Subgroups — #32976804

Order 18688.

554 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988666	C₂	1	1	yes	maximal, characteristic
2	#22609798	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14556513	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988316	C₂	1	1	yes	maximal, characteristic
5	#22607771	C₂	1	1	yes	maximal, characteristic, complemented
6	#22607766	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈²×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11924011	—	2	1	no	—
9	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
10	#17988666	—	2	2	no	—
11	#17988666	C₄	1	4	yes	complemented
12	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	C₂×#14556513	C₄	1	2	yes	—
15	C₂×#14556503	—	2	1	no	—
16	C₂×#14556501	—	2	1	no	—
17	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
18	#17988350	C₄	1	2	yes	—
19	#17988341	—	2	1	no	—
20	#17988316	C₄	1	4	yes	complemented
21	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₈²×C₇₃	—	2	2	no	—
25	C₂²×D₂₉₂	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₇₃	—	2	1	no	—
27	C₂×#11924011	—	2	4	no	—
28	C₂×#11924011	—	4	2	no	—
29	C₂×#11924011	—	4	2	no	—
30	C₂²×#10596831	—	2	1	no	—
31	C₂²×#10596831	—	2	1	no	—
32	C₄×D₂₉₂	—	2	1	no	—
33	C₄×D₂₉₂	—	2	1	no	—
34	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
35	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×#11924005	—	2	1	no	—
39	C₂×C₄×Dic₇₃	—	2	1	no	—
40	C₄⋊#10596831	—	2	1	no	—
41	C₄⋊#10596831	—	2	1	no	—
42	C₄⋊#10596831	C₂×C₄	1	2	yes	—
43	C₄⋊#10596831	C₂×C₄	1	2	yes	—
44	#14556513	—	2	4	no	—
45	#14556503	—	2	4	no	—
46	#14556501	—	4	2	no	—
47	#14556498	—	2	1	no	—
48	C₂²×C₈×C₇₃	—	2	1	no	—
49	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₇₃	—	2	1	no	—
54	C₄×C₈×C₇₃	—	2	1	no	—
55	C₄×C₈×C₇₃	—	2	1	no	—
56	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₇₃	D₄	1	2	yes	—
59	C₂³×D₇₃	—	2	1	no	—
60	C₂×D₂₉₂	—	2	2	no	—
61	C₂×D₂₉₂	—	2	2	no	—
62	C₂×D₂₉₂	—	2	1	no	—
63	C₂×D₂₉₂	C₄²	1	2	yes	—
64	C₂²×Dic₇₃	—	2	1	no	—
65	C₂²×Dic₇₃	—	2	2	no	—
66	C₂×C₄×D₇₃	—	2	2	no	—
67	C₂×C₄×D₇₃	—	2	2	no	—
68	C₂×C₄×D₇₃	—	4	1	no	—
69	C₂×C₄×D₇₃	—	4	1	no	—
70	C₂×C₄×D₇₃	—	4	1	no	—
71	C₂×C₄×D₇₃	—	4	1	no	—
72	#11924011	—	4	8	no	—
73	C₂×#10596831	—	2	2	no	—
74	C₂×#10596831	—	2	2	no	—
75	C₂×#10596831	—	4	1	no	—
76	C₂×#10596831	—	4	1	no	—
77	C₂×#10596831	—	4	1	no	—
78	C₂×#10596831	—	4	1	no	—
79	C₄⋊Dic₇₃	—	2	1	no	—
80	C₄⋊Dic₇₃	—	2	2	no	—
81	C₄⋊Dic₇₃	—	2	2	no	—
82	C₄⋊Dic₇₃	C₄²	1	2	yes	—
83	#11924005	—	2	2	no	—
84	#11924005	—	2	2	no	—
85	C₄×Dic₇₃	—	4	1	no	—
86	C₄×Dic₇₃	—	4	1	no	—
87	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
