Subgroups — #33104264

Order 18688.

1122 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×#17988699	C₂	1	2	yes	maximal
2	D₄×C₄⋊Dic₇₃	C₂	1	2	yes	maximal, complemented
3	#22610670	C₂	1	2	yes	maximal, complemented
4	#22610194	C₂	1	4	yes	maximal, complemented
5	#22610088	C₂	1	1	yes	maximal, characteristic, complemented
6	#22609931	C₂	1	2	yes	maximal, complemented
7	#22609798	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×D₄×Dic₇₃	C₂²	1	4	yes	—
10	C₂²×C₄⋊Dic₇₃	—	2	2	no	—
11	#17988699	—	2	8	no	—
12	C₂×#14556523	C₂²	1	2	yes	—
13	C₂×#14556523	C₂²	1	2	yes	—
14	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
15	C₂×#14556512	C₂²	1	4	yes	—
16	C₂×#14556501	C₂²	1	2	yes	—
17	#17988486	C₂²	1	4	yes	—
18	#17988462	—	2	4	no	—
19	#17988457	C₂²	1	2	yes	—
20	#17988447	—	2	2	no	—
21	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
22	#17988349	C₂²	1	4	yes	—
23	#17988341	C₂²	1	2	yes	—
24	C₂²×D₈×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₇₃	C₂²	1	2	yes	—
26	C₄×D₈×C₇₃	C₄	1	8	yes	complemented
27	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
28	C₂×#102×C₇₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
30	C₂³×Dic₇₃	—	2	4	no	—
31	C₂²×#10596831	C₂³	1	2	yes	—
32	D₄×Dic₇₃	—	2	8	no	—
33	D₄×Dic₇₃	—	4	4	no	—
34	C₂×C₄⋊Dic₇₃	—	2	4	no	—
35	C₂×C₄⋊Dic₇₃	—	2	2	no	—
36	C₂×C₄⋊Dic₇₃	—	2	2	no	—
37	C₂×C₄⋊Dic₇₃	—	4	8	no	—
38	C₂×C₄⋊Dic₇₃	D₄	1	2	yes	—
39	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
40	C₂×C₂²⋊Dic₇₃	—	2	4	no	—
41	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
42	#14556523	—	2	4	no	—
43	#14556523	—	4	2	no	—
44	C₄⋊#10596831	—	2	2	no	—
45	#14556512	—	2	8	no	—
46	#14556501	—	2	4	no	—
47	#14556498	—	2	4	no	—
48	#14556498	D₄	1	2	yes	—
49	C₂²×D₄×C₇₃	C₂³	1	2	yes	—
50	C₂³×C₄×C₇₃	—	2	2	no	—
51	C₂×D₈×C₇₃	C₂×C₄	1	4	yes	—
52	C₂×D₈×C₇₃	D₄	1	4	yes	—
53	C₂×D₈×C₇₃	Q₈	1	4	yes	—
54	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
56	C₄×D₄×C₇₃	—	2	4	no	—
57	C₄×D₄×C₇₃	C₂×C₄	1	8	yes	complemented
58	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
59	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
60	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
61	#107×C₇₃	C₂×C₄	1	4	yes	complemented
62	#102×C₇₃	C₂×C₄	1	8	yes	complemented
63	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	complemented
64	C₂²×Dic₇₃	—	2	4	no	—
65	C₂²×Dic₇₃	—	4	8	no	—
66	C₂²×Dic₇₃	—	4	8	no	—
67	C₂²×Dic₇₃	—	8	4	no	—
68	C₂²×Dic₇₃	C₂×D₄	1	4	yes	—
69	C₂×#10596831	—	2	2	no	—
70	C₂×#10596831	—	2	2	no	—
71	C₂×#10596831	—	2	2	no	—
72	C₄⋊Dic₇₃	—	2	4	no	—
73	C₄⋊Dic₇₃	—	2	4	no	—
74	C₄⋊Dic₇₃	—	2	4	no	—
75	C₄⋊Dic₇₃	—	4	8	no	—
76	C₄⋊Dic₇₃	—	4	4	no	—
77	C₄⋊Dic₇₃	—	8	2	no	—
78	C₂²⋊Dic₇₃	—	4	8	no	—
79	C₄×Dic₇₃	—	2	2	no	—
80	C₄×Dic₇₃	—	2	2	no	—
81	C₂⁴×C₇₃	—	2	2	no	—
