Subgroups — #32977206

Order 18688.

394 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22609786	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#22609676	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988340	C₂	1	1	yes	maximal, characteristic
5	#22607765	C₂	1	1	yes	maximal, characteristic
6	#22607765	C₂	1	1	yes	maximal, characteristic
7	C₂×#334×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
10	C₂×#14556501	—	2	1	no	—
11	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	—	2	1	no	—
13	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
14	#17988341	—	2	1	no	—
15	#17988341	—	2	1	no	—
16	#17988341	C₄	1	2	yes	—
17	#17988341	C₄	1	2	yes	—
18	#17988340	C₄	1	4	yes	complemented
19	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	#334×C₇₃	C₄	1	4	yes	complemented
23	C₂²×#10596831	—	2	1	no	—
24	C₂²×#10596831	—	2	1	no	—
25	C₂²×#10596831	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
28	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
29	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
30	C₄⋊#10596831	—	2	1	no	—
31	C₄⋊#10596831	—	2	1	no	—
32	C₄⋊#10596831	C₂×C₄	1	2	yes	—
33	C₄⋊#10596831	C₂×C₄	1	2	yes	—
34	#14556501	—	4	2	no	—
35	C₄×#10596831	—	2	1	no	—
36	C₄×#10596831	—	2	1	no	—
37	C₈×Dic₇₃	—	4	2	no	—
38	C₂²×C₈×C₇₃	—	2	1	no	—
39	C₂²×C₈×C₇₃	—	2	1	no	—
40	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₇₃	—	2	1	no	—
44	C₄⋊C₈×C₇₃	—	2	1	no	—
45	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
46	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
49	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
50	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
51	C₂×#10596831	—	2	2	no	—
52	C₂×#10596831	—	2	2	no	—
53	C₂×#10596831	—	2	2	no	—
54	C₂×#10596831	—	2	2	no	—
55	C₂×#10596831	—	2	2	no	—
56	C₂×#10596831	—	4	1	no	—
57	C₂×#10596831	—	4	1	no	—
58	C₂×#10596831	—	4	1	no	—
59	C₂×#10596831	—	4	1	no	—
60	C₄×Dic₇₃	—	2	2	no	—
61	C₄×Dic₇₃	—	2	2	no	—
62	C₄×Dic₇₃	—	2	1	no	—
63	C₄×Dic₇₃	—	2	1	no	—
64	C₄×Dic₇₃	—	2	1	no	—
65	C₄×Dic₇₃	—	2	1	no	—
66	C₄×Dic₇₃	—	2	4	no	—
67	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈×C₇₃	—	2	2	no	—
71	C₂×C₈×C₇₃	—	2	2	no	—
72	C₂×C₈×C₇₃	—	2	2	no	—
73	C₂×C₈×C₇₃	—	2	2	no	—
74	C₂×C₈×C₇₃	—	4	1	no	—
75	C₂×C₈×C₇₃	—	4	1	no	—
76	C₂×C₈×C₇₃	—	4	1	no	—
77	C₂×C₈×C₇₃	—	4	1	no	—
78	C₂×C₈×C₇₃	C₄²	1	2	yes	—
79	C₂×C₈×C₇₃	C₄²	1	2	yes	—
80	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₇₃	#41	1	1	yes	characteristic
83	C₄²×C₇₃	#41	1	1	yes	characteristic
84	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₇₃	—	2	2	no	—
86	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
88	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₇₃	#131	1	2	yes	—
93	C₂×C₄×C₇₃	#131	1	2	yes	—
94	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
97	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
98	C₂×Dic₇₃	—	2	2	no	—
99	C₂×Dic₇₃	—	2	2	no	—
100	C₂×Dic₇₃	—	2	2	no	—
101	C₂×Dic₇₃	—	2	2	no	—
102	C₂×Dic₇₃	—	2	2	no	—
103	C₂×Dic₇₃	#108	1	4	yes	—
104	C₈×C₇₃	—	2	2	no	—
105	C₈×C₇₃	—	2	2	no	—
106	C₈×C₇₃	—	4	2	no	—
107	C₈×C₇₃	—	4	2	no	—
108	#10596831	—	4	2	no	—
109	#10596831	—	4	2	no	—
110	#10596831	—	4	2	no	—
111	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₇₃	#424	1	2	yes	—
113	C₂²×C₇₃	#428	1	1	yes	characteristic
114	C₂²×C₇₃	#428	1	1	yes	characteristic
115	C₂²×C₇₃	C₂×#108	1	2	yes	—
116	C₄×C₇₃	—	2	1	no	—
117	C₄×C₇₃	—	2	1	no	—
118	C₄×C₇₃	#405	1	1	yes	characteristic
119	C₄×C₇₃	#405	1	1	yes	characteristic
120	C₄×C₇₃	#433	1	2	yes	—
121	C₄×C₇₃	#440	1	1	yes	characteristic
122	C₄×C₇₃	#440	1	1	yes	characteristic
123	C₄×C₇₃	#441	1	1	yes	characteristic
124	C₄×C₇₃	#442	1	1	yes	characteristic
125	Dic₇₃	—	2	4	no	—
126	Dic₇₃	—	4	2	no	—
127	#8279	—	73	1	no	maximal, Sylow
128	C₂×C₇₃	#1573	1	2	yes	—
129	C₂×C₇₃	#1574	1	2	yes	—
130	C₂×C₇₃	#1768	1	1	yes	characteristic
131	C₂×C₇₃	#1776	1	1	yes	characteristic
132	C₂×C₇₃	C₄×#108	1	1	yes	characteristic
133	#1767	—	73	1	no	—
134	C₄×C₄⋊C₈	—	73	1	no	—
135	#1744	—	73	1	no	—
136	C₂×#335	—	73	1	no	—
137	C₂×#334	D₇₃	1	1	yes	characteristic
138	#1277	—	73	1	no	—
139	#1277	—	73	1	no	—
140	C₇₃	#8279	1	1	yes	characteristic, complemented, Sylow
