Subgroups — #33105746

Order 18688.

766 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612896	C₂	1	2	yes	maximal
2	#22610793	C₂	1	2	yes	maximal
3	#22610567	C₂	1	4	yes	maximal
4	#22610095	C₂	1	1	yes	maximal, characteristic
5	#22609946	C₂	1	2	yes	maximal
6	#107×Dic₇₃	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556525	C₂²	1	2	yes	—
10	C₂×#14556523	C₂²	1	2	yes	—
11	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
12	C₂×#14556515	C₂²	1	4	yes	—
13	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
14	#17988494	C₂²	1	4	yes	—
15	#17988473	—	2	4	no	—
16	#17988455	—	2	2	no	—
17	#17988453	C₂²	1	2	yes	—
18	C₄⋊C₄×Dic₇₃	C₂²	1	4	yes	—
19	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
20	#17988433	—	2	2	no	—
21	#17988349	C₂²	1	4	yes	—
22	#17988341	C₂²	1	2	yes	—
23	C₂²×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₇₃	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₇₃	C₄	1	8	yes	complemented
26	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×#103×C₇₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
29	C₂²×#10596831	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₇₃	D₄	1	2	yes	—
31	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
32	C₂×C₄×Dic₇₃	—	2	4	no	—
33	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
34	#14556525	—	2	4	no	—
35	#14556523	—	4	2	no	—
36	C₄⋊#10596831	—	2	2	no	—
37	#14556515	—	4	4	no	—
38	C₈×Dic₇₃	—	2	4	no	—
39	#14556498	—	2	4	no	—
40	#14556498	—	2	4	no	—
41	#14556498	—	2	4	no	—
42	#14556498	—	2	2	no	—
43	#14556498	—	2	2	no	—
44	#14556498	D₄	1	1	yes	characteristic
45	#14556498	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₇₃	C₂³	1	2	yes	—
47	C₂×Q₁₆×C₇₃	—	2	2	no	—
48	C₂×Q₁₆×C₇₃	—	2	2	no	—
49	C₂×Q₁₆×C₇₃	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
52	C₄×Q₈×C₇₃	—	2	4	no	—
53	C₄×Q₈×C₇₃	C₂×C₄	1	8	yes	—
54	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₇₃	—	2	2	no	—
57	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
58	#107×C₇₃	C₂×C₄	1	4	yes	—
59	#103×C₇₃	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
61	C₂²×Dic₇₃	—	2	4	no	—
62	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
63	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
64	C₂×#10596831	—	2	2	no	—
65	C₂×#10596831	—	2	2	no	—
66	C₂×#10596831	—	2	2	no	—
67	C₄⋊Dic₇₃	—	2	4	no	—
68	C₄⋊Dic₇₃	—	2	4	no	—
69	C₄⋊Dic₇₃	—	2	4	no	—
70	C₄×Dic₇₃	—	2	2	no	—
71	C₄×Dic₇₃	—	2	2	no	—
72	C₄×Dic₇₃	—	4	8	no	—
73	C₂×Q₈×C₇₃	—	2	2	no	—
74	C₂×Q₈×C₇₃	—	2	2	no	—
75	C₂×Q₈×C₇₃	—	2	2	no	—
76	C₂×Q₈×C₇₃	—	2	2	no	—
77	C₂×Q₈×C₇₃	—	2	2	no	—
78	C₂×Q₈×C₇₃	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₇₃	—	2	2	no	—
80	C₂²×C₄×C₇₃	—	2	2	no	—
81	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
84	Q₁₆×C₇₃	—	2	8	no	—
85	C₂×C₈×C₇₃	—	2	1	no	—
86	C₂×C₈×C₇₃	—	2	1	no	—
87	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₇₃	#41	1	2	yes	—
90	C₂×C₈×C₇₃	#41	1	2	yes	—
91	C₄⋊C₄×C₇₃	—	2	8	no	—
92	C₄⋊C₄×C₇₃	—	2	2	no	—
93	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