82	C₂×D₄×C₇₃	—	2	2	no	—
83	C₂×D₄×C₇₃	—	2	2	no	—
84	C₂×D₄×C₇₃	—	2	2	no	—
85	C₂×D₄×C₇₃	C₂²×C₄	1	4	yes	—
86	C₂×D₄×C₇₃	C₂×D₄	1	4	yes	—
87	C₂×D₄×C₇₃	C₂×Q₈	1	4	yes	—
88	C₂²×C₄×C₇₃	—	2	2	no	—
89	C₂²×C₄×C₇₃	—	2	8	no	—
90	C₂²×C₄×C₇₃	—	4	4	no	—
91	C₂²×C₄×C₇₃	—	4	4	no	—
92	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
95	D₈×C₇₃	C₄⋊C₄	1	16	yes	complemented
96	C₂×C₈×C₇₃	—	2	1	no	—
97	C₂×C₈×C₇₃	—	2	1	no	—
98	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
102	C₄⋊C₄×C₇₃	—	2	2	no	—
103	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
104	C₂²⋊C₄×C₇₃	—	2	8	no	—
105	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₇₃	#41	1	2	yes	—
107	C₂³×C₇₃	—	2	4	no	—
108	C₂³×C₇₃	—	2	4	no	—
109	C₂³×C₇₃	—	2	4	no	—
110	C₂³×C₇₃	—	4	2	no	—
111	C₂³×C₇₃	—	4	2	no	—
112	C₂³×C₇₃	—	4	2	no	—
113	C₂³×C₇₃	—	4	2	no	—
114	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
115	D₄×C₇₃	—	2	4	no	—
116	D₄×C₇₃	—	2	4	no	—
117	D₄×C₇₃	—	2	4	no	—
118	D₄×C₇₃	C₂×C₄⋊C₄	1	8	yes	complemented
119	C₂×C₄×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	—	2	4	no	—
121	C₂×C₄×C₇₃	—	4	2	no	—
122	C₂×C₄×C₇₃	—	4	2	no	—
123	C₂×C₄×C₇₃	—	4	16	no	—
124	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
127	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×Dic₇₃	—	2	4	no	—
134	C₂×Dic₇₃	—	2	4	no	—
135	C₂×Dic₇₃	—	2	4	no	—
136	C₂×Dic₇₃	—	4	4	no	—
137	C₂×Dic₇₃	—	4	4	no	—
138	C₂×Dic₇₃	—	4	4	no	—
139	C₂×Dic₇₃	—	8	16	no	—
140	C₈×C₇₃	—	2	2	no	—
141	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	complemented
142	#10596831	—	4	2	no	—
143	C₂²×C₇₃	—	2	8	no	—
144	C₂²×C₇₃	—	4	4	no	—
145	C₂²×C₇₃	—	4	4	no	—
146	C₂²×C₇₃	—	4	4	no	—
147	C₂²×C₇₃	—	4	4	no	—
148	C₂²×C₇₃	—	4	4	no	—
149	C₂²×C₇₃	—	4	4	no	—
150	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂²×C₇₃	D₄²	1	1	yes	characteristic
152	C₂²×C₇₃	D₄×Q₈	1	1	yes	characteristic
153	C₂²×C₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
154	C₂²×C₇₃	#577	1	2	yes	—
155	C₄×C₇₃	—	2	2	no	—
156	C₄×C₇₃	—	4	4	no	—
157	C₄×C₇₃	#442	1	4	yes	complemented
158	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₄×C₇₃	#520	1	1	yes	characteristic
160	C₄×C₇₃	#547	1	1	yes	characteristic
161	C₄×C₇₃	#554	1	1	yes	characteristic
162	Dic₇₃	—	4	4	no	—
163	Dic₇₃	—	8	4	no	—
164	#22670	—	73	1	no	maximal, Sylow
165	C₂×C₇₃	—	4	8	no	—
166	C₂×C₇₃	D₄×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₇₃	C₂×#442	1	1	yes	characteristic
168	C₂×C₇₃	#2952	1	1	yes	characteristic
169	C₂×C₇₃	#3272	1	2	yes	—
170	C₂×C₇₃	#3384	1	2	yes	—
171	C₂×#442	—	73	2	no	—
172	C₂×C₄×D₈	D₇₃	1	1	yes	characteristic
173	D₄×C₄⋊C₄	—	73	2	no	—
174	#1943	—	73	2	no	—
175	#1864	—	73	4	no	—
176	#1844	—	73	1	no	—
177	#1802	—	73	2	no	—
178	#1770	—	73	1	no	—
179	C₇₃	#22670	1	1	yes	characteristic, complemented, Sylow
