Subgroups — #34012129

Order 18688.

670 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989643	C₂	1	2	yes	maximal
2	#22624976	C₂	1	2	yes	maximal, complemented
3	#22624975	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556666	—	2	2	no	—
6	#17989643	—	2	4	no	—
7	#17989643	C₄	1	8	yes	complemented
8	C₂×#14556665	C₂²	1	4	yes	—
9	C₂×#14556664	—	2	2	no	—
10	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
11	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₄²×D₇₃	C₄	1	1	yes	characteristic
13	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₇₃	—	2	4	no	—
17	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
18	#14556666	—	2	8	no	—
19	#14556665	—	2	4	no	—
20	#14556665	C₂×C₄	1	8	yes	complemented
21	#14556664	—	2	8	no	—
22	C₂×#11924031	—	2	4	no	—
23	C₂²×C₄×D₇₃	—	2	1	no	—
24	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10596831	C₂³	1	1	yes	characteristic
27	C₂×C₈×D₇₃	—	2	2	no	—
28	C₂×C₈×D₇₃	—	2	2	no	—
29	C₂×C₈×D₇₃	C₈	1	2	yes	—
30	C₂×C₈×D₇₃	C₈	1	2	yes	—
31	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
32	C₂×C₈×D₇₃	D₄	1	4	yes	—
33	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
34	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
35	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
36	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
37	C₂×C₄×Dic₇₃	—	2	1	no	—
38	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×D₇₃	—	2	2	no	—
40	C₄²×D₇₃	C₈	1	4	yes	—
41	C₄×#10596831	—	2	2	no	—
42	C₈×Dic₇₃	—	2	2	no	—
43	C₈×Dic₇₃	—	2	2	no	—
44	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₇₃	—	2	2	no	—
48	#11924031	—	2	8	no	—
49	C₂³×D₇₃	M₁₆	1	1	yes	characteristic
50	C₂²×Dic₇₃	—	2	1	no	—
51	C₂²×Dic₇₃	M₁₆	1	1	yes	characteristic
52	C₂×C₄×D₇₃	—	2	8	no	—
53	C₂×C₄×D₇₃	—	2	4	no	—
54	C₂×C₄×D₇₃	C₄²	1	2	yes	—
55	C₂×C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
59	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
60	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
61	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
62	C₂×#10596831	—	2	1	no	—
63	C₂×#10596831	—	2	1	no	—
64	C₂×#10596831	C₂×C₈	1	2	yes	—
65	C₂×#10596831	C₂²×C₄	1	2	yes	—
66	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
67	C₂×#10596831	C₂×D₄	1	2	yes	—
68	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
69	C₈×D₇₃	—	2	4	no	—
70	C₈×D₇₃	—	2	4	no	—
71	C₈×D₇₃	C₂²⋊C₄	1	4	yes	—
72	C₈×D₇₃	C₂²⋊C₄	1	4	yes	—
73	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
74	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
75	C₄×Dic₇₃	—	2	1	no	—
76	C₄×Dic₇₃	—	2	4	no	—
77	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
78	C₂²×C₄×C₇₃	—	2	1	no	—
79	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₃	—	2	1	no	—
81	C₂×C₈×C₇₃	—	2	1	no	—
82	C₂×C₈×C₇₃	C₄²	1	2	yes	—
83	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
87	C₄²×C₇₃	—	2	1	no	—
88	C₄²×C₇₃	C₂×C₈	1	2	yes	—
89	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂²×D₇₃	—	2	4	no	—
91	C₂²×D₇₃	#97	1	2	yes	—
92	C₂²×D₇₃	C₂²⋊C₈	1	2	yes	—
