Subgroups — C₇₃×#10770 #32920174

Order 18688.

1162 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#519×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#101×C₇₃	C₂	1	2	yes	maximal
3	C₂×#344×C₇₃	C₂	1	4	yes	maximal
4	C₂³×Q₈×C₇₃	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₇₃	C₂²	1	4	yes	—
6	#519×C₇₃	—	2	8	no	—
7	C₂×C₄×Q₈×C₇₃	—	2	4	no	—
8	C₂×#117×C₇₃	C₄	1	4	yes	—
9	C₂×#117×C₇₃	C₂²	1	2	yes	—
10	C₂×#101×C₇₃	—	2	4	no	—
11	C₂×#101×C₇₃	—	2	4	no	—
12	C₂×#101×C₇₃	C₄	1	8	yes	complemented
13	#344×C₇₃	—	2	8	no	—
14	C₂²×Q₈×C₇₃	—	2	4	no	—
15	C₂²×Q₈×C₇₃	—	2	4	no	—
16	C₂²×Q₈×C₇₃	C₂×C₄	1	4	yes	—
17	C₂²×Q₈×C₇₃	D₄	1	4	yes	—
18	C₂²×Q₈×C₇₃	D₄	1	2	yes	—
19	C₂²×Q₈×C₇₃	Q₈	1	2	yes	—
20	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₇₃	—	2	4	no	—
23	C₂×M₁₆×C₇₃	—	2	4	no	—
24	C₂×M₁₆×C₇₃	—	2	4	no	—
25	C₂×M₁₆×C₇₃	—	2	4	no	—
26	C₂×M₁₆×C₇₃	—	2	4	no	—
27	C₂×M₁₆×C₇₃	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₇₃	—	2	4	no	—
29	C₄×Q₈×C₇₃	—	2	32	no	—
30	#117×C₇₃	—	2	4	no	—
31	#117×C₇₃	—	2	16	no	—
32	#117×C₇₃	D₄	1	8	yes	—
33	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
34	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
35	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₇₃	—	2	4	no	—
37	C₂×C₄²×C₇₃	—	2	2	no	—
38	#101×C₇₃	—	2	16	no	—
39	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×Q₈×C₇₃	—	2	8	no	—
41	C₂×Q₈×C₇₃	—	2	8	no	—
42	C₂×Q₈×C₇₃	—	2	8	no	—
43	C₂×Q₈×C₇₃	—	2	8	no	—
44	C₂×Q₈×C₇₃	—	2	8	no	—
45	C₂×Q₈×C₇₃	—	2	8	no	—
46	C₂×Q₈×C₇₃	C₂²⋊C₄	1	8	yes	—
47	C₂×Q₈×C₇₃	C₄⋊C₄	1	8	yes	—
48	C₂²×C₄×C₇₃	—	2	4	no	—
49	C₂²×C₄×C₇₃	—	2	4	no	—
50	C₂²×C₄×C₇₃	—	2	4	no	—
51	C₂²×C₄×C₇₃	C₄²	1	4	yes	—
52	C₂²×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
53	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	#41	1	4	yes	—
60	M₁₆×C₇₃	—	2	8	no	—
61	M₁₆×C₇₃	—	2	8	no	—
62	M₁₆×C₇₃	—	4	4	no	—
63	M₁₆×C₇₃	—	4	4	no	—
64	C₂×C₈×C₇₃	—	2	8	no	—
65	C₂×C₈×C₇₃	—	4	4	no	—
66	C₂×C₈×C₇₃	—	4	4	no	—
67	C₄⋊C₄×C₇₃	—	2	16	no	—
68	C₄⋊C₄×C₇₃	—	2	8	no	—
69	C₂²⋊C₄×C₇₃	—	2	4	no	—
70	C₂²⋊C₄×C₇₃	C₂²⋊C₄	1	8	yes	—
71	C₄²×C₇₃	—	2	16	no	—
72	C₄²×C₇₃	—	2	8	no	—
73	C₂³×C₇₃	—	4	2	no	—
74	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
75	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂³×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
77	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₇₃	#117	1	2	yes	—
79	Q₈×C₇₃	—	2	16	no	—
80	Q₈×C₇₃	—	2	16	no	—
81	C₂×C₄×C₇₃	—	2	8	no	—
