Subgroups — #34033284

Order 18688.

502 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#10596842	C₂	1	1	yes	maximal, characteristic
2	C₄²×#10596842	C₂	1	1	yes	maximal, characteristic
3	C₂×#17989671	C₂	1	2	yes	maximal
4	(C₂×#10596842)⋊C₈	C₂	1	2	yes	maximal
5	C₄⋊C₈×#64408	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596842	C₄	1	2	yes	—
7	C₂×C₄×#10596842	C₄	1	2	yes	—
8	C₂×C₄×#10596842	C₄	1	2	yes	—
9	C₂×C₄×#10596842	C₂²	1	1	yes	characteristic
10	C₂×C₄×#10596842	C₂²	1	2	yes	—
11	C₂×C₄×#10596842	C₂²	1	1	yes	characteristic
12	(C₂×#10596842)⋊C₄	C₄	1	2	yes	—
13	#17989671	—	2	4	no	—
14	#17989588	C₄	1	1	yes	characteristic
15	C₂×C₈×#64408	C₂²	1	2	yes	—
16	C₄²×#64408	C₂²	1	1	yes	characteristic
17	#17989452	C₄	1	2	yes	—
18	C₄⋊C₈×D₇₃	C₄	1	1	yes	characteristic
19	C₂²×#10596842	C₂×C₄	1	4	yes	—
20	C₂²×#10596842	C₂×C₄	1	2	yes	—
21	C₂²×#10596842	C₂×C₄	1	2	yes	—
22	C₄⋊#10596842	—	2	4	no	—
23	C₄×#10596842	—	2	4	no	—
24	C₄×#10596842	—	2	4	no	—
25	C₄×#10596842	—	2	2	no	—
26	C₄×#10596842	—	2	1	no	—
27	C₄×#10596842	—	2	1	no	—
28	C₄×#10596842	—	2	2	no	—
29	C₄×#10596842	D₄	1	2	yes	—
30	C₄×#10596842	Q₈	1	2	yes	—
31	C₂×#11924036	C₂×C₄	1	2	yes	—
32	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×#64408	C₂³	1	1	yes	characteristic
38	#14556658	C₈	1	2	yes	complemented
39	C₄⋊#10596833	C₈	1	2	yes	complemented
40	C₈×#64408	—	2	2	no	—
41	C₈×#64408	—	2	2	no	—
42	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
43	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
44	C₄⋊#10596831	C₈	1	1	yes	characteristic, complemented
45	#14556520	C₈	1	2	yes	complemented
46	C₄⋊C₈×C₇₃	C₈	1	1	yes	characteristic, complemented
47	C₂×#10596842	—	2	2	no	—
48	C₂×#10596842	—	2	2	no	—
49	C₂×#10596842	—	2	2	no	—
50	C₂×#10596842	—	2	2	no	—
51	C₂×#10596842	—	2	4	no	—
52	C₂×#10596842	—	2	4	no	—
53	C₂×#10596842	—	2	4	no	—
54	C₂×#10596842	C₄⋊C₄	1	4	yes	—
55	C₂×#10596842	M₁₆	1	4	yes	—
56	C₂²×#64408	C₄²	1	1	yes	characteristic
57	C₂²×#64408	C₄²	1	1	yes	characteristic
58	C₂²×#64408	C₂²×C₄	1	1	yes	characteristic
59	C₂²×#64408	C₂²×C₄	1	1	yes	characteristic
60	#11924036	—	2	2	no	—
61	#11924036	—	2	2	no	—
62	C₂×#10596833	C₂×C₈	1	4	yes	—
63	C₄×#64408	—	2	2	no	—
64	C₄×#64408	—	2	2	no	—
65	C₄×#64408	—	2	1	no	—
66	C₄×#64408	—	2	1	no	—
67	C₄×#64408	M₁₆	1	4	yes	—
68	C₄×#64408	M₁₆	1	4	yes	—
69	C₄×#64408	C₂×D₄	1	1	yes	characteristic
