Subgroups — #33238604

Order 18688.

806 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988997	C₂	1	2	yes	maximal
2	C₂×#17988992	C₂	1	1	yes	maximal, characteristic
3	#22616052	C₂	1	2	yes	maximal
4	#22610096	C₂	1	4	yes	maximal
5	#22610094	C₂	1	2	yes	maximal
6	#22610089	C₂	1	2	yes	maximal
7	#22609617	C₂	1	1	yes	maximal, characteristic
8	C₂×#499×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556591	C₂²	1	2	yes	—
10	#17988997	—	2	8	no	—
11	#17988992	—	2	4	no	—
12	C₂×#14556525	C₂²	1	2	yes	—
13	C₂×#14556525	C₂²	1	2	yes	—
14	C₂×#14556524	C₂²	1	4	yes	—
15	C₂×#14556522	C₂²	1	4	yes	—
16	C₂×#14556502	C₂²	1	1	yes	characteristic
17	C₂×#14556501	C₂²	1	2	yes	—
18	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
19	#17988517	—	2	4	no	—
20	#17988507	—	2	4	no	—
21	#17988456	C₂²	1	4	yes	—
22	#17988452	C₂²	1	4	yes	—
23	#17988433	C₂²	1	1	yes	characteristic
24	C₂×#125×C₇₃	C₂²	1	2	yes	—
25	#499×C₇₃	—	2	4	no	—
26	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
28	C₂×#106×C₇₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
30	C₂²×Dic₁₄₆	—	2	2	no	—
31	C₂²×#10596831	C₂³	1	2	yes	—
32	#14556591	—	2	4	no	—
33	#14556591	—	2	4	no	—
34	C₂×C₄⋊Dic₇₃	—	2	2	no	—
35	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
36	C₂×#11924007	—	2	8	no	—
37	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
38	#14556525	—	2	4	no	—
39	#14556525	—	2	4	no	—
40	#14556524	—	2	8	no	—
41	#14556522	—	2	8	no	—
42	#14556502	D₄	1	4	yes	—
43	#14556501	Q₈	1	8	yes	complemented
44	C₈×Dic₇₃	Q₈	1	8	yes	complemented
45	#14556498	—	2	8	no	—
46	#14556498	D₄	1	2	yes	—
47	#14556498	D₄	1	2	yes	—
48	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
50	#125×C₇₃	—	2	4	no	—
51	#125×C₇₃	—	2	4	no	—
52	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
57	#107×C₇₃	—	2	2	no	—
58	#107×C₇₃	—	2	2	no	—
59	#106×C₇₃	—	2	4	no	—
60	C₄×C₈×C₇₃	D₄	1	4	yes	—
61	C₂×Dic₁₄₆	—	4	4	no	—
62	C₂×Dic₁₄₆	—	8	4	no	—
63	C₂²×Dic₇₃	—	2	4	no	—
64	C₂²×Dic₇₃	C₂×D₄	1	4	yes	—
65	C₂×#10596831	C₂×D₄	1	4	yes	—
66	C₂×#10596831	C₂×Q₈	1	4	yes	—
67	C₂×#10596831	C₂×Q₈	1	4	yes	—
68	C₄⋊Dic₇₃	—	2	4	no	—
69	C₄⋊Dic₇₃	—	2	4	no	—
70	C₄⋊Dic₇₃	—	4	4	no	—
71	#11924007	—	8	8	no	—
72	C₄×Dic₇₃	C₂×Q₈	1	4	yes	—
73	C₄×Dic₇₃	C₂×Q₈	1	4	yes	—
74	C₂²×C₄×C₇₃	—	2	2	no	—
75	C₂²×C₄×C₇₃	—	2	2	no	—
76	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
82	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
83	C₄⋊C₄×C₇₃	—	2	2	no	—
84	C₄⋊C₄×C₇₃	—	2	2	no	—
85	C₄⋊C₄×C₇₃	—	2	2	no	—
86	C₄⋊C₄×C₇₃	—	2	2	no	—
87	C₄⋊C₄×C₇₃	—	4	4	no	—
88	C₄⋊C₄×C₇₃	—	4	4	no	—
89	C₄²×C₇₃	C₂×D₄	1	2	yes	—
90	C₄²×C₇₃	C₂×D₄	1	2	yes	—
