Subgroups — C₇₃×#3193 #22607346

Order 9344.

462 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#121×C₇₃	C₂	1	1	yes	maximal, characteristic
3	#484×C₇₃	C₂	1	2	yes	maximal, complemented
4	#480×C₇₃	C₂	1	2	yes	maximal, complemented
5	#468×C₇₃	C₂	1	4	yes	maximal, complemented
6	#450×C₇₃	C₂	1	2	yes	maximal, complemented
7	#448×C₇₃	C₂	1	2	yes	maximal, complemented
8	#407×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₇₃	C₂²	1	1	yes	characteristic, complemented
10	C₂²×D₄×C₇₃	—	2	1	no	—
11	C₂²×D₄×C₇₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₇₃	C₂²	1	4	yes	complemented
14	C₂×M₁₆×C₇₃	C₂²	1	2	yes	complemented
15	C₂²⋊Q₈×C₇₃	—	2	1	no	—
16	C₂²⋊Q₈×C₇₃	—	2	1	no	—
17	C₂²⋊Q₈×C₇₃	C₂²	1	2	yes	—
18	C₂²⋊Q₈×C₇₃	C₂²	1	2	yes	complemented
19	#121×C₇₃	—	2	1	no	—
20	#121×C₇₃	—	2	1	no	—
21	#121×C₇₃	C₂²	1	2	yes	—
22	#121×C₇₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
24	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
27	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
28	#107×C₇₃	C₂²	1	4	yes	complemented
29	#103×C₇₃	C₂²	1	4	yes	complemented
30	#102×C₇₃	C₂²	1	4	yes	complemented
31	C₂²⋊C₈×C₇₃	C₂²	1	2	yes	—
32	C₂²⋊C₈×C₇₃	C₂²	1	2	yes	complemented
33	C₂⁴×C₇₃	—	2	1	no	—
34	C₂⁴×C₇₃	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₇₃	—	2	1	no	—
36	C₂×Q₈×C₇₃	—	4	2	no	—
37	C₂×Q₈×C₇₃	C₂³	1	2	yes	—
38	C₂×D₄×C₇₃	—	2	2	no	—
39	C₂×D₄×C₇₃	—	2	2	no	—
40	C₂×D₄×C₇₃	—	2	1	no	—
41	C₂×D₄×C₇₃	—	4	2	no	—
42	C₂×D₄×C₇₃	—	4	4	no	—
43	C₂×D₄×C₇₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₇₃	—	2	1	no	—
45	C₂²×C₄×C₇₃	—	2	1	no	—
46	C₂²×C₄×C₇₃	—	2	1	no	—
47	C₂²×C₄×C₇₃	—	4	2	no	—
48	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
54	SD₁₆×C₇₃	—	4	4	no	—
55	M₁₆×C₇₃	—	4	2	no	—
56	C₂×C₈×C₇₃	C₂³	1	4	yes	—
57	C₄⋊C₄×C₇₃	—	2	2	no	—
58	C₄⋊C₄×C₇₃	—	2	2	no	—
59	C₄⋊C₄×C₇₃	—	2	1	no	—
60	C₄⋊C₄×C₇₃	—	2	1	no	—
61	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
62	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
63	C₂²⋊C₄×C₇₃	—	2	2	no	—
64	C₂²⋊C₄×C₇₃	—	2	2	no	—
65	C₂²⋊C₄×C₇₃	—	2	2	no	—
66	C₂²⋊C₄×C₇₃	—	2	2	no	—
67	C₂³×C₇₃	—	2	2	no	—
68	C₂³×C₇₃	—	2	2	no	—
69	C₂³×C₇₃	—	2	2	no	—
70	C₂³×C₇₃	—	4	1	no	—
71	C₂³×C₇₃	—	4	2	no	—
72	C₂³×C₇₃	—	4	2	no	—
73	C₂³×C₇₃	—	4	2	no	—
74	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
77	Q₈×C₇₃	—	2	4	no	—
78	Q₈×C₇₃	—	4	2	no	—
79	D₄×C₇₃	—	2	4	no	—
80	D₄×C₇₃	—	4	2	no	—
81	D₄×C₇₃	—	4	4	no	—
82	D₄×C₇₃	—	4	4	no	—
83	C₂×C₄×C₇₃	—	2	2	no	—
84	C₂×C₄×C₇₃	—	2	2	no	—
85	C₂×C₄×C₇₃	—	2	2	no	—
86	C₂×C₄×C₇₃	—	2	2	no	—
87	C₂×C₄×C₇₃	—	2	2	no	—
88	C₂×C₄×C₇₃	—	2	2	no	—
89	C₂×C₄×C₇₃	—	4	1	no	—
90	C₂×C₄×C₇₃	—	4	1	no	—
91	C₂×C₄×C₇₃	—	4	1	no	—
92	C₂×C₄×C₇₃	—	4	2	no	—