70	C₄×#64408	C₂×Q₈	1	1	yes	characteristic
71	C₄×#64408	#41	1	1	yes	characteristic
72	C₄×#64408	#41	1	1	yes	characteristic
73	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
74	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
75	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×#10596831	C₂×C₈	1	2	yes	—
77	C₈×D₇₃	—	2	2	no	—
78	C₈×D₇₃	—	2	2	no	—
79	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
80	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
81	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
82	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
83	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
84	C₂²×D₇₃	C₂×C₄²	1	1	yes	characteristic
85	C₂×#64408	—	2	1	no	—
86	C₂×#64408	—	2	1	no	—
87	C₂×#64408	—	2	1	no	—
88	C₂×#64408	—	2	1	no	—
89	C₂×#64408	#97	1	2	yes	—
90	C₂×#64408	#97	1	2	yes	—
91	C₂×#64408	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×#64408	#117	1	1	yes	characteristic
93	C₂×#64408	C₄×D₄	1	1	yes	characteristic
94	C₂×#64408	C₄×Q₈	1	1	yes	characteristic
95	C₂×#64408	C₂×M₁₆	1	2	yes	—
96	C₂×#64408	C₂×M₁₆	1	2	yes	—
97	C₂×#64408	C₂×M₁₆	1	2	yes	—
98	C₂×#64408	#131	1	2	yes	—
99	#10596842	—	2	4	no	—
100	#10596842	—	2	4	no	—
101	#10596842	—	4	4	no	—
102	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
106	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
107	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
108	C₄×D₇₃	—	2	1	no	—
109	C₄×D₇₃	—	2	1	no	—
110	C₄×D₇₃	#97	1	2	yes	—
111	C₄×D₇₃	#97	1	2	yes	—
112	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
113	C₄×D₇₃	C₄×Q₈	1	1	yes	characteristic
114	C₄×D₇₃	#131	1	1	yes	characteristic
115	C₄×D₇₃	#131	1	1	yes	characteristic
116	#10596833	—	2	4	no	—
117	C₈×C₇₃	—	2	2	no	—
118	#10596831	—	2	2	no	—
119	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
120	C₂×D₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₂×D₇₃	C₂×#97	1	1	yes	characteristic
122	C₂×D₇₃	C₂×#97	1	1	yes	characteristic
123	C₂×D₇₃	C₄×M₁₆	1	1	yes	characteristic
124	C₂×D₇₃	#405	1	1	yes	characteristic
125	C₂×D₇₃	#405	1	1	yes	characteristic
126	#64408	—	2	2	no	—
127	#64408	—	2	2	no	—
128	#64408	#423	1	2	yes	—
129	#64408	#432	1	2	yes	—
130	#64408	#436	1	2	yes	—
131	#64408	#446	1	2	yes	—
132	C₄×C₇₃	—	2	1	no	—
133	C₄×C₇₃	#322	1	2	yes	complemented
134	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