82	C₂×C₄×C₇₃	—	2	4	no	—
83	C₂×C₄×C₇₃	—	2	8	no	—
84	C₂×C₄×C₇₃	—	2	4	no	—
85	C₂×C₄×C₇₃	—	2	8	no	—
86	C₂×C₄×C₇₃	—	4	8	no	—
87	C₂×C₄×C₇₃	#95	1	8	yes	—
88	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₇₃	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₇₃	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₇₃	#120	1	4	yes	—
93	C₂×C₄×C₇₃	#121	1	4	yes	—
94	C₂×C₄×C₇₃	C₂²⋊Q₈	1	4	yes	—
95	C₂×C₄×C₇₃	#123	1	4	yes	—
96	C₈×C₇₃	—	4	8	no	—
97	C₂²×C₇₃	—	2	2	no	—
98	C₂²×C₇₃	—	2	4	no	—
99	C₂²×C₇₃	—	4	1	no	—
100	C₂²×C₇₃	—	4	2	no	—
101	C₂²×C₇₃	—	4	1	no	—
102	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
103	C₂²×C₇₃	C₄×C₂²⋊C₄	1	2	yes	—
104	C₂²×C₇₃	#379	1	1	yes	characteristic
105	C₂²×C₇₃	#380	1	2	yes	—
106	C₂²×C₇₃	#381	1	1	yes	characteristic
107	C₄×C₇₃	—	2	8	no	—
108	C₄×C₇₃	—	4	8	no	—
109	C₄×C₇₃	#385	1	4	yes	—
110	C₄×C₇₃	#386	1	4	yes	—
111	#10770	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₇₃	—	2	1	no	—
113	C₂×C₇₃	—	2	1	no	—
114	C₂×C₇₃	—	2	2	no	—
115	C₂×C₇₃	—	2	2	no	—
116	C₂×C₇₃	C₂²⋊#95	1	1	yes	characteristic
117	C₂×C₇₃	#1830	1	2	yes	—
118	C₂×#519	C₂×C₇₃	1	1	yes	characteristic
119	C₂²×#101	C₂×C₇₃	1	2	yes	—
120	C₂×#344	C₂×C₇₃	1	4	yes	—
121	C₇₃	#10770	1	1	yes	characteristic, direct product factor, Sylow
122	C₂³×Q₈	C₂²×C₇₃	1	1	yes	characteristic
123	C₂²×M₁₆	C₂²×C₇₃	1	4	yes	—
124	#519	—	2	8	no	—
125	C₂×C₄×Q₈	—	2	4	no	—
126	C₂×#117	C₄×C₇₃	1	4	yes	—
127	C₂×#117	C₂²×C₇₃	1	2	yes	—
128	C₂×#101	—	2	4	no	—
129	C₂×#101	—	2	4	no	—
130	C₂×#101	C₄×C₇₃	1	8	yes	complemented
131	#344	—	2	8	no	—
132	C₂²×Q₈	—	2	4	no	—
133	C₂²×Q₈	—	2	4	no	—
134	C₂²×Q₈	C₂×C₄×C₇₃	1	4	yes	—
135	C₂²×Q₈	D₄×C₇₃	1	4	yes	—
136	C₂²×Q₈	D₄×C₇₃	1	2	yes	—
137	C₂²×Q₈	Q₈×C₇₃	1	2	yes	—
138	C₂³×C₄	C₂×C₄×C₇₃	1	2	yes	—
139	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
140	C₂×M₁₆	—	2	4	no	—
141	C₂×M₁₆	—	2	4	no	—
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	—	2	4	no	—
144	C₂×M₁₆	—	2	4	no	—
145	C₂×M₁₆	C₂×C₄×C₇₃	1	8	yes	—
146	C₂²×C₈	—	2	4	no	—
147	C₄×Q₈	—	2	32	no	—
148	#117	—	2	4	no	—
149	#117	—	2	16	no	—
150	#117	D₄×C₇₃	1	8	yes	—
151	C₂×C₄⋊C₄	—	2	4	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₂²⋊C₄	C₂×C₄×C₇₃	1	4	yes	—
154	C₂×C₄²	—	2	4	no	—
155	C₂×C₄²	—	2	2	no	—
156	#101	—	2	16	no	—
157	C₂⁴	C₂²×C₄×C₇₃	1	1	yes	characteristic
158	C₂×Q₈	—	2	8	no	—
159	C₂×Q₈	—	2	8	no	—
160	C₂×Q₈	—	2	8	no	—
161	C₂×Q₈	—	2	8	no	—
162	C₂×Q₈	—	2	8	no	—
163	C₂×Q₈	—	2	8	no	—
164	C₂×Q₈	C₂²⋊C₄×C₇₃	1	8	yes	—