93	C₂×C₄×C₇₃	—	4	2	no	—
94	C₂×C₄×C₇₃	—	4	2	no	—
95	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
99	C₈×C₇₃	—	2	4	no	—
100	C₂²×C₇₃	—	2	1	no	—
101	C₂²×C₇₃	—	2	1	no	—
102	C₂²×C₇₃	—	2	1	no	—
103	C₂²×C₇₃	—	2	1	no	—
104	C₂²×C₇₃	—	2	1	no	—
105	C₂²×C₇₃	—	2	1	no	—
106	C₂²×C₇₃	—	2	1	no	—
107	C₂²×C₇₃	—	4	4	no	—
108	C₂²×C₇₃	—	4	2	no	—
109	C₂²×C₇₃	—	4	2	no	—
110	C₂²×C₇₃	—	4	2	no	—
111	C₂²×C₇₃	—	8	4	no	—
112	C₂²×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
113	C₂²×C₇₃	#137	1	2	yes	—
114	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₇₃	—	2	1	no	—
116	C₄×C₇₃	—	2	1	no	—
117	C₄×C₇₃	—	2	1	no	—
118	C₄×C₇₃	—	4	2	no	—
119	C₄×C₇₃	—	4	2	no	—
120	C₄×C₇₃	—	4	2	no	—
121	C₄×C₇₃	H(5,𝔽₂)	1	2	yes	—
122	C₂×C₇₃	—	2	2	no	—
123	C₂×C₇₃	—	2	2	no	—
124	C₂×C₇₃	—	4	1	no	—
125	C₂×C₇₃	—	8	2	no	—
126	C₂×C₇₃	#534	1	1	yes	characteristic
127	C₂×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₇₃	C₂×#137	1	1	yes	characteristic
129	#3193	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₇₃	#3193	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×C₂²⋊Q₈	C₂×C₇₃	1	1	yes	characteristic, complemented
132	C₂×#121	C₂×C₇₃	1	1	yes	characteristic
133	#484	C₂×C₇₃	1	2	yes	complemented
134	#480	C₂×C₇₃	1	2	yes	complemented
135	#468	C₂×C₇₃	1	4	yes	complemented
136	#450	C₂×C₇₃	1	2	yes	complemented
137	#448	C₂×C₇₃	1	2	yes	complemented
138	#407	C₂×C₇₃	1	1	yes	characteristic, complemented
139	C₂²×Q₈	C₂²×C₇₃	1	1	yes	characteristic, complemented
140	C₂²×D₄	—	2	1	no	—
141	C₂²×D₄	C₂²×C₇₃	1	1	yes	characteristic
142	C₂³×C₄	C₂²×C₇₃	1	1	yes	characteristic
143	C₂×SD₁₆	C₂²×C₇₃	1	4	yes	complemented
144	C₂×M₁₆	C₂²×C₇₃	1	2	yes	complemented
145	C₂²⋊Q₈	—	2	1	no	—
146	C₂²⋊Q₈	—	2	1	no	—
147	C₂²⋊Q₈	C₂²×C₇₃	1	2	yes	—
148	C₂²⋊Q₈	C₂²×C₇₃	1	2	yes	complemented
149	#121	—	2	1	no	—
150	#121	—	2	1	no	—
151	#121	C₂²×C₇₃	1	2	yes	—
152	#121	C₂²×C₇₃	1	2	yes	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	C₂²×C₇₃	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂²×C₇₃	1	1	yes	characteristic, complemented
156	C₂×C₂²⋊C₄	—	2	1	no	—
157	C₂×C₂²⋊C₄	—	2	1	no	—
158	#107	C₂²×C₇₃	1	4	yes	complemented
159	#103	C₂²×C₇₃	1	4	yes	complemented
160	#102	C₂²×C₇₃	1	4	yes	complemented
161	C₂²⋊C₈	C₂²×C₇₃	1	2	yes	—
162	C₂²⋊C₈	C₂²×C₇₃	1	2	yes	complemented
163	C₂⁴	—	2	1	no	—
164	C₂⁴	D₄×C₇₃	1	1	yes	characteristic
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×Q₈	C₂³×C₇₃	1	2	yes	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	4	2	no	—
172	C₂×D₄	—	4	4	no	—
173	C₂×D₄	C₂³×C₇₃	1	2	yes	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
179	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
