Subgroups — #33096622

Order 18688.

658 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	#22618569	C₂	1	1	yes	maximal, characteristic
2	#22618495	C₂	1	1	yes	maximal, characteristic
3	#22610707	C₂	1	4	yes	maximal, complemented
4	#22610138	C₂	1	4	yes	maximal
5	#22610106	C₂	1	1	yes	maximal, characteristic, complemented
6	#22610103	C₂	1	1	yes	maximal, characteristic, complemented
7	#22610072	C₂	1	1	yes	maximal, characteristic, complemented
8	#22610069	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#433×C₇₃	C₂	1	1	yes	maximal, characteristic
10	C₂²×#11924012	C₂²	1	2	yes	—
11	#17988648	—	2	4	no	—
12	C₂×#14556527	C₂²	1	4	yes	—
13	C₂×#14556527	C₂²	1	4	yes	—
14	#17988614	—	2	4	no	—
15	#17988611	C₂²	1	2	yes	—
16	#17988611	C₂²	1	1	yes	characteristic
17	#17988611	C₂²	1	1	yes	characteristic
18	#17988611	C₂²	1	1	yes	characteristic
19	#17988611	C₂²	1	1	yes	characteristic
20	#17988551	C₂²	1	2	yes	—
21	C₄×#11924012	C₂²	1	2	yes	—
22	#17988352	C₂²	1	4	yes	—
23	#17988352	C₂²	1	4	yes	—
24	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
25	#433×C₇₃	C₄	1	8	yes	—
26	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₇₃	C₂²	1	2	yes	—
28	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
30	C₂×#11924012	—	2	2	no	—
31	C₂×#11924012	—	4	4	no	—
32	C₂×#11924012	C₂³	1	2	yes	—
33	C₂×#11924012	C₂³	1	2	yes	—
34	C₂×#11924012	C₂³	1	2	yes	—
35	C₂×#11924012	C₂³	1	2	yes	—
36	C₂²×#10596831	—	2	2	no	—
37	#14556527	—	4	4	no	—
38	#14556527	—	4	4	no	—
39	C₄⋊#10596831	—	2	4	no	—
40	C₄⋊#10596831	—	2	2	no	—
41	C₄⋊#10596831	—	2	2	no	—
42	C₄⋊#10596831	—	2	2	no	—
43	C₄⋊#10596831	—	2	2	no	—
44	C₂²⋊#10596831	—	2	4	no	—
45	#14556502	—	2	2	no	—
46	C₄×#10596831	—	2	2	no	—
47	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
50	#117×C₇₃	C₂×C₄	1	4	yes	—
51	#117×C₇₃	D₄	1	4	yes	—
52	C₂×C₄⋊C₄×C₇₃	Q₈	1	2	yes	—
53	C₂×C₂²⋊C₄×C₇₃	Q₈	1	2	yes	—
54	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₇₃	C₂×C₄	1	8	yes	—
57	C₂²⋊C₈×C₇₃	C₂×C₄	1	8	yes	—
58	#97×C₇₃	C₂×C₄	1	4	yes	—
59	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
60	#11924012	—	2	4	no	—
61	#11924012	—	2	4	no	—
62	#11924012	—	2	4	no	—
63	#11924012	—	2	4	no	—
64	#11924012	—	4	4	no	—
65	C₂×#10596831	—	2	2	no	—
66	C₂×#10596831	—	2	2	no	—
67	C₂×#10596831	—	2	2	no	—
68	C₂×#10596831	—	2	2	no	—
69	C₂×#10596831	—	4	4	no	—
70	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	—	2	2	no	—
72	C₂²×C₄×C₇₃	—	2	2	no	—
73	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₇₃	—	2	2	no	—
80	C₂×C₈×C₇₃	—	2	2	no	—
81	C₂×C₈×C₇₃	—	2	2	no	—
82	C₂×C₈×C₇₃	—	2	2	no	—
83	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
85	C₄⋊C₄×C₇₃	C₄⋊C₄	1	8	yes	—
86	C₂²⋊C₄×C₇₃	C₄⋊C₄	1	8	yes	—
