₆×C₇₃	C₂³	semidirect
41. C₂³×D₇₃	C₁₆	semidirect
74. C₂²×D₇₃	C₂×C₁₆	semidirect
90. C₂²×C₇₃	C₂²⋊C₁₆	semidirect
98. C₂×D₇₃	C₂²×C₁₆	semidirect
104. C₂×C₇₃	C₂×C₂²⋊C₁₆	semidirect
111. C₂³×C₁₆	D₇₃	semidirect
113. C₇₃	C₂²×C₂²⋊C₁₆	semidirect
124. C₂²×C₁₆	C₂×D₇₃	semidirect
147. C₂×C₁₆	C₂²×D₇₃	semidirect
203. C₂²	#17988366	direct (N × H)
213. C₂	C₂×#17988366	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×D₇₃	2
C₂³×#11924003	2
C₂×#17988366	2
C₂³×C₁₆×C₇₃	2
C₂²×C₂²⋊C₁₆	73

Subgroups — C22×#17988366 #33238648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988366 #33238648