Subgroups — #33073941

Order 18688.

750 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22621084	C₂	1	2	yes	maximal
2	#22620517	C₂	1	1	yes	maximal, characteristic
3	#22620477	C₂	1	2	yes	maximal
4	C₂×#17988967	C₂	1	1	yes	maximal, characteristic
5	#22609608	C₂	1	4	yes	maximal, complemented
6	#22609549	C₂	1	2	yes	maximal, complemented
7	#22609528	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₉₂	—	2	2	no	—
10	#17989144	C₂²	1	2	yes	—
11	C₂×#14556591	C₂²	1	2	yes	—
12	#17988996	—	2	4	no	—
13	#17988992	C₂²	1	4	yes	—
14	#17988992	C₂²	1	1	yes	characteristic
15	#17988992	C₂²	1	4	yes	—
16	#17988985	C₂²	1	4	yes	—
17	#17988967	—	2	2	no	—
18	#17988967	C₂²	1	1	yes	characteristic
19	#17988967	C₂²	1	2	yes	—
20	#17988967	C₂²	1	2	yes	—
21	#17988938	C₂²	1	2	yes	—
22	#17988938	C₂²	1	4	yes	—
23	#17988923	—	2	4	no	—
24	#17988883	—	2	4	no	—
25	#17988828	—	2	2	no	—
26	C₄×Dic₂₉₂	—	2	2	no	—
27	#17988629	—	2	4	no	—
28	#431×C₇₃	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
31	#322×C₇₃	C₂²	1	2	yes	—
32	C₈²×C₇₃	C₂²	1	2	yes	—
33	C₂²×Dic₁₄₆	—	2	2	no	—
34	C₂×Dic₂₉₂	—	2	2	no	—
35	C₂×Dic₂₉₂	—	2	4	no	—
36	C₂×Dic₂₉₂	—	2	4	no	—
37	C₂×Dic₂₉₂	—	4	8	no	—
38	#14556591	—	2	2	no	—
39	#14556591	—	2	2	no	—
40	#14556591	—	2	2	no	—
41	#14556591	C₂³	1	4	yes	—
42	#14556591	C₂³	1	2	yes	—
43	#14556591	C₂³	1	2	yes	—
44	#14556583	—	2	2	no	—
45	#14556570	—	2	2	no	—
46	#14556570	—	2	2	no	—
47	C₄×Dic₁₄₆	—	2	2	no	—
48	C₄×Dic₁₄₆	—	2	2	no	—
49	#14556550	—	2	2	no	—
50	C₂×C₄⋊Dic₇₃	—	2	2	no	—
51	C₂×C₄⋊Dic₇₃	—	2	2	no	—
52	#14556525	—	2	8	no	—
53	#14556525	—	2	8	no	—
54	#14556525	—	2	2	no	—
55	#14556523	—	2	8	no	—
56	#14556516	—	2	8	no	—
57	#14556516	—	2	4	no	—
58	C₂×M₁₆×C₇₃	D₄	1	2	yes	—
59	C₂²×C₈×C₇₃	D₄	1	2	yes	—
60	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
62	C₂²⋊C₈×C₇₃	D₄	1	2	yes	—
63	#97×C₇₃	D₄	1	2	yes	—
64	C₄×C₈×C₇₃	D₄	1	2	yes	—
65	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₇₃	C₂³	1	2	yes	—
68	C₄×C₈×C₇₃	C₂³	1	2	yes	—
69	C₂×Dic₁₄₆	—	2	2	no	—
70	C₂×Dic₁₄₆	—	2	2	no	—
71	C₂×Dic₁₄₆	—	2	4	no	—
72	C₂×Dic₁₄₆	—	4	8	no	—
73	C₂×Dic₁₄₆	—	4	2	no	—
74	C₂²×Dic₇₃	—	4	2	no	—
75	Dic₂₉₂	—	4	8	no	—
76	Dic₂₉₂	—	4	4	no	—
77	Dic₂₉₂	—	4	8	no	—
78	C₄⋊Dic₇₃	—	2	4	no	—
79	C₄⋊Dic₇₃	—	2	2	no	—
80	C₄⋊Dic₇₃	—	2	2	no	—
81	C₄⋊Dic₇₃	—	2	4	no	—
82	C₄⋊Dic₇₃	—	2	2	no	—
83	C₄⋊Dic₇₃	—	2	2	no	—
84	C₄⋊Dic₇₃	—	4	2	no	—
85	C₄⋊Dic₇₃	—	4	2	no	—
86	#11924007	—	4	2	no	—
87	#11924007	—	4	2	no	—
88	C₂²⋊Dic₇₃	—	4	2	no	—
89	C₄×Dic₇₃	—	4	2	no	—
90	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
93	M₁₆×C₇₃	Q₁₆	1	8	yes	complemented
94	C₂×C₈×C₇₃	—	2	4	no	—
95	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
97	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
98	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
99	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
103	C₂³×C₇₃	#127	1	1	yes	characteristic
