Subgroups — #33001048

Order 18688.

466 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988542	C₂	1	1	yes	maximal, characteristic
2	#22611542	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988364	C₂	1	2	yes	maximal, complemented
4	#22608010	C₂	1	2	yes	maximal, complemented
5	C₇₃×#2112	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924012	C₄	1	1	yes	characteristic
7	C₂×#14556550	C₄	1	1	yes	characteristic
8	C₂×#14556531	C₂²	1	2	yes	—
9	#17988542	—	2	4	no	—
10	C₂×#14556501	C₂²	1	1	yes	characteristic
11	C₂×#14556501	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
14	C₂²⋊#11924003	C₄	1	4	yes	complemented
15	#17988364	—	2	4	no	—
16	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
18	C₂²⋊C₁₆×C₇₃	C₄	1	4	yes	complemented
19	C₂×#11924012	—	2	2	no	—
20	C₂×#11924012	—	2	2	no	—
21	C₂×#11924012	—	2	2	no	—
22	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
23	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
24	#14556550	—	2	2	no	—
25	#14556550	—	2	2	no	—
26	C₂×C₄⋊Dic₇₃	—	2	1	no	—
27	C₂×C₂²⋊Dic₇₃	—	2	1	no	—
28	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	#14556531	—	2	2	no	—
31	#14556531	—	2	2	no	—
32	C₂×#11924003	C₂×C₄	1	4	yes	—
33	#14556501	—	2	2	no	—
34	#14556501	D₄	1	4	yes	—
35	C₈×Dic₇₃	—	2	2	no	—
36	C₈×Dic₇₃	—	2	2	no	—
37	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₃	—	2	2	no	—
39	C₂²×C₈×C₇₃	—	2	2	no	—
40	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
44	M₃₂×C₇₃	—	2	2	no	—
45	M₃₂×C₇₃	—	2	2	no	—
46	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	—
47	C₂²×Dic₇₃	—	2	1	no	—
48	C₂²×Dic₇₃	—	2	1	no	—
49	#11924012	—	2	8	no	—
50	C₂×#10596831	—	2	1	no	—
51	C₂×#10596831	—	2	1	no	—
52	C₂×#10596831	—	2	1	no	—
53	C₂×#10596831	—	2	1	no	—
54	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
55	C₂×#10596831	M₁₆	1	2	yes	—
56	C₄⋊Dic₇₃	—	2	4	no	—
57	C₂²⋊Dic₇₃	—	2	4	no	—
58	C₄×Dic₇₃	—	2	1	no	—
59	C₄×Dic₇₃	—	2	1	no	—
60	C₄×Dic₇₃	C₂²⋊C₄	1	2	yes	—
61	C₄×Dic₇₃	M₁₆	1	2	yes	—
62	#11924003	—	2	4	no	—
63	C₂⁴×C₇₃	M₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	—	2	1	no	—
65	C₂²×C₄×C₇₃	—	2	1	no	—
66	C₂²×C₄×C₇₃	—	2	1	no	—
67	C₂²×C₄×C₇₃	—	2	1	no	—
68	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
69	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
71	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₇₃	—	2	1	no	—
73	C₂×C₈×C₇₃	—	2	1	no	—
74	C₂×C₈×C₇₃	—	2	4	no	—
75	C₂×C₈×C₇₃	—	2	4	no	—
76	C₂×C₈×C₇₃	C₄²	1	2	yes	—
77	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₇₃	#41	1	2	yes	—
81	C₁₆×C₇₃	—	2	4	no	—
82	C₂³×C₇₃	—	2	1	no	—
83	C₂³×C₇₃	—	2	1	no	—
84	C₂³×C₇₃	—	4	2	no	—
85	C₂³×C₇₃	#97	1	2	yes	—
86	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇₃	—	2	1	no	—
88	C₂×C₄×C₇₃	—	2	1	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	#97	1	2	yes	—
94	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
97	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×Dic₇₃	—	2	2	no	—
101	C₂×Dic₇₃	—	2	2	no	—
102	C₂×Dic₇₃	—	4	2	no	—
103	C₂×Dic₇₃	—	4	2	no	—
104	C₈×C₇₃	—	2	2	no	—
105	C₈×C₇₃	C₄×D₄	1	2	yes	—
106	C₈×C₇₃	C₄×D₄	1	2	yes	—
107	#10596831	—	2	2	no	—
108	#10596831	—	2	2	no	—
109	C₂²×C₇₃	—	2	2	no	—
110	C₂²×C₇₃	—	2	4	no	—
111	C₂²×C₇₃	—	2	2	no	—
112	C₂²×C₇₃	—	4	2	no	—
113	C₂²×C₇₃	—	4	2	no	—
114	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
115	C₂²×C₇₃	#407	1	1	yes	characteristic
116	C₂²×C₇₃	#432	1	1	yes	characteristic
117	C₄×C₇₃	—	2	2	no	—
118	C₄×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
120	C₄×C₇₃	#407	1	1	yes	characteristic
121	C₄×C₇₃	#432	1	1	yes	characteristic
122	Dic₇₃	—	4	2	no	—
123	Dic₇₃	—	4	2	no	—
124	#12606	—	73	1	no	maximal, Sylow
125	C₂×C₇₃	—	2	1	no	—
126	C₂×C₇₃	—	2	1	no	—
127	C₂×C₇₃	—	4	2	no	—
128	C₂×C₇₃	#1749	1	1	yes	characteristic
129	C₂×C₇₃	#2110	1	2	yes	—
130	C₂×#405	—	73	1	no	—
131	#2112	—	73	1	no	—
132	#2112	D₇₃	1	1	yes	characteristic
133	C₂×#347	—	73	2	no	—
134	#1375	—	73	2	no	—
135	C₇₃	#12606	1	1	yes	characteristic, complemented, Sylow