141	C₂×C₄⋊C₈	—	73	1	no	—
142	C₂×C₄⋊C₈	—	73	1	no	—
143	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
144	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
145	C₂×#97	—	146	1	no	—
146	C₂×C₄×C₈	—	73	1	no	—
147	C₂×C₄×C₈	—	146	1	no	—
148	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
149	C₄³	—	73	1	no	—
150	#336	—	73	2	no	—
151	#336	—	73	2	no	—
152	#336	—	146	1	no	—
153	#336	—	146	1	no	—
154	#335	—	73	4	no	—
155	#334	Dic₇₃	1	4	yes	complemented
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	73	2	no	—
159	C₂²×C₈	—	146	1	no	—
160	C₂²×C₈	—	146	1	no	—
161	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
162	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
163	C₂×C₄²	—	73	2	no	—
164	C₂×C₄²	—	73	1	no	—
165	C₂×C₄²	—	73	1	no	—
166	C₂×C₄²	—	73	2	no	—
167	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
168	C₄⋊C₈	—	2	1	no	—
169	C₄⋊C₈	—	2	1	no	—
170	C₄⋊C₈	—	73	2	no	—
171	C₄⋊C₈	—	73	2	no	—
172	C₄⋊C₈	—	146	1	no	—
173	C₄⋊C₈	—	146	1	no	—
174	C₄⋊C₈	C₂×Dic₇₃	1	2	yes	—
175	C₄⋊C₈	C₂×Dic₇₃	1	2	yes	—
176	#97	—	292	2	no	—
177	C₄×C₈	—	292	2	no	—
178	C₄×C₈	—	146	1	no	—
179	C₄×C₈	—	146	1	no	—
180	C₄×C₈	C₄×D₇₃	1	2	yes	—
181	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
182	C₂²×C₄	—	73	2	no	—
183	C₂²×C₄	—	73	2	no	—
184	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
185	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
186	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	292	1	no	—
196	C₂×C₈	—	292	1	no	—
197	C₂×C₈	—	292	1	no	—
198	C₂×C₈	—	292	1	no	—
199	C₂×C₈	—	146	2	no	—
200	C₂×C₈	—	146	2	no	—
201	C₂×C₈	—	146	2	no	—
202	C₂×C₈	—	146	2	no	—
203	C₂×C₈	—	146	2	no	—
204	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
205	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
206	C₄²	—	146	4	no	—
207	C₄²	—	146	2	no	—
208	C₄²	—	146	2	no	—
209	C₄²	—	146	1	no	—
210	C₄²	—	146	1	no	—
211	C₄²	—	146	1	no	—
212	C₄²	—	146	1	no	—
213	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
214	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
215	C₄²	#11924027	1	1	yes	characteristic
216	C₄²	#11924029	1	1	yes	characteristic
217	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	73	4	no	—
220	C₂×C₄	—	146	2	no	—
221	C₂×C₄	—	146	2	no	—
222	C₂×C₄	—	146	2	no	—
223	C₂×C₄	—	146	2	no	—
224	C₂×C₄	—	146	2	no	—
225	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
226	C₂×C₄	#14556549	1	1	yes	characteristic
227	C₂×C₄	#14556553	1	1	yes	characteristic
228	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
229	C₂×C₄	#14556556	1	1	yes	characteristic
230	C₂×C₄	Q₈×Dic₇₃	1	1	yes	characteristic
231	C₂×C₄	#14556597	1	2	yes	—
232	C₂×C₄	#14556599	1	2	yes	—
233	C₂×C₄	#14556618	1	1	yes	characteristic
234	C₂×C₄	#14556618	1	1	yes	characteristic
235	C₂×C₄	#14556620	1	1	yes	characteristic
236	C₂×C₄	#14556620	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	2	2	no	—
239	C₈	—	4	2	no	—
240	C₈	—	4	2	no	—
241	C₈	—	292	2	no	—
242	C₈	—	292	2	no	—
243	C₈	—	292	2	no	—
244	C₂²	C₄⋊C₄×Dic₇₃	1	1	yes	characteristic
245	C₂²	#17988612	1	2	yes	—
246	C₂²	#17988626	1	1	yes	characteristic
247	C₂²	#17988626	1	1	yes	characteristic
248	C₂²	#108×D₇₃	1	2	yes	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	292	2	no	—
252	C₄	—	146	4	no	—
253	C₄	#17988542	1	1	yes	characteristic
254	C₄	#17988545	1	1	yes	characteristic
255	C₄	#17988647	1	2	yes	—
256	C₄	#17988688	1	1	yes	characteristic
257	C₄	#17988688	1	1	yes	characteristic
258	C₄	#17988693	1	1	yes	characteristic
259	C₄	#17988698	1	1	yes	characteristic
260	C₂	#22608713	1	2	yes	—
261	C₂	#22608718	1	2	yes	—
262	C₂	#22609790	1	1	yes	characteristic
263	C₂	#22609831	1	1	yes	characteristic
264	C₂	#108×Dic₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#334×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#334	D₇₃
hypocenter	C₇₃	#8279
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
18. #17988340	C₄	semidirect
22. #334×C₇₃	C₄	semidirect
140. C₇₃	#8279	semidirect
155. #334	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22609786	2
C₄⋊C₈×Dic₇₃	2
#22609676	2
C₂×#17988340	2
#22607765	2
C₂×#334×C₇₃	2
#8279	73