Subgroups — #33001910

Order 18688.

486 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11924011	C₂	1	1	yes	maximal, characteristic
2	#22611687	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17988368	C₂	1	2	yes	maximal
4	#22608014	C₂	1	2	yes	maximal, complemented
5	C₇₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11924011	C₄	1	1	yes	characteristic
7	C₂²×#11924011	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₇₃	C₄	1	1	yes	characteristic
9	C₂×#14556531	C₂²	1	2	yes	—
10	C₄×#11924011	—	2	2	no	—
11	C₄×#11924011	—	2	2	no	—
12	C₂×#14556501	C₄	1	1	yes	characteristic
13	C₂×#14556501	C₂²	1	1	yes	characteristic
14	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
15	C₄⋊#11924003	—	2	2	no	—
16	#17988368	—	2	4	no	—
17	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₁₆×C₇₃	—	2	2	no	—
20	C₂²×C₄×D₇₃	C₈	1	2	yes	—
21	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#11924011	—	2	2	no	—
23	C₂×#11924011	—	2	2	no	—
24	C₂×#11924011	—	2	2	no	—
25	C₂×#11924011	—	2	1	no	—
26	C₂×#11924011	—	2	1	no	—
27	C₂×#11924011	—	2	1	no	—
28	C₂×#11924011	—	2	1	no	—
29	C₂×#11924011	D₄	1	2	yes	—
30	C₂×#11924011	Q₈	1	2	yes	—
31	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
32	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
34	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₇₃	—	2	2	no	—
36	C₄²×D₇₃	—	2	1	no	—
37	C₄²×D₇₃	—	2	1	no	—
38	#14556531	—	2	2	no	—
39	#14556531	—	2	2	no	—
40	C₂×#11924003	—	2	2	no	—
41	#14556501	—	2	2	no	—
42	#14556501	—	2	1	no	—
43	#14556501	—	2	1	no	—
44	C₄×#10596831	—	2	2	no	—
45	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₇₃	—	2	2	no	—
49	M₃₂×C₇₃	—	2	2	no	—
50	C₂×C₁₆×C₇₃	—	2	2	no	—
51	C₄×C₈×C₇₃	D₄	1	2	yes	—
52	C₄×C₈×C₇₃	D₄	1	2	yes	—
53	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
56	C₂×C₄×D₇₃	—	2	1	no	—
57	C₂×C₄×D₇₃	—	2	1	no	—
58	C₂×C₄×D₇₃	—	2	1	no	—
59	C₂×C₄×D₇₃	—	2	1	no	—
60	C₂×C₄×D₇₃	—	2	2	no	—
61	C₂×C₄×D₇₃	—	2	2	no	—
62	C₂×C₄×D₇₃	—	4	4	no	—
63	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
65	#11924011	—	2	8	no	—
66	#11924011	—	2	2	no	—
67	#11924011	—	2	2	no	—
68	#11924011	—	2	2	no	—
69	#11924011	—	2	2	no	—
70	C₂×#10596831	—	2	1	no	—
71	C₂×#10596831	—	2	1	no	—
72	C₂×#10596831	—	2	1	no	—
73	C₂×#10596831	—	2	1	no	—
74	C₂×#10596831	C₄⋊C₄	1	2	yes	—
75	C₂×#10596831	M₁₆	1	2	yes	—
76	C₄×Dic₇₃	—	2	2	no	—
77	C₄×Dic₇₃	—	2	2	no	—
78	C₄×Dic₇₃	—	2	1	no	—
79	C₄×Dic₇₃	—	2	1	no	—
80	#11924003	—	4	2	no	—
81	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
82	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₃	—	2	1	no	—
84	C₂×C₈×C₇₃	—	2	1	no	—
85	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
91	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
92	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
94	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
95	C₁₆×C₇₃	—	4	2	no	—
96	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
97	C₂²×D₇₃	—	2	1	no	—
98	C₂²×D₇₃	—	2	1	no	—
99	C₂²×D₇₃	—	4	2	no	—
100	C₂²×D₇₃	C₄⋊C₈	1	2	yes	—
101	C₂×C₄×C₇₃	—	2	4	no	—