90	C₂×C₈×C₇₃	—	2	4	no	—
91	C₂×C₈×C₇₃	—	2	2	no	—
92	C₂×C₈×C₇₃	—	2	2	no	—
93	C₂×C₈×C₇₃	—	2	2	no	—
94	C₂×C₈×C₇₃	—	2	1	no	—
95	C₂×C₈×C₇₃	—	2	1	no	—
96	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
97	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
98	C₂×C₈×C₇₃	M₁₆	1	4	yes	complemented
99	C₂×C₈×C₇₃	D₈	1	2	yes	—
100	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
101	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₇₃	#41	1	1	yes	characteristic
105	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
106	C₂²×D₇₃	—	2	2	no	—
107	C₂²×D₇₃	—	4	2	no	—
108	C₂²×D₇₃	—	4	1	no	—
109	C₂²×D₇₃	—	4	1	no	—
110	C₂²×D₇₃	—	4	2	no	—
111	C₂×C₄×C₇₃	—	2	2	no	—
112	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₇₃	#131	1	2	yes	—
120	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
123	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
124	C₂×Dic₇₃	—	2	2	no	—
125	C₂×Dic₇₃	—	4	1	no	—
126	C₂×Dic₇₃	—	4	1	no	—
127	C₂×Dic₇₃	—	4	2	no	—
128	C₂×Dic₇₃	—	4	2	no	—
129	C₂×Dic₇₃	—	4	2	no	—
130	D₂₉₂	—	2	2	no	—
131	D₂₉₂	—	2	2	no	—
132	D₂₉₂	—	4	2	no	—
133	C₄×D₇₃	—	4	2	no	—
134	C₄×D₇₃	—	4	2	no	—
135	C₄×D₇₃	—	4	2	no	—
136	C₄×D₇₃	—	4	2	no	—
137	C₈×C₇₃	—	2	4	no	—
138	C₈×C₇₃	—	2	2	no	—
139	C₈×C₇₃	—	2	4	no	—
140	C₈×C₇₃	#102	1	4	yes	complemented
141	#10596831	—	4	2	no	—
142	#10596831	—	4	2	no	—
143	C₂²×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₇₃	#436	1	2	yes	—
145	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
146	C₂²×C₇₃	C₄×SD₁₆	1	1	yes	characteristic
147	C₂²×C₇₃	#443	1	2	yes	—
148	C₂×D₇₃	—	4	2	no	—
149	C₂×D₇₃	—	4	2	no	—
150	C₂×D₇₃	—	4	2	no	—
151	C₂×D₇₃	—	8	2	no	—
152	C₂×D₇₃	—	8	2	no	—
153	C₄×C₇₃	—	2	1	no	—
154	C₄×C₇₃	—	2	1	no	—
155	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
156	C₄×C₇₃	#405	1	1	yes	characteristic
157	C₄×C₇₃	C₂×#102	1	1	yes	characteristic
158	C₄×C₇₃	#416	1	1	yes	characteristic
159	C₄×C₇₃	#435	1	2	yes	—
160	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
161	C₄×C₇₃	C₄×SD₁₆	1	1	yes	characteristic
162	Dic₇₃	—	4	2	no	—
163	Dic₇₃	—	8	2	no	—
164	#8206	—	73	1	no	maximal, Sylow
165	C₂×C₇₃	#1589	1	2	yes	—
166	C₂×C₇₃	#1590	1	2	yes	—
167	C₂×C₇₃	#1753	1	1	yes	characteristic
168	C₂×C₇₃	C₄×#102	1	1	yes	characteristic
169	C₂×C₇₃	#1764	1	1	yes	characteristic
170	D₇₃	—	8	2	no	—
171	C₂×#436	—	73	1	no	—
172	#1770	—	73	1	no	—
173	C₄×#102	—	73	1	no	—
174	C₂×D₄⋊C₈	—	73	1	no	—
175	C₂×C₈²	D₇₃	1	1	yes	characteristic, complemented
176	#1279	—	73	1	no	—
177	#1277	—	73	1	no	—
178	C₇₃	#8206	1	1	yes	characteristic, complemented, Sylow
179	C₂²×M₁₆	—	146	1	no	—
180	C₂×C₄×D₄	—	73	1	no	—
181	#436	—	73	4	no	—