165	C₂×Q₈	C₄⋊C₄×C₇₃	1	8	yes	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	C₄²×C₇₃	1	4	yes	—
170	C₂²×C₄	C₂²⋊C₄×C₇₃	1	4	yes	—
171	C₂²×C₄	C₄⋊C₄×C₇₃	1	4	yes	—
172	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
173	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
174	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
175	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
176	C₂²×C₄	C₂×Q₈×C₇₃	1	1	yes	characteristic
177	C₂²×C₄	#41×C₇₃	1	4	yes	—
178	M₁₆	—	2	8	no	—
179	M₁₆	—	2	8	no	—
180	M₁₆	—	4	4	no	—
181	M₁₆	—	4	4	no	—
182	C₂×C₈	—	2	8	no	—
183	C₂×C₈	—	4	4	no	—
184	C₂×C₈	—	4	4	no	—
185	C₄⋊C₄	—	2	16	no	—
186	C₄⋊C₄	—	2	8	no	—
187	C₂²⋊C₄	—	2	4	no	—
188	C₂²⋊C₄	C₂²⋊C₄×C₇₃	1	8	yes	—
189	C₄²	—	2	16	no	—
190	C₄²	—	2	8	no	—
191	C₂³	—	4	2	no	—
192	C₂³	C₂×C₄²×C₇₃	1	1	yes	characteristic
193	C₂³	C₂×C₂²⋊C₄×C₇₃	1	1	yes	characteristic
194	C₂³	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
195	C₂³	C₂×C₄⋊C₄×C₇₃	1	1	yes	characteristic
196	C₂³	#117×C₇₃	1	2	yes	—
197	Q₈	—	2	16	no	—
198	Q₈	—	2	16	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	#95×C₇₃	1	8	yes	—
206	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	2	yes	—
208	C₂×C₄	C₄×D₄×C₇₃	1	4	yes	—
209	C₂×C₄	C₄×D₄×C₇₃	1	4	yes	—
210	C₂×C₄	#120×C₇₃	1	4	yes	—
211	C₂×C₄	#121×C₇₃	1	4	yes	—
212	C₂×C₄	C₂²⋊Q₈×C₇₃	1	4	yes	—
213	C₂×C₄	#123×C₇₃	1	4	yes	—
214	C₈	—	4	8	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	4	1	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	1	no	—
220	C₂²	C₂×#95×C₇₃	1	1	yes	characteristic
221	C₂²	C₄×C₂²⋊C₄×C₇₃	1	2	yes	—
222	C₂²	#379×C₇₃	1	1	yes	characteristic
223	C₂²	#380×C₇₃	1	2	yes	—
224	C₂²	#381×C₇₃	1	1	yes	characteristic
225	C₄	—	2	8	no	—
226	C₄	—	4	8	no	—
227	C₄	#385×C₇₃	1	4	yes	—
228	C₄	#386×C₇₃	1	4	yes	—
229	C₂	—	2	1	no	—
230	C₂	—	2	2	no	—
231	C₂	—	2	2	no	—
232	C₂	—	2	1	no	—
233	C₂	C₇₃×C₂²⋊#95	1	1	yes	characteristic
234	C₂	C₇₃×#1830	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇₃	#381
commutator	C₂³	C₂×C₄²×C₇₃
Frattini, cosocle	C₂³×C₄	C₂³×C₇₃
prime radical	C₂⁴×C₇₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂³×C₄×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×#101×C₇₃	C₄	semidirect
111. #10770	C₇₃	direct (N × H)
121. C₇₃	#10770	direct (N × H)
130. C₂×#101	C₄×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#519×C₇₃	2
C₂²×#101×C₇₃	2
C₂×#344×C₇₃	2
#10770	73

Subgroups — C73×#10770 #32920174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#10770 #32920174