Subgroups — #34008780

Order 18688.

982 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556660	C₂	1	2	yes	maximal
2	C₂×#17989461	C₂	1	2	yes	maximal
3	#22625109	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊#64408	C₂²	1	4	yes	—
6	C₂×#14556660	—	2	4	no	—
7	C₂×#14556660	—	2	4	no	—
8	C₂×#14556660	C₄	1	8	yes	complemented
9	C₂×(C₂×#64408)⋊C₄	—	2	4	no	—
10	#17989461	—	2	4	no	—
11	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×#10596831	C₄	1	1	yes	characteristic, complemented
15	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic, complemented
16	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic, complemented
17	C₄×C₈×D₇₃	—	2	4	no	—
18	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic, complemented
19	C₂³×#64408	—	2	4	no	—
20	C₂×C₄⋊#64408	—	2	4	no	—
21	C₂×C₄⋊#64408	—	2	4	no	—
22	C₂×C₄⋊#64408	—	2	4	no	—
23	C₂×C₄⋊#64408	—	2	4	no	—
24	C₂×C₄⋊#64408	—	2	4	no	—
25	C₂×C₄⋊#64408	C₂×C₄	1	8	yes	—
26	#14556660	—	2	16	no	—
27	(C₂×#64408)⋊C₄	—	4	8	no	—
28	C₂²×C₄×D₇₃	D₄	1	2	yes	—
29	C₂²×C₄×D₇₃	C₂³	1	1	yes	characteristic
30	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×D₇₃	—	2	2	no	—
33	C₂×C₈×D₇₃	—	2	2	no	—
34	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
35	C₂×C₈×D₇₃	D₄	1	4	yes	—
36	C₂×C₈×D₇₃	D₄	1	2	yes	—
37	C₂×C₈×D₇₃	D₄	1	2	yes	—
38	C₂×C₈×D₇₃	Q₈	1	2	yes	—
39	C₂×C₈×D₇₃	Q₈	1	2	yes	—
40	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
42	C₄²×D₇₃	—	2	2	no	—
43	C₄²×D₇₃	—	2	2	no	—
44	C₄×#10596831	—	2	2	no	—
45	C₈×Dic₇₃	—	2	2	no	—
46	C₈×Dic₇₃	—	2	2	no	—
47	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₇₃	—	2	2	no	—
51	C₂²×#64408	—	2	8	no	—
52	C₂²×#64408	—	4	4	no	—
53	C₂²×#64408	—	4	4	no	—
54	C₂²×#64408	—	4	8	no	—
55	C₂²×#64408	—	4	8	no	—
56	C₄⋊#64408	—	2	8	no	—
57	C₄⋊#64408	—	2	8	no	—
58	C₄⋊#64408	—	4	4	no	—
59	C₄⋊#64408	—	4	4	no	—
60	C₂³×D₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
62	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₄×D₇₃	—	2	4	no	—
64	C₂×C₄×D₇₃	—	2	4	no	—
65	C₂×C₄×D₇₃	D₈	1	4	yes	—
66	C₂×C₄×D₇₃	Q₁₆	1	4	yes	—
67	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₄×D₇₃	#41	1	2	yes	—
74	C₂×C₄×D₇₃	#41	1	2	yes	—
75	C₂×#10596831	—	2	1	no	—
76	C₂×#10596831	—	2	1	no	—
77	C₂×#10596831	C₄²	1	2	yes	—
78	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
79	C₂×#10596831	C₄⋊C₄	1	2	yes	—
80	C₂×#10596831	C₄⋊C₄	1	2	yes	—
81	C₈×D₇₃	—	2	4	no	—
82	C₈×D₇₃	—	2	4	no	—
83	C₈×D₇₃	C₂²⋊C₄	1	8	yes	—
84	C₈×D₇₃	C₄⋊C₄	1	8	yes	—
85	C₄×Dic₇₃	—	2	1	no	—
86	C₄×Dic₇₃	—	2	1	no	—
87	C₄×Dic₇₃	—	2	4	no	—
88	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
89	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₇₃	—	2	1	no	—
