Subgroups — #33238603

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×#17988996	C₂	1	2	yes	maximal
2	C₂×#17988992	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988989	C₂	1	1	yes	maximal, characteristic
4	#22616050	C₂	1	2	yes	maximal
5	#22610099	C₂	1	4	yes	maximal
6	#22610095	C₂	1	2	yes	maximal
7	#22610091	C₂	1	2	yes	maximal
8	C₇₃×#1726	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556591	C₂²	1	2	yes	—
10	C₂×#14556583	C₂²	1	2	yes	—
11	#17988996	—	2	8	no	—
12	#17988992	—	2	4	no	—
13	#17988989	—	2	4	no	—
14	C₂×#14556525	C₂²	1	1	yes	characteristic
15	C₂×#14556525	C₂²	1	1	yes	characteristic
16	C₂×#14556525	C₂²	1	2	yes	—
17	C₂×#14556525	C₂²	1	2	yes	—
18	C₂×#14556525	C₂²	1	4	yes	—
19	C₂×#14556523	C₂²	1	2	yes	—
20	C₂×#14556523	C₂²	1	4	yes	—
21	#17988519	—	2	4	no	—
22	#17988508	—	2	4	no	—
23	#17988457	C₂²	1	4	yes	—
24	#17988453	C₂²	1	4	yes	—
25	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
26	C₂×#97×C₇₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
29	#376×C₇₃	C₂²	1	1	yes	characteristic
30	C₂²×Dic₁₄₆	—	2	2	no	—
31	#14556591	—	2	4	no	—
32	#14556591	—	2	4	no	—
33	#14556583	—	2	4	no	—
34	#14556583	—	2	4	no	—
35	C₂×C₄⋊Dic₇₃	—	2	2	no	—
36	C₂×C₄⋊Dic₇₃	—	2	8	no	—
37	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
39	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
40	C₂×#11924007	—	2	2	no	—
41	C₂×#11924007	—	2	2	no	—
42	#14556525	—	2	2	no	—
43	#14556525	—	2	2	no	—
44	#14556525	—	2	4	no	—
45	#14556525	—	2	4	no	—
46	#14556525	—	2	8	no	—
47	#14556523	—	2	4	no	—
48	#14556523	—	2	8	no	—
49	#14556498	—	2	8	no	—
50	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
55	#97×C₇₃	D₄	1	4	yes	—
56	#97×C₇₃	Q₈	1	8	yes	complemented
57	C₄×C₈×C₇₃	D₄	1	4	yes	—
58	C₄×C₈×C₇₃	Q₈	1	8	yes	complemented
59	#95×C₇₃	D₄	1	2	yes	—
60	#95×C₇₃	D₄	1	2	yes	—
61	C₂×Dic₁₄₆	—	4	4	no	—
62	C₂×Dic₁₄₆	—	8	4	no	—
63	C₂²×Dic₇₃	—	2	2	no	—
64	C₂²×Dic₇₃	—	2	4	no	—
65	C₂²×Dic₇₃	—	2	2	no	—
66	C₄⋊Dic₇₃	—	2	2	no	—
67	C₄⋊Dic₇₃	—	2	2	no	—
68	C₄⋊Dic₇₃	—	2	2	no	—
69	C₄⋊Dic₇₃	—	2	4	no	—
70	C₄⋊Dic₇₃	—	2	2	no	—
71	C₄⋊Dic₇₃	—	2	4	no	—
72	C₄⋊Dic₇₃	—	4	4	no	—
73	C₄⋊Dic₇₃	—	8	8	no	—
74	#11924007	—	4	4	no	—
75	#11924007	—	4	4	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	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
84	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
85	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
86	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
87	C₄²×C₇₃	C₂×D₄	1	2	yes	—
88	C₄²×C₇₃	C₂×D₄	1	2	yes	—
89	C₄²×C₇₃	C₂×Q₈	1	4	yes	—
90	C₄²×C₇₃	C₂×Q₈	1	4	yes	—
91	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇₃	—	2	4	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	#127	1	2	yes	—
97	C₂×C₄×C₇₃	#127	1	2	yes	—
98	C₂×C₄×C₇₃	C₄⋊Q₈	1	8	yes	—
99	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
102	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
103	C₂×Dic₇₃	—	4	2	no	—
104	C₂×Dic₇₃	—	4	2	no	—
105	C₂×Dic₇₃	—	4	4	no	—
106	C₂×Dic₇₃	—	4	4	no	—
107	C₂×Dic₇₃	—	4	8	no	—
108	C₂×Dic₇₃	—	4	4	no	—
109	Dic₁₄₆	—	8	8	no	—
110	C₈×C₇₃	C₄⋊Q₈	1	4	yes	—
111	C₈×C₇₃	C₄⋊Q₈	1	4	yes	—
112	C₈×C₇₃	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	—	2	8	no	—
120	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₄×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
123	Dic₇₃	—	8	4	no	—
124	Dic₇₃	—	8	2	no	—
125	Dic₇₃	—	8	2	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	—	73	1	no	—
134	#2660	—	73	2	no	—
135	#1847	—	73	4	no	—
136	#1846	—	73	2	no	—
137	#1845	—	73	2	no	—
138	#1726	D₇₃	1	1	yes	characteristic
139	C₇₃	#33319	1	1	yes	characteristic, complemented, Sylow
140	C₂×C₄⋊Q₈	—	73	2	no	—
141	C₂×#125	—	73	2	no	—
142	#501	—	146	8	no	—
143	#500	—	146	4	no	—
144	#499	—	146	4	no	—
145	C₂×#107	—	73	1	no	—