94	C₄²×C₇₃	—	2	8	no	—
95	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₇₃	#41	1	2	yes	—
97	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₇₃	—	2	4	no	—
99	Q₈×C₇₃	—	2	4	no	—
100	Q₈×C₇₃	—	4	2	no	—
101	Q₈×C₇₃	—	4	2	no	—
102	C₂×C₄×C₇₃	—	2	2	no	—
103	C₂×C₄×C₇₃	—	2	2	no	—
104	C₂×C₄×C₇₃	—	2	4	no	—
105	C₂×C₄×C₇₃	—	2	4	no	—
106	C₂×C₄×C₇₃	—	4	2	no	—
107	C₂×C₄×C₇₃	—	4	2	no	—
108	C₂×C₄×C₇₃	—	4	4	no	—
109	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂×Dic₇₃	—	2	2	no	—
118	C₂×Dic₇₃	—	2	2	no	—
119	C₂×Dic₇₃	—	2	2	no	—
120	C₂×Dic₇₃	—	2	2	no	—
121	C₂×Dic₇₃	—	2	2	no	—
122	C₂×Dic₇₃	—	4	4	no	—
123	C₂×Dic₇₃	—	4	4	no	—
124	C₂×Dic₇₃	—	4	4	no	—
125	C₂×Dic₇₃	#135	1	4	yes	—
126	C₈×C₇₃	—	2	2	no	—
127	C₈×C₇₃	#117	1	4	yes	—
128	#10596831	—	4	2	no	—
129	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂²×C₇₃	#547	1	1	yes	characteristic
131	C₂²×C₇₃	#550	1	1	yes	characteristic
132	C₂²×C₇₃	C₂×#135	1	2	yes	—
133	C₂²×C₇₃	#578	1	2	yes	—
134	C₄×C₇₃	—	2	2	no	—
135	C₄×C₇₃	—	2	2	no	—
136	C₄×C₇₃	—	4	4	no	—
137	C₄×C₇₃	—	4	4	no	—
138	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
139	C₄×C₇₃	#520	1	1	yes	characteristic
140	C₄×C₇₃	#555	1	1	yes	characteristic
141	C₄×C₇₃	#556	1	1	yes	characteristic
142	Dic₇₃	—	2	4	no	—
143	Dic₇₃	—	4	2	no	—
144	Dic₇₃	—	8	4	no	—
145	#22809	—	73	1	no	maximal, Sylow
146	C₂×C₇₃	#2354	1	1	yes	characteristic
147	C₂×C₇₃	C₄×#135	1	1	yes	characteristic
148	C₂×C₇₃	#2950	1	1	yes	characteristic
149	C₂×C₇₃	#3336	1	2	yes	—
150	C₂×C₇₃	#3393	1	2	yes	—
151	C₂×C₄×Q₁₆	D₇₃	1	1	yes	characteristic
152	#2356	—	73	2	no	—
153	#1969	—	73	2	no	—
154	#1927	—	73	4	no	—
155	#1846	—	73	1	no	—
156	#1805	—	73	2	no	—
157	C₄×#107	—	73	2	no	—
158	C₈×C₄⋊C₄	—	73	1	no	—
159	C₇₃	#22809	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
161	C₂×C₄×Q₈	C₂×D₇₃	1	2	yes	—
162	C₄×Q₁₆	Dic₇₃	1	8	yes	complemented
163	C₂×#107	—	73	2	no	—
164	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
165	C₂×#106	—	73	2	no	—
166	C₂×C₄⋊C₈	—	73	1	no	—
167	C₂×#103	—	73	4	no	—
168	C₂×#103	C₂×D₇₃	1	2	yes	—
169	C₂×C₄×C₈	—	73	2	no	—
170	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
171	#391	—	73	4	no	—
172	#387	—	146	4	no	—
173	#383	—	146	2	no	—
174	#382	—	73	2	no	—
175	C₄×C₄⋊C₄	—	73	1	no	—
176	C₄×C₄⋊C₄	—	73	4	no	—
177	#376	—	146	2	no	—
178	#340	—	73	4	no	—
179	#336	—	73	2	no	—
180	C₂²×Q₈	C₂²×D₇₃	1	2	yes	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	C₂×Dic₇₃	1	4	yes	—
184	C₂²×C₈	—	73	2	no	—
185	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
187	C₄×Q₈	—	2	4	no	—
188	C₄×Q₈	C₂×Dic₇₃	1	8	yes	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	73	2	no	—
191	C₂×C₄⋊C₄	—	73	4	no	—
192	C₂×C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	73	2	no	—
195	C₂×C₄²	—	146	4	no	—
196	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
197	#107	—	146	4	no	—
198	#107	C₂×Dic₇₃	1	4	yes	—
199	#106	—	292	2	no	—
200	C₄⋊C₈	—	146	2	no	—
201	#103	—	292	4	no	—
202	#103	C₂×Dic₇₃	1	8	yes	—
