Subgroups — C₇₃×#3544 #22607697

Order 9344.

1004 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#563×C₇₃	C₂	1	4	yes	maximal, complemented
2	Q₈²×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#555×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#554×C₇₃	C₂	1	2	yes	maximal, complemented
5	#552×C₇₃	C₂	1	4	yes	maximal, complemented
6	#547×C₇₃	C₂	1	4	yes	maximal, complemented
7	#544×C₇₃	C₂	1	4	yes	maximal, complemented
8	#542×C₇₃	C₂	1	4	yes	maximal, complemented
9	#536×C₇₃	C₂	1	4	yes	maximal, complemented
10	C₂×C₄⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#529×C₇₃	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₇₃	C₂²	1	4	yes	—
13	C₂²×Q₈×C₇₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₇₃	C₂²	1	4	yes	—
15	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
18	C₄⋊Q₈×C₇₃	C₂²	1	4	yes	—
19	C₄⋊Q₈×C₇₃	C₂²	1	4	yes	—
20	C₄⋊Q₈×C₇₃	C₂²	1	4	yes	—
21	#126×C₇₃	C₂²	1	4	yes	—
22	#126×C₇₃	C₂²	1	8	yes	—
23	#125×C₇₃	C₂²	1	2	yes	—
24	#125×C₇₃	C₂²	1	4	yes	—
25	#125×C₇₃	C₂²	1	8	yes	—
26	#124×C₇₃	C₂²	1	2	yes	—
27	#124×C₇₃	C₂²	1	4	yes	—
28	#124×C₇₃	C₂²	1	4	yes	—
29	#123×C₇₃	C₂²	1	4	yes	—
30	#123×C₇₃	C₂²	1	8	yes	—
31	#123×C₇₃	C₂²	1	8	yes	—
32	C₂²⋊Q₈×C₇₃	C₂²	1	2	yes	—
33	C₂²⋊Q₈×C₇₃	C₂²	1	8	yes	—
34	C₂²⋊Q₈×C₇₃	C₂²	1	8	yes	—
35	C₂²⋊Q₈×C₇₃	C₂²	1	4	yes	—
36	C₂²⋊Q₈×C₇₃	C₂²	1	8	yes	—
37	#121×C₇₃	C₂²	1	2	yes	—
38	#121×C₇₃	C₂²	1	4	yes	—
39	C₄×Q₈×C₇₃	C₂²	1	2	yes	—
40	C₄×Q₈×C₇₃	C₂²	1	4	yes	—
41	C₄×D₄×C₇₃	C₂²	1	4	yes	—
42	C₄×D₄×C₇₃	C₂²	1	2	yes	—
43	C₄×D₄×C₇₃	C₂²	1	4	yes	—
44	#117×C₇₃	C₂²	1	4	yes	—
45	#117×C₇₃	C₂²	1	4	yes	—
46	#117×C₇₃	C₂²	1	4	yes	—
47	C₂×C₄⋊C₄×C₇₃	C₂²	1	4	yes	—
48	C₂×C₄⋊C₄×C₇₃	C₂²	1	4	yes	—
49	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
50	#41×C₇₃	—	4	8	no	—
51	C₂×Q₈×C₇₃	—	4	4	no	—
52	C₂×Q₈×C₇₃	C₂³	1	2	yes	—
53	C₂×Q₈×C₇₃	C₂³	1	2	yes	—
54	C₂×Q₈×C₇₃	C₂³	1	4	yes	—
55	C₂×Q₈×C₇₃	C₂³	1	4	yes	—
56	C₂×Q₈×C₇₃	C₂³	1	2	yes	—
57	C₂×D₄×C₇₃	C₂³	1	2	yes	—
58	C₂×D₄×C₇₃	C₂³	1	4	yes	—
59	C₂×D₄×C₇₃	C₂³	1	4	yes	—
60	C₂²×C₄×C₇₃	C₂³	1	2	yes	—
61	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	C₂³	1	4	yes	—
63	C₂²×C₄×C₇₃	C₂³	1	4	yes	—
64	C₂²×C₄×C₇₃	C₂³	1	8	yes	—
65	C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
66	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