104	C₂×C₄×C₇₃	—	2	2	no	—
105	C₂×C₄×C₇₃	—	2	2	no	—
106	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
107	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
108	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂×Q₁₆	1	4	yes	—
110	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×Dic₇₃	—	4	2	no	—
114	C₂×Dic₇₃	—	4	2	no	—
115	C₂×Dic₇₃	—	4	4	no	—
116	C₂×Dic₇₃	—	8	4	no	—
117	Dic₁₄₆	—	4	8	no	—
118	Dic₁₄₆	—	4	4	no	—
119	Dic₁₄₆	—	4	4	no	—
120	Dic₁₄₆	—	8	4	no	—
121	C₈×C₇₃	—	2	2	no	—
122	C₈×C₇₃	—	2	4	no	—
123	C₈×C₇₃	C₂×Q₁₆	1	8	yes	complemented
124	C₈×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
125	C₂²×C₇₃	—	2	1	no	—
126	C₂²×C₇₃	—	2	1	no	—
127	C₂²×C₇₃	#494	1	2	yes	—
128	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
129	C₄×C₇₃	—	2	1	no	—
130	C₄×C₇₃	—	2	1	no	—
131	C₄×C₇₃	#494	1	2	yes	—
132	C₄×C₇₃	C₂²×Q₁₆	1	2	yes	—
133	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
134	C₄×C₇₃	#578	1	2	yes	—
135	Dic₇₃	—	8	4	no	—
136	Dic₇₃	—	8	2	no	—
137	Dic₇₃	—	8	2	no	—
138	#19632	—	73	1	no	maximal, Sylow
139	C₂×C₇₃	—	2	2	no	—
140	C₂×C₇₃	C₂×#494	1	1	yes	characteristic
141	C₂×C₇₃	#3151	1	1	yes	characteristic
142	C₂×C₇₃	#3156	1	1	yes	characteristic
143	#3230	—	73	2	no	—
144	#3170	—	73	1	no	—
145	#3165	—	73	2	no	—
146	C₂×#494	—	73	1	no	—
147	#1723	—	73	4	no	—
148	#1713	—	73	2	no	—
149	#1707	—	73	2	no	—
150	C₈×M₁₆	D₇₃	1	1	yes	characteristic
151	C₇₃	#19632	1	1	yes	characteristic, complemented, Sylow
152	C₂²×Q₁₆	—	146	2	no	—
153	#533	—	73	2	no	—
154	C₂×C₄⋊Q₈	—	73	2	no	—
155	#501	—	146	4	no	—
156	#500	—	73	4	no	—
157	#500	—	73	1	no	—
158	#500	—	73	4	no	—
159	#498	—	73	4	no	—
160	#494	—	73	2	no	—
161	#494	—	73	2	no	—
162	#494	—	73	1	no	—
163	#494	—	146	2	no	—
164	#487	—	73	2	no	—
165	#487	—	73	4	no	—
166	#484	—	146	4	no	—
167	#477	—	146	4	no	—
168	#467	—	146	2	no	—
169	C₄×Q₁₆	—	146	2	no	—
170	#431	C₂×D₇₃	1	1	yes	characteristic
171	#428	—	146	4	no	—
172	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
174	#322	C₂×D₇₃	1	2	yes	—
175	C₈²	C₂×D₇₃	1	2	yes	—
176	C₂²×Q₈	—	146	2	no	—
177	C₂×Q₁₆	—	292	8	no	—
178	C₂×Q₁₆	—	146	4	no	—
179	C₂×Q₁₆	—	146	4	no	—
180	C₂×Q₁₆	—	146	2	no	—
181	C₂×M₁₆	D₂₉₂	1	2	yes	—
182	C₂²×C₈	D₂₉₂	1	2	yes	—
183	C₄⋊Q₈	—	73	4	no	—
184	C₄⋊Q₈	—	73	2	no	—
185	C₄⋊Q₈	—	73	2	no	—
186	C₄⋊Q₈	—	146	2	no	—
187	C₄⋊Q₈	—	146	2	no	—
188	C₄⋊Q₈	—	146	2	no	—
189	#125	—	146	2	no	—
190	C₂²⋊Q₈	—	146	2	no	—
191	C₂²⋊Q₈	—	146	2	no	—
192	C₄×Q₈	—	146	2	no	—
193	C₄×Q₈	—	146	2	no	—
194	#117	—	146	2	no	—
195	C₂×C₄⋊C₄	—	146	2	no	—
196	C₂×C₄⋊C₄	—	146	2	no	—
197	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
198	#107	—	146	8	no	—
199	#107	—	146	2	no	—
200	#107	—	146	8	no	—
201	#106	—	146	8	no	—
202	C₄⋊C₈	D₂₉₂	1	2	yes	—
203	#103	—	146	4	no	—
204	#103	—	146	8	no	—
205	C₂²⋊C₈	D₂₉₂	1	2	yes	—
206	#97	D₂₉₂	1	2	yes	—
207	C₄×C₈	D₂₉₂	1	2	yes	—