180	C₂²×D₈	C₂×D₇₃	1	1	yes	characteristic
181	C₂×C₄×D₄	—	73	4	no	—
182	C₂×C₄×D₄	C₂×D₇₃	1	2	yes	—
183	C₂²×C₄⋊C₄	—	146	2	no	—
184	#442	—	146	8	no	—
185	C₄×D₈	Dic₇₃	1	8	yes	complemented
186	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
187	C₂×#106	—	73	2	no	—
188	C₂×#106	—	73	2	no	—
189	C₂×C₄⋊C₈	—	73	1	no	—
190	C₂×#102	—	73	4	no	—
191	C₂×#102	C₂×D₇₃	1	2	yes	—
192	C₂×#97	—	73	2	no	—
193	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
194	#389	—	73	4	no	—
195	#385	—	146	4	no	—
196	#384	—	73	2	no	—
197	#380	—	146	2	no	—
198	C₄×C₄⋊C₄	—	73	1	no	—
199	#340	—	73	4	no	—
200	#336	—	73	2	no	—
201	C₂²×D₄	C₂²×D₇₃	1	2	yes	—
202	C₂³×C₄	—	2	2	no	—
203	C₂³×C₄	—	146	4	no	—
204	C₂×D₈	Dic₁₄₆	1	4	yes	—
205	C₂×D₈	D₂₉₂	1	4	yes	—
206	C₂×D₈	C₂×Dic₇₃	1	4	yes	—
207	C₂²×C₈	—	73	2	no	—
208	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
210	C₄×D₄	—	2	4	no	—
211	C₄×D₄	—	292	4	no	—
212	C₄×D₄	—	146	8	no	—
213	C₄×D₄	C₂×Dic₇₃	1	8	yes	complemented
214	C₂×C₄⋊C₄	—	292	8	no	—
215	C₂×C₄⋊C₄	—	73	2	no	—
216	C₂×C₄⋊C₄	—	73	4	no	—
217	C₂×C₄⋊C₄	—	146	4	no	—
218	C₂×C₄⋊C₄	—	146	2	no	—
219	C₂×C₄⋊C₄	—	146	2	no	—
220	C₂×C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
221	C₂×C₂²⋊C₄	—	2	2	no	—
222	C₂×C₂²⋊C₄	—	146	4	no	—
223	C₂×C₄²	—	73	2	no	—
224	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
225	#107	C₂×Dic₇₃	1	4	yes	complemented
226	#106	—	292	2	no	—
227	#106	—	146	4	no	—
228	C₄⋊C₈	—	146	2	no	—
229	#102	—	146	8	no	—
230	#102	C₂×Dic₇₃	1	8	yes	complemented
231	#97	—	146	4	no	—
232	C₄×C₈	C₂×Dic₇₃	1	4	yes	complemented
233	#95	—	73	2	no	—
234	#95	—	146	4	no	—
235	C₂⁴	—	2	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	C₂²×Dic₇₃	1	4	yes	—
240	C₂×D₄	C₂×D₂₉₂	1	4	yes	—
241	C₂×D₄	C₂×Dic₁₄₆	1	4	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	8	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	292	8	no	—
247	C₂²×C₄	—	292	8	no	—
248	C₂²×C₄	—	584	4	no	—
249	C₂²×C₄	—	73	4	no	—
250	C₂²×C₄	—	146	4	no	—
251	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
253	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
254	D₈	C₄⋊Dic₇₃	1	16	yes	complemented
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	146	2	no	—
258	C₂×C₈	—	146	2	no	—
259	C₂×C₈	—	146	2	no	—
260	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
261	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
262	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
263	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	292	8	no	—
266	C₄⋊C₄	—	292	4	no	—
267	C₄⋊C₄	—	584	2	no	—
268	C₄⋊C₄	—	146	4	no	—
269	C₄⋊C₄	—	146	4	no	—
270	C₄⋊C₄	—	146	4	no	—
271	C₄⋊C₄	C₂²×Dic₇₃	1	4	yes	—
272	C₂²⋊C₄	—	2	8	no	—
273	C₂²⋊C₄	—	292	8	no	—
274	C₄²	—	146	2	no	—
275	C₄²	—	146	2	no	—
276	C₄²	C₂²×Dic₇₃	1	2	yes	—
