Subgroups — #34030152

Order 18688.

590 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626298	C₂	1	4	yes	maximal, complemented
2	#22626294	C₂	1	2	yes	maximal, complemented
3	#22626290	C₂	1	2	yes	maximal, complemented
4	#22626265	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17989508	C₂	1	1	yes	maximal, characteristic, complemented
6	#22626163	C₂	1	2	yes	maximal
7	#22626140	C₂	1	2	yes	maximal, complemented
8	#22618353	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556665	C₂²	1	2	yes	—
10	C₂²×#11924031	C₂²	1	2	yes	—
11	#17989508	C₂²	1	1	yes	characteristic
12	#17989508	C₂²	1	1	yes	characteristic
13	#17989508	C₂²	1	2	yes	—
14	#17989508	C₂²	1	2	yes	—
15	#17989508	C₂²	1	2	yes	—
16	#17989482	C₂²	1	4	yes	—
17	#17989482	C₂²	1	4	yes	—
18	#17989475	C₂²	1	4	yes	—
19	C₄×#11924031	C₂²	1	4	yes	—
20	C₂×#14556599	C₂²	1	1	yes	characteristic
21	C₂×#14556597	C₄	1	1	yes	characteristic
22	#17989055	C₄	1	1	yes	characteristic
23	#17988544	C₂²	1	2	yes	—
24	#17988543	C₂²	1	1	yes	characteristic
25	#17988541	C₄	1	2	yes	—
26	#17988539	C₄	1	2	yes	—
27	M₁₆×Dic₇₃	C₄	1	1	yes	characteristic
28	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
29	C₄×C₈×D₇₃	C₂²	1	2	yes	—
30	#405×C₇₃	C₄	1	1	yes	characteristic
31	#14556665	—	2	4	no	—
32	C₂×#11924031	—	2	4	no	—
33	C₂×#11924031	C₂³	1	2	yes	—
34	C₂×#11924031	C₂³	1	2	yes	—
35	C₂×#11924031	C₂³	1	4	yes	—
36	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
37	#14556599	—	2	2	no	—
38	#14556599	—	2	2	no	—
39	#14556597	—	2	4	no	—
40	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
41	C₂×#11924011	C₂×C₄	1	2	yes	—
42	C₂×#11924011	C₂×C₄	1	2	yes	—
43	C₂²×#10596831	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₇₃	C₂³	1	2	yes	—
46	#14556556	C₈	1	1	yes	characteristic
47	#14556556	C₈	1	1	yes	characteristic
48	#14556553	C₈	1	1	yes	characteristic
49	#14556553	C₈	1	1	yes	characteristic
50	#14556551	C₈	1	2	yes	—
51	#14556551	C₈	1	2	yes	—
52	#14556548	C₂×C₄	1	2	yes	—
53	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
54	C₄²×D₇₃	C₂×C₄	1	2	yes	—
55	#14556502	C₂×C₄	1	2	yes	—
56	#14556501	C₂×C₄	1	2	yes	—
57	C₄×#10596831	C₂³	1	2	yes	—
58	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
59	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
60	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
61	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
62	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
63	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
65	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
66	#97×C₇₃	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
68	#11924031	—	2	2	no	—
69	#11924031	—	2	2	no	—
70	#11924031	—	2	4	no	—
71	#11924028	—	2	2	no	—
72	#11924028	—	2	2	no	—
73	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
74	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
75	C₂×C₂²⋊D₇₃	C₂×C₈	1	2	yes	—
76	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
77	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
78	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
79	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
81	#11924012	M₁₆	1	4	yes	—
82	#11924011	—	2	4	no	—
83	#11924011	—	2	4	no	—
84	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
85	C₂×#10596831	C₂²×C₄	1	2	yes	—
86	C₂×#10596831	#41	1	2	yes	—
87	C₂×#10596831	#41	1	2	yes	—
88	C₂×#10596831	C₂⁴	1	1	yes	characteristic
89	C₈×D₇₃	—	2	4	no	—
90	C₈×D₇₃	—	2	2	no	—
91	C₈×D₇₃	—	2	2	no	—
