Subgroups — #34019642

Order 18688.

578 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626727	C₂	1	2	yes	maximal
2	#22625577	C₂	1	2	yes	maximal, complemented
3	#22625555	C₂	1	4	yes	maximal, complemented
4	#22625526	C₂	1	2	yes	maximal, complemented
5	#22625473	C₂	1	1	yes	maximal, characteristic
6	#22625241	C₂	1	2	yes	maximal
7	#22625141	C₂	1	1	yes	maximal, characteristic
8	#22620518	C₂	1	1	yes	maximal, characteristic
9	#17989638	—	2	4	no	—
10	#17989637	C₂²	1	4	yes	—
11	#17989636	C₂²	1	4	yes	—
12	#17989606	C₂²	1	1	yes	characteristic
13	#17989594	—	2	2	no	—
14	#17989588	C₂²	1	2	yes	—
15	#17989552	—	2	2	no	—
16	C₄×#11924038	C₂²	1	1	yes	characteristic
17	#17989449	C₂²	1	2	yes	—
18	#17989448	C₂²	1	1	yes	characteristic
19	#17989448	C₂²	1	1	yes	characteristic
20	#17989437	C₂²	1	4	yes	—
21	#17989437	C₂²	1	4	yes	—
22	#17989429	C₂²	1	2	yes	—
23	#17989428	C₂²	1	2	yes	—
24	#17989142	C₂²	1	2	yes	—
25	#17988967	C₄	1	1	yes	characteristic
26	#17988939	C₄	1	2	yes	—
27	#17988829	C₄	1	2	yes	—
28	#17988829	C₄	1	2	yes	—
29	#17988681	C₂²	1	2	yes	—
30	#17988681	C₂²	1	2	yes	—
31	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
32	#500×C₇₃	C₄	1	1	yes	characteristic
33	C₂×#11924038	D₄	1	2	yes	—
34	C₂×#11924036	—	2	2	no	—
35	#14556658	—	2	4	no	—
36	#14556658	—	2	2	no	—
37	C₄⋊#10596833	—	2	4	no	—
38	C₄⋊#10596833	—	2	1	no	—
39	C₄⋊#10596833	C₂³	1	2	yes	—
40	C₄⋊#10596833	C₂³	1	2	yes	—
41	#14556648	—	2	1	no	—
42	#14556646	—	2	4	no	—
43	#14556646	D₄	1	2	yes	—
44	C₄×#10596833	—	2	4	no	—
45	C₄×#10596833	C₂³	1	2	yes	—
46	C₄×#10596833	C₂³	1	2	yes	—
47	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
48	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
49	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
51	#14556591	C₂×C₄	1	2	yes	—
52	#14556582	—	2	2	no	—
53	#14556571	C₂×C₄	1	4	yes	—
54	#14556567	—	2	2	no	—
55	Q₈×Dic₇₃	—	2	2	no	—
56	#14556556	C₂³	1	4	yes	—
57	#14556549	—	2	2	no	—
58	C₄²×D₇₃	C₂³	1	1	yes	characteristic
59	#14556514	C₂×C₄	1	4	yes	—
60	#14556514	C₂×C₄	1	4	yes	—
61	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
62	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
63	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
65	#107×C₇₃	C₂×C₄	1	2	yes	—
66	#107×C₇₃	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
68	#11924038	—	2	4	no	—
69	#11924036	—	4	4	no	—
70	C₂×#10596833	—	2	2	no	—
71	C₂×#10596833	—	2	2	no	—
72	C₂×#10596833	—	2	2	no	—
73	C₂×#10596833	C₂×D₄	1	4	yes	—
74	C₂×Dic₁₄₆	C₂²×C₄	1	4	yes	—
75	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
78	Dic₂₉₂	—	2	4	no	—
79	Dic₂₉₂	—	4	2	no	—
80	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
81	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
82	C₈×D₇₃	—	2	2	no	—
83	C₈×D₇₃	—	4	1	no	—
84	C₄⋊Dic₇₃	—	2	2	no	—
85	C₄⋊Dic₇₃	—	2	2	no	—
86	C₄⋊Dic₇₃	—	2	4	no	—