208	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₇₃	1	2	yes	—
210	C₄×C₈	C₂²×D₇₃	1	2	yes	—
211	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
212	C₂×Q₈	—	292	8	no	—
213	C₂×Q₈	—	292	2	no	—
214	C₂×Q₈	—	146	2	no	—
215	C₂×Q₈	—	146	2	no	—
216	C₂×Q₈	—	146	4	no	—
217	C₂²×C₄	—	292	2	no	—
218	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
221	Q₁₆	—	292	8	no	—
222	Q₁₆	—	292	8	no	—
223	Q₁₆	—	292	4	no	—
224	M₁₆	Dic₂₉₂	1	8	yes	complemented
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
227	C₂×C₈	C₂×D₂₉₂	1	4	yes	—
228	C₂×C₈	C₂×D₂₉₂	1	4	yes	—
229	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
230	C₄⋊C₄	—	292	2	no	—
231	C₄⋊C₄	—	292	2	no	—
232	C₄⋊C₄	—	292	2	no	—
233	C₄⋊C₄	—	292	2	no	—
234	C₄⋊C₄	—	146	4	no	—
235	C₄⋊C₄	—	146	2	no	—
236	C₄⋊C₄	—	146	2	no	—
237	C₄⋊C₄	—	146	4	no	—
238	C₄⋊C₄	—	146	2	no	—
239	C₄⋊C₄	—	146	2	no	—
240	C₂²⋊C₄	—	292	2	no	—
241	C₄²	—	292	2	no	—
242	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
243	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
244	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
245	C₄²	C₂³×D₇₃	1	1	yes	characteristic
246	C₂³	#14556588	1	1	yes	characteristic
247	Q₈	—	292	8	no	—
248	Q₈	—	292	4	no	—
249	Q₈	—	292	4	no	—
250	Q₈	—	584	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	292	2	no	—
254	C₂×C₄	—	292	2	no	—
255	C₂×C₄	—	292	4	no	—
256	C₂×C₄	—	584	4	no	—
257	C₂×C₄	#14556588	1	1	yes	characteristic
258	C₂×C₄	#14556588	1	1	yes	characteristic
259	C₂×C₄	#14556588	1	1	yes	characteristic
260	C₂×C₄	C₂×Dic₂₉₂	1	4	yes	—
261	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	2	4	no	—
266	C₈	C₂×Dic₂₉₂	1	8	yes	complemented
267	C₈	#14556619	1	4	yes	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	#17988967	1	2	yes	—
271	C₂²	C₂×#14556588	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	584	2	no	—
275	C₄	—	584	4	no	—
276	C₄	—	584	2	no	—
277	C₄	#17988967	1	2	yes	—
278	C₄	C₂²×Dic₂₉₂	1	2	yes	—
279	C₄	C₂×#14556619	1	2	yes	—
280	C₄	#17989404	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	C₂×#17988967	1	1	yes	characteristic
283	C₂	#22620383	1	1	yes	characteristic
284	C₂	#22620425	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556588
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₈×M₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#127
prime radical	C₂³×C₇₃	#127
hypercenter	C₈×M₁₆	D₇₃
hypocenter	C₇₃	#19632
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
5. #22609608	C₂	semidirect
6. #22609549	C₂	semidirect
7. #22609528	C₂	semidirect
93. M₁₆×C₇₃	Q₁₆	semidirect
123. C₈×C₇₃	C₂×Q₁₆	semidirect
151. C₇₃	#19632	semidirect
224. M₁₆	Dic₂₉₂	semidirect
266. C₈	C₂×Dic₂₉₂	semidirect

Maximal subgroups

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

subgroup	index
#22621084	2
#22620517	2
#22620477	2
C₂×#17988967	2
#22609608	2
#22609549	2
#22609528	2
C₈×M₁₆×C₇₃	2
#19632	73