Subgroups — #22618927

Order 9344.

372 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17988718	C₂	1	2	yes	maximal, complemented
2	#17988659	C₂	1	2	yes	maximal, complemented
3	#17988659	C₂	1	2	yes	maximal, complemented
4	D₄×#10596831	C₂	1	2	yes	maximal, complemented
5	#17988648	C₂	1	2	yes	maximal, complemented
6	#17988640	C₂	1	1	yes	maximal, characteristic, complemented
7	#17988543	C₂	1	2	yes	maximal, complemented
8	C₂×#14556502	C₂	1	1	yes	maximal, characteristic, complemented
9	#529×C₇₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#11924012	C₂²	1	2	yes	—
11	C₂²×#10596831	C₂²	1	2	yes	—
12	C₂²×#10596831	C₂²	1	2	yes	—
13	C₄⋊#10596831	C₂²	1	2	yes	—
14	C₄⋊#10596831	C₂²	1	2	yes	—
15	C₄⋊#10596831	C₂²	1	2	yes	—
16	C₂²⋊#10596831	C₂²	1	1	yes	characteristic
17	C₂²⋊#10596831	C₂²	1	1	yes	characteristic
18	C₂²⋊#10596831	C₂²	1	2	yes	—
19	C₂²⋊#10596831	C₂²	1	2	yes	—
20	#14556502	C₂²	1	1	yes	characteristic
21	#14556502	C₂²	1	2	yes	—
22	#14556502	C₂²	1	2	yes	—
23	#14556502	C₂²	1	1	yes	characteristic
24	C₄×#10596831	C₂²	1	2	yes	—
25	C₄×#10596831	C₂²	1	2	yes	—
26	#126×C₇₃	C₄	1	2	yes	—
27	#125×C₇₃	C₄	1	1	yes	characteristic
28	#124×C₇₃	C₄	1	1	yes	characteristic
29	#123×C₇₃	C₄	1	2	yes	—
30	C₂²⋊Q₈×C₇₃	C₄	1	1	yes	characteristic
31	#121×C₇₃	C₄	1	1	yes	characteristic
32	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
33	C₄×D₄×C₇₃	C₂²	1	2	yes	—
34	C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
35	#117×C₇₃	C₂²	1	2	yes	—
36	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
37	#11924012	—	2	4	no	—
38	C₂×#10596831	—	2	4	no	—
39	C₂×#10596831	—	2	4	no	—
40	C₂×#10596831	C₂³	1	2	yes	—
41	C₂×#10596831	C₂³	1	2	yes	—
42	C₂×#10596831	C₂³	1	2	yes	—
43	C₂×#10596831	C₂³	1	2	yes	—
44	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂×D₄×C₇₃	C₂×C₄	1	2	yes	—
46	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	C₂³	1	2	yes	—
51	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
52	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
53	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
55	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
57	C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
58	C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
59	C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
60	C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
61	C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
62	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
63	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₇₃	C₂³	1	1	yes	characteristic
65	C₄²×C₇₃	C₂³	1	1	yes	characteristic
66	C₄²×C₇₃	C₂³	1	2	yes	—
67	C₂³×C₇₃	C₂²×C₄	1	2	yes	—
68	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	Q₈×C₇₃	—	4	1	no	—
70	D₄×C₇₃	—	2	4	no	—
71	D₄×C₇₃	—	4	1	no	—
72	C₂×C₄×C₇₃	—	2	2	no	—
73	C₂×C₄×C₇₃	—	2	2	no	—
74	C₂×C₄×C₇₃	—	2	2	no	—
75	C₂×C₄×C₇₃	—	2	2	no	—
76	C₂×C₄×C₇₃	—	4	1	no	—
77	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
86	#10596831	—	2	2	no	—
87	#10596831	—	2	2	no	—
88	#10596831	—	2	2	no	—
89	#10596831	#41	1	4	yes	—
90	C₂²×C₇₃	—	2	1	no	—
91	C₂²×C₇₃	—	2	1	no	—
92	C₂²×C₇₃	—	2	2	no	—
93	C₂²×C₇₃	—	2	2	no	—
94	C₂²×C₇₃	—	2	2	no	—
95	C₂²×C₇₃	#131	1	2	yes	—
96	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
97	C₄×C₇₃	—	2	1	no	—
98	C₄×C₇₃	—	2	2	no	—
99	C₄×C₇₃	—	2	1	no	—
100	C₄×C₇₃	—	2	1	no	—
101	C₄×C₇₃	—	2	2	no	—
102	C₄×C₇₃	—	2	1	no	—
103	C₄×C₇₃	—	2	1	no	—
104	C₄×C₇₃	—	2	2	no	—
105	C₄×C₇₃	C₂×#41	1	2	yes	—
106	C₂×C₇₃	—	2	2	no	—
107	C₂×C₇₃	—	4	2	no	—
108	C₂×C₇₃	C₄×#41	1	1	yes	characteristic
109	C₂×C₇₃	C₂×#131	1	1	yes	characteristic