91	C₂×C₈×C₇₃	—	2	1	no	—
92	C₂×C₈×C₇₃	C₄²	1	2	yes	—
93	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
94	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
95	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
96	C₄²×C₇₃	—	2	1	no	—
97	C₄²×C₇₃	—	2	1	no	—
98	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×D₇₃	C₄×D₄	1	2	yes	—
100	C₂²×D₇₃	#121	1	2	yes	—
101	C₂²×D₇₃	#127	1	1	yes	characteristic
102	C₂²×D₇₃	C₄⋊Q₈	1	1	yes	characteristic
103	C₂²×D₇₃	C₂×D₈	1	2	yes	—
104	C₂²×D₇₃	C₂×Q₁₆	1	2	yes	—
105	C₂²×D₇₃	#135	1	4	yes	—
106	C₂×#64408	—	4	8	no	—
107	C₂×#64408	—	4	16	no	—
108	C₂×#64408	—	8	8	no	—
109	C₂×#64408	—	8	4	no	—
110	C₂×#64408	—	8	4	no	—
111	C₂×C₄×C₇₃	—	2	2	no	—
112	C₂×C₄×C₇₃	—	2	2	no	—
113	C₂×C₄×C₇₃	#107	1	4	yes	complemented
114	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
119	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
120	C₂×Dic₇₃	—	2	2	no	—
121	C₂×Dic₇₃	—	2	2	no	—
122	C₂×Dic₇₃	#107	1	4	yes	complemented
123	C₂×Dic₇₃	C₄×Q₈	1	2	yes	—
124	C₂×Dic₇₃	C₂²⋊Q₈	1	2	yes	—
125	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
126	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
127	C₄×D₇₃	—	2	8	no	—
128	C₄×D₇₃	—	2	8	no	—
129	C₄×D₇₃	C₂×C₂²⋊C₄	1	2	yes	—
130	C₄×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₄×D₇₃	C₄×D₄	1	2	yes	—
132	C₄×D₇₃	C₄×Q₈	1	2	yes	—
133	C₄×D₇₃	#121	1	2	yes	—
134	C₄×D₇₃	C₂²⋊Q₈	1	2	yes	—
135	C₄×D₇₃	#124	1	2	yes	—
136	C₄×D₇₃	#125	1	2	yes	—
137	C₈×C₇₃	—	2	2	no	—
138	C₈×C₇₃	#95	1	4	yes	complemented
139	#10596831	—	2	2	no	—
140	#10596831	#95	1	4	yes	complemented
141	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×C₇₃	#380	1	1	yes	characteristic
143	C₂²×C₇₃	#384	1	1	yes	characteristic
144	C₂²×C₇₃	C₂×#107	1	2	yes	—
145	C₂²×C₇₃	#427	1	2	yes	—
146	C₂×D₇₃	#389	1	2	yes	—
147	C₂×D₇₃	#390	1	2	yes	—
148	C₂×D₇₃	C₄×D₈	1	4	yes	—
149	C₂×D₇₃	C₄×Q₁₆	1	4	yes	—
150	C₂×D₇₃	#466	1	4	yes	—
151	C₂×D₇₃	#467	1	4	yes	—
152	C₂×D₇₃	#493	1	1	yes	characteristic
153	C₂×D₇₃	#494	1	1	yes	characteristic
154	C₂×D₇₃	#495	1	2	yes	—
155	C₂×D₇₃	#499	1	2	yes	—
156	C₂×D₇₃	#500	1	2	yes	—
157	#64408	—	8	8	no	—
158	C₄×C₇₃	—	2	4	no	—
159	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
160	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₇₃	#380	1	1	yes	characteristic
162	C₄×C₇₃	#382	1	1	yes	characteristic
163	Dic₇₃	—	2	4	no	—
164	Dic₇₃	#389	1	2	yes	—
165	Dic₇₃	#391	1	2	yes	—
166	#10862	—	73	1	no	maximal, Sylow
167	C₂×C₇₃	C₄⋊#95	1	1	yes	characteristic
168	C₂×C₇₃	C₄×#107	1	2	yes	—
169	C₂×C₇₃	#1808	1	2	yes	—
170	C₂×C₇₃	#1845	1	1	yes	characteristic
171	C₂×C₇₃	#1846	1	1	yes	characteristic
172	D₇₃	#1942	1	4	yes	—
173	D₇₃	#1969	1	2	yes	—