91	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇₃	—	2	2	no	—
93	C₂×C₄×C₇₃	—	2	2	no	—
94	C₂×C₄×C₇₃	—	4	2	no	—
95	C₂×C₄×C₇₃	—	4	2	no	—
96	C₂×C₄×C₇₃	—	4	4	no	—
97	C₂×C₄×C₇₃	—	4	4	no	—
98	C₂×C₄×C₇₃	#127	1	2	yes	—
99	C₂×C₄×C₇₃	#127	1	2	yes	—
100	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
103	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
104	C₂×Dic₇₃	—	2	4	no	—
105	C₂×Dic₇₃	—	2	4	no	—
106	C₂×Dic₇₃	—	4	4	no	—
107	C₂×Dic₇₃	—	4	8	no	—
108	C₂×Dic₇₃	C₄⋊Q₈	1	8	yes	—
109	Dic₁₄₆	—	8	8	no	—
110	C₈×C₇₃	C₄⋊Q₈	1	4	yes	—
111	C₈×C₇₃	C₄⋊Q₈	1	4	yes	—
112	#10596831	C₄⋊Q₈	1	8	yes	—
113	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
114	C₂²×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
115	C₂²×C₇₃	#576	1	2	yes	—
116	C₂²×C₇₃	#578	1	2	yes	—
117	C₄×C₇₃	—	2	2	no	—
118	C₄×C₇₃	—	2	2	no	—
119	C₄×C₇₃	—	8	2	no	—
120	C₄×C₇₃	—	8	2	no	—
121	C₄×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
122	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
124	Dic₇₃	—	2	8	no	—
125	Dic₇₃	—	8	4	no	—
126	#33319	—	73	1	no	maximal, Sylow
127	C₂×C₇₃	#2868	1	1	yes	characteristic
128	C₂×C₇₃	#3155	1	1	yes	characteristic
129	C₂×C₇₃	#3156	1	1	yes	characteristic
130	C₂×C₇₃	#3164	1	4	yes	—
131	C₂×#501	—	73	2	no	—
132	C₂×#500	—	73	1	no	—
133	C₂×#499	D₇₃	1	1	yes	characteristic
134	#2660	—	73	2	no	—
135	#1847	—	73	4	no	—
136	#1846	—	73	2	no	—
137	#1845	—	73	2	no	—
138	#1726	—	73	1	no	—
139	C₇₃	#33319	1	1	yes	characteristic, complemented, Sylow
140	C₂×C₄⋊Q₈	—	73	2	no	—
141	C₂×#125	C₂×D₇₃	1	2	yes	—
142	#501	—	146	8	no	—
143	#500	—	146	4	no	—
144	#499	—	2	4	no	—
145	C₂×#107	—	73	2	no	—
146	C₂×#107	—	73	2	no	—
147	C₂×#107	—	73	4	no	—
148	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
149	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
150	C₂×#106	—	73	4	no	—
151	C₂×#106	C₂×D₇₃	1	2	yes	—
152	C₂×#97	—	73	2	no	—
153	C₂×#97	—	73	1	no	—
154	C₂×C₄×C₈	—	73	2	no	—
155	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
156	#398	—	146	4	no	—
157	#395	—	146	4	no	—
158	#384	—	73	4	no	—
159	#382	—	73	4	no	—
160	#376	—	73	1	no	—
161	C₂²×Q₈	—	146	2	no	—
162	C₂²×C₈	—	73	2	no	—
163	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
165	C₄⋊Q₈	—	146	4	no	—
166	C₄⋊Q₈	—	146	4	no	—
167	#125	—	2	4	no	—
168	#125	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	73	4	no	—
172	C₂×C₄⋊C₄	—	146	2	no	—
173	C₂×C₄⋊C₄	—	146	8	no	—
174	C₂×C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
175	C₂×C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
176	C₂×C₄²	—	73	2	no	—
177	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