93	C₂²×D₇₃	C₄⋊C₈	1	1	yes	characteristic
94	C₂²×D₇₃	C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₇₃	—	2	4	no	—
96	C₂×C₄×C₇₃	—	2	2	no	—
97	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×Dic₇₃	—	2	4	no	—
104	C₂×Dic₇₃	—	2	2	no	—
105	C₂×Dic₇₃	#97	1	2	yes	—
106	C₂×Dic₇₃	C₂²⋊C₈	1	2	yes	—
107	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
108	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
109	C₄×D₇₃	—	2	16	no	—
110	C₄×D₇₃	#95	1	2	yes	—
111	C₄×D₇₃	#95	1	2	yes	—
112	C₄×D₇₃	C₄×C₈	1	2	yes	—
113	C₄×D₇₃	C₄×C₈	1	2	yes	—
114	C₄×D₇₃	C₂²⋊C₈	1	2	yes	—
115	C₄×D₇₃	C₂²⋊C₈	1	2	yes	—
116	C₄×D₇₃	C₄⋊C₈	1	2	yes	—
117	C₄×D₇₃	C₄⋊C₈	1	2	yes	—
118	C₈×C₇₃	—	2	2	no	—
119	C₈×C₇₃	#95	1	2	yes	—
120	C₈×C₇₃	#95	1	2	yes	—
121	#10596831	—	2	2	no	—
122	#10596831	C₂×C₂²⋊C₄	1	2	yes	—
123	#10596831	C₂×C₄⋊C₄	1	2	yes	—
124	C₂²×C₇₃	—	2	2	no	—
125	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
126	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
128	C₂×D₇₃	—	2	2	no	—
129	C₂×D₇₃	—	2	4	no	—
130	C₂×D₇₃	—	2	4	no	—
131	C₂×D₇₃	#336	1	2	yes	—
132	C₂×D₇₃	#336	1	2	yes	—
133	C₂×D₇₃	#364	1	4	yes	—
134	C₄×C₇₃	—	2	4	no	—
135	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
136	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
137	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
139	Dic₇₃	—	2	4	no	—
140	Dic₇₃	#336	1	2	yes	—
141	Dic₇₃	#336	1	2	yes	—
142	C₂×#1376	—	73	1	no	maximal, Sylow
143	C₂×C₇₃	—	2	1	no	—
144	C₂×C₇₃	—	2	1	no	—
145	C₂×C₇₃	C₂×#336	1	1	yes	characteristic
146	C₂×C₇₃	C₂×#364	1	2	yes	—
147	D₇₃	—	2	2	no	—
148	D₇₃	#1376	1	4	yes	complemented
149	C₂²×M₃₂	—	73	2	no	—
150	C₂²×C₄×C₈	—	73	1	no	—
151	#1376	—	73	2	no	—
152	#1376	—	73	2	no	—
153	C₇₃	C₂×#1376	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₈	—	73	1	no	—
155	C₂³×C₈	—	73	1	no	—
156	C₂²×C₄²	—	73	1	no	—
157	C₂×M₃₂	—	73	4	no	—
158	C₂×M₃₂	—	73	8	no	—
159	C₂×M₃₂	—	146	4	no	—
160	C₂×M₃₂	—	146	2	no	—
161	C₂²×C₁₆	—	146	2	no	—
162	C₂×C₄×C₈	—	73	1	no	—
163	C₂×C₄×C₈	—	73	1	no	—
164	C₂×C₄×C₈	—	73	1	no	—
165	C₂×C₄×C₈	—	146	4	no	—
166	C₂×C₄×C₈	#64408	1	1	yes	characteristic
167	C₂³×C₄	—	73	1	no	—
168	C₂³×C₄	—	146	1	no	—
169	C₂²×C₈	—	73	1	no	—
170	C₂²×C₈	—	73	4	no	—
171	C₂²×C₈	—	73	4	no	—
172	C₂²×C₈	—	73	1	no	—
173	C₂²×C₈	—	73	1	no	—
174	C₂²×C₈	—	73	1	no	—
175	C₂²×C₈	—	73	1	no	—
176	C₂²×C₈	—	73	1	no	—
177	C₂²×C₈	—	73	2	no	—
178	C₂²×C₈	—	73	2	no	—
179	C₂²×C₈	—	146	2	no	—
180	C₂²×C₈	—	146	2	no	—
181	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
182	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
183	C₂×C₄²	—	73	1	no	—
184	C₂×C₄²	—	73	4	no	—
185	C₂×C₄²	—	146	2	no	—
186	C₂×C₄²	—	146	1	no	—
187	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
188	M₃₂	—	73	8	no	—
189	M₃₂	—	146	4	no	—
190	M₃₂	—	146	8	no	—
191	C₂×C₁₆	—	146	4	no	—
192	C₂×C₁₆	—	146	8	no	—
193	C₄×C₈	—	2	2	no	—
194	C₄×C₈	—	146	2	no	—
195	C₄×C₈	—	146	2	no	—
196	C₄×C₈	—	146	2	no	—
197	C₂⁴	—	73	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	73	1	no	—
200	C₂²×C₄	—	73	2	no	—
201	C₂²×C₄	—	73	1	no	—
