Subgroups — #33239405

Order 18688.

642 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22623067	C₂	1	1	yes	maximal, characteristic
2	#22618849	C₂	1	2	yes	maximal, complemented
3	#22611818	C₂	1	4	yes	maximal, complemented
4	#22611813	C₂	1	4	yes	maximal, complemented
5	#22611726	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988395	C₂	1	2	yes	maximal, complemented
7	C₇₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556613	C₂²	1	2	yes	—
9	#17989353	—	2	2	no	—
10	#17989030	C₄	1	4	yes	complemented
11	#17989025	C₄	1	4	yes	complemented
12	#17989019	C₂²	1	4	yes	—
13	#17988710	—	2	4	no	—
14	#17988705	C₂²	1	8	yes	complemented
15	C₂×#14556527	C₂²	1	2	yes	—
16	#17988605	—	2	2	no	—
17	#17988417	C₂²	1	4	yes	complemented
18	#17988395	C₂²	1	4	yes	complemented
19	#17988395	C₂²	1	4	yes	complemented
20	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
22	#405×C₇₃	C₂²	1	1	yes	characteristic
23	#346×C₇₃	C₂²	1	2	yes	complemented
24	C₄×C₁₆×C₇₃	C₂²	1	2	yes	complemented
25	C₂×#11924028	—	2	2	no	—
26	#14556613	—	2	2	no	—
27	#14556613	—	2	2	no	—
28	#14556613	C₂×C₄	1	4	yes	—
29	#14556613	C₂³	1	4	yes	—
30	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
31	C₂×#11924017	—	2	2	no	—
32	C₂×D₅₈₄	C₂×C₄	1	2	yes	—
33	#14556599	—	4	4	no	—
34	#14556597	—	2	8	no	—
35	C₂×#11924012	—	2	2	no	—
36	M₁₆×D₇₃	—	4	4	no	—
37	Dic₅₈₄	C₂×C₄	1	4	yes	—
38	#14556536	C₂×C₄	1	8	yes	—
39	D₁₁₆₈	C₂×C₄	1	4	yes	—
40	#14556527	—	2	2	no	—
41	#14556527	C₂³	1	4	yes	—
42	#14556519	—	2	8	no	—
43	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
45	#117×C₇₃	D₄	1	1	yes	characteristic
46	M₃₂×C₇₃	C₂×C₄	1	4	yes	—
47	C₂×C₁₆×C₇₃	—	2	2	no	—
48	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
51	#97×C₇₃	D₄	1	2	yes	—
52	C₄×C₈×C₇₃	C₂³	1	2	yes	—
53	#11924028	—	2	4	no	—
54	#11924028	—	2	4	no	—
55	#11924028	—	4	2	no	—
56	#11924028	—	4	2	no	—
57	C₂×Dic₁₄₆	—	2	2	no	—
58	C₂×D₂₉₂	—	2	2	no	—
59	C₂×C₂²⋊D₇₃	—	4	2	no	—
60	C₂×C₄×D₇₃	—	4	2	no	—
61	Dic₂₉₂	—	2	2	no	—
62	Dic₂₉₂	C₂²×C₄	1	4	yes	—
63	#11924017	—	2	4	no	—
64	#11924017	—	2	4	no	—
65	D₅₈₄	—	2	2	no	—
66	D₅₈₄	C₂²×C₄	1	4	yes	—
67	#11924012	—	2	4	no	—
68	#11924012	—	4	2	no	—
69	#11924011	—	4	8	no	—
70	C₂×#10596831	—	4	2	no	—
71	C₈×D₇₃	—	4	8	no	—
72	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
73	M₁₆×C₇₃	D₈	1	4	yes	—
74	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₇₃	#41	1	2	yes	—
78	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₇₃	—	2	1	no	—
80	C₂²⋊C₄×C₇₃	—	2	1	no	—
81	C₄²×C₇₃	C₂×D₄	1	2	yes	—
82	C₁₆×C₇₃	—	2	1	no	—
83	C₁₆×C₇₃	—	2	1	no	—
84	C₁₆×C₇₃	C₂²×C₄	1	4	yes	—
85	C₂³×C₇₃	#135	1	1	yes	characteristic
86	C₂²×D₇₃	—	4	2	no	—
87	C₂×C₄×C₇₃	—	2	2	no	—