180	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₇₃	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₇₃	1	1	yes	characteristic
183	C₂²×C₄	C₂³×C₇₃	1	1	yes	characteristic
184	SD₁₆	—	4	4	no	—
185	M₁₆	—	4	2	no	—
186	C₂×C₈	C₂³×C₇₃	1	4	yes	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
192	C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	4	1	no	—
201	C₂³	—	4	2	no	—
202	C₂³	—	4	2	no	—
203	C₂³	—	4	2	no	—
204	C₂³	C₂×D₄×C₇₃	1	1	yes	characteristic
205	C₂³	C₂×D₄×C₇₃	1	1	yes	characteristic
206	C₂³	C₂×D₄×C₇₃	1	1	yes	characteristic
207	Q₈	—	2	4	no	—
208	Q₈	—	4	2	no	—
209	D₄	—	2	4	no	—
210	D₄	—	4	2	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
226	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
227	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
228	C₂×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
229	C₈	—	2	4	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	8	4	no	—
242	C₂²	AGL(1,ℤ/8)×C₇₃	1	2	yes	—
243	C₂²	#137×C₇₃	1	2	yes	—
244	C₂²	C₂²×D₄×C₇₃	1	1	yes	characteristic
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	H(5,𝔽₂)×C₇₃	1	2	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	2	2	no	—
254	C₂	—	4	1	no	—
255	C₂	—	8	2	no	—
256	C₂	#534×C₇₃	1	1	yes	characteristic
257	C₂	C₂×AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic
258	C₂	C₂×#137×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇₃
prime radical	C₂×#121×C₇₃	C₂
LCS	G ▷ C₂×C₄ ▷ 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
1. C₂×C₂²⋊Q₈×C₇₃	C₂	semidirect
3. #484×C₇₃	C₂	semidirect
4. #480×C₇₃	C₂	semidirect
5. #468×C₇₃	C₂	semidirect
6. #450×C₇₃	C₂	semidirect
7. #448×C₇₃	C₂	semidirect
8. #407×C₇₃	C₂	semidirect
9. C₂²×Q₈×C₇₃	C₂²	semidirect
13. C₂×SD₁₆×C₇₃	C₂²	semidirect
14. C₂×M₁₆×C₇₃	C₂²	semidirect
18. C₂²⋊Q₈×C₇₃	C₂²	semidirect
25. C₂×C₄⋊C₄×C₇₃	C₂²	semidirect
28. #107×C₇₃	C₂²	semidirect
29. #103×C₇₃	C₂²	semidirect
30. #102×C₇₃	C₂²	semidirect
32. C₂²⋊C₈×C₇₃	C₂²	semidirect
129. #3193	C₇₃	direct (N × H)
130. C₇₃	#3193	direct (N × H)
131. C₂×C₂²⋊Q₈	C₂×C₇₃	semidirect
133. #484	C₂×C₇₃	semidirect
134. #480	C₂×C₇₃	semidirect
135. #468	C₂×C₇₃	semidirect
136. #450	C₂×C₇₃	semidirect
137. #448	C₂×C₇₃	semidirect
138. #407	C₂×C₇₃	semidirect
139. C₂²×Q₈	C₂²×C₇₃	semidirect
143. C₂×SD₁₆	C₂²×C₇₃	semidirect
144. C₂×M₁₆	C₂²×C₇₃	semidirect
148. C₂²⋊Q₈	C₂²×C₇₃	semidirect
155. C₂×C₄⋊C₄	C₂²×C₇₃	semidirect
158. #107	C₂²×C₇₃	semidirect
159. #103	C₂²×C₇₃	semidirect
160. #102	C₂²×C₇₃	semidirect
162. C₂²⋊C₈	C₂²×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Q₈×C₇₃	2
C₂×#121×C₇₃	2
#484×C₇₃	2
#480×C₇₃	2
#468×C₇₃	2
#450×C₇₃	2
#448×C₇₃	2
#407×C₇₃	2
#3193	73

Subgroups — C73×#3193 #22607346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#3193 #22607346