Subgroups — #33056544

Order 18688.

598 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620517	C₂	1	1	yes	maximal, characteristic
2	#22618865	C₂	1	2	yes	maximal
3	#22608841	C₂	1	4	yes	maximal, complemented
4	#22608751	C₂	1	4	yes	maximal, complemented
5	#22608687	C₂	1	1	yes	maximal, characteristic
6	#22608387	C₂	1	2	yes	maximal
7	C₂×#322×C₇₃	C₂	1	1	yes	maximal, characteristic
8	#17989144	C₂²	1	2	yes	—
9	#17988992	C₄	1	1	yes	characteristic
10	#17988967	C₄	1	1	yes	characteristic
11	#17988938	C₄	1	2	yes	—
12	#17988828	C₄	1	4	yes	—
13	#17988717	—	2	2	no	—
14	#17988717	C₂²	1	4	yes	—
15	C₈×Dic₁₄₆	—	2	2	no	—
16	C₈×Dic₁₄₆	C₂²	1	4	yes	—
17	C₄×Dic₂₉₂	C₂²	1	4	yes	—
18	#17988613	—	2	2	no	—
19	#17988611	C₂²	1	2	yes	—
20	#17988542	—	2	2	no	—
21	#17988338	C₂²	1	4	yes	—
22	#17988319	C₂²	1	8	yes	—
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
26	#322×C₇₃	C₂²	1	2	yes	—
27	#322×C₇₃	C₂²	1	2	yes	—
28	C₂×Dic₂₉₂	C₂×C₄	1	4	yes	—
29	C₂×#11924012	—	2	2	no	—
30	#14556591	C₂×C₄	1	2	yes	—
31	#14556591	C₂×C₄	1	2	yes	—
32	#14556583	—	2	2	no	—
33	#14556570	—	2	2	no	—
34	#14556570	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₄₆	—	2	2	no	—
36	C₄×Dic₁₄₆	C₂³	1	4	yes	—
37	#14556550	—	2	2	no	—
38	#14556525	C₂×C₄	1	4	yes	—
39	C₄⋊#10596831	—	2	2	no	—
40	C₄⋊#10596831	C₂³	1	4	yes	—
41	#14556520	—	2	4	no	—
42	#14556520	—	2	4	no	—
43	#14556516	C₂×C₄	1	8	yes	—
44	#14556501	—	2	4	no	—
45	C₈×Dic₇₃	—	2	4	no	—
46	C₂²×C₈×C₇₃	—	2	1	no	—
47	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₇₃	D₄	1	2	yes	—
49	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₇₃	—	2	1	no	—
51	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₇₃	D₄	1	2	yes	—
54	C₄×C₈×C₇₃	C₂³	1	2	yes	—
55	C₄×C₈×C₇₃	C₂³	1	2	yes	—
56	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₄₆	—	2	2	no	—
59	C₂×Dic₁₄₆	C₂²×C₄	1	4	yes	—
60	Dic₂₉₂	—	2	8	no	—
61	#11924012	—	4	4	no	—
62	C₂×#10596831	—	2	4	no	—
63	C₄⋊Dic₇₃	—	2	2	no	—
64	C₄⋊Dic₇₃	—	2	2	no	—
65	C₄⋊Dic₇₃	—	2	2	no	—
66	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
67	#11924007	—	2	4	no	—
68	#11924007	—	2	4	no	—
69	C₂²⋊Dic₇₃	—	2	4	no	—
70	C₄×Dic₇₃	—	2	4	no	—
71	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
74	C₂×C₈×C₇₃	—	2	4	no	—
75	C₂×C₈×C₇₃	—	2	4	no	—
76	C₂×C₈×C₇₃	—	2	2	no	—
77	C₂×C₈×C₇₃	—	4	2	no	—
78	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
82	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₇₃	#41	1	1	yes	characteristic
85	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
86	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₇₃	—	2	2	no	—
88	C₂×C₄×C₇₃	—	2	2	no	—
89	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇₃	#137	1	4	yes	—
93	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
96	C₂×Dic₇₃	—	2	4	no	—
97	C₂×Dic₇₃	—	2	4	no	—
98	Dic₁₄₆	—	2	8	no	—
99	Dic₁₄₆	—	4	4	no	—
100	C₈×C₇₃	—	2	2	no	—
101	C₈×C₇₃	—	2	2	no	—
102	C₈×C₇₃	—	2	2	no	—
103	C₈×C₇₃	—	4	2	no	—
104	C₈×C₇₃	#131	1	4	yes	—
105	#10596831	—	4	4	no	—
106	C₂²×C₇₃	—	2	1	no	—
107	C₂²×C₇₃	—	2	1	no	—
108	C₂²×C₇₃	#441	1	2	yes	—
109	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
110	C₄×C₇₃	—	2	1	no	—
111	C₄×C₇₃	—	2	1	no	—
112	C₄×C₇₃	#441	1	2	yes	—
113	C₄×C₇₃	#444	1	2	yes	—
114	C₄×C₇₃	C₂×#131	1	2	yes	—
115	C₄×C₇₃	C₂×#137	1	2	yes	—
116	Dic₇₃	—	4	4	no	—
117	Dic₇₃	—	4	4	no	—
118	#17961	—	73	1	no	maximal, Sylow
119	C₂×C₇₃	—	2	2	no	—
120	C₂×C₇₃	#2931	1	1	yes	characteristic
121	C₂×C₇₃	C₂×#441	1	1	yes	characteristic
122	C₂×C₇₃	C₂×#444	1	1	yes	characteristic
123	#3170	—	73	1	no	—
124	#2960	—	73	2	no	—
125	#1593	—	73	4	no	—
126	#1579	—	73	4	no	—
127	#1566	—	73	1	no	—
128	#1493	—	73	2	no	—
129	C₂×#322	D₇₃	1	1	yes	characteristic
130	C₇₃	#17961	1	1	yes	characteristic, complemented, Sylow
