Subgroups — #33001027

Order 18688.

490 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988544	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988368	C₂	1	2	yes	maximal
3	M₃₂×Dic₇₃	C₂	1	2	yes	maximal, complemented
4	#22608015	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1377	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11924011	C₂²	1	1	yes	characteristic
7	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
8	C₂×#14556548	C₄	1	1	yes	characteristic
9	C₂×#14556531	C₂²	1	2	yes	—
10	#17988544	—	2	2	no	—
11	#17988544	—	2	2	no	—
12	C₂×#14556501	C₄	1	1	yes	characteristic
13	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	#17988368	C₄	1	8	yes	complemented
16	#17988362	—	2	2	no	—
17	C₁₆×Dic₇₃	—	2	2	no	—
18	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
19	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×#11924011	—	2	2	no	—
22	C₂×#11924011	—	2	1	no	—
23	C₂×#11924011	—	2	1	no	—
24	C₂×#11924011	C₂×C₄	1	4	yes	—
25	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₇₃	—	2	1	no	—
28	C₂×C₈×D₇₃	—	2	1	no	—
29	C₂×C₈×D₇₃	—	2	2	no	—
30	C₂×C₈×D₇₃	C₈	1	4	yes	—
31	#14556548	—	2	2	no	—
32	#14556548	—	2	2	no	—
33	C₂×#11924007	C₈	1	2	yes	—
34	C₂×#11924005	C₈	1	2	yes	—
35	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
36	#14556531	—	2	2	no	—
37	#14556531	C₂×C₄	1	4	yes	—
38	C₂×#11924003	—	2	2	no	—
39	#14556501	—	2	2	no	—
40	C₄×#10596831	—	2	1	no	—
41	C₄×#10596831	—	2	1	no	—
42	C₈×Dic₇₃	D₄	1	2	yes	—
43	C₈×Dic₇₃	D₄	1	2	yes	—
44	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
47	M₃₂×C₇₃	—	2	2	no	—
48	M₃₂×C₇₃	C₂×C₄	1	4	yes	—
49	C₂×C₁₆×C₇₃	—	2	2	no	—
50	#97×C₇₃	—	2	2	no	—
51	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
54	C₂×C₄×D₇₃	—	2	1	no	—
55	C₂×C₄×D₇₃	—	2	1	no	—
56	C₂×C₄×D₇₃	—	2	1	no	—
57	C₂×C₄×D₇₃	—	2	1	no	—
58	C₂×C₄×D₇₃	C₄²	1	2	yes	—
59	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
60	#11924011	—	2	4	no	—
61	#11924011	—	2	4	no	—
62	C₂×#10596831	—	2	1	no	—
63	C₂×#10596831	—	2	1	no	—
64	C₂×#10596831	C₄²	1	2	yes	—
65	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×#10596831	C₂×C₈	1	2	yes	—
68	C₂×#10596831	M₁₆	1	2	yes	—
69	C₈×D₇₃	—	2	4	no	—
70	C₈×D₇₃	—	2	4	no	—
71	#11924007	—	2	4	no	—
72	#11924005	—	2	4	no	—
73	C₄×Dic₇₃	C₂²⋊C₄	1	2	yes	—
74	C₄×Dic₇₃	M₁₆	1	2	yes	—
75	#11924003	—	2	4	no	—
76	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
77	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₇₃	—	2	1	no	—
79	C₂×C₈×C₇₃	—	2	1	no	—
80	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
81	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
85	C₄²×C₇₃	—	2	1	no	—
86	C₄²×C₇₃	—	2	1	no	—
87	C₁₆×C₇₃	—	2	4	no	—
88	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
89	C₂²×D₇₃	—	2	1	no	—
90	C₂²×D₇₃	—	2	1	no	—