67	C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
68	C₄⋊C₄×C₇₃	C₂³	1	8	yes	—
69	C₄⋊C₄×C₇₃	C₂³	1	8	yes	—
70	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
71	C₄⋊C₄×C₇₃	C₂³	1	8	yes	—
72	C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
73	C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
74	C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
75	C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
76	C₄⋊C₄×C₇₃	C₂³	1	8	yes	—
77	C₂²⋊C₄×C₇₃	C₂³	1	4	yes	—
78	C₂²⋊C₄×C₇₃	C₂³	1	4	yes	—
79	C₂²⋊C₄×C₇₃	C₂³	1	8	yes	—
80	C₂²⋊C₄×C₇₃	C₂³	1	4	yes	—
81	C₂²⋊C₄×C₇₃	C₂³	1	8	yes	—
82	C₂²⋊C₄×C₇₃	C₂³	1	4	yes	—
83	C₂²⋊C₄×C₇₃	C₂³	1	4	yes	—
84	C₄²×C₇₃	C₂³	1	2	yes	—
85	C₄²×C₇₃	C₂³	1	1	yes	characteristic
86	C₄²×C₇₃	C₂³	1	1	yes	characteristic
87	C₄²×C₇₃	C₂³	1	4	yes	—
88	C₄²×C₇₃	C₂³	1	4	yes	—
89	C₄²×C₇₃	C₂³	1	4	yes	—
90	C₂³×C₇₃	C₂⁴	1	4	yes	—
91	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
92	Q₈×C₇₃	—	2	4	no	—
93	Q₈×C₇₃	—	4	2	no	—
94	Q₈×C₇₃	—	4	4	no	—
95	Q₈×C₇₃	—	4	2	no	—
96	Q₈×C₇₃	—	4	4	no	—
97	D₄×C₇₃	—	2	8	no	—
98	D₄×C₇₃	—	2	4	no	—
99	D₄×C₇₃	—	4	4	no	—
100	C₂×C₄×C₇₃	—	2	2	no	—
101	C₂×C₄×C₇₃	—	2	8	no	—
102	C₂×C₄×C₇₃	—	2	4	no	—
103	C₂×C₄×C₇₃	—	2	8	no	—
104	C₂×C₄×C₇₃	—	4	8	no	—
105	C₂×C₄×C₇₃	C₂⁴	1	4	yes	—
106	C₂×C₄×C₇₃	C₂⁴	1	2	yes	—
107	C₂×C₄×C₇₃	C₂⁴	1	4	yes	—
108	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂⁴	1	4	yes	—
111	C₂×C₄×C₇₃	C₂⁴	1	4	yes	—
112	C₂×C₄×C₇₃	C₂⁴	1	4	yes	—
113	C₂×C₄×C₇₃	C₂⁴	1	2	yes	—
114	C₂²×C₇₃	—	2	2	no	—
115	C₂²×C₇₃	—	2	4	no	—
116	C₂²×C₇₃	—	2	4	no	—
117	C₂²×C₇₃	—	2	4	no	—
118	C₂²×C₇₃	H(5,𝔽₂)	1	2	yes	—
119	C₂²×C₇₃	C₂⁵	1	1	yes	characteristic
120	C₄×C₇₃	—	2	4	no	—
121	C₄×C₇₃	—	2	4	no	—
122	C₄×C₇₃	—	2	1	no	—
123	C₄×C₇₃	—	2	1	no	—
124	C₄×C₇₃	—	2	4	no	—
125	C₄×C₇₃	—	2	2	no	—
126	C₄×C₇₃	—	2	4	no	—
127	C₄×C₇₃	—	2	4	no	—
128	C₄×C₇₃	2₋¹⁺⁴	1	4	yes	—
129	C₂×C₇₃	—	2	2	no	—
130	C₂×C₇₃	—	4	4	no	—
131	C₂×C₇₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₇₃	C₂×2₋¹⁺⁴	1	2	yes	—
133	#3544	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₇₃	#3544	1	1	yes	characteristic, direct product factor, Sylow
135	#563	C₂×C₇₃	1	4	yes	complemented
136	Q₈²	C₂×C₇₃	1	1	yes	characteristic, complemented
137	#555	C₂×C₇₃	1	1	yes	characteristic, complemented
138	#554	C₂×C₇₃	1	2	yes	complemented
139	#552	C₂×C₇₃	1	4	yes	complemented