87	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₇₃	C₂×D₄	1	2	yes	—
90	C₄²×C₇₃	C₂×D₄	1	2	yes	—
91	C₂³×C₇₃	—	2	2	no	—
92	C₂³×C₇₃	—	4	2	no	—
93	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₇₃	—	2	2	no	—
96	C₂×C₄×C₇₃	—	2	4	no	—
97	C₂×C₄×C₇₃	—	2	2	no	—
98	C₂×C₄×C₇₃	—	2	2	no	—
99	C₂×C₄×C₇₃	—	4	2	no	—
100	C₂×C₄×C₇₃	—	4	2	no	—
101	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₃	H(5,𝔽₂)	1	2	yes	—
107	C₂×C₄×C₇₃	2₋¹⁺⁴	1	2	yes	—
108	C₈×C₇₃	—	2	4	no	—
109	C₈×C₇₃	—	2	4	no	—
110	#10596831	—	4	2	no	—
111	#10596831	—	4	2	no	—
112	#10596831	—	4	2	no	—
113	#10596831	—	4	2	no	—
114	C₂²×C₇₃	—	2	2	no	—
115	C₂²×C₇₃	—	2	2	no	—
116	C₂²×C₇₃	—	2	4	no	—
117	C₂²×C₇₃	—	8	2	no	—
118	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₇₃	#544	1	2	yes	—
120	C₄×C₇₃	—	2	2	no	—
121	C₄×C₇₃	—	2	4	no	—
122	C₄×C₇₃	—	2	4	no	—
123	C₄×C₇₃	#520	1	2	yes	—
124	C₄×C₇₃	#535	1	1	yes	characteristic
125	C₄×C₇₃	#536	1	1	yes	characteristic
126	#21995	—	73	1	no	maximal, Sylow
127	C₂×C₇₃	—	2	2	no	—
128	C₂×C₇₃	—	4	2	no	—
129	C₂×C₇₃	#2317	1	1	yes	characteristic
130	C₂×C₇₃	#2916	1	2	yes	—
131	#2921	—	73	1	no	—
132	C₂×#433	D₇₃	1	1	yes	characteristic
133	#2905	—	73	1	no	—
134	#1950	—	73	4	no	—
135	#1855	—	73	4	no	—
136	#1849	—	73	1	no	—
137	#1848	—	73	1	no	—
138	#1838	—	73	1	no	—
139	#1837	—	73	1	no	—
140	C₇₃	#21995	1	1	yes	characteristic, complemented, Sylow
141	C₂²×M₁₆	—	73	2	no	—
142	C₂×#117	C₂×D₇₃	1	1	yes	characteristic
143	#433	—	146	4	no	—
144	#433	Dic₇₃	1	8	yes	—
145	C₂×#108	—	73	4	no	—
146	C₂×#108	—	73	4	no	—
147	#424	—	146	4	no	—
148	#423	—	73	1	no	—
149	#423	—	73	1	no	—
150	#423	—	73	1	no	—
151	#423	—	73	1	no	—
152	#423	—	73	2	no	—
153	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
154	#407	—	73	2	no	—
155	C₂×C₂²⋊C₈	C₂×D₇₃	1	2	yes	—
156	C₄×M₁₆	—	73	2	no	—
157	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
159	#341	—	73	4	no	—
160	#341	—	73	4	no	—
161	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
162	C₂×M₁₆	—	292	4	no	—
163	C₂×M₁₆	—	73	2	no	—
164	C₂×M₁₆	—	73	2	no	—
165	C₂×M₁₆	—	73	2	no	—
166	C₂×M₁₆	—	73	2	no	—
167	C₂×M₁₆	—	146	2	no	—
168	C₂²×C₈	—	146	2	no	—
169	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
170	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
171	#117	D₂₉₂	1	4	yes	—
172	#117	C₂×Dic₇₃	1	4	yes	—
173	C₂×C₄⋊C₄	Dic₁₄₆	1	2	yes	—
174	C₂×C₂²⋊C₄	Dic₁₄₆	1	2	yes	—
175	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
176	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
177	#108	—	292	4	no	—
178	#108	—	292	4	no	—
179	C₄⋊C₈	—	146	4	no	—
180	C₄⋊C₈	—	146	2	no	—
181	C₄⋊C₈	—	146	2	no	—
182	C₄⋊C₈	—	146	2	no	—
183	C₄⋊C₈	—	146	2	no	—
184	C₄⋊C₈	C₂×Dic₇₃	1	8	yes	—