91	C₂²×D₇₃	—	4	2	no	—
92	C₂²×D₇₃	C₄×C₈	1	2	yes	—
93	C₂×C₄×C₇₃	—	2	2	no	—
94	C₂×C₄×C₇₃	—	4	2	no	—
95	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₇₃	—	2	1	no	—
102	C₂×Dic₇₃	—	2	1	no	—
103	C₂×Dic₇₃	—	4	2	no	—
104	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
105	C₂×Dic₇₃	C₂²⋊C₈	1	4	yes	—
106	C₄×D₇₃	—	2	2	no	—
107	C₄×D₇₃	—	2	2	no	—
108	C₄×D₇₃	—	2	2	no	—
109	C₄×D₇₃	—	2	2	no	—
110	C₈×C₇₃	—	2	2	no	—
111	C₈×C₇₃	C₄×D₄	1	2	yes	—
112	C₈×C₇₃	C₄×D₄	1	2	yes	—
113	#10596831	—	2	2	no	—
114	#10596831	C₂²⋊C₈	1	2	yes	—
115	#10596831	C₂²⋊C₈	1	2	yes	—
116	C₂²×C₇₃	—	2	2	no	—
117	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
118	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₇₃	#431	1	1	yes	characteristic
120	C₂×D₇₃	—	2	2	no	—
121	C₂×D₇₃	—	4	2	no	—
122	C₂×D₇₃	—	4	2	no	—
123	C₂×D₇₃	—	4	2	no	—
124	C₄×C₇₃	—	4	2	no	—
125	C₄×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
127	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₄×C₇₃	#431	1	1	yes	characteristic
129	Dic₇₃	—	2	4	no	—
130	Dic₇₃	—	2	2	no	—
131	#12602	—	73	1	no	maximal, Sylow
132	C₂×C₇₃	—	2	1	no	—
133	C₂×C₇₃	—	2	1	no	—
134	C₂×C₇₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₇₃	#2115	1	2	yes	—
136	D₇₃	—	4	2	no	—
137	C₂×#405	—	73	1	no	—
138	C₂×#349	—	73	2	no	—
139	C₄×M₃₂	—	73	2	no	—
140	#1377	—	73	1	no	—
141	#1377	D₇₃	1	1	yes	characteristic, complemented
142	C₇₃	#12602	1	1	yes	characteristic, complemented, Sylow
143	C₂²×M₁₆	—	73	1	no	—
144	C₂³×C₈	—	73	1	no	—
145	C₂×#117	—	73	1	no	—
146	C₂×M₃₂	—	73	2	no	—
147	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
148	#405	—	146	2	no	—
149	#405	—	146	2	no	—
150	C₂×#97	—	73	1	no	—
151	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
152	C₂×C₄×C₈	—	73	1	no	—
153	C₂×C₄×C₈	—	73	1	no	—
154	#349	—	73	8	no	—
155	#346	—	146	2	no	—
156	C₄×C₁₆	—	146	2	no	—
157	C₂³×C₄	—	73	1	no	—
158	C₂×M₁₆	—	73	4	no	—
159	C₂×M₁₆	—	146	2	no	—
160	C₂×M₁₆	—	146	1	no	—
161	C₂×M₁₆	—	146	1	no	—
162	C₂²×C₈	—	73	1	no	—
163	C₂²×C₈	—	73	1	no	—
164	C₂²×C₈	—	73	4	no	—
165	C₂²×C₈	—	146	1	no	—
166	C₂²×C₈	—	146	1	no	—
167	C₂²×C₈	—	146	2	no	—
168	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
170	#117	—	146	2	no	—
171	#117	—	146	2	no	—
172	C₂×C₄⋊C₄	—	73	2	no	—
173	C₂×C₂²⋊C₄	—	73	2	no	—
174	C₂×C₄²	—	73	1	no	—
175	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
176	M₃₂	—	2	2	no	—
177	M₃₂	—	73	4	no	—
178	M₃₂	—	146	2	no	—
179	M₃₂	C₄×D₇₃	1	4	yes	—
180	C₂×C₁₆	—	2	2	no	—
181	C₂×C₁₆	—	146	2	no	—
182	#97	—	2	2	no	—
183	#97	—	146	2	no	—
184	C₄×C₈	—	73	2	no	—
185	C₄×C₈	—	73	2	no	—
186	C₄×C₈	—	146	1	no	—
187	C₄×C₈	—	146	1	no	—
188	C₂⁴	—	73	1	no	—
189	C₂²×C₄	—	73	2	no	—
190	C₂²×C₄	—	73	1	no	—
191	C₂²×C₄	—	73	2	no	—
192	C₂²×C₄	—	73	2	no	—
193	C₂²×C₄	—	146	1	no	—
194	C₂²×C₄	—	146	1	no	—
