Subgroups — #33090687

Order 18688.

686 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22610868	C₂	1	2	yes	maximal
2	#22610650	C₂	1	2	yes	maximal
3	#22610490	C₂	1	4	yes	maximal
4	#22609788	C₂	1	2	yes	maximal
5	#22609784	C₂	1	1	yes	maximal, characteristic
6	#22609683	C₂	1	2	yes	maximal
7	C₄²×#10596831	C₂	1	1	yes	maximal, characteristic
8	C₇₃×#2295	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊#10596831	C₂²	1	4	yes	—
10	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
11	C₂×#14556502	C₂²	1	2	yes	—
12	C₂×C₄×#10596831	C₂²	1	2	yes	—
13	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
14	C₂×C₄×#10596831	C₂²	1	2	yes	—
15	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
16	#17988342	C₂²	1	4	yes	—
17	#17988342	C₂²	1	4	yes	—
18	#17988342	C₂²	1	2	yes	—
19	#17988342	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
21	#391×C₇₃	C₄	1	2	yes	—
22	#389×C₇₃	C₄	1	2	yes	—
23	#387×C₇₃	C₄	1	4	yes	—
24	C₄×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
26	#376×C₇₃	C₂²	1	2	yes	—
27	C₄³×C₇₃	C₂²	1	1	yes	characteristic
28	C₂²×#10596831	C₂³	1	4	yes	—
29	C₂²×#10596831	C₂³	1	2	yes	—
30	C₂²×#10596831	C₂³	1	2	yes	—
31	C₄⋊#10596831	—	2	8	no	—
32	C₄⋊#10596831	—	2	4	no	—
33	#14556502	—	2	4	no	—
34	C₄×#10596831	—	2	4	no	—
35	C₄×#10596831	—	2	4	no	—
36	C₄×#10596831	—	2	2	no	—
37	C₄×#10596831	Q₈	1	4	yes	complemented
38	C₂²×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₇₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
47	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
51	#95×C₇₃	C₂×C₄	1	2	yes	—
52	#95×C₇₃	C₂×C₄	1	4	yes	—
53	#95×C₇₃	C₂×C₄	1	4	yes	—
54	#95×C₇₃	C₂×C₄	1	4	yes	—
55	C₂×#10596831	—	2	2	no	—
56	C₂×#10596831	—	2	2	no	—
57	C₂×#10596831	—	2	2	no	—
58	C₂×#10596831	—	2	2	no	—
59	C₂×#10596831	—	2	4	no	—
60	C₂×#10596831	—	2	4	no	—
61	C₂×#10596831	—	2	4	no	—
62	C₂×#10596831	C₂×Q₈	1	4	yes	—
63	C₂×#10596831	#41	1	4	yes	—
64	C₂×Q₈×C₇₃	—	2	2	no	—
65	C₂×Q₈×C₇₃	—	2	4	no	—
66	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₇₃	—	2	2	no	—
75	C₄⋊C₄×C₇₃	—	2	4	no	—
76	C₄⋊C₄×C₇₃	—	2	8	no	—
77	C₄⋊C₄×C₇₃	—	2	4	no	—
78	C₄²×C₇₃	—	2	4	no	—
79	C₄²×C₇₃	—	2	4	no	—
80	C₄²×C₇₃	—	2	4	no	—
81	C₄²×C₇₃	—	2	2	no	—
82	C₄²×C₇₃	—	2	4	no	—
83	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
84	C₄²×C₇₃	#41	1	4	yes	—
85	C₄²×C₇₃	#41	1	2	yes	—
86	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
87	Q₈×C₇₃	—	4	4	no	—
88	C₂×C₄×C₇₃	—	2	2	no	—
89	C₂×C₄×C₇₃	—	2	2	no	—
90	C₂×C₄×C₇₃	—	2	2	no	—
91	C₂×C₄×C₇₃	—	2	2	no	—
92	C₂×C₄×C₇₃	—	2	2	no	—
93	C₂×C₄×C₇₃	—	2	2	no	—
94	C₂×C₄×C₇₃	—	2	2	no	—
95	C₂×C₄×C₇₃	—	2	4	no	—
96	C₂×C₄×C₇₃	—	2	4	no	—
97	C₂×C₄×C₇₃	—	2	4	no	—