131	#533	—	73	2	no	—
132	#500	—	73	1	no	—
133	#494	—	73	1	no	—
134	#487	—	73	2	no	—
135	#467	—	73	4	no	—
136	#446	—	73	4	no	—
137	#446	—	146	2	no	—
138	C₈×Q₈	—	73	4	no	—
139	C₈×Q₈	—	146	2	no	—
140	C₄×Q₁₆	—	73	4	no	—
141	#424	—	146	2	no	—
142	#423	—	73	2	no	—
143	#405	—	146	2	no	—
144	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
147	#334	—	73	4	no	—
148	Q₈⋊C₈	—	73	8	no	—
149	#322	C₂×D₇₃	1	2	yes	—
150	#322	C₂×D₇₃	1	2	yes	—
151	C₂×Q₁₆	—	73	4	no	—
152	C₂×M₁₆	—	146	2	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	C₄×D₇₃	1	2	yes	—
155	C₂²×C₈	D₂₉₂	1	2	yes	—
156	C₄⋊Q₈	—	73	2	no	—
157	C₄⋊Q₈	—	73	2	no	—
158	#125	—	146	2	no	—
159	C₂²⋊Q₈	—	73	4	no	—
160	C₂²⋊Q₈	—	146	2	no	—
161	C₄×Q₈	—	73	4	no	—
162	C₄×Q₈	—	146	2	no	—
163	#117	—	146	2	no	—
164	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
165	#107	—	73	4	no	—
166	C₄⋊C₈	—	73	4	no	—
167	C₄⋊C₈	—	146	4	no	—
168	C₄⋊C₈	—	146	4	no	—
169	C₄⋊C₈	—	146	2	no	—
170	#103	—	73	8	no	—
171	#97	—	146	4	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	—	146	4	no	—
174	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
175	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
176	C₄×C₈	D₂₉₂	1	2	yes	—
177	C₄×C₈	C₂²×D₇₃	1	2	yes	—
178	C₄×C₈	C₂²×D₇₃	1	2	yes	—
179	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
181	C₂×Q₈	—	73	4	no	—
182	C₂×Q₈	—	146	2	no	—
183	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
185	C₂²×C₄	#11924028	1	1	yes	characteristic
186	Q₁₆	—	146	8	no	—
187	M₁₆	—	292	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	146	4	no	—
193	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
194	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
195	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
196	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
197	C₄⋊C₄	—	73	4	no	—
198	C₄⋊C₄	—	146	4	no	—
199	C₄⋊C₄	—	146	2	no	—
200	C₄⋊C₄	—	146	2	no	—
201	C₄⋊C₄	—	146	2	no	—
202	C₄⋊C₄	—	146	4	no	—
203	C₂²⋊C₄	—	146	4	no	—
204	C₄²	—	146	4	no	—
205	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
206	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
207	C₄²	#11924028	1	1	yes	characteristic
208	C₄²	C₂³×D₇₃	1	1	yes	characteristic
209	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
210	Q₈	—	292	4	no	—
211	Q₈	—	146	8	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	146	4	no	—
215	C₂×C₄	—	146	4	no	—
216	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
217	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
218	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
219	C₂×C₄	#14556624	1	4	yes	—
220	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
222	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	4	2	no	—
227	C₈	—	292	4	no	—
228	C₈	#14556597	1	4	yes	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#17988695	1	2	yes	—
232	C₂²	C₂×C₄×D₂₉₂	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	292	4	no	—
236	C₄	—	292	4	no	—
237	C₄	#17988695	1	2	yes	—
238	C₄	#17988710	1	2	yes	—
239	C₄	C₂×#14556597	1	2	yes	—
240	C₄	C₂×#14556624	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	#22618644	1	1	yes	characteristic
243	C₂	C₂×#17988695	1	1	yes	characteristic
244	C₂	C₂×#17988710	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×#322×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₂×#322	D₇₃
hypocenter	C₇₃	#17961
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #22608841	C₂	semidirect
4. #22608751	C₂	semidirect
130. C₇₃	#17961	semidirect

Maximal subgroups

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

subgroup	index
#22620517	2
#22618865	2
#22608841	2
#22608751	2
#22608687	2
#22608387	2
C₂×#322×C₇₃	2
#17961	73