178	#107	—	2	2	no	—
179	#107	—	2	2	no	—
180	#107	—	146	8	no	—
181	#107	—	146	4	no	—
182	#107	—	146	4	no	—
183	#106	—	2	4	no	—
184	#106	—	146	8	no	—
185	#97	—	73	8	no	—
186	#97	—	73	4	no	—
187	C₄×C₈	—	73	8	no	—
188	C₄×C₈	C₂²⋊D₇₃	1	4	yes	—
189	#95	—	73	2	no	—
190	#95	—	73	2	no	—
191	#95	—	146	8	no	—
192	C₂×Q₈	—	292	4	no	—
193	C₂×Q₈	—	584	4	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	73	4	no	—
197	C₂²×C₄	—	146	4	no	—
198	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
199	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
200	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
201	C₂×C₈	—	73	4	no	—
202	C₂×C₈	—	73	4	no	—
203	C₂×C₈	—	73	4	no	—
204	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
205	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
206	C₂×C₈	Q₈×D₇₃	1	4	yes	—
207	C₂×C₈	Q₈×D₇₃	1	4	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	292	4	no	—
215	C₄⋊C₄	—	584	8	no	—
216	C₄⋊C₄	—	146	4	no	—
217	C₄⋊C₄	—	146	4	no	—
218	C₄²	—	73	4	no	—
219	C₄²	—	73	4	no	—
220	C₄²	D₄×D₇₃	1	2	yes	—
221	C₄²	C₂×C₂²⋊D₇₃	1	2	yes	—
222	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
223	Q₈	—	584	8	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	292	4	no	—
231	C₂×C₄	—	292	8	no	—
232	C₂×C₄	—	73	8	no	—
233	C₂×C₄	—	146	4	no	—
234	C₂×C₄	—	146	4	no	—
235	C₂×C₄	#14556587	1	2	yes	—
236	C₂×C₄	#14556587	1	2	yes	—
237	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
239	C₂×C₄	C₂×Q₈×D₇₃	1	2	yes	—
240	C₂×C₄	C₂×Q₈×D₇₃	1	2	yes	—
241	C₈	—	73	8	no	—
242	C₈	#14556589	1	4	yes	—
243	C₈	#14556589	1	4	yes	—
244	C₂²	C₂×#14556587	1	1	yes	characteristic
245	C₂²	C₄⋊Q₈×D₇₃	1	2	yes	—
246	C₂²	#17989389	1	2	yes	—
247	C₂²	#17989401	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	8	2	no	—
251	C₄	—	8	2	no	—
252	C₄	—	584	4	no	—
253	C₄	—	146	8	no	—
254	C₄	C₄⋊Q₈×D₇₃	1	2	yes	—
255	C₄	C₂×#14556589	1	1	yes	characteristic
256	C₄	C₂×#14556589	1	1	yes	characteristic
257	C₂	#22618327	1	1	yes	characteristic
258	C₂	#22620418	1	1	yes	characteristic
259	C₂	#22620426	1	1	yes	characteristic
260	C₂	#22620468	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₃
commutator, cosocle	C₂²×C₄×C₇₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×#499×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
hypercenter	C₂×#499	D₇₃
hypocenter	C₇₃	#33319
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
43. #14556501	Q₈	semidirect
44. C₈×Dic₇₃	Q₈	semidirect
139. C₇₃	#33319	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988997	2
C₂×#17988992	2
#22616052	2
#22610096	2
#22610094	2
#22610089	2
#22609617	2
C₂×#499×C₇₃	2
#33319	73