98	C₂×C₄×C₇₃	—	2	4	no	—
99	C₂×C₄×C₇₃	—	2	4	no	—
100	C₂×C₄×C₇₃	#117	1	2	yes	—
101	C₂×C₄×C₇₃	#117	1	2	yes	—
102	C₂×C₄×C₇₃	C₄×Q₈	1	4	yes	—
103	C₂×C₄×C₇₃	#131	1	4	yes	—
104	C₂×C₄×C₇₃	#131	1	4	yes	—
105	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
107	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
108	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
109	#10596831	—	2	4	no	—
110	#10596831	—	2	4	no	—
111	#10596831	—	4	4	no	—
112	C₂²×C₇₃	C₂×#117	1	1	yes	characteristic
113	C₂²×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₂²×C₇₃	C₄×#41	1	1	yes	characteristic
115	C₂²×C₇₃	C₂×#131	1	2	yes	—
116	C₂²×C₇₃	C₂×#131	1	2	yes	—
117	C₄×C₇₃	—	2	2	no	—
118	C₄×C₇₃	—	2	4	no	—
119	C₄×C₇₃	—	2	4	no	—
120	C₄×C₇₃	—	4	2	no	—
121	C₄×C₇₃	—	4	2	no	—
122	C₄×C₇₃	—	4	4	no	—
123	C₄×C₇₃	#433	1	4	yes	—
124	C₄×C₇₃	#529	1	2	yes	—
125	C₄×C₇₃	#533	1	2	yes	—
126	#21341	—	73	1	no	maximal, Sylow
127	C₂×C₇₃	#2295	1	1	yes	characteristic
128	C₂×C₇₃	C₂×#433	1	1	yes	characteristic
129	C₂×C₇₃	#2922	1	1	yes	characteristic
130	C₂×C₇₃	#2960	1	2	yes	—
131	C₂×C₇₃	#2966	1	2	yes	—
132	#2295	D₇₃	1	1	yes	characteristic
133	#1984	—	73	2	no	—
134	#1939	—	73	2	no	—
135	#1916	—	73	4	no	—
136	#1768	—	73	2	no	—
137	#1767	—	73	1	no	—
138	#1746	—	73	2	no	—
139	C₄²×C₈	—	73	1	no	—
140	C₇₃	#21341	1	1	yes	characteristic, complemented, Sylow
141	C₂×C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
142	C₂×C₄⋊C₈	—	73	4	no	—
143	C₂×C₄⋊C₈	—	73	2	no	—
144	C₂×#97	—	73	2	no	—
145	C₂×C₄×C₈	—	73	1	no	—
146	C₂×C₄×C₈	—	73	2	no	—
147	C₂×C₄×C₈	—	73	2	no	—
148	C₂×C₄×C₈	—	73	1	no	—
149	#391	Dic₇₃	1	2	yes	—
150	#389	Dic₇₃	1	2	yes	—
151	#387	Dic₇₃	1	4	yes	—
152	C₄×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
153	C₄×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
154	#376	C₂×D₇₃	1	2	yes	—
155	C₄³	C₂×D₇₃	1	1	yes	characteristic
156	#336	—	73	2	no	—
157	#336	—	73	4	no	—
158	#336	—	73	4	no	—
159	#336	—	73	4	no	—
160	C₂²×Q₈	C₂×Dic₇₃	1	1	yes	characteristic
161	C₂²×C₈	—	73	2	no	—
162	C₂²×C₈	—	73	2	no	—
163	C₂²×C₈	—	73	4	no	—
164	C₄×Q₈	—	2	4	no	—
165	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
166	C₂×C₄⋊C₄	C₂×Dic₇₃	1	1	yes	characteristic
167	C₂×C₄⋊C₄	C₂×Dic₇₃	1	4	yes	—
168	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
169	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
170	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
171	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
172	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
174	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
176	C₄⋊C₈	—	146	8	no	—
177	C₄⋊C₈	—	146	4	no	—
178	#97	—	146	4	no	—
179	C₄×C₈	—	73	4	no	—
