Subgroups — #32994663

Order 18688.

694 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988636	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988614	C₂	1	2	yes	maximal
3	#22609641	C₂	1	2	yes	maximal
4	#22609629	C₂	1	1	yes	maximal, characteristic
5	C₂×H(3,ℤ/4)×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924012	C₂²	1	2	yes	—
7	C₂³×#10596831	C₂²	1	1	yes	characteristic
8	C₂×#14556550	C₂²	1	1	yes	characteristic
9	#17988636	—	2	4	no	—
10	C₂×#14556527	—	2	2	no	—
11	#17988614	—	2	4	no	—
12	#17988614	—	2	4	no	—
13	C₂×C₄⋊#10596831	—	2	4	no	—
14	#17988347	—	2	4	no	—
15	#17988341	—	2	2	no	—
16	C₂²×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×#117×C₇₃	C₂²	1	2	yes	—
18	H(3,ℤ/4)×C₇₃	C₄	1	4	yes	complemented
19	C₂×#11924012	—	2	2	no	—
20	C₂×#11924012	—	2	2	no	—
21	C₂×#11924012	—	2	2	no	—
22	C₂×#11924012	—	2	2	no	—
23	C₂×#11924012	—	2	2	no	—
24	C₂×#11924012	D₄	1	4	yes	—
25	C₂²×#10596831	—	2	2	no	—
26	C₂²×#10596831	—	2	2	no	—
27	C₂²×#10596831	—	2	2	no	—
28	C₂²×#10596831	—	2	1	no	—
29	C₂²×#10596831	—	2	1	no	—
30	C₂²×#10596831	D₄	1	2	yes	—
31	#14556550	—	2	2	no	—
32	#14556550	—	2	2	no	—
33	C₂×C₄⋊Dic₇₃	Q₈	1	2	yes	—
34	C₂×C₂²⋊Dic₇₃	—	2	1	no	—
35	C₂×C₄×Dic₇₃	—	2	1	no	—
36	#14556527	—	2	8	no	—
37	C₄⋊#10596831	—	2	16	no	—
38	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₇₃	—	2	2	no	—
40	C₂×M₁₆×C₇₃	—	2	2	no	—
41	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
42	C₂×M₁₆×C₇₃	Q₈	1	2	yes	—
43	C₂²×C₈×C₇₃	D₄	1	2	yes	—
44	#117×C₇₃	—	2	2	no	—
45	#117×C₇₃	—	2	2	no	—
46	#117×C₇₃	C₂×C₄	1	4	yes	—
47	#117×C₇₃	Q₈	1	4	yes	—
48	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
50	C₂×C₄²×C₇₃	D₄	1	4	yes	—
51	C₂²×Dic₇₃	—	2	2	no	—
52	#11924012	—	2	4	no	—
53	#11924012	—	2	4	no	—
54	#11924012	—	4	2	no	—
55	#11924012	—	4	2	no	—
56	C₂×#10596831	—	2	4	no	—
57	C₂×#10596831	—	2	4	no	—
58	C₂×#10596831	—	2	2	no	—
59	C₂×#10596831	—	4	1	no	—
60	C₂×#10596831	—	4	2	no	—
61	C₂×#10596831	—	4	2	no	—
62	C₂×#10596831	—	4	1	no	—
63	C₂×#10596831	—	4	2	no	—
64	C₄⋊Dic₇₃	—	2	4	no	—
65	C₂²⋊Dic₇₃	—	4	2	no	—
66	C₄×Dic₇₃	—	4	1	no	—
67	C₄×Dic₇₃	—	4	1	no	—
68	C₂⁴×C₇₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	—	2	4	no	—
70	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₇₃	#41	1	2	yes	—
79	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
80	M₁₆×C₇₃	—	2	4	no	—
81	M₁₆×C₇₃	—	2	4	no	—
82	C₂×C₈×C₇₃	—	2	2	no	—
83	C₂×C₈×C₇₃	—	2	2	no	—
84	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
85	C₄⋊C₄×C₇₃	—	2	4	no	—
86	C₄⋊C₄×C₇₃	—	2	4	no	—
87	C₂²⋊C₄×C₇₃	—	2	4	no	—
88	C₂²⋊C₄×C₇₃	—	2	4	no	—
89	C₄²×C₇₃	—	2	4	no	—
90	C₄²×C₇₃	C₄⋊C₄	1	8	yes	—
91	C₂³×C₇₃	—	4	2	no	—
92	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
93	C₂³×C₇₃	C₂²⋊Q₈	1	2	yes	—
94	C₂³×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂³×C₇₃	#123	1	2	yes	—
96	C₂³×C₇₃	#125	1	1	yes	characteristic
97	C₂×C₄×C₇₃	—	2	8	no	—
98	C₂×C₄×C₇₃	—	4	8	no	—