135	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
136	Dic₇₃	—	2	1	no	—
137	Dic₇₃	#322	1	2	yes	complemented
138	Dic₇₃	C₈×D₄	1	1	yes	characteristic
139	Dic₇₃	C₈×Q₈	1	1	yes	characteristic
140	#8261	—	73	1	no	maximal, Sylow
141	C₂×C₇₃	C₂×#322	1	1	yes	characteristic
142	C₂×C₇₃	C₈×M₁₆	1	1	yes	characteristic
143	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
144	D₇₃	#1451	1	1	yes	characteristic
145	D₇₃	#1452	1	1	yes	characteristic
146	D₇₃	#1767	1	1	yes	characteristic
147	D₇₃	#1770	1	1	yes	characteristic
148	C₄×C₄⋊C₈	—	73	1	no	—
149	C₄×C₄⋊C₈	—	73	1	no	—
150	C₄²×C₈	—	73	1	no	—
151	C₂×#322	—	73	2	no	—
152	#1273	—	73	2	no	—
153	C₇₃	#8261	1	1	yes	characteristic, complemented, Sylow
154	C₂×C₄⋊C₈	—	73	1	no	—
155	C₂×C₄⋊C₈	—	73	1	no	—
156	C₂×C₄⋊C₈	—	73	2	no	—
157	C₂×C₄×C₈	—	73	1	no	—
158	C₂×C₄×C₈	—	73	2	no	—
159	C₂×C₄×C₈	—	73	2	no	—
160	C₂×C₄×C₈	—	73	2	no	—
161	C₂×C₄×C₈	—	73	1	no	—
162	C₂×C₄×C₈	—	73	2	no	—
163	C₄³	—	73	1	no	—
164	#336	—	73	2	no	—
165	#336	—	73	2	no	—
166	#322	—	146	4	no	—
167	C₂²×C₈	—	73	2	no	—
168	C₂²×C₈	—	73	2	no	—
169	C₂²×C₈	—	73	2	no	—
170	C₂²×C₈	—	73	2	no	—
171	C₂²×C₈	—	73	4	no	—
172	C₂×C₄²	—	73	1	no	—
173	C₂×C₄²	—	73	1	no	—
174	C₂×C₄²	—	73	1	no	—
175	C₂×C₄²	—	73	1	no	—
176	C₂×C₄²	—	73	1	no	—
177	C₂×C₄²	—	73	1	no	—
178	C₂×C₄²	—	73	1	no	—
179	C₄⋊C₈	—	73	2	no	—
180	C₄⋊C₈	—	73	2	no	—
181	C₄⋊C₈	—	73	2	no	—
182	C₄⋊C₈	—	73	1	no	—
183	C₄⋊C₈	—	146	4	no	—
184	C₄⋊C₈	#10596842	1	1	yes	characteristic, complemented
185	C₄×C₈	—	73	2	no	—
186	C₄×C₈	—	73	2	no	—
187	C₄×C₈	—	146	2	no	—
188	C₄×C₈	—	146	4	no	—
189	C₄×C₈	—	146	2	no	—
190	C₄×C₈	—	146	1	no	—
191	C₄×C₈	—	146	1	no	—
192	C₄×C₈	—	146	2	no	—
193	C₄×C₈	—	146	2	no	—
194	C₄×C₈	—	146	4	no	—
195	C₂²×C₄	—	73	1	no	—
196	C₂²×C₄	—	73	1	no	—
197	C₂²×C₄	—	73	1	no	—
198	C₂²×C₄	—	73	1	no	—
199	C₂²×C₄	—	73	1	no	—
200	C₂²×C₄	—	73	1	no	—
201	C₂²×C₄	—	73	1	no	—
202	C₂×C₈	—	73	4	no	—
203	C₂×C₈	—	73	4	no	—
204	C₂×C₈	—	73	4	no	—
205	C₂×C₈	—	73	2	no	—
206	C₂×C₈	—	146	2	no	—
207	C₂×C₈	—	146	2	no	—
208	C₂×C₈	—	146	4	no	—
209	C₂×C₈	—	146	4	no	—
210	C₂×C₈	—	146	4	no	—
211	C₂×C₈	—	146	2	no	—
212	C₂×C₈	—	146	2	no	—
213	C₂×C₈	—	146	2	no	—
214	C₂×C₈	—	146	2	no	—
215	C₂×C₈	—	146	2	no	—
216	C₂×C₈	—	146	2	no	—
217	C₂×C₈	C₂×#10596842	1	2	yes	—
218	C₄²	—	73	4	no	—
219	C₄²	—	73	4	no	—
220	C₄²	—	73	1	no	—
221	C₄²	—	73	1	no	—
222	C₄²	—	73	1	no	—
223	C₄²	—	73	1	no	—
224	C₄²	—	73	1	no	—