180	C₄×C₈	—	146	4	no	—
181	C₄×C₈	—	146	4	no	—
182	C₄×C₈	—	146	2	no	—
183	#95	C₂×Dic₇₃	1	4	yes	—
184	#95	C₂×Dic₇₃	1	4	yes	—
185	#95	C₂×Dic₇₃	1	2	yes	—
186	#95	C₂×Dic₇₃	1	4	yes	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂²×C₄	C₂²×Dic₇₃	1	4	yes	—
190	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
191	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
192	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
193	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
194	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
195	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
196	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
197	C₂×C₈	—	73	4	no	—
198	C₂×C₈	—	73	4	no	—
199	C₂×C₈	—	146	4	no	—
200	C₂×C₈	—	146	4	no	—
201	C₂×C₈	—	146	4	no	—
202	C₂×C₈	—	146	2	no	—
203	C₂×C₈	—	146	2	no	—
204	C₂×C₈	—	146	2	no	—
205	C₂×C₈	—	146	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄²	—	2	4	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	4	no	—
214	C₄²	—	2	4	no	—
215	C₄²	Q₈×D₇₃	1	2	yes	—
216	C₄²	#11924028	1	4	yes	—
217	C₄²	#11924029	1	2	yes	—
218	C₂³	C₂³×Dic₇₃	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	#14556550	1	2	yes	—
233	C₂×C₄	#14556550	1	2	yes	—
234	C₂×C₄	Q₈×Dic₇₃	1	4	yes	—
235	C₂×C₄	#14556598	1	4	yes	—
236	C₂×C₄	#14556598	1	4	yes	—
237	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
238	C₂×C₄	#41×D₇₃	1	2	yes	—
239	C₂×C₄	C₂×#11924028	1	2	yes	—
240	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
241	C₈	—	292	4	no	—
242	C₈	—	146	4	no	—
243	C₈	—	146	4	no	—
244	C₂²	C₂×#14556550	1	1	yes	characteristic
245	C₂²	C₂×Q₈×Dic₇₃	1	1	yes	characteristic
246	C₂²	#41×Dic₇₃	1	1	yes	characteristic
247	C₂²	C₂×#14556598	1	2	yes	—
248	C₂²	C₂×#14556598	1	2	yes	—
249	C₄	—	2	4	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	4	no	—
252	C₄	—	4	4	no	—
253	C₄	—	4	2	no	—
254	C₄	—	4	2	no	—
255	C₄	#17988648	1	4	yes	—
256	C₄	#17989124	1	2	yes	—
257	C₄	#17989141	1	2	yes	—
258	C₂	#22612409	1	1	yes	characteristic
259	C₂	C₂×#17988648	1	1	yes	characteristic
260	C₂	#22618575	1	1	yes	characteristic
261	C₂	#22618866	1	2	yes	—
262	C₂	#22618907	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₃
commutator, socle, prime radical	C₂³×C₇₃	C₂³×C₄
Fitting	C₇₃×#2295	C₂
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#2295	D₇₃
hypocenter	C₇₃	#21341
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
37. C₄×#10596831	Q₈	semidirect
140. C₇₃	#21341	semidirect

Maximal subgroups

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

subgroup	index
#22610868	2
#22610650	2
#22610490	2
#22609788	2
#22609784	2
#22609683	2
C₄²×#10596831	2
C₇₃×#2295	2
#21341	73