174	D₇₃	#1970	1	2	yes	—
175	C₂²×#107	—	73	2	no	—
176	C₂²×C₄×C₈	—	73	1	no	—
177	C₂×#340	—	73	2	no	—
178	C₄⋊#95	—	73	2	no	—
179	C₇₃	#10862	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₈	—	73	1	no	—
181	C₂³×C₈	—	73	1	no	—
182	C₂²×C₄⋊C₄	—	73	4	no	—
183	C₂²×C₄²	—	73	1	no	—
184	C₂×#107	—	73	8	no	—
185	C₂×#107	—	146	4	no	—
186	C₂×#107	—	146	4	no	—
187	C₂×C₄×C₈	—	73	1	no	—
188	C₂×C₄×C₈	—	73	1	no	—
189	C₂×C₄×C₈	—	73	1	no	—
190	C₂×C₄×C₈	—	146	4	no	—
191	C₂×C₄×C₈	#64408	1	1	yes	characteristic, complemented
192	C₂×#95	—	146	4	no	—
193	#340	—	146	4	no	—
194	C₂³×C₄	—	73	1	no	—
195	C₂³×C₄	—	73	2	no	—
196	C₂³×C₄	—	146	4	no	—
197	C₂²×C₈	—	73	1	no	—
198	C₂²×C₈	—	73	4	no	—
199	C₂²×C₈	—	73	4	no	—
200	C₂²×C₈	—	73	2	no	—
201	C₂²×C₈	—	73	2	no	—
202	C₂²×C₈	—	73	1	no	—
203	C₂²×C₈	—	73	2	no	—
204	C₂²×C₈	—	73	2	no	—
205	C₂²×C₈	—	146	2	no	—
206	C₂²×C₈	—	146	2	no	—
207	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
208	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
209	C₂×C₄⋊C₄	—	73	8	no	—
210	C₂×C₄⋊C₄	—	146	4	no	—
211	C₂×C₄⋊C₄	—	146	4	no	—
212	C₂×C₄⋊C₄	—	146	4	no	—
213	C₂×C₄⋊C₄	—	146	4	no	—
214	C₂×C₄⋊C₄	—	146	4	no	—
215	C₂×C₄²	—	73	2	no	—
216	C₂×C₄²	—	73	1	no	—
217	C₂×C₄²	—	146	2	no	—
218	C₂×C₄²	—	146	2	no	—
219	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
220	#107	—	146	16	no	—
221	C₄×C₈	—	2	2	no	—
222	C₄×C₈	—	146	2	no	—
223	C₄×C₈	—	146	2	no	—
224	C₄×C₈	—	146	2	no	—
225	#95	—	292	8	no	—
226	C₂⁴	—	73	1	no	—
227	C₂²×C₄	—	292	8	no	—
228	C₂²×C₄	—	292	4	no	—
229	C₂²×C₄	—	292	4	no	—
230	C₂²×C₄	—	292	8	no	—
231	C₂²×C₄	—	73	1	no	—
232	C₂²×C₄	—	73	2	no	—
233	C₂²×C₄	—	73	2	no	—
234	C₂²×C₄	—	73	1	no	—
235	C₂²×C₄	—	73	1	no	—
236	C₂²×C₄	—	73	1	no	—
237	C₂²×C₄	—	73	1	no	—
238	C₂²×C₄	—	73	2	no	—
239	C₂²×C₄	—	73	2	no	—
240	C₂²×C₄	—	73	2	no	—
241	C₂²×C₄	—	73	4	no	—
242	C₂²×C₄	—	73	4	no	—
243	C₂²×C₄	—	146	4	no	—
244	C₂²×C₄	—	146	4	no	—
245	C₂²×C₄	—	146	8	no	—
246	C₂²×C₄	C₄⋊#64408	1	2	yes	—
247	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	73	2	no	—
251	C₂×C₈	—	73	2	no	—
252	C₂×C₈	—	73	2	no	—
253	C₂×C₈	—	73	8	no	—
254	C₂×C₈	—	73	8	no	—
255	C₂×C₈	—	73	2	no	—
256	C₂×C₈	—	146	4	no	—
257	C₂×C₈	—	146	4	no	—
258	C₂×C₈	—	146	1	no	—
259	C₂×C₈	—	146	1	no	—
260	C₂×C₈	C₄×#64408	1	2	yes	—
261	C₂×C₈	C₂²⋊#64408	1	2	yes	—
262	C₂×C₈	C₄⋊#64408	1	2	yes	—
263	C₂×C₈	C₄⋊#64408	1	2	yes	—
264	C₄⋊C₄	—	292	4	no	—
265	C₄⋊C₄	—	292	4	no	—
266	C₄⋊C₄	—	146	8	no	—
267	C₄⋊C₄	—	146	8	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	146	4	no	—
271	C₄²	—	146	1	no	—
272	C₄²	—	146	1	no	—
273	C₂³	—	73	2	no	—
274	C₂³	—	73	2	no	—
275	C₂³	—	73	1	no	—
276	C₂³	—	73	1	no	—
277	C₂³	—	73	2	no	—
278	C₂³	—	73	2	no	—
