Subgroups — #34012120

Order 18688.

630 conjugacy classes of subgroups.

The subgroup lattice has 292 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	#22624973	C₂	1	2	yes	maximal, complemented
3	#22624972	C₂	1	2	yes	maximal, complemented
4	C₂×#97×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556666	C₄	1	4	yes	complemented
6	#17989643	—	2	4	no	—
7	#17989643	—	2	4	no	—
8	C₂×#14556665	C₂²	1	4	yes	—
9	C₂×#14556664	C₄	1	4	yes	complemented
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×#14556502	C₂²	1	1	yes	characteristic
14	C₂×#14556501	C₄	1	1	yes	characteristic
15	C₂×#14556501	C₂²	1	1	yes	characteristic
16	#97×D₇₃	—	2	4	no	—
17	C₂×#97×C₇₃	C₄	1	1	yes	characteristic
18	#14556666	—	2	8	no	—
19	#14556665	—	2	4	no	—
20	#14556665	—	2	4	no	—
21	#14556664	—	2	8	no	—
22	C₂×#11924031	C₂×C₄	1	8	yes	complemented
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	1	no	—
29	C₂×C₈×D₇₃	—	2	1	no	—
30	C₂×C₈×D₇₃	C₈	1	4	yes	—
31	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₇₃	D₄	1	4	yes	—
34	C₂×C₈×D₇₃	D₄	1	2	yes	—
35	C₂×C₈×D₇₃	Q₈	1	2	yes	—
36	C₂×C₄×Dic₇₃	—	2	1	no	—
37	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄²×D₇₃	—	2	2	no	—
39	C₄²×D₇₃	C₈	1	4	yes	—
40	#14556502	—	2	2	no	—
41	#14556501	—	2	2	no	—
42	#14556501	—	2	2	no	—
43	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
46	#97×C₇₃	—	2	2	no	—
47	#11924031	—	2	8	no	—
48	C₂³×D₇₃	M₁₆	1	1	yes	characteristic
49	C₂²×Dic₇₃	—	2	1	no	—
50	C₂²×Dic₇₃	M₁₆	1	1	yes	characteristic
51	C₂×C₄×D₇₃	—	2	4	no	—
52	C₂×C₄×D₇₃	—	4	4	no	—
53	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
54	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
55	C₂×C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
58	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
59	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
60	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
61	C₂×#10596831	—	2	1	no	—
62	C₂×#10596831	—	2	1	no	—
63	C₂×#10596831	C₂×C₈	1	2	yes	—
64	C₂×#10596831	C₂²×C₄	1	2	yes	—
65	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
66	C₂×#10596831	C₂×D₄	1	2	yes	—
67	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
68	C₈×D₇₃	—	2	4	no	—
69	C₈×D₇₃	—	2	4	no	—
70	C₈×D₇₃	C₂²⋊C₄	1	4	yes	—
71	C₈×D₇₃	C₂²⋊C₄	1	4	yes	—
72	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
73	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
74	C₄×Dic₇₃	—	2	1	no	—
75	C₄×Dic₇₃	—	2	4	no	—
76	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
77	C₂²×C₄×C₇₃	—	2	1	no	—
78	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₇₃	—	2	1	no	—
80	C₂×C₈×C₇₃	—	2	1	no	—
81	C₂×C₈×C₇₃	C₄²	1	2	yes	—
82	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
85	C₄²×C₇₃	—	2	1	no	—
86	C₄²×C₇₃	C₂×C₈	1	2	yes	—
87	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂²×D₇₃	—	2	4	no	—
89	C₂²×D₇₃	#97	1	1	yes	characteristic