146	C₂×#107	—	73	2	no	—
147	C₂×#107	—	73	4	no	—
148	C₂×#107	—	73	1	no	—
149	C₂×#107	—	73	2	no	—
150	C₂×#106	—	73	4	no	—
151	C₂×#106	—	73	2	no	—
152	C₂×#97	C₂×D₇₃	1	2	yes	—
153	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
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	C₂×D₇₃	1	1	yes	characteristic
161	C₂²×Q₈	—	146	2	no	—
162	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
165	C₄⋊Q₈	—	146	4	no	—
166	C₄⋊Q₈	—	146	4	no	—
167	#125	—	146	4	no	—
168	#125	—	146	4	no	—
169	C₂×C₄⋊C₄	—	73	2	no	—
170	C₂×C₄⋊C₄	—	73	2	no	—
171	C₂×C₄⋊C₄	—	73	4	no	—
172	C₂×C₄⋊C₄	—	146	2	no	—
173	C₂×C₄⋊C₄	—	146	2	no	—
174	C₂×C₄⋊C₄	—	146	2	no	—
175	C₂×C₄⋊C₄	—	146	8	no	—
176	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
178	#107	—	146	2	no	—
179	#107	—	146	2	no	—
180	#107	—	146	4	no	—
181	#107	—	146	4	no	—
182	#107	—	146	8	no	—
183	#106	—	146	4	no	—
184	#106	—	146	8	no	—
185	#97	Dic₁₄₆	1	8	yes	complemented
186	#97	D₂₉₂	1	4	yes	—
187	C₄×C₈	Dic₁₄₆	1	8	yes	complemented
188	C₄×C₈	D₂₉₂	1	4	yes	—
189	#95	—	146	8	no	—
190	#95	D₂₉₂	1	2	yes	—
191	#95	D₂₉₂	1	2	yes	—
192	C₂×Q₈	—	292	4	no	—
193	C₂×Q₈	—	584	4	no	—
194	C₂²×C₄	—	146	2	no	—
195	C₂²×C₄	—	146	4	no	—
196	C₂²×C₄	—	146	2	no	—
197	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₂₉₂	1	4	yes	—
200	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
201	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
202	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
203	C₂×C₈	C₂×D₂₉₂	1	4	yes	—
204	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
205	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
206	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
207	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
208	C₄⋊C₄	—	292	4	no	—
209	C₄⋊C₄	—	292	4	no	—
210	C₄⋊C₄	—	292	4	no	—
211	C₄⋊C₄	—	584	8	no	—
212	C₄⋊C₄	—	146	2	no	—
213	C₄⋊C₄	—	146	2	no	—
214	C₄⋊C₄	—	146	2	no	—
215	C₄⋊C₄	—	146	4	no	—
216	C₄⋊C₄	—	146	2	no	—
217	C₄⋊C₄	—	146	4	no	—
218	C₄²	C₂×D₂₉₂	1	2	yes	—
219	C₄²	C₂×D₂₉₂	1	2	yes	—
220	C₄²	C₂×Dic₁₄₆	1	4	yes	—
221	C₄²	C₂×Dic₁₄₆	1	4	yes	—
222	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
223	Q₈	—	584	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	292	2	no	—
229	C₂×C₄	—	292	2	no	—
230	C₂×C₄	—	292	4	no	—
231	C₂×C₄	—	292	4	no	—
232	C₂×C₄	—	292	8	no	—
233	C₂×C₄	—	292	4	no	—
234	C₂×C₄	#14556588	1	2	yes	—
235	C₂×C₄	#14556588	1	2	yes	—
236	C₂×C₄	#14556591	1	8	yes	—
237	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
238	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
239	C₂×C₄	C₂²×Dic₁₄₆	1	2	yes	—
240	C₂×C₄	C₂²×Dic₁₄₆	1	2	yes	—
241	C₈	#14556591	1	4	yes	—
242	C₈	#14556591	1	4	yes	—
243	C₈	#14556591	1	8	yes	—
244	C₂²	C₂×#14556588	1	1	yes	characteristic
245	C₂²	C₂×#14556591	1	2	yes	—
246	C₂²	#17989392	1	2	yes	—
247	C₂²	#17989404	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	8	no	—
251	C₄	—	584	4	no	—
252	C₄	—	584	2	no	—
253	C₄	—	584	2	no	—
254	C₄	C₂×#14556591	1	1	yes	characteristic
255	C₄	C₂×#14556591	1	1	yes	characteristic
256	C₄	C₂×#14556591	1	2	yes	—
257	C₂	#22618329	1	1	yes	characteristic
258	C₂	#22620417	1	1	yes	characteristic
259	C₂	#22620425	1	1	yes	characteristic
260	C₂	#22620470	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₉₂
commutator, cosocle	C₂²×C₄×C₇₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1726	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
hypercenter	#1726	D₇₃
hypocenter	C₇₃	#33319
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
56. #97×C₇₃	Q₈	semidirect
58. C₄×C₈×C₇₃	Q₈	semidirect
139. C₇₃	#33319	semidirect
185. #97	Dic₁₄₆	semidirect
187. C₄×C₈	Dic₁₄₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988996	2
C₂×#17988992	2
C₂×#17988989	2
#22616050	2
#22610099	2
#22610095	2
#22610091	2
C₇₃×#1726	2
#33319	73