195	C₂²×C₄	—	146	1	no	—
196	C₂²×C₄	—	146	1	no	—
197	C₂²×C₄	C₈×D₇₃	1	2	yes	—
198	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
199	M₁₆	—	146	4	no	—
200	M₁₆	—	146	4	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	73	1	no	—
204	C₂×C₈	—	73	1	no	—
205	C₂×C₈	—	73	2	no	—
206	C₂×C₈	—	73	2	no	—
207	C₂×C₈	—	73	2	no	—
208	C₂×C₈	—	146	4	no	—
209	C₂×C₈	—	146	4	no	—
210	C₂×C₈	—	146	1	no	—
211	C₂×C₈	—	146	1	no	—
212	C₂×C₈	C₈×D₇₃	1	2	yes	—
213	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
214	C₂×C₈	D₄×D₇₃	1	2	yes	—
215	C₂×C₈	#11924027	1	1	yes	characteristic
216	C₂×C₈	#11924029	1	1	yes	characteristic
217	C₄⋊C₄	—	146	4	no	—
218	C₂²⋊C₄	—	146	4	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	73	2	no	—
222	C₄²	—	73	2	no	—
223	C₁₆	—	2	4	no	—
224	C₁₆	—	146	4	no	—
225	C₂³	—	292	2	no	—
226	C₂³	—	73	2	no	—
227	C₂³	—	146	1	no	—
228	C₂³	—	146	1	no	—
229	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	292	2	no	—
233	C₂×C₄	—	73	2	no	—
234	C₂×C₄	—	73	4	no	—
235	C₂×C₄	—	146	1	no	—
236	C₂×C₄	—	146	1	no	—
237	C₂×C₄	—	146	2	no	—
238	C₂×C₄	—	146	2	no	—
239	C₂×C₄	—	146	2	no	—
240	C₂×C₄	—	146	2	no	—
241	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
242	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
243	C₂×C₄	#14556549	1	1	yes	characteristic
244	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
245	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
246	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	73	2	no	—
249	C₈	—	73	2	no	—
250	C₈	—	146	2	no	—
251	C₈	#14556551	1	2	yes	—
252	C₈	#14556553	1	2	yes	—
253	C₂²	—	2	2	no	—
254	C₂²	—	292	2	no	—
255	C₂²	—	292	2	no	—
256	C₂²	—	292	2	no	—
257	C₂²	—	146	2	no	—
258	C₂²	C₄×C₈×D₇₃	1	1	yes	characteristic
259	C₂²	C₂²⋊C₈×D₇₃	1	1	yes	characteristic
260	C₂²	#17988639	1	1	yes	characteristic
261	C₄	—	4	2	no	—
262	C₄	—	146	2	no	—
263	C₄	—	146	4	no	—
264	C₄	#17988434	1	1	yes	characteristic
265	C₄	#17988539	1	1	yes	characteristic
266	C₄	C₂²⋊C₈×D₇₃	1	1	yes	characteristic
267	C₄	#17988639	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	292	2	no	—
271	C₂	#22609688	1	1	yes	characteristic
272	C₂	#22611550	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₇₃
commutator	C₂²×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1377	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1377	D₇₃
hypocenter	C₇₃	#12602
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1377

Complements

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

N	H ≅ G/N	type
3. M₃₂×Dic₇₃	C₂	semidirect
4. #22608015	C₂	semidirect
5. C₇₃×#1377	C₂	semidirect
15. #17988368	C₄	semidirect
141. #1377	D₇₃	semidirect
142. C₇₃	#12602	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988544	2
C₂×#17988368	2
M₃₂×Dic₇₃	2
#22608015	2
C₇₃×#1377	2
#12602	73