92	#11924007	C₂×C₈	1	2	yes	—
93	#11924007	C₂×C₈	1	2	yes	—
94	#11924005	C₂×C₈	1	2	yes	—
95	#11924005	C₂×C₈	1	2	yes	—
96	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
97	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
98	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
99	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
100	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
103	M₁₆×C₇₃	—	2	2	no	—
104	C₂×C₈×C₇₃	—	2	2	no	—
105	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
109	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
110	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
111	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
112	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
113	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₇₃	#117	1	2	yes	—
115	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
120	C₂²⋊D₇₃	—	2	4	no	—
121	C₂×Dic₇₃	—	2	2	no	—
122	C₂×Dic₇₃	—	4	1	no	—
123	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
125	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
128	D₂₉₂	—	2	2	no	—
129	C₄×D₇₃	—	2	2	no	—
130	C₄×D₇₃	—	2	2	no	—
131	C₄×D₇₃	—	2	4	no	—
132	C₄×D₇₃	—	4	2	no	—
133	Dic₁₄₆	—	2	2	no	—
134	C₈×C₇₃	—	2	2	no	—
135	C₈×C₇₃	—	2	1	no	—
136	C₈×C₇₃	—	2	1	no	—
137	#10596831	#117	1	2	yes	—
138	#10596831	C₂×M₁₆	1	4	yes	—
139	#10596831	C₂×#41	1	1	yes	characteristic
140	#10596831	C₂×#41	1	1	yes	characteristic
141	C₂²×C₇₃	—	2	1	no	—
142	C₂²×C₇₃	—	2	1	no	—
143	C₂²×C₇₃	#431	1	2	yes	—
144	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
145	C₂×D₇₃	—	2	2	no	—
146	C₂×D₇₃	—	2	2	no	—
147	C₂×D₇₃	—	2	2	no	—
148	C₄×C₇₃	—	2	2	no	—
149	C₄×C₇₃	—	2	2	no	—
150	C₄×C₇₃	#431	1	2	yes	—
151	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
152	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
153	Dic₇₃	—	2	1	no	—
154	Dic₇₃	—	2	1	no	—
155	Dic₇₃	—	2	1	no	—
156	Dic₇₃	—	2	2	no	—
157	Dic₇₃	#504	1	2	yes	—
158	#33423	—	73	1	no	maximal, Sylow
159	C₂×C₇₃	—	2	2	no	—
160	C₂×C₇₃	C₂×#431	1	1	yes	characteristic
161	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
162	C₂×C₇₃	#3407	1	1	yes	characteristic
163	D₇₃	—	4	2	no	—
164	C₄×#131	—	73	1	no	—
165	#2164	—	73	4	no	—
166	#2163	—	73	2	no	—
167	#2162	—	73	2	no	—
168	#2150	—	73	1	no	—
169	C₂×C₄⋊C₁₆	—	73	1	no	—
170	#2113	—	73	2	no	—
171	#2108	—	73	2	no	—
172	C₇₃	#33423	1	1	yes	characteristic, complemented, Sylow
173	C₂×#131	—	73	1	no	—
174	C₂×#131	—	73	1	no	—
175	C₄×#41	—	73	1	no	—
176	C₂×M₃₂	—	73	2	no	—
177	C₂²×C₁₆	—	73	2	no	—
178	#405	—	73	1	no	—
179	#405	—	73	2	no	—
180	#405	—	73	2	no	—
181	#405	#64408	1	1	yes	characteristic
182	C₄×M₁₆	—	73	1	no	—
183	C₄×M₁₆	—	73	2	no	—
184	C₂×C₄×C₈	—	73	1	no	—
185	C₂×C₄×C₈	—	73	2	no	—
186	C₄⋊C₁₆	—	73	1	no	—
187	C₄⋊C₁₆	—	73	1	no	—
188	C₄⋊C₁₆	—	73	2	no	—
189	C₄⋊C₁₆	—	73	2	no	—
190	C₄⋊C₁₆	—	73	2	no	—
191	C₂²⋊C₁₆	—	73	4	no	—
192	C₂²⋊C₁₆	—	73	4	no	—
193	#346	—	73	4	no	—
194	C₄×C₁₆	—	73	4	no	—
195	C₂×#41	—	73	1	no	—
196	#131	—	146	4	no	—
197	#131	—	146	2	no	—
198	#131	—	146	2	no	—
199	C₂×M₁₆	—	73	1	no	—
200	C₂×M₁₆	—	73	2	no	—
201	C₂×M₁₆	—	73	2	no	—
202	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
203	C₂²×C₈	—	73	1	no	—
204	C₂²×C₈	—	73	2	no	—
205	C₂²×C₈	—	73	2	no	—
206	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
207	C₄×Q₈	—	73	1	no	—
208	C₄×Q₈	—	73	1	no	—
209	C₄×D₄	—	73	1	no	—
210	C₄×D₄	—	73	1	no	—
211	C₄×D₄	—	73	2	no	—
212	C₄×D₄	—	73	2	no	—
213	#117	—	73	2	no	—
214	#117	C₂×#64408	1	1	yes	characteristic