279	C₂³	—	73	4	no	—
280	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	292	8	no	—
284	C₂×C₄	—	292	16	no	—
285	C₂×C₄	—	584	8	no	—
286	C₂×C₄	—	584	4	no	—
287	C₂×C₄	—	584	4	no	—
288	C₂×C₄	—	73	2	no	—
289	C₂×C₄	—	73	2	no	—
290	C₂×C₄	—	73	1	no	—
291	C₂×C₄	—	73	1	no	—
292	C₂×C₄	—	73	2	no	—
293	C₂×C₄	—	73	2	no	—
294	C₂×C₄	—	73	2	no	—
295	C₂×C₄	—	73	2	no	—
296	C₂×C₄	—	73	2	no	—
297	C₂×C₄	—	73	2	no	—
298	C₂×C₄	—	73	2	no	—
299	C₂×C₄	—	73	2	no	—
300	C₂×C₄	—	73	4	no	—
301	C₂×C₄	—	146	8	no	—
302	C₂×C₄	—	146	8	no	—
303	C₂×C₄	—	146	2	no	—
304	C₂×C₄	—	146	2	no	—
305	C₂×C₄	#14556660	1	4	yes	complemented
306	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
307	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
308	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
309	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
310	C₂×C₄	#14556670	1	1	yes	characteristic
311	C₂×C₄	#14556670	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	73	4	no	—
314	C₈	—	146	2	no	—
315	C₈	(C₂×#64408)⋊C₄	1	4	yes	complemented
316	C₂²	—	73	4	no	—
317	C₂²	—	73	4	no	—
318	C₂²	—	73	1	no	—
319	C₂²	—	73	1	no	—
320	C₂²	—	73	2	no	—
321	C₂²	—	73	2	no	—
322	C₂²	—	73	2	no	—
323	C₂²	—	73	2	no	—
324	C₂²	—	73	2	no	—
325	C₂²	—	73	4	no	—
326	C₂²	—	73	4	no	—
327	C₂²	C₄×C₄⋊#64408	1	1	yes	characteristic
328	C₂²	#17989529	1	1	yes	characteristic
329	C₂²	#17989533	1	1	yes	characteristic
330	C₂²	C₂×#14556660	1	2	yes	—
331	C₂²	#17989597	1	2	yes	—
332	C₄	—	2	4	no	—
333	C₄	—	584	8	no	—
334	C₄	—	73	2	no	—
335	C₄	—	73	2	no	—
336	C₄	—	146	4	no	—
337	C₄	C₂×(C₂×#64408)⋊C₄	1	1	yes	characteristic
338	C₄	C₄×C₄⋊#64408	1	1	yes	characteristic
339	C₄	#17989529	1	1	yes	characteristic
340	C₄	#17989531	1	1	yes	characteristic
341	C₂	—	73	2	no	—
342	C₂	—	73	4	no	—
343	C₂	—	73	2	no	—
344	C₂	#22625109	1	1	yes	characteristic
345	C₂	C₄×#14556660	1	2	yes	—
346	C₂	#22625808	1	2	yes	—
347	C₂	#22625867	1	1	yes	characteristic
348	C₂	#22625868	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂×C₄⋊C₄
prime radical	C₂³×D₇₃	C₂×D₄
cosocle	C₂²×C₄×D₇₃	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#10862
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. C₂×#14556660	C₄	semidirect
14. C₂×C₄×#10596831	C₄	semidirect
15. C₂×C₈×Dic₇₃	C₄	semidirect
16. C₂×C₈×Dic₇₃	C₄	semidirect
18. C₂×C₄×C₈×C₇₃	C₄	semidirect
113. C₂×C₄×C₇₃	#107	semidirect
122. C₂×Dic₇₃	#107	semidirect
138. C₈×C₇₃	#95	semidirect
140. #10596831	#95	semidirect
179. C₇₃	#10862	semidirect
191. C₂×C₄×C₈	#64408	semidirect
305. C₂×C₄	#14556660	semidirect
315. C₈	(C₂×#64408)⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556660	2
C₂×#17989461	2
#22625109	2
C₂×C₄×C₈×D₇₃	2
#10862	73