225	C₄²	—	73	1	no	—
226	C₄²	—	73	1	no	—
227	C₄²	—	146	2	no	—
228	C₄²	—	146	2	no	—
229	C₄²	—	146	1	no	—
230	C₄²	—	146	1	no	—
231	C₄²	C₂×#10596842	1	1	yes	characteristic
232	C₂³	—	73	1	no	—
233	C₂×C₄	—	73	1	no	—
234	C₂×C₄	—	73	1	no	—
235	C₂×C₄	—	73	1	no	—
236	C₂×C₄	—	73	2	no	—
237	C₂×C₄	—	73	2	no	—
238	C₂×C₄	—	73	1	no	—
239	C₂×C₄	—	73	2	no	—
240	C₂×C₄	—	73	2	no	—
241	C₂×C₄	—	73	1	no	—
242	C₂×C₄	—	73	1	no	—
243	C₂×C₄	—	73	2	no	—
244	C₂×C₄	—	73	2	no	—
245	C₂×C₄	—	73	1	no	—
246	C₂×C₄	—	73	1	no	—
247	C₂×C₄	—	73	1	no	—
248	C₂×C₄	—	73	1	no	—
249	C₂×C₄	—	73	2	no	—
250	C₂×C₄	—	73	2	no	—
251	C₂×C₄	—	73	1	no	—
252	C₂×C₄	—	146	1	no	—
253	C₂×C₄	—	146	1	no	—
254	C₂×C₄	—	146	1	no	—
255	C₂×C₄	—	146	1	no	—
256	C₂×C₄	—	146	1	no	—
257	C₂×C₄	—	146	1	no	—
258	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
259	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
260	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	292	4	no	—
263	C₈	—	146	4	no	—
264	C₈	—	146	4	no	—
265	C₈	—	146	4	no	—
266	C₈	—	146	2	no	—
267	C₂²	—	73	1	no	—
268	C₂²	—	73	1	no	—
269	C₂²	—	73	1	no	—
270	C₂²	—	73	1	no	—
271	C₂²	—	73	1	no	—
272	C₂²	—	73	1	no	—
273	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	73	2	no	—
276	C₄	—	73	2	no	—
277	C₄	—	73	2	no	—
278	C₄	—	73	2	no	—
279	C₄	—	73	2	no	—
280	C₄	—	73	1	no	—
281	C₄	—	73	1	no	—
282	C₄	—	146	2	no	—
283	C₄	—	146	2	no	—
284	C₄	—	146	1	no	—
285	C₄	#17989671	1	2	yes	complemented
286	C₄	D₄×#10596842	1	1	yes	characteristic
287	C₄	Q₈×#10596842	1	1	yes	characteristic
288	C₂	—	73	1	no	—
289	C₂	—	73	1	no	—
290	C₂	—	73	1	no	—
291	C₂	—	73	1	no	—
292	C₂	C₂×#17989671	1	1	yes	characteristic
293	C₂	M₁₆×#10596842	1	1	yes	characteristic
294	C₂	C₄⋊C₄×#10596842	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₄⋊C₈×C₇₃	C₈
prime radical	C₂²×D₇₃	C₂×C₄²
cosocle	C₂×C₄×#64408	C₂³
hypercenter	C₄⋊C₈	#10596842
hypocenter	C₇₃	#8261
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
38. #14556658	C₈	semidirect
39. C₄⋊#10596833	C₈	semidirect
44. C₄⋊#10596831	C₈	semidirect
45. #14556520	C₈	semidirect
46. C₄⋊C₈×C₇₃	C₈	semidirect
133. C₄×C₇₃	#322	semidirect
137. Dic₇₃	#322	semidirect
153. C₇₃	#8261	semidirect
184. C₄⋊C₈	#10596842	semidirect
285. C₄	#17989671	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#10596842	2
C₄²×#10596842	2
C₂×#17989671	2
(C₂×#10596842)⋊C₈	2
C₄⋊C₈×#64408	2
#8261	73