136	C₂²×M₁₆	—	73	1	no	—
137	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
138	C₂×#117	—	73	1	no	—
139	C₂×M₃₂	—	73	2	no	—
140	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
141	#405	—	146	4	no	—
142	C₂×#97	—	73	1	no	—
143	C₂×#97	—	73	1	no	—
144	C₂×C₄×C₈	—	73	1	no	—
145	C₂×C₄×C₈	—	73	1	no	—
146	C₂²⋊C₁₆	—	73	4	no	—
147	C₂²⋊C₁₆	Dic₇₃	1	4	yes	complemented
148	#347	—	146	4	no	—
149	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
150	C₂×M₁₆	—	146	2	no	—
151	C₂×M₁₆	—	146	2	no	—
152	C₂×M₁₆	—	146	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	73	1	no	—
156	C₂²×C₈	—	73	1	no	—
157	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
158	C₂²×C₈	C₄×D₇₃	1	2	yes	—
159	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
160	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
161	#117	—	146	2	no	—
162	#117	—	146	2	no	—
163	C₂×C₄⋊C₄	—	146	1	no	—
164	C₂×C₂²⋊C₄	—	146	1	no	—
165	C₂×C₄²	—	73	1	no	—
166	C₂×C₄²	—	73	1	no	—
167	M₃₂	—	2	2	no	—
168	M₃₂	—	2	2	no	—
169	M₃₂	—	146	2	no	—
170	M₃₂	—	146	2	no	—
171	C₂×C₁₆	—	73	4	no	—
172	C₂×C₁₆	C₂×Dic₇₃	1	4	yes	—
173	#97	—	73	4	no	—
174	#97	—	146	2	no	—
175	C₄×C₈	—	146	2	no	—
176	C₄×C₈	—	146	2	no	—
177	C₂⁴	#11924011	1	1	yes	characteristic
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	146	1	no	—
183	C₂²×C₄	—	146	1	no	—
184	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
185	C₂²×C₄	#11924011	1	1	yes	characteristic
186	C₂²×C₄	#11924011	1	2	yes	—
187	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
188	M₁₆	—	146	8	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	73	2	no	—
194	C₂×C₈	—	73	2	no	—
195	C₂×C₈	—	146	1	no	—
196	C₂×C₈	—	146	1	no	—
197	C₂×C₈	—	146	1	no	—
198	C₂×C₈	—	146	1	no	—
199	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
200	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
201	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
202	C₂×C₈	D₄×D₇₃	1	2	yes	—
203	C₂×C₈	#11924027	1	2	yes	—
204	C₄⋊C₄	—	146	4	no	—
205	C₂²⋊C₄	—	146	4	no	—
206	C₄²	—	73	2	no	—
207	C₄²	—	73	2	no	—
208	C₄²	—	146	1	no	—
209	C₄²	—	146	1	no	—
210	C₁₆	—	2	4	no	—
211	C₁₆	—	146	4	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	4	2	no	—
215	C₂³	#14556501	1	2	yes	—
216	C₂³	C₂×#11924011	1	1	yes	characteristic
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	292	2	no	—
224	C₂×C₄	—	292	2	no	—
225	C₂×C₄	—	146	2	no	—
226	C₂×C₄	—	146	2	no	—
227	C₂×C₄	#14556501	1	2	yes	—
228	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
229	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
230	C₂×C₄	#14556547	1	1	yes	characteristic
231	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
232	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
233	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	146	2	no	—
236	C₈	—	146	2	no	—
237	C₈	#14556551	1	2	yes	—
238	C₈	D₄×Dic₇₃	1	2	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	C₂×#14556501	1	1	yes	characteristic
245	C₂²	#17988549	1	1	yes	characteristic
246	C₂²	#17988641	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	292	2	no	—
249	C₄	—	292	2	no	—
250	C₄	C₂²⋊C₄×Dic₇₃	1	1	yes	characteristic
251	C₄	M₁₆×Dic₇₃	1	1	yes	characteristic
252	C₄	#17988549	1	1	yes	characteristic
253	C₄	#17988641	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	#22609695	1	1	yes	characteristic
258	C₂	#22611535	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#2112	C₂
socle	C₂²×C₇₃	C₂×#97
prime radical	C₂⁴×C₇₃	M₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#2112	D₇₃
hypocenter	C₇₃	#12606
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #2112

Complements

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

N	H ≅ G/N	type
3. C₂×#17988364	C₂	semidirect
4. #22608010	C₂	semidirect
14. C₂²⋊#11924003	C₄	semidirect
18. C₂²⋊C₁₆×C₇₃	C₄	semidirect
135. C₇₃	#12606	semidirect
147. C₂²⋊C₁₆	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988542	2
#22611542	2
C₂×#17988364	2
#22608010	2
C₇₃×#2112	2
#12606	73