90	C₂²×D₇₃	#97	1	1	yes	characteristic
91	C₂²×D₇₃	C₂²⋊C₈	1	2	yes	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₇₃	—	2	2	no	—
102	C₂×Dic₇₃	—	4	2	no	—
103	C₂×Dic₇₃	#97	1	1	yes	characteristic
104	C₂×Dic₇₃	#97	1	1	yes	characteristic
105	C₂×Dic₇₃	C₂²⋊C₈	1	2	yes	—
106	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
107	C₄×D₇₃	—	4	8	no	—
108	C₄×D₇₃	#95	1	2	yes	—
109	C₄×D₇₃	#95	1	2	yes	—
110	C₄×D₇₃	C₄×C₈	1	2	yes	—
111	C₄×D₇₃	C₄×C₈	1	2	yes	—
112	C₄×D₇₃	C₂²⋊C₈	1	2	yes	—
113	C₄×D₇₃	C₂²⋊C₈	1	2	yes	—
114	C₄×D₇₃	C₄⋊C₈	1	2	yes	—
115	C₄×D₇₃	C₄⋊C₈	1	2	yes	—
116	C₈×C₇₃	—	2	2	no	—
117	C₈×C₇₃	#95	1	4	yes	—
118	#10596831	—	2	2	no	—
119	#10596831	C₂×C₂²⋊C₄	1	2	yes	—
120	#10596831	C₂×C₄⋊C₄	1	2	yes	—
121	C₂²×C₇₃	—	2	2	no	—
122	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
123	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
125	C₂×D₇₃	—	2	2	no	—
126	C₂×D₇₃	—	2	4	no	—
127	C₂×D₇₃	—	2	4	no	—
128	C₂×D₇₃	#336	1	2	yes	—
129	C₂×D₇₃	#336	1	2	yes	—
130	C₂×D₇₃	#347	1	4	yes	—
131	C₄×C₇₃	—	4	2	no	—
132	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
133	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
134	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
136	Dic₇₃	—	4	2	no	—
137	Dic₇₃	#336	1	2	yes	—
138	Dic₇₃	#336	1	2	yes	—
139	C₂×#1375	—	73	1	no	maximal, Sylow
140	C₂×C₇₃	—	2	1	no	—
141	C₂×C₇₃	—	2	1	no	—
142	C₂×C₇₃	C₂×#336	1	1	yes	characteristic
143	C₂×C₇₃	C₂×#347	1	2	yes	—
144	D₇₃	—	2	2	no	—
145	D₇₃	#1375	1	4	yes	complemented
146	C₂²×M₃₂	—	73	2	no	—
147	C₂²×#97	—	73	1	no	—
148	#1375	—	73	2	no	—
149	#1375	—	73	2	no	—
150	C₇₃	C₂×#1375	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₈	—	73	1	no	—
152	C₂³×C₈	—	73	1	no	—
153	C₂²×C₄²	—	73	1	no	—
154	C₂×M₃₂	—	73	4	no	—
155	C₂×M₃₂	—	73	4	no	—
156	C₂×M₃₂	—	146	4	no	—
157	C₂×M₃₂	—	146	4	no	—
158	C₂²×C₁₆	—	73	4	no	—
159	C₂×#97	—	73	1	no	—
160	C₂×#97	—	73	1	no	—
161	C₂×#97	—	73	1	no	—
162	C₂×#97	—	146	4	no	—
163	C₂×#97	#64408	1	1	yes	characteristic
164	C₂³×C₄	—	73	1	no	—
165	C₂³×C₄	—	146	1	no	—
166	C₂²×C₈	—	73	1	no	—
167	C₂²×C₈	—	73	2	no	—
168	C₂²×C₈	—	73	2	no	—
169	C₂²×C₈	—	73	4	no	—
170	C₂²×C₈	—	73	2	no	—
171	C₂²×C₈	—	73	2	no	—
172	C₂²×C₈	—	73	1	no	—
173	C₂²×C₈	—	73	4	no	—
174	C₂²×C₈	—	146	2	no	—
175	C₂²×C₈	—	146	1	no	—
176	C₂²×C₈	—	146	1	no	—
177	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
178	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
179	C₂×C₄²	—	73	1	no	—
180	C₂×C₄²	—	73	4	no	—
181	C₂×C₄²	—	146	2	no	—
182	C₂×C₄²	—	146	1	no	—
183	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
184	M₃₂	—	146	4	no	—
185	M₃₂	—	146	4	no	—
186	M₃₂	—	146	8	no	—
187	C₂×C₁₆	—	73	8	no	—
188	C₂×C₁₆	—	146	8	no	—
189	#97	—	2	2	no	—
190	#97	—	146	2	no	—
191	#97	—	146	2	no	—
192	#97	—	146	2	no	—
193	C₂⁴	—	73	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	292	4	no	—
196	C₂²×C₄	—	73	1	no	—