99	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
104	C₂×C₄×C₇₃	#120	1	2	yes	—
105	C₂×C₄×C₇₃	#121	1	2	yes	—
106	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
111	C₂×C₄×C₇₃	#125	1	2	yes	—
112	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
114	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×Dic₇₃	—	4	4	no	—
118	C₂×Dic₇₃	—	4	2	no	—
119	C₈×C₇₃	—	2	4	no	—
120	#10596831	—	4	2	no	—
121	#10596831	—	4	4	no	—
122	C₂²×C₇₃	—	2	2	no	—
123	C₂²×C₇₃	—	2	4	no	—
124	C₂²×C₇₃	—	4	1	no	—
125	C₂²×C₇₃	—	4	2	no	—
126	C₂²×C₇₃	—	4	1	no	—
127	C₂²×C₇₃	#385	1	2	yes	—
128	C₂²×C₇₃	#389	1	1	yes	characteristic
129	C₂²×C₇₃	#395	1	1	yes	characteristic
130	C₂²×C₇₃	#398	1	2	yes	—
131	C₂²×C₇₃	#399	1	1	yes	characteristic
132	C₄×C₇₃	—	4	8	no	—
133	C₄×C₇₃	#384	1	1	yes	characteristic
134	C₄×C₇₃	#385	1	1	yes	characteristic
135	C₄×C₇₃	#389	1	2	yes	—
136	C₄×C₇₃	#392	1	1	yes	characteristic
137	C₄×C₇₃	#395	1	2	yes	—
138	C₄×C₇₃	#398	1	1	yes	characteristic
139	Dic₇₃	—	8	2	no	—
140	#11336	—	73	1	no	maximal, Sylow
141	C₂×C₇₃	—	2	2	no	—
142	C₂×C₇₃	—	2	1	no	—
143	C₂×C₇₃	—	2	1	no	—
144	C₂×C₇₃	—	2	2	no	—
145	C₂×C₇₃	#1444	1	1	yes	characteristic
146	C₂×C₇₃	#2098	1	2	yes	—
147	C₂×#430	—	73	1	no	—
148	C₂×#424	—	73	2	no	—
149	#1733	—	73	2	no	—
150	C₂×H(3,ℤ/4)	D₇₃	1	1	yes	characteristic
151	#1729	—	73	1	no	—
152	C₇₃	#11336	1	1	yes	characteristic, complemented, Sylow
153	C₂²×M₁₆	—	73	2	no	—
154	C₂²×M₁₆	C₂×D₇₃	1	1	yes	characteristic
155	C₂³×C₈	—	73	1	no	—
156	C₂×#117	—	73	1	no	—
157	C₂×#117	C₂×D₇₃	1	2	yes	—
158	#430	—	146	4	no	—
159	C₂×#108	—	146	2	no	—
160	#424	—	146	4	no	—
161	#424	—	146	4	no	—
162	C₂×C₄⋊C₈	—	146	4	no	—
163	#339	—	146	4	no	—
164	H(3,ℤ/4)	Dic₇₃	1	4	yes	complemented
165	#336	—	146	2	no	—
166	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	73	4	no	—
170	C₂×M₁₆	—	146	2	no	—
171	C₂×M₁₆	—	146	2	no	—
172	C₂×M₁₆	—	146	2	no	—
173	C₂×M₁₆	—	146	2	no	—
174	C₂×M₁₆	—	146	2	no	—
175	C₂×M₁₆	Dic₁₄₆	1	2	yes	—
176	C₂×M₁₆	C₂×Dic₇₃	1	2	yes	—
177	C₂²×C₈	—	73	2	no	—
178	C₂²×C₈	—	146	2	no	—
179	C₂²×C₈	—	146	1	no	—
180	C₂²×C₈	—	146	2	no	—
181	C₂²×C₈	—	146	2	no	—
182	C₂²×C₈	—	146	1	no	—
183	C₂²×C₈	D₂₉₂	1	2	yes	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	146	2	no	—
187	#117	—	146	2	no	—
188	#117	Dic₁₄₆	1	4	yes	—
189	#117	C₂×Dic₇₃	1	4	yes	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	73	2	no	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₂²⋊C₄	—	146	1	no	—
194	C₂×C₄²	—	146	1	no	—
195	C₂×C₄²	D₂₉₂	1	4	yes	—
196	#108	—	146	8	no	—
197	C₄⋊C₈	—	146	16	no	—
198	C₂⁴	#11924027	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	146	2	no	—
201	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
202	C₂²×C₄	D₄×D₇₃	1	2	yes	—
203	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
205	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
206	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
208	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