182	#436	—	146	2	no	—
183	C₂×C₄⋊C₈	—	73	1	no	—
184	C₂×C₄⋊C₈	—	73	1	no	—
185	C₂×#102	—	73	2	no	—
186	C₂×C₂²⋊C₈	—	146	1	no	—
187	C₂×#97	—	146	1	no	—
188	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
190	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
191	C₄×C₄⋊C₄	—	73	1	no	—
192	#340	—	73	2	no	—
193	#336	—	146	1	no	—
194	D₄⋊C₈	—	73	4	no	—
195	C₈²	—	2	2	no	—
196	C₂²×D₄	—	73	1	no	—
197	C₂³×C₄	—	146	1	no	—
198	C₂×M₁₆	—	292	2	no	—
199	C₂×M₁₆	—	292	2	no	—
200	C₂×M₁₆	—	146	4	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	146	1	no	—
203	C₂²×C₈	—	146	1	no	—
204	C₂²×C₈	C₄×D₇₃	1	2	yes	—
205	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
206	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
207	C₄×D₄	—	73	2	no	—
208	C₄×D₄	—	73	2	no	—
209	C₄×D₄	—	146	1	no	—
210	C₄×D₄	—	146	1	no	—
211	C₂×C₄⋊C₄	—	73	1	no	—
212	C₂×C₄⋊C₄	—	73	2	no	—
213	C₂×C₂²⋊C₄	—	146	1	no	—
214	C₂×C₄²	—	146	1	no	—
215	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
216	C₄⋊C₈	—	73	2	no	—
217	C₄⋊C₈	—	73	2	no	—
218	C₄⋊C₈	—	146	1	no	—
219	C₄⋊C₈	—	146	1	no	—
220	#102	—	146	4	no	—
221	C₂²⋊C₈	—	146	4	no	—
222	#97	—	292	2	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	C₄×D₇₃	1	2	yes	—
227	C₄×C₈	C₄×D₇₃	1	2	yes	—
228	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
229	#95	—	146	1	no	—
230	C₂⁴	—	146	1	no	—
231	C₂×D₄	—	73	2	no	—
232	C₂×D₄	—	146	1	no	—
233	C₂×D₄	—	146	2	no	—
234	C₂×D₄	—	146	2	no	—
235	C₂²×C₄	—	292	1	no	—
236	C₂²×C₄	—	292	1	no	—
237	C₂²×C₄	—	292	1	no	—
238	C₂²×C₄	—	292	1	no	—
239	C₂²×C₄	—	146	2	no	—
240	C₂²×C₄	—	146	1	no	—
241	C₂²×C₄	—	146	2	no	—
242	C₂²×C₄	—	146	2	no	—
243	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
245	C₂²×C₄	#11924028	1	1	yes	characteristic
246	M₁₆	—	292	8	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	292	1	no	—
254	C₂×C₈	—	292	1	no	—
255	C₂×C₈	—	292	1	no	—
256	C₂×C₈	—	292	1	no	—
257	C₂×C₈	—	146	2	no	—
258	C₂×C₈	—	146	2	no	—
259	C₂×C₈	#11924005	1	2	yes	—
260	C₂×C₈	#11924005	1	2	yes	—
261	C₂×C₈	#11924011	1	4	yes	complemented
262	C₂×C₈	D₅₈₄	1	2	yes	—
263	C₂×C₈	#11924017	1	2	yes	—
264	C₄⋊C₄	—	73	2	no	—
265	C₄⋊C₄	—	146	2	no	—
266	C₄⋊C₄	—	146	2	no	—
267	C₄⋊C₄	—	146	1	no	—
268	C₂²⋊C₄	—	146	2	no	—
269	C₂²⋊C₄	—	146	2	no	—
270	C₄²	—	292	1	no	—
271	C₄²	—	292	1	no	—
272	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
274	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
275	C₄²	#11924028	1	1	yes	characteristic
276	C₂³	—	292	2	no	—
277	C₂³	—	292	1	no	—
278	C₂³	—	292	1	no	—
279	C₂³	—	292	2	no	—
280	C₂³	—	146	2	no	—
281	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
282	D₄	—	292	2	no	—