202	C₂²×C₄	—	73	1	no	—
203	C₂²×C₄	—	73	2	no	—
204	C₂²×C₄	—	73	2	no	—
205	C₂²×C₄	—	73	2	no	—
206	C₂²×C₄	—	73	1	no	—
207	C₂²×C₄	—	73	1	no	—
208	C₂²×C₄	—	146	4	no	—
209	C₂²×C₄	—	146	8	no	—
210	C₂²×C₄	—	146	1	no	—
211	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	73	2	no	—
215	C₂×C₈	—	73	2	no	—
216	C₂×C₈	—	73	1	no	—
217	C₂×C₈	—	73	1	no	—
218	C₂×C₈	—	73	4	no	—
219	C₂×C₈	—	73	4	no	—
220	C₂×C₈	—	73	4	no	—
221	C₂×C₈	—	73	4	no	—
222	C₂×C₈	—	73	2	no	—
223	C₂×C₈	—	146	4	no	—
224	C₂×C₈	—	146	4	no	—
225	C₂×C₈	—	146	1	no	—
226	C₂×C₈	—	146	1	no	—
227	C₂×C₈	C₄×#64408	1	2	yes	—
228	C₂×C₈	C₂²⋊#64408	1	2	yes	—
229	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
230	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
231	C₂×C₈	#11924036	1	2	yes	—
232	C₄²	—	2	1	no	—
233	C₄²	—	73	2	no	—
234	C₄²	—	146	4	no	—
235	C₄²	—	146	1	no	—
236	C₄²	#11924036	1	2	yes	—
237	C₁₆	—	146	8	no	—
238	C₂³	—	73	1	no	—
239	C₂³	—	73	2	no	—
240	C₂³	—	73	2	no	—
241	C₂³	—	73	1	no	—
242	C₂³	—	146	4	no	—
243	C₂³	D₇₃⋊M₁₆	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	73	2	no	—
247	C₂×C₄	—	73	2	no	—
248	C₂×C₄	—	73	1	no	—
249	C₂×C₄	—	73	1	no	—
250	C₂×C₄	—	73	2	no	—
251	C₂×C₄	—	73	2	no	—
252	C₂×C₄	—	73	2	no	—
253	C₂×C₄	—	73	2	no	—
254	C₂×C₄	—	73	2	no	—
255	C₂×C₄	—	73	2	no	—
256	C₂×C₄	—	73	2	no	—
257	C₂×C₄	—	73	2	no	—
258	C₂×C₄	—	146	16	no	—
259	C₂×C₄	—	146	4	no	—
260	C₂×C₄	—	146	2	no	—
261	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
262	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
264	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
265	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
266	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	73	2	no	—
269	C₈	—	73	2	no	—
270	C₈	—	146	2	no	—
271	C₈	(C₂×#64408)⋊C₄	1	2	yes	—
272	C₈	(C₂×#64408)⋊C₄	1	2	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	73	2	no	—
275	C₂²	—	73	4	no	—
276	C₂²	—	73	2	no	—
277	C₂²	—	146	4	no	—
278	C₂²	—	146	4	no	—
279	C₂²	—	146	2	no	—
280	C₂²	#17989546	1	1	yes	characteristic
281	C₂²	#17989555	1	1	yes	characteristic
282	C₂²	#17989588	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	73	2	no	—
285	C₄	—	73	2	no	—
286	C₄	—	146	4	no	—
287	C₄	C₂×(C₂×#64408)⋊C₄	1	1	yes	characteristic
288	C₄	C₄×#11924036	1	1	yes	characteristic
289	C₄	#17989555	1	1	yes	characteristic
290	C₄	#17989588	1	1	yes	characteristic
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	—	73	4	no	—
294	C₂	—	146	2	no	—
295	C₂	#22625592	1	1	yes	characteristic
296	C₂	#22626270	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄⋊#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂³×D₇₃	M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₂×#1376
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22624976	C₂	semidirect
3. #22624975	C₂	semidirect
7. #17989643	C₄	semidirect
20. #14556665	C₂×C₄	semidirect
148. D₇₃	#1376	semidirect
153. C₇₃	C₂×#1376	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989643	2
#22624976	2
#22624975	2
C₂×C₄×C₈×D₇₃	2
C₂×#1376	73