209	C₂²×C₄	#11924027	1	2	yes	—
210	C₂²×C₄	#11924029	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	292	2	no	—
214	M₁₆	—	292	2	no	—
215	M₁₆	—	146	4	no	—
216	M₁₆	—	146	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	292	1	no	—
220	C₂×C₈	—	292	2	no	—
221	C₂×C₈	—	292	2	no	—
222	C₂×C₈	—	292	1	no	—
223	C₂×C₈	—	292	2	no	—
224	C₂×C₈	—	146	4	no	—
225	C₂×C₈	—	146	4	no	—
226	C₂×C₈	—	146	2	no	—
227	C₂×C₈	C₄⋊Dic₇₃	1	4	yes	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	146	4	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	292	2	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	292	1	no	—
236	C₄²	—	292	1	no	—
237	C₄²	C₄⋊Dic₇₃	1	8	yes	—
238	C₂³	—	4	2	no	—
239	C₂³	D₄×Dic₇₃	1	1	yes	characteristic
240	C₂³	#14556568	1	2	yes	—
241	C₂³	#14556571	1	1	yes	characteristic
242	C₂³	#14556575	1	2	yes	—
243	C₂³	#14556582	1	1	yes	characteristic
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	4	8	no	—
246	C₂×C₄	—	292	4	no	—
247	C₂×C₄	—	292	2	no	—
248	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
249	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
250	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
251	C₂×C₄	Q₈×Dic₇₃	1	1	yes	characteristic
252	C₂×C₄	#14556561	1	1	yes	characteristic
253	C₂×C₄	#14556561	1	2	yes	—
254	C₂×C₄	#14556566	1	2	yes	—
255	C₂×C₄	#14556566	1	1	yes	characteristic
256	C₂×C₄	#14556568	1	2	yes	—
257	C₂×C₄	#14556568	1	2	yes	—
258	C₂×C₄	#14556571	1	2	yes	—
259	C₂×C₄	#14556575	1	1	yes	characteristic
260	C₂×C₄	#14556582	1	2	yes	—
261	C₂×C₄	#14556588	1	1	yes	characteristic
262	C₂×C₄	#14556590	1	2	yes	—
263	C₂×C₄	#14556590	1	1	yes	characteristic
264	C₂×C₄	#14556591	1	2	yes	—
265	C₂×C₄	#14556591	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	—	292	2	no	—
268	C₈	—	292	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	1	no	—
274	C₂²	#17988462	1	2	yes	—
275	C₂²	#17988486	1	1	yes	characteristic
276	C₂²	#17988508	1	1	yes	characteristic
277	C₂²	#17988519	1	2	yes	—
278	C₂²	#17988524	1	1	yes	characteristic
279	C₄	—	4	8	no	—
280	C₄	—	584	2	no	—
281	C₄	#17988457	1	1	yes	characteristic
282	C₄	#17988462	1	1	yes	characteristic
283	C₄	#17988486	1	2	yes	—
284	C₄	#17988497	1	1	yes	characteristic
285	C₄	#17988508	1	2	yes	—
286	C₄	#17988519	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	#22608196	1	1	yes	characteristic
292	C₂	#22611485	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14556588
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₂×H(3,ℤ/4)×C₇₃	C₂
socle	C₂²×C₇₃	#399
prime radical	C₂⁴×C₇₃	#41
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₇₃
hypocenter	C₇₃	#11336
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
18. H(3,ℤ/4)×C₇₃	C₄	semidirect
152. C₇₃	#11336	semidirect
164. H(3,ℤ/4)	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988636	2
C₂×#17988614	2
#22609641	2
#22609629	2
C₂×H(3,ℤ/4)×C₇₃	2
#11336	73