87	#11924007	—	2	4	no	—
88	#11924005	—	2	4	no	—
89	C₄×Dic₇₃	—	2	4	no	—
90	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
91	C₄×Dic₇₃	#41	1	1	yes	characteristic
92	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
93	C₂×Q₈×C₇₃	—	2	2	no	—
94	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₄⋊C₄×C₇₃	—	2	2	no	—
97	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
98	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
100	Q₈×C₇₃	—	8	2	no	—
101	C₂×C₄×C₇₃	—	2	4	no	—
102	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×Dic₇₃	—	2	4	no	—
106	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
107	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×Dic₇₃	C₂×#41	1	1	yes	characteristic
109	C₄×D₇₃	—	2	1	no	—
110	C₄×D₇₃	—	2	1	no	—
111	C₄×D₇₃	—	4	1	no	—
112	C₄×D₇₃	#137	1	4	yes	—
113	Dic₁₄₆	—	2	4	no	—
114	Dic₁₄₆	—	2	4	no	—
115	#10596833	—	2	4	no	—
116	#10596833	—	4	2	no	—
117	#10596833	—	4	2	no	—
118	#10596833	—	4	2	no	—
119	C₈×C₇₃	—	2	1	no	—
120	C₈×C₇₃	#131	1	2	yes	—
121	#10596831	—	2	1	no	—
122	#10596831	—	2	1	no	—
123	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂×D₇₃	—	2	1	no	—
125	C₂×D₇₃	—	2	1	no	—
126	C₂×D₇₃	#441	1	2	yes	—
127	C₄×C₇₃	—	2	1	no	—
128	C₄×C₇₃	—	4	4	no	—
129	C₄×C₇₃	#441	1	2	yes	—
130	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
131	C₄×C₇₃	#568	1	1	yes	characteristic
132	Dic₇₃	—	2	1	no	—
133	Dic₇₃	—	2	1	no	—
134	Dic₇₃	—	4	4	no	—
135	Dic₇₃	C₂×#137	1	2	yes	—
136	#18072	—	73	1	no	maximal, Sylow
137	C₂×C₇₃	#2935	1	1	yes	characteristic
138	C₂×C₇₃	C₂×#441	1	1	yes	characteristic
139	C₂×C₇₃	#2957	1	1	yes	characteristic
140	D₇₃	—	2	2	no	—
141	#3170	—	73	1	no	—
142	#2962	—	73	2	no	—
143	#1593	—	73	2	no	—
144	#1587	—	73	4	no	—
145	#1579	—	73	2	no	—
146	C₂×#334	—	73	1	no	—
147	#1481	—	73	2	no	—
148	#1451	—	73	1	no	—
149	C₇₃	#18072	1	1	yes	characteristic, complemented, Sylow
150	#533	—	73	2	no	—
151	#500	#64408	1	1	yes	characteristic
152	#494	—	73	1	no	—
153	#487	—	73	2	no	—
154	#467	—	73	2	no	—
155	#467	—	73	2	no	—
156	#446	—	73	4	no	—
157	#446	—	146	4	no	—
158	C₈×Q₈	—	73	4	no	—
159	C₄×Q₁₆	—	73	2	no	—
160	C₄×Q₁₆	—	73	2	no	—
161	#431	—	73	1	no	—
162	#424	—	146	2	no	—
163	C₂×C₄⋊C₈	—	73	2	no	—
164	#405	—	146	2	no	—
165	C₄×M₁₆	—	73	1	no	—
166	C₂×C₄×C₈	—	73	1	no	—
167	#334	—	73	1	no	—
168	#334	—	73	2	no	—
169	#334	—	73	1	no	—
170	Q₈⋊C₈	—	73	4	no	—
171	Q₈⋊C₈	—	73	4	no	—
172	#322	—	73	2	no	—
173	#322	—	73	2	no	—
174	C₂×Q₁₆	—	73	2	no	—
175	C₂×Q₁₆	—	73	2	no	—
176	C₂×M₁₆	—	73	2	no	—
177	C₂²×C₈	—	73	1	no	—
178	C₂²×C₈	—	73	1	no	—
179	C₂²×C₈	—	146	2	no	—
180	C₄⋊Q₈	—	73	2	no	—
181	C₄⋊Q₈	C₂×#64408	1	2	yes	—
182	#125	—	146	2	no	—
183	C₂²⋊Q₈	—	73	4	no	—
184	C₂²⋊Q₈	—	146	2	no	—
185	C₄×Q₈	—	73	4	no	—
186	C₄×Q₈	—	146	2	no	—
187	#117	—	146	2	no	—
188	C₂×C₄²	—	73	1	no	—
189	#107	C₂×#64408	1	2	yes	—
190	#107	C₂×#64408	1	2	yes	—