203	C₄×C₈	—	146	4	no	—
204	C₄×C₈	C₂×Dic₇₃	1	4	yes	—
205	#95	—	73	1	no	—
206	#95	—	73	1	no	—
207	#95	—	146	4	no	—
208	#95	—	146	4	no	—
209	#95	—	146	4	no	—
210	#95	—	146	2	no	—
211	#95	—	146	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	C₂²×Dic₇₃	1	4	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	73	2	no	—
221	C₂²×C₄	—	73	2	no	—
222	C₂²×C₄	—	146	4	no	—
223	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
226	Q₁₆	—	2	8	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	146	2	no	—
230	C₂×C₈	—	146	2	no	—
231	C₂×C₈	—	146	2	no	—
232	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
233	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
234	C₂×C₈	#11924028	1	2	yes	—
235	C₂×C₈	#11924028	1	2	yes	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	146	4	no	—
239	C₄⋊C₄	—	146	4	no	—
240	C₄⋊C₄	—	146	4	no	—
241	C₄⋊C₄	C₂²×Dic₇₃	1	4	yes	—
242	C₄²	—	2	8	no	—
243	C₄²	—	292	8	no	—
244	C₄²	—	146	2	no	—
245	C₄²	—	146	2	no	—
246	C₄²	C₂²×Dic₇₃	1	2	yes	—
247	C₄²	#11924027	1	2	yes	—
248	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
249	Q₈	—	2	4	no	—
250	Q₈	—	2	4	no	—
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	292	4	no	—
261	C₂×C₄	—	292	4	no	—
262	C₂×C₄	—	292	4	no	—
263	C₂×C₄	—	73	4	no	—
264	C₂×C₄	—	146	2	no	—
265	C₂×C₄	—	146	2	no	—
266	C₂×C₄	—	146	2	no	—
267	C₂×C₄	—	146	2	no	—
268	C₂×C₄	—	146	2	no	—
269	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
270	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
271	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
272	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
273	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
274	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
275	C₂×C₄	#14556637	1	1	yes	characteristic
276	C₂×C₄	#14556638	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	292	2	no	—
279	C₈	#14556550	1	4	yes	—
280	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
281	C₂²	#17989242	1	1	yes	characteristic
282	C₂²	#17989258	1	1	yes	characteristic
283	C₂²	#135×D₇₃	1	2	yes	—
284	C₂²	#17989400	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	4	no	—
289	C₄	—	292	2	no	—
290	C₄	—	584	4	no	—
291	C₄	—	146	4	no	—
292	C₄	C₂×#14556550	1	1	yes	characteristic
293	C₄	#17989070	1	1	yes	characteristic
294	C₄	#17989302	1	1	yes	characteristic
295	C₄	#17989307	1	1	yes	characteristic
296	C₂	#22612880	1	1	yes	characteristic
297	C₂	#135×Dic₇₃	1	1	yes	characteristic
298	C₂	#22618799	1	1	yes	characteristic
299	C₂	#22622301	1	2	yes	—
300	C₂	#22622919	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×C₄×Q₁₆×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₇₃
hypocenter	C₇₃	#22809
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
25. C₄×Q₁₆×C₇₃	C₄	semidirect
159. C₇₃	#22809	semidirect
162. C₄×Q₁₆	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22612896	2
#22610793	2
#22610567	2
#22610095	2
#22609946	2
#107×Dic₇₃	2
C₈×C₄⋊Dic₇₃	2
C₂×C₄×Q₁₆×C₇₃	2
#22809	73