102	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×Dic₇₃	—	2	1	no	—
111	C₂×Dic₇₃	—	2	1	no	—
112	C₂×Dic₇₃	—	2	2	no	—
113	C₂×Dic₇₃	—	4	2	no	—
114	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
115	C₄×D₇₃	—	2	2	no	—
116	C₄×D₇₃	—	2	2	no	—
117	C₄×D₇₃	—	2	2	no	—
118	C₄×D₇₃	—	2	2	no	—
119	C₄×D₇₃	—	4	4	no	—
120	C₄×D₇₃	—	4	4	no	—
121	C₈×C₇₃	—	2	2	no	—
122	C₈×C₇₃	#121	1	1	yes	characteristic
123	C₈×C₇₃	#121	1	1	yes	characteristic
124	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
126	#10596831	—	2	2	no	—
127	#10596831	—	2	2	no	—
128	C₂²×C₇₃	—	2	2	no	—
129	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
131	C₂²×C₇₃	#431	1	1	yes	characteristic
132	C₂×D₇₃	—	2	2	no	—
133	C₂×D₇₃	—	4	2	no	—
134	C₂×D₇₃	—	4	2	no	—
135	C₂×D₇₃	—	4	2	no	—
136	C₄×C₇₃	—	2	2	no	—
137	C₄×C₇₃	#349	1	4	yes	complemented
138	C₄×C₇₃	#380	1	1	yes	characteristic
139	C₄×C₇₃	#407	1	1	yes	characteristic
140	C₄×C₇₃	#423	1	1	yes	characteristic
141	C₄×C₇₃	#432	1	1	yes	characteristic
142	Dic₇₃	—	2	2	no	—
143	Dic₇₃	—	4	2	no	—
144	#12785	—	73	1	no	maximal, Sylow
145	C₂×C₇₃	—	2	1	no	—
146	C₂×C₇₃	—	2	1	no	—
147	C₂×C₇₃	#1797	1	1	yes	characteristic
148	C₂×C₇₃	C₂×#349	1	2	yes	—
149	D₇₃	—	4	2	no	—
150	C₂×C₄×M₁₆	—	73	1	no	—
151	#2150	—	73	1	no	—
152	#2150	D₇₃	1	1	yes	characteristic, complemented
153	C₂×#349	—	73	2	no	—
154	#1377	—	73	2	no	—
155	C₇₃	#12785	1	1	yes	characteristic, complemented, Sylow
156	C₂²×M₁₆	—	73	1	no	—
157	C₂²×M₁₆	—	73	1	no	—
158	C₂²×C₄²	—	73	1	no	—
159	C₂×M₃₂	—	73	2	no	—
160	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
161	C₄×M₁₆	—	146	2	no	—
162	C₄×M₁₆	—	146	2	no	—
163	C₂×#97	—	73	1	no	—
164	C₂×#97	—	73	1	no	—
165	C₂×C₄×C₈	—	73	1	no	—
166	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
167	C₄⋊C₁₆	—	2	2	no	—
168	C₄⋊C₁₆	—	146	2	no	—
169	#349	—	146	4	no	—
170	C₂³×C₄	—	73	2	no	—
171	C₂³×C₄	—	73	1	no	—
172	C₂×M₁₆	—	73	2	no	—
173	C₂×M₁₆	—	73	2	no	—
174	C₂×M₁₆	—	146	2	no	—
175	C₂×M₁₆	—	146	2	no	—
176	C₂×M₁₆	—	146	2	no	—
177	C₂×M₁₆	—	146	1	no	—
178	C₂×M₁₆	—	146	1	no	—
179	C₂×M₁₆	—	146	1	no	—
180	C₂×M₁₆	—	146	1	no	—
181	C₂²×C₈	—	73	1	no	—
182	C₂²×C₈	—	73	1	no	—
183	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
185	C₂×C₄²	—	73	2	no	—
186	C₂×C₄²	—	73	1	no	—
187	C₂×C₄²	—	146	2	no	—
188	C₂×C₄²	—	146	1	no	—
189	C₂×C₄²	—	146	1	no	—
190	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
191	M₃₂	—	2	2	no	—
192	M₃₂	—	2	2	no	—
193	M₃₂	—	146	2	no	—
194	M₃₂	—	146	2	no	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	—	146	2	no	—
197	#97	—	146	2	no	—
198	#97	—	146	1	no	—
199	#97	—	146	1	no	—
200	C₄×C₈	—	146	2	no	—
201	C₄×C₈	D₂₉₂	1	2	yes	—
202	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
203	C₂⁴	—	73	1	no	—
204	C₂²×C₄	—	292	4	no	—
205	C₂²×C₄	—	73	2	no	—
206	C₂²×C₄	—	73	1	no	—
207	C₂²×C₄	—	73	2	no	—
208	C₂²×C₄	—	73	2	no	—
209	C₂²×C₄	—	146	2	no	—
210	C₂²×C₄	—	146	2	no	—
211	C₂²×C₄	—	146	1	no	—
212	C₂²×C₄	—	146	1	no	—
213	C₂²×C₄	—	146	1	no	—
214	C₂²×C₄	—	146	1	no	—
215	C₂²×C₄	C₈×D₇₃	1	2	yes	—
216	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