283	D₄	—	146	2	no	—
284	D₄	—	146	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	292	1	no	—
287	C₂×C₄	—	292	1	no	—
288	C₂×C₄	—	292	2	no	—
289	C₂×C₄	—	292	2	no	—
290	C₂×C₄	—	292	2	no	—
291	C₂×C₄	—	292	2	no	—
292	C₂×C₄	—	292	2	no	—
293	C₂×C₄	—	292	2	no	—
294	C₂×C₄	—	292	2	no	—
295	C₂×C₄	—	146	2	no	—
296	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
297	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
298	C₂×C₄	#14556548	1	1	yes	characteristic
299	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
300	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
301	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
302	C₂×C₄	C₂×#11924011	1	2	yes	—
303	C₂×C₄	#14556597	1	2	yes	—
304	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
305	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
306	C₂×C₄	#14556613	1	1	yes	characteristic
307	C₂×C₄	#14556613	1	1	yes	characteristic
308	C₈	—	2	4	no	—
309	C₈	—	2	4	no	—
310	C₈	—	2	2	no	—
311	C₈	—	292	2	no	—
312	C₈	—	292	2	no	—
313	C₈	#14556513	1	4	yes	complemented
314	C₂²	—	292	2	no	—
315	C₂²	—	292	2	no	—
316	C₂²	—	292	2	no	—
317	C₂²	—	584	2	no	—
318	C₂²	—	584	2	no	—
319	C₂²	C₄×#11924005	1	1	yes	characteristic
320	C₂²	#17988666	1	2	yes	—
321	C₂²	C₄×D₅₈₄	1	1	yes	characteristic
322	C₂²	C₄×#11924017	1	1	yes	characteristic
323	C₂²	#17988705	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	292	2	no	—
327	C₄	—	584	2	no	—
328	C₄	C₄×#11924011	1	1	yes	characteristic
329	C₄	#17988541	1	1	yes	characteristic
330	C₄	C₂×#14556513	1	1	yes	characteristic
331	C₄	#17988584	1	1	yes	characteristic
332	C₄	#17988658	1	2	yes	—
333	C₄	C₄×D₅₈₄	1	1	yes	characteristic
334	C₄	C₄×#11924017	1	1	yes	characteristic
335	C₂	—	584	2	no	—
336	C₂	#22608817	1	2	yes	—
337	C₂	#22608823	1	2	yes	—
338	C₂	#22609714	1	1	yes	characteristic
339	C₂	C₄×#14556513	1	1	yes	characteristic
340	C₂	#22609768	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×C₈²×C₇₃	C₂
socle	C₂³×C₇₃	C₄×D₄
prime radical	C₂²×D₂₉₂	C₂×C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×C₈²	D₇₃
hypocenter	C₇₃	#8206
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #22609798	C₂	semidirect
5. #22607771	C₂	semidirect
6. #22607766	C₂	semidirect
7. C₂×C₈²×C₇₃	C₂	semidirect
11. #17988666	C₄	semidirect
20. #17988316	C₄	semidirect
98. C₂×C₈×C₇₃	M₁₆	semidirect
140. C₈×C₇₃	#102	semidirect
175. C₂×C₈²	D₇₃	semidirect
178. C₇₃	#8206	semidirect
261. C₂×C₈	#11924011	semidirect
313. C₈	#14556513	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988666	2
#22609798	2
C₄×#14556513	2
C₂×#17988316	2
#22607771	2
#22607766	2
C₂×C₈²×C₇₃	2
#8206	73