191	C₄⋊C₈	—	73	2	no	—
192	C₄⋊C₈	—	73	2	no	—
193	C₄⋊C₈	—	146	2	no	—
194	C₄⋊C₈	—	146	1	no	—
195	C₄⋊C₈	—	146	4	no	—
196	C₄⋊C₈	—	146	4	no	—
197	#103	—	73	4	no	—
198	#103	—	73	4	no	—
199	C₂²⋊C₈	—	146	1	no	—
200	#97	—	73	2	no	—
201	#97	—	146	4	no	—
202	C₄×C₈	—	73	2	no	—
203	C₄×C₈	—	73	1	no	—
204	C₄×C₈	—	73	1	no	—
205	C₄×C₈	—	73	2	no	—
206	C₄×C₈	—	73	1	no	—
207	C₄×C₈	—	146	4	no	—
208	C₄×C₈	C₂×#64408	1	1	yes	characteristic
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	73	4	no	—
211	C₂²×C₄	—	73	1	no	—
212	C₂²×C₄	—	73	1	no	—
213	C₂²×C₄	—	73	1	no	—
214	Q₁₆	—	292	2	no	—
215	Q₁₆	—	146	4	no	—
216	M₁₆	—	146	4	no	—
217	C₂×C₈	—	292	1	no	—
218	C₂×C₈	—	292	4	no	—
219	C₂×C₈	—	73	4	no	—
220	C₂×C₈	—	73	1	no	—
221	C₂×C₈	—	73	1	no	—
222	C₂×C₈	—	146	2	no	—
223	C₂×C₈	—	146	2	no	—
224	C₂×C₈	—	146	2	no	—
225	C₂×C₈	—	146	2	no	—
226	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
227	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	146	4	no	—
230	C₄⋊C₄	—	146	4	no	—
231	C₄⋊C₄	—	146	2	no	—
232	C₄⋊C₄	—	146	2	no	—
233	C₄⋊C₄	C₂²×#64408	1	4	yes	—
234	C₂²⋊C₄	—	146	4	no	—
235	C₄²	—	73	1	no	—
236	C₄²	—	73	1	no	—
237	C₄²	—	73	1	no	—
238	C₄²	—	146	4	no	—
239	C₄²	C₂²×#64408	1	1	yes	characteristic
240	C₂³	—	73	1	no	—
241	Q₈	—	8	2	no	—
242	Q₈	—	146	4	no	—
243	Q₈	—	146	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	292	1	no	—
246	C₂×C₄	—	73	1	no	—
247	C₂×C₄	—	73	4	no	—
248	C₂×C₄	—	73	1	no	—
249	C₂×C₄	—	73	1	no	—
250	C₂×C₄	—	146	4	no	—
251	C₂×C₄	—	146	1	no	—
252	C₂×C₄	—	146	1	no	—
253	C₂×C₄	D₄×#64408	1	1	yes	characteristic
254	C₂×C₄	D₄×#64408	1	1	yes	characteristic
255	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	292	2	no	—
258	C₈	—	292	2	no	—
259	C₈	—	292	2	no	—
260	C₈	—	146	4	no	—
261	C₈	—	146	1	no	—
262	C₈	—	146	1	no	—
263	C₈	#14556679	1	2	yes	—
264	C₂²	—	73	2	no	—
265	C₂²	—	146	1	no	—
266	C₂²	—	146	1	no	—
267	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	4	4	no	—
270	C₄	—	292	4	no	—
271	C₄	—	73	2	no	—
272	C₄	—	146	1	no	—
273	C₄	—	146	1	no	—
274	C₄	#17989627	1	2	yes	—
275	C₄	C₂×#14556679	1	1	yes	characteristic
276	C₄	#17989667	1	1	yes	characteristic
277	C₂	—	146	2	no	—
278	C₂	#22626677	1	1	yes	characteristic
279	C₂	C₂×#17989627	1	1	yes	characteristic
280	C₂	#22626707	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	#500×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂²×D₇₃	C₄×D₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	#500	#64408
hypocenter	C₇₃	#18072
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
2. #22625577	C₂	semidirect
3. #22625555	C₂	semidirect
4. #22625526	C₂	semidirect
149. C₇₃	#18072	semidirect

Maximal subgroups

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

subgroup	index
#22626727	2
#22625577	2
#22625555	2
#22625526	2
#22625473	2
#22625241	2
#22625141	2
#22620518	2
#18072	73