140	#547	C₂×C₇₃	1	4	yes	complemented
141	#544	C₂×C₇₃	1	4	yes	complemented
142	#542	C₂×C₇₃	1	4	yes	complemented
143	#536	C₂×C₇₃	1	4	yes	complemented
144	C₂×C₄⋊Q₈	C₂×C₇₃	1	1	yes	characteristic, complemented
145	#529	C₂×C₇₃	1	2	yes	complemented
146	C₂×#41	C₂²×C₇₃	1	4	yes	—
147	C₂²×Q₈	C₂²×C₇₃	1	2	yes	—
148	C₄⋊Q₈	C₂²×C₇₃	1	4	yes	—
149	C₄⋊Q₈	C₂²×C₇₃	1	1	yes	characteristic
150	C₄⋊Q₈	C₂²×C₇₃	1	1	yes	characteristic
151	C₄⋊Q₈	C₂²×C₇₃	1	2	yes	—
152	C₄⋊Q₈	C₂²×C₇₃	1	4	yes	—
153	C₄⋊Q₈	C₂²×C₇₃	1	4	yes	—
154	C₄⋊Q₈	C₂²×C₇₃	1	4	yes	—
155	#126	C₂²×C₇₃	1	4	yes	—
156	#126	C₂²×C₇₃	1	8	yes	—
157	#125	C₂²×C₇₃	1	2	yes	—
158	#125	C₂²×C₇₃	1	4	yes	—
159	#125	C₂²×C₇₃	1	8	yes	—
160	#124	C₂²×C₇₃	1	2	yes	—
161	#124	C₂²×C₇₃	1	4	yes	—
162	#124	C₂²×C₇₃	1	4	yes	—
163	#123	C₂²×C₇₃	1	4	yes	—
164	#123	C₂²×C₇₃	1	8	yes	—
165	#123	C₂²×C₇₃	1	8	yes	—
166	C₂²⋊Q₈	C₂²×C₇₃	1	2	yes	—
167	C₂²⋊Q₈	C₂²×C₇₃	1	8	yes	—
168	C₂²⋊Q₈	C₂²×C₇₃	1	8	yes	—
169	C₂²⋊Q₈	C₂²×C₇₃	1	4	yes	—
170	C₂²⋊Q₈	C₂²×C₇₃	1	8	yes	—
171	#121	C₂²×C₇₃	1	2	yes	—
172	#121	C₂²×C₇₃	1	4	yes	—
173	C₄×Q₈	C₂²×C₇₃	1	2	yes	—
174	C₄×Q₈	C₂²×C₇₃	1	4	yes	—
175	C₄×D₄	C₂²×C₇₃	1	4	yes	—
176	C₄×D₄	C₂²×C₇₃	1	2	yes	—
177	C₄×D₄	C₂²×C₇₃	1	4	yes	—
178	#117	C₂²×C₇₃	1	4	yes	—
179	#117	C₂²×C₇₃	1	4	yes	—
180	#117	C₂²×C₇₃	1	4	yes	—
181	C₂×C₄⋊C₄	C₂²×C₇₃	1	4	yes	—
182	C₂×C₄⋊C₄	C₂²×C₇₃	1	4	yes	—
183	C₂×C₄²	C₂²×C₇₃	1	1	yes	characteristic
184	#41	—	4	8	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	C₂³×C₇₃	1	2	yes	—
187	C₂×Q₈	C₂³×C₇₃	1	2	yes	—
188	C₂×Q₈	C₂³×C₇₃	1	4	yes	—
189	C₂×Q₈	C₂³×C₇₃	1	4	yes	—
190	C₂×Q₈	C₂³×C₇₃	1	2	yes	—
191	C₂×D₄	C₂³×C₇₃	1	2	yes	—
192	C₂×D₄	C₂³×C₇₃	1	4	yes	—
193	C₂×D₄	C₂³×C₇₃	1	4	yes	—
194	C₂²×C₄	C₂³×C₇₃	1	2	yes	—
195	C₂²×C₄	C₂³×C₇₃	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₇₃	1	4	yes	—
197	C₂²×C₄	C₂³×C₇₃	1	4	yes	—
198	C₂²×C₄	C₂³×C₇₃	1	8	yes	—
199	C₄⋊C₄	C₂³×C₇₃	1	4	yes	—
200	C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
201	C₄⋊C₄	C₂³×C₇₃	1	4	yes	—
202	C₄⋊C₄	C₂³×C₇₃	1	8	yes	—
203	C₄⋊C₄	C₂³×C₇₃	1	8	yes	—
204	C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
205	C₄⋊C₄	C₂³×C₇₃	1	8	yes	—
206	C₄⋊C₄	C₂³×C₇₃	1	4	yes	—
207	C₄⋊C₄	C₂³×C₇₃	1	4	yes	—
208	C₄⋊C₄	C₂³×C₇₃	1	4	yes	—
209	C₄⋊C₄	C₂³×C₇₃	1	4	yes	—
210	C₄⋊C₄	C₂³×C₇₃	1	8	yes	—
211	C₂²⋊C₄	C₂³×C₇₃	1	4	yes	—
212	C₂²⋊C₄	C₂³×C₇₃	1	4	yes	—