110	C₂×C₇₃	#568	1	1	yes	characteristic
111	#2968	—	73	1	no	maximal, Sylow
112	C₇₃	#2968	1	1	yes	characteristic, complemented, Sylow
113	#529	D₇₃	1	1	yes	characteristic
114	#446	—	73	2	no	—
115	#435	—	73	2	no	—
116	#435	—	73	2	no	—
117	C₈×D₄	—	73	2	no	—
118	#433	—	73	2	no	—
119	#431	—	73	1	no	—
120	#405	—	73	2	no	—
121	C₂×#97	—	73	1	no	—
122	C₂×M₁₆	—	73	2	no	—
123	C₂²×C₈	—	73	2	no	—
124	C₂²×C₈	—	73	2	no	—
125	#126	Dic₇₃	1	2	yes	—
126	#125	Dic₇₃	1	1	yes	characteristic
127	#124	Dic₇₃	1	1	yes	characteristic
128	#123	Dic₇₃	1	2	yes	—
129	C₂²⋊Q₈	Dic₇₃	1	1	yes	characteristic
130	#121	Dic₇₃	1	1	yes	characteristic
131	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
132	C₄×D₄	C₂×D₇₃	1	2	yes	—
133	C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
134	#117	C₂×D₇₃	1	2	yes	—
135	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
136	C₄⋊C₈	—	73	2	no	—
137	C₄⋊C₈	—	73	2	no	—
138	C₄⋊C₈	—	73	2	no	—
139	C₂²⋊C₈	—	73	1	no	—
140	C₂²⋊C₈	—	73	1	no	—
141	C₂²⋊C₈	—	73	2	no	—
142	C₂²⋊C₈	—	73	2	no	—
143	#97	—	73	1	no	—
144	#97	—	73	1	no	—
145	#97	—	73	2	no	—
146	#97	—	73	2	no	—
147	C₄×C₈	—	73	2	no	—
148	C₄×C₈	—	73	2	no	—
149	C₂×Q₈	C₂×Dic₇₃	1	1	yes	characteristic
150	C₂×D₄	C₂×Dic₇₃	1	2	yes	—
151	C₂×D₄	C₂×Dic₇₃	1	1	yes	characteristic
152	C₂²×C₄	C₂×Dic₇₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×Dic₇₃	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₇₃	1	2	yes	—
156	M₁₆	—	146	4	no	—
157	C₂×C₈	—	73	2	no	—
158	C₂×C₈	—	73	2	no	—
159	C₂×C₈	—	73	2	no	—
160	C₂×C₈	—	73	2	no	—
161	C₂×C₈	—	146	4	no	—
162	C₂×C₈	—	146	4	no	—
163	C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
164	C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
165	C₄⋊C₄	C₂×Dic₇₃	1	1	yes	characteristic
166	C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
167	C₄⋊C₄	C₂×Dic₇₃	1	1	yes	characteristic
168	C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
169	C₂²⋊C₄	C₂×Dic₇₃	1	2	yes	—
170	C₂²⋊C₄	C₂×Dic₇₃	1	2	yes	—
171	C₂²⋊C₄	C₂×Dic₇₃	1	2	yes	—
172	C₂²⋊C₄	C₂×Dic₇₃	1	2	yes	—
173	C₂²⋊C₄	C₂×Dic₇₃	1	2	yes	—
174	C₄²	C₂×Dic₇₃	1	1	yes	characteristic
175	C₄²	C₂×Dic₇₃	1	1	yes	characteristic
176	C₄²	C₂²×D₇₃	1	1	yes	characteristic
177	C₄²	C₂²×D₇₃	1	2	yes	—
178	C₄²	C₂²×D₇₃	1	1	yes	characteristic
179	C₂³	C₂²×Dic₇₃	1	2	yes	—
180	C₂³	C₂²×Dic₇₃	1	1	yes	characteristic
181	Q₈	—	4	1	no	—
182	D₄	—	2	4	no	—
183	D₄	—	4	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
191	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
192	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
193	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
194	C₂×C₄	C₂²×Dic₇₃	1	2	yes	—
195	C₂×C₄	C₂²×Dic₇₃	1	2	yes	—
196	C₂×C₄	C₂²×Dic₇₃	1	2	yes	—
197	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
198	C₈	—	73	4	no	—
199	C₈	—	146	2	no	—
200	C₈	—	146	2	no	—
201	C₈	—	146	2	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	#14556598	1	2	yes	—
208	C₂²	C₂³×Dic₇₃	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	#41×D₇₃	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	4	2	no	—
220	C₂	#41×Dic₇₃	1	1	yes	characteristic
221	C₂	C₂×#14556598	1	1	yes	characteristic
222	C₂	#17989354	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₃
commutator, socle	C₂²×C₇₃	C₂³×C₄
Fitting	#529×C₇₃	C₂
hypercenter	#529	D₇₃
hypocenter	C₇₃	#2968
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #17988718	C₂	semidirect
2. #17988659	C₂	semidirect
3. #17988659	C₂	semidirect
4. D₄×#10596831	C₂	semidirect
5. #17988648	C₂	semidirect
6. #17988640	C₂	semidirect
7. #17988543	C₂	semidirect
8. C₂×#14556502	C₂	semidirect
112. C₇₃	#2968	semidirect