88	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
91	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
92	C₂²⋊D₇₃	—	4	4	no	—
93	C₂²⋊D₇₃	—	4	4	no	—
94	C₂×Dic₇₃	—	4	2	no	—
95	D₂₉₂	—	2	4	no	—
96	D₂₉₂	—	4	2	no	—
97	C₄×D₇₃	—	4	4	no	—
98	C₄×D₇₃	—	4	4	no	—
99	Dic₁₄₆	—	2	4	no	—
100	Dic₁₄₆	—	4	2	no	—
101	C₈×C₇₃	C₄×D₄	1	2	yes	—
102	C₈×C₇₃	C₂×D₈	1	4	yes	—
103	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
104	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
105	#10596831	—	4	4	no	—
106	C₂²×C₇₃	—	4	1	no	—
107	C₂²×C₇₃	C₄×D₈	1	2	yes	—
108	C₂²×C₇₃	C₂×#135	1	1	yes	characteristic
109	C₂×D₇₃	—	4	4	no	—
110	C₂×D₇₃	—	16	2	no	—
111	C₄×C₇₃	—	4	2	no	—
112	C₄×C₇₃	C₄×D₈	1	2	yes	—
113	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
114	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
115	Dic₇₃	—	4	4	no	—
116	#33464	—	73	1	no	maximal, Sylow
117	C₂×C₇₃	—	2	1	no	—
118	C₂×C₇₃	—	2	2	no	—
119	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
120	D₇₃	—	8	4	no	—
121	#3414	—	73	1	no	—
122	#2957	—	73	2	no	—
123	#2179	—	73	4	no	—
124	#2178	—	73	4	no	—
125	#2161	—	73	1	no	—
126	C₂×#359	—	73	2	no	—
127	#2108	D₇₃	1	1	yes	characteristic, complemented
128	C₇₃	#33464	1	1	yes	characteristic, complemented, Sylow
129	C₂×#135	—	73	2	no	—
130	#568	—	146	2	no	—
131	#510	—	73	4	no	—
132	#509	—	73	4	no	—
133	#508	—	73	4	no	—
134	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
135	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
136	#444	—	146	4	no	—
137	#443	—	73	8	no	—
138	C₂×#108	—	73	2	no	—
139	#421	—	146	2	no	—
140	#405	C₂×D₇₃	1	1	yes	characteristic
141	#368	—	73	4	no	—
142	#359	—	73	4	no	—
143	#359	—	73	4	no	—
144	#346	C₂×D₇₃	1	2	yes	complemented
145	C₄×C₁₆	C₂×D₇₃	1	2	yes	complemented
146	C₂×#41	—	146	2	no	—
147	#135	—	73	4	no	—
148	#135	—	73	4	no	—
149	#135	—	146	2	no	—
150	#135	—	146	2	no	—
151	C₂×Q₁₆	—	73	2	no	—
152	C₂×SD₁₆	—	146	2	no	—
153	C₂×D₈	—	73	2	no	—
154	#131	—	292	4	no	—
155	#131	—	146	8	no	—
156	C₂×M₁₆	—	292	4	no	—
157	C₂×M₁₆	—	146	2	no	—
158	C₂×M₁₆	D₂₉₂	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
160	#117	D₂₉₂	1	1	yes	characteristic
161	Q₃₂	—	73	4	no	—
162	SD₃₂	—	73	8	no	—
163	D₁₆	—	73	4	no	—
164	M₃₂	C₄×D₇₃	1	4	yes	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
167	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
168	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
169	#108	—	73	4	no	—
170	#108	—	146	2	no	—
171	#104	—	146	8	no	—
172	#97	D₂₉₂	1	2	yes	—
173	C₄×C₈	C₂²×D₇₃	1	2	yes	—
174	#41	—	292	2	no	—
175	#41	—	292	2	no	—
176	#41	—	146	4	no	—
177	#41	—	146	4	no	—
178	C₂×Q₈	—	146	2	no	—
179	C₂×D₄	—	292	2	no	—
180	C₂×D₄	—	146	2	no	—