185	C₂²⋊C₈	—	146	4	no	—
186	C₂²⋊C₈	C₂×Dic₇₃	1	8	yes	—
187	#97	—	146	2	no	—
188	#97	C₂×Dic₇₃	1	4	yes	—
189	C₄×C₈	—	146	2	no	—
190	C₄×C₈	C₂×Dic₇₃	1	4	yes	—
191	C₂⁴	C₂×Dic₁₄₆	1	1	yes	characteristic
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
195	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
196	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
197	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
198	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
199	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
200	M₁₆	—	292	4	no	—
201	M₁₆	—	146	4	no	—
202	M₁₆	—	146	4	no	—
203	M₁₆	—	146	4	no	—
204	M₁₆	—	146	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	292	4	no	—
210	C₂×C₈	—	146	2	no	—
211	C₂×C₈	—	146	2	no	—
212	C₂×C₈	—	146	2	no	—
213	C₂×C₈	—	146	2	no	—
214	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
215	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
216	C₄⋊C₄	C₄⋊Dic₇₃	1	8	yes	—
217	C₂²⋊C₄	C₄⋊Dic₇₃	1	8	yes	—
218	C₄²	C₂²×Dic₇₃	1	2	yes	—
219	C₄²	C₂²×Dic₇₃	1	2	yes	—
220	C₄²	C₂×D₂₉₂	1	2	yes	—
221	C₄²	C₂×D₂₉₂	1	2	yes	—
222	C₂³	—	2	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	C₂×C₄⋊Dic₇₃	1	2	yes	—
225	C₂³	C₂²×Dic₁₄₆	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
233	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
234	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
235	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
237	C₂×C₄	#14556637	1	2	yes	—
238	C₂×C₄	#14556638	1	2	yes	—
239	C₈	—	2	4	no	—
240	C₈	—	2	4	no	—
241	C₈	—	292	2	no	—
242	C₈	—	292	2	no	—
243	C₈	—	292	2	no	—
244	C₈	—	292	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
250	C₂²	#17989215	1	2	yes	—
251	C₄	—	2	2	no	—
252	C₄	—	2	4	no	—
253	C₄	—	2	4	no	—
254	C₄	#17989070	1	2	yes	—
255	C₄	#17989156	1	1	yes	characteristic
256	C₄	#17989163	1	1	yes	characteristic
257	C₂	—	2	2	no	—
258	C₂	—	4	2	no	—
259	C₂	#22612554	1	1	yes	characteristic
260	C₂	#22618542	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×#433×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇₃	C₂×Q₈
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#433	D₇₃
hypocenter	C₇₃	#21995
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#433

Complements

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

N	H ≅ G/N	type
3. #22610707	C₂	semidirect
5. #22610106	C₂	semidirect
6. #22610103	C₂	semidirect
7. #22610072	C₂	semidirect
8. #22610069	C₂	semidirect
140. C₇₃	#21995	semidirect

Maximal subgroups

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

subgroup	index
#22618569	2
#22618495	2
#22610707	2
#22610138	2
#22610106	2
#22610103	2
#22610072	2
#22610069	2
C₂×#433×C₇₃	2
#21995	73