197	C₂²×C₄	—	73	2	no	—
198	C₂²×C₄	—	73	2	no	—
199	C₂²×C₄	—	73	1	no	—
200	C₂²×C₄	—	73	1	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	—	146	4	no	—
206	C₂²×C₄	—	146	1	no	—
207	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	73	2	no	—
211	C₂×C₈	—	73	2	no	—
212	C₂×C₈	—	73	1	no	—
213	C₂×C₈	—	73	1	no	—
214	C₂×C₈	—	73	4	no	—
215	C₂×C₈	—	73	4	no	—
216	C₂×C₈	—	73	4	no	—
217	C₂×C₈	—	73	4	no	—
218	C₂×C₈	—	73	2	no	—
219	C₂×C₈	—	146	4	no	—
220	C₂×C₈	—	146	4	no	—
221	C₂×C₈	—	146	1	no	—
222	C₂×C₈	—	146	1	no	—
223	C₂×C₈	C₄×#64408	1	2	yes	—
224	C₂×C₈	C₂²⋊#64408	1	2	yes	—
225	C₂×C₈	C₄⋊#64408	1	2	yes	—
226	C₂×C₈	#11924036	1	2	yes	—
227	C₄²	—	2	1	no	—
228	C₄²	—	73	2	no	—
229	C₄²	—	146	4	no	—
230	C₄²	—	146	1	no	—
231	C₄²	#11924036	1	2	yes	—
232	C₁₆	—	146	8	no	—
233	C₂³	—	73	1	no	—
234	C₂³	—	73	1	no	—
235	C₂³	—	73	2	no	—
236	C₂³	—	73	2	no	—
237	C₂³	—	146	4	no	—
238	C₂³	D₇₃⋊M₁₆	1	1	yes	characteristic
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	292	8	no	—
242	C₂×C₄	—	292	2	no	—
243	C₂×C₄	—	73	1	no	—
244	C₂×C₄	—	73	1	no	—
245	C₂×C₄	—	73	2	no	—
246	C₂×C₄	—	73	2	no	—
247	C₂×C₄	—	73	2	no	—
248	C₂×C₄	—	73	2	no	—
249	C₂×C₄	—	73	2	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	—	146	2	no	—
256	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
259	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
260	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
261	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
262	C₈	—	2	2	no	—
263	C₈	—	73	2	no	—
264	C₈	—	73	2	no	—
265	C₈	—	146	2	no	—
266	C₈	(C₂×#64408)⋊C₄	1	4	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	73	2	no	—
269	C₂²	—	73	4	no	—
270	C₂²	—	73	2	no	—
271	C₂²	—	146	4	no	—
272	C₂²	—	146	4	no	—
273	C₂²	—	146	2	no	—
274	C₂²	#17989546	1	1	yes	characteristic
275	C₂²	#17989555	1	1	yes	characteristic
276	C₂²	#17989588	1	1	yes	characteristic
277	C₄	—	4	2	no	—
278	C₄	—	292	2	no	—
279	C₄	—	73	2	no	—
280	C₄	—	73	2	no	—
281	C₄	C₂×(C₂×#64408)⋊C₄	1	1	yes	characteristic
282	C₄	C₄×#11924036	1	1	yes	characteristic
283	C₄	#17989555	1	1	yes	characteristic
284	C₄	#17989588	1	1	yes	characteristic
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	73	4	no	—
288	C₂	—	146	2	no	—
289	C₂	#22625592	1	1	yes	characteristic
290	C₂	#22626148	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#97×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#97
prime radical	C₂³×D₇₃	M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×#97	#64408
hypocenter	C₇₃	C₂×#1375
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #22624973	C₂	semidirect
3. #22624972	C₂	semidirect
5. C₂×#14556666	C₄	semidirect
9. C₂×#14556664	C₄	semidirect
22. C₂×#11924031	C₂×C₄	semidirect
145. D₇₃	#1375	semidirect
150. C₇₃	C₂×#1375	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
#22624973	2
#22624972	2
C₂×#97×D₇₃	2
C₂×#1375	73