181	C₂²×C₄	—	292	2	no	—
182	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
183	Q₁₆	—	73	4	no	—
184	Q₁₆	—	146	2	no	—
185	SD₁₆	—	146	4	no	—
186	SD₁₆	—	146	4	no	—
187	D₈	—	73	4	no	—
188	D₈	—	146	2	no	—
189	M₁₆	—	292	2	no	—
190	M₁₆	—	292	8	no	—
191	M₁₆	—	146	4	no	—
192	M₁₆	D₅₈₄	1	4	yes	—
193	C₂×C₈	—	292	2	no	—
194	C₂×C₈	—	292	8	no	—
195	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
196	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
197	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
198	C₂×C₈	#11924028	1	2	yes	—
199	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	C₂×D₂₉₂	1	2	yes	—
203	C₁₆	—	2	1	no	—
204	C₁₆	—	2	1	no	—
205	C₁₆	C₂×C₄×D₇₃	1	4	yes	—
206	C₂³	—	292	2	no	—
207	C₂³	#14556613	1	1	yes	characteristic
208	Q₈	—	292	2	no	—
209	Q₈	—	146	4	no	—
210	D₄	—	292	4	no	—
211	D₄	—	292	4	no	—
212	D₄	—	292	2	no	—
213	D₄	—	146	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	292	2	no	—
216	C₂×C₄	—	292	4	no	—
217	C₂×C₄	—	292	4	no	—
218	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
219	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
220	C₂×C₄	#14556613	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
222	C₈	—	292	4	no	—
223	C₈	C₄×D₂₉₂	1	2	yes	—
224	C₈	C₂×D₅₈₄	1	4	yes	—
225	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
226	C₈	C₂×#11924028	1	1	yes	characteristic
227	C₂²	—	4	1	no	—
228	C₂²	—	292	4	no	—
229	C₂²	—	1168	2	no	—
230	C₂²	C₄×D₅₈₄	1	2	yes	—
231	C₂²	C₂×#14556613	1	1	yes	characteristic
232	C₄	—	4	2	no	—
233	C₄	—	292	4	no	—
234	C₄	C₄×D₅₈₄	1	2	yes	—
235	C₄	C₂×C₄×D₂₉₂	1	1	yes	characteristic
236	C₄	C₂²×D₅₈₄	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	—	584	4	no	—
240	C₂	C₂×C₄×D₅₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₅₈₄
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2108	C₂
socle	C₂×C₇₃	C₂×C₄×D₈
prime radical	#22623067	C₂
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2108	D₇₃
hypocenter	C₇₃	#33464
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #22618849	C₂	semidirect
3. #22611818	C₂	semidirect
4. #22611813	C₂	semidirect
5. #22611726	C₂	semidirect
6. C₂×#17988395	C₂	semidirect
7. C₇₃×#2108	C₂	semidirect
10. #17989030	C₄	semidirect
11. #17989025	C₄	semidirect
14. #17988705	C₂²	semidirect
17. #17988417	C₂²	semidirect
18. #17988395	C₂²	semidirect
19. #17988395	C₂²	semidirect
23. #346×C₇₃	C₂²	semidirect
24. C₄×C₁₆×C₇₃	C₂²	semidirect
127. #2108	D₇₃	semidirect
128. C₇₃	#33464	semidirect
144. #346	C₂×D₇₃	semidirect
145. C₄×C₁₆	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22623067	2
#22618849	2
#22611818	2
#22611813	2
#22611726	2
C₂×#17988395	2
C₇₃×#2108	2
#33464	73