217	M₁₆	—	146	8	no	—
218	M₁₆	—	146	2	no	—
219	M₁₆	—	146	2	no	—
220	M₁₆	—	146	2	no	—
221	M₁₆	—	146	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	73	2	no	—
225	C₂×C₈	—	73	2	no	—
226	C₂×C₈	—	146	1	no	—
227	C₂×C₈	—	146	1	no	—
228	C₂×C₈	—	146	1	no	—
229	C₂×C₈	—	146	1	no	—
230	C₂×C₈	#11924005	1	2	yes	—
231	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
232	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
233	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
234	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
235	C₂×C₈	#11924027	1	1	yes	characteristic
236	C₂×C₈	#11924029	1	1	yes	characteristic
237	C₄²	—	146	2	no	—
238	C₄²	—	146	2	no	—
239	C₄²	—	146	1	no	—
240	C₄²	—	146	1	no	—
241	C₄²	#11924005	1	2	yes	—
242	C₄²	#11924011	1	1	yes	characteristic
243	C₄²	#11924011	1	1	yes	characteristic
244	C₁₆	—	4	2	no	—
245	C₁₆	—	292	2	no	—
246	C₂³	—	292	2	no	—
247	C₂³	—	73	2	no	—
248	C₂³	—	146	1	no	—
249	C₂³	—	146	1	no	—
250	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	292	4	no	—
253	C₂×C₄	—	292	4	no	—
254	C₂×C₄	—	292	2	no	—
255	C₂×C₄	—	73	2	no	—
256	C₂×C₄	—	146	2	no	—
257	C₂×C₄	—	146	1	no	—
258	C₂×C₄	—	146	1	no	—
259	C₂×C₄	—	146	2	no	—
260	C₂×C₄	—	146	2	no	—
261	C₂×C₄	—	146	2	no	—
262	C₂×C₄	—	146	2	no	—
263	C₂×C₄	#14556503	1	2	yes	—
264	C₂×C₄	#14556503	1	2	yes	—
265	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
266	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
267	C₂×C₄	#14556549	1	1	yes	characteristic
268	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
269	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
270	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	146	2	no	—
273	C₈	—	146	2	no	—
274	C₈	#14556562	1	1	yes	characteristic
275	C₈	#14556566	1	1	yes	characteristic
276	C₈	#14556567	1	1	yes	characteristic
277	C₈	#14556571	1	1	yes	characteristic
278	C₂²	—	2	2	no	—
279	C₂²	—	292	2	no	—
280	C₂²	—	292	2	no	—
281	C₂²	—	292	2	no	—
282	C₂²	—	146	2	no	—
283	C₂²	C₂×#14556503	1	1	yes	characteristic
284	C₂²	C₄⋊C₈×D₇₃	1	1	yes	characteristic
285	C₂²	#17988639	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	292	2	no	—
288	C₄	—	146	2	no	—
289	C₄	#17988368	1	4	yes	complemented
290	C₄	#17988444	1	1	yes	characteristic
291	C₄	#17988550	1	1	yes	characteristic
292	C₄	#17988609	1	1	yes	characteristic
293	C₄	#17988643	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	292	2	no	—
297	C₂	#22609908	1	1	yes	characteristic
298	C₂	C₂×#17988368	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11924005
commutator	C₂³×C₇₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#2150	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#2150	D₇₃
hypocenter	C₇₃	#12785
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #22611687	C₂	semidirect
4. #22608014	C₂	semidirect
5. C₇₃×#2150	C₂	semidirect
137. C₄×C₇₃	#349	semidirect
152. #2150	D₇₃	semidirect
155. C₇₃	#12785	semidirect
289. C₄	#17988368	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11924011	2
#22611687	2
C₂×#17988368	2
#22608014	2
C₇₃×#2150	2
#12785	73