215	C₂×C₄²	—	73	1	no	—
216	C₂×C₄²	—	73	2	no	—
217	M₃₂	—	146	4	no	—
218	C₂×C₁₆	—	73	2	no	—
219	C₂×C₁₆	—	73	4	no	—
220	C₂×C₁₆	—	73	2	no	—
221	C₂×C₁₆	—	146	4	no	—
222	#97	—	73	2	no	—
223	#97	—	73	2	no	—
224	#97	C₂×#64408	1	2	yes	—
225	C₄×C₈	—	73	1	no	—
226	C₄×C₈	—	73	2	no	—
227	C₄×C₈	—	73	1	no	—
228	C₄×C₈	—	73	1	no	—
229	C₄×C₈	—	73	1	no	—
230	C₄×C₈	—	73	2	no	—
231	C₄×C₈	C₂×#64408	1	2	yes	—
232	#41	—	146	2	no	—
233	#41	—	146	2	no	—
234	C₂×Q₈	—	73	1	no	—
235	C₂×D₄	—	73	2	no	—
236	C₂×D₄	—	73	1	no	—
237	C₂²×C₄	—	73	1	no	—
238	C₂²×C₄	—	73	2	no	—
239	C₂²×C₄	—	73	1	no	—
240	C₂²×C₄	—	73	2	no	—
241	C₂²×C₄	—	73	2	no	—
242	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
243	M₁₆	—	2	2	no	—
244	M₁₆	—	73	4	no	—
245	M₁₆	—	146	4	no	—
246	M₁₆	—	146	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	73	1	no	—
249	C₂×C₈	—	73	2	no	—
250	C₂×C₈	—	73	1	no	—
251	C₂×C₈	—	73	2	no	—
252	C₂×C₈	—	73	2	no	—
253	C₂×C₈	—	146	2	no	—
254	C₂×C₈	—	146	2	no	—
255	C₂×C₈	—	146	4	no	—
256	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
257	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
258	C₂×C₈	C₂²×#64408	1	2	yes	—
259	C₄⋊C₄	—	73	2	no	—
260	C₄⋊C₄	—	73	2	no	—
261	C₄⋊C₄	#11924036	1	1	yes	characteristic
262	C₄⋊C₄	#11924036	1	1	yes	characteristic
263	C₂²⋊C₄	—	73	2	no	—
264	C₂²⋊C₄	—	73	2	no	—
265	C₂²⋊C₄	#11924036	1	2	yes	—
266	C₄²	—	73	1	no	—
267	C₄²	—	73	2	no	—
268	C₄²	—	73	1	no	—
269	C₄²	—	73	2	no	—
270	C₄²	—	73	1	no	—
271	C₄²	—	73	1	no	—
272	C₄²	C₂²×#64408	1	2	yes	—
273	C₁₆	—	146	2	no	—
274	C₁₆	—	146	2	no	—
275	C₁₆	—	146	4	no	—
276	C₂³	—	73	2	no	—
277	C₂³	C₂×#11924036	1	1	yes	characteristic
278	Q₈	—	146	2	no	—
279	D₄	—	146	4	no	—
280	D₄	—	146	2	no	—
281	C₂×C₄	—	292	2	no	—
282	C₂×C₄	—	292	1	no	—
283	C₂×C₄	—	73	1	no	—
284	C₂×C₄	—	73	1	no	—
285	C₂×C₄	—	73	1	no	—
286	C₂×C₄	—	73	1	no	—
287	C₂×C₄	—	73	2	no	—
288	C₂×C₄	—	146	4	no	—
289	C₂×C₄	—	146	2	no	—
290	C₂×C₄	—	146	2	no	—
291	C₂×C₄	—	146	2	no	—
292	C₂×C₄	#14556670	1	2	yes	—
293	C₂×C₄	C₂×#11924036	1	2	yes	—
294	C₂×C₄	C₂×#11924036	1	2	yes	—
295	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
296	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
297	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	73	1	no	—
302	C₈	—	73	1	no	—
303	C₈	—	73	2	no	—
304	C₈	—	73	4	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	146	2	no	—
308	C₂²	—	146	2	no	—
309	C₂²	—	146	2	no	—
310	C₂²	#17989605	1	2	yes	—
311	C₂²	C₂²×#11924036	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	73	2	no	—
315	C₄	—	146	2	no	—
316	C₄	—	146	1	no	—
317	C₄	—	146	1	no	—
318	C₄	—	146	1	no	—
319	C₄	#17989605	1	2	yes	—
320	C₄	C₂×#14556670	1	1	yes	characteristic
321	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	292	2	no	—
324	C₂	C₂×#17989605	1	1	yes	characteristic
325	C₂	#22626896	1	1	yes	characteristic
326	C₂	#22626899	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	#405×C₇₃	C₄
prime radical	C₂×#11924028	C₂×C₄
cosocle	C₂×#10596831	C₂⁴
hypercenter	#405	#64408
hypocenter	C₇₃	#33423
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
1. #22626298	C₂	semidirect
2. #22626294	C₂	semidirect
3. #22626290	C₂	semidirect
4. #22626265	C₂	semidirect
5. C₂×#17989508	C₂	semidirect
7. #22626140	C₂	semidirect
172. C₇₃	#33423	semidirect

Maximal subgroups

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

subgroup	index
#22626298	2
#22626294	2
#22626290	2
#22626265	2
C₂×#17989508	2
#22626163	2
#22626140	2
#22618353	2
#33423	73