213	C₂²⋊C₄	C₂³×C₇₃	1	8	yes	—
214	C₂²⋊C₄	C₂³×C₇₃	1	4	yes	—
215	C₂²⋊C₄	C₂³×C₇₃	1	8	yes	—
216	C₂²⋊C₄	C₂³×C₇₃	1	4	yes	—
217	C₂²⋊C₄	C₂³×C₇₃	1	4	yes	—
218	C₄²	C₂³×C₇₃	1	2	yes	—
219	C₄²	C₂³×C₇₃	1	1	yes	characteristic
220	C₄²	C₂³×C₇₃	1	1	yes	characteristic
221	C₄²	C₂³×C₇₃	1	4	yes	—
222	C₄²	C₂³×C₇₃	1	4	yes	—
223	C₄²	C₂³×C₇₃	1	4	yes	—
224	C₂³	C₂⁴×C₇₃	1	4	yes	—
225	C₂³	C₂⁴×C₇₃	1	1	yes	characteristic
226	Q₈	—	2	4	no	—
227	Q₈	—	4	2	no	—
228	Q₈	—	4	4	no	—
229	Q₈	—	4	2	no	—
230	Q₈	—	4	4	no	—
231	D₄	—	2	8	no	—
232	D₄	—	2	4	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	C₂⁴×C₇₃	1	4	yes	—
240	C₂×C₄	C₂⁴×C₇₃	1	2	yes	—
241	C₂×C₄	C₂⁴×C₇₃	1	4	yes	—
242	C₂×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
243	C₂×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
244	C₂×C₄	C₂⁴×C₇₃	1	4	yes	—
245	C₂×C₄	C₂⁴×C₇₃	1	4	yes	—
246	C₂×C₄	C₂⁴×C₇₃	1	4	yes	—
247	C₂×C₄	C₂⁴×C₇₃	1	2	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	H(5,𝔽₂)×C₇₃	1	2	yes	—
253	C₂²	C₂⁵×C₇₃	1	1	yes	characteristic
254	C₄	—	2	4	no	—
255	C₄	—	2	4	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	4	no	—
261	C₄	—	2	4	no	—
262	C₄	2₋¹⁺⁴×C₇₃	1	4	yes	—
263	C₂	—	2	2	no	—
264	C₂	—	4	4	no	—
265	C₂	C₂×H(5,𝔽₂)×C₇₃	1	1	yes	characteristic
266	C₂	C₂×2₋¹⁺⁴×C₇₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₇₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #563×C₇₃	C₂	semidirect
2. Q₈²×C₇₃	C₂	semidirect
3. #555×C₇₃	C₂	semidirect
4. #554×C₇₃	C₂	semidirect
5. #552×C₇₃	C₂	semidirect
6. #547×C₇₃	C₂	semidirect
7. #544×C₇₃	C₂	semidirect
8. #542×C₇₃	C₂	semidirect
9. #536×C₇₃	C₂	semidirect
10. C₂×C₄⋊Q₈×C₇₃	C₂	semidirect
11. #529×C₇₃	C₂	semidirect
133. #3544	C₇₃	direct (N × H)
134. C₇₃	#3544	direct (N × H)
135. #563	C₂×C₇₃	semidirect
136. Q₈²	C₂×C₇₃	semidirect
137. #555	C₂×C₇₃	semidirect
138. #554	C₂×C₇₃	semidirect
139. #552	C₂×C₇₃	semidirect
140. #547	C₂×C₇₃	semidirect
141. #544	C₂×C₇₃	semidirect
142. #542	C₂×C₇₃	semidirect
143. #536	C₂×C₇₃	semidirect
144. C₂×C₄⋊Q₈	C₂×C₇₃	semidirect
145. #529	C₂×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
#563×C₇₃	2
Q₈²×C₇₃	2
#555×C₇₃	2
#554×C₇₃	2
#552×C₇₃	2
#547×C₇₃	2
#544×C₇₃	2
#542×C₇₃	2
#536×C₇₃	2
C₂×C₄⋊Q₈×C₇₃	2
#529×C₇₃	2
#3544	73

Subgroups — C73×#3544 #22607697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#3544 #22607697