277	C₄²	#11924027	1	2	yes	—
278	C₂³	—	2	4	no	—
279	C₂³	—	2	4	no	—
280	C₂³	—	2	4	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	4	2	no	—
283	C₂³	—	4	2	no	—
284	C₂³	—	4	2	no	—
285	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
286	D₄	—	2	4	no	—
287	D₄	—	2	4	no	—
288	D₄	—	2	4	no	—
289	D₄	C₂×C₄⋊Dic₇₃	1	8	yes	complemented
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	16	no	—
295	C₂×C₄	—	292	4	no	—
296	C₂×C₄	—	292	4	no	—
297	C₂×C₄	—	292	4	no	—
298	C₂×C₄	—	584	16	no	—
299	C₂×C₄	—	146	4	no	—
300	C₂×C₄	—	146	4	no	—
301	C₂×C₄	—	146	4	no	—
302	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
303	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
304	C₂×C₄	#14556615	1	4	yes	—
305	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
307	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
308	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
309	C₂×C₄	#14556637	1	1	yes	characteristic
310	C₂×C₄	#14556638	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	292	2	no	—
313	C₈	C₂×C₄⋊Dic₇₃	1	4	yes	complemented
314	C₂²	—	2	8	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	4	4	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	4	no	—
320	C₂²	—	4	4	no	—
321	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
322	C₂²	D₄×D₂₉₂	1	1	yes	characteristic
323	C₂²	D₄×Dic₁₄₆	1	1	yes	characteristic
324	C₂²	C₂×#14556615	1	2	yes	—
325	C₂²	#17989394	1	2	yes	—
326	C₄	—	2	2	no	—
327	C₄	—	4	4	no	—
328	C₄	—	292	4	no	—
329	C₄	—	584	4	no	—
330	C₄	#17988699	1	4	yes	complemented
331	C₄	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
332	C₄	#17989070	1	1	yes	characteristic
333	C₄	#17989241	1	1	yes	characteristic
334	C₄	#17989294	1	1	yes	characteristic
335	C₂	—	4	8	no	—
336	C₂	D₄×C₄⋊Dic₇₃	1	1	yes	characteristic
337	C₂	C₂×#17988699	1	1	yes	characteristic
338	C₂	#22618815	1	1	yes	characteristic
339	C₂	#22621532	1	2	yes	—
340	C₂	#22622828	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×C₄×D₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×D₄
prime radical	C₂²×D₈×C₇₃	C₂²
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×C₄×D₈	D₇₃
hypocenter	C₇₃	#22670
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. D₄×C₄⋊Dic₇₃	C₂	semidirect
3. #22610670	C₂	semidirect
4. #22610194	C₂	semidirect
5. #22610088	C₂	semidirect
6. #22609931	C₂	semidirect
7. #22609798	C₂	semidirect
26. C₄×D₈×C₇₃	C₄	semidirect
57. C₄×D₄×C₇₃	C₂×C₄	semidirect
61. #107×C₇₃	C₂×C₄	semidirect
62. #102×C₇₃	C₂×C₄	semidirect
63. C₄×C₈×C₇₃	C₂×C₄	semidirect
95. D₈×C₇₃	C₄⋊C₄	semidirect
118. D₄×C₇₃	C₂×C₄⋊C₄	semidirect
141. C₈×C₇₃	C₂×C₄⋊C₄	semidirect
157. C₄×C₇₃	#442	semidirect
179. C₇₃	#22670	semidirect
185. C₄×D₈	Dic₇₃	semidirect
213. C₄×D₄	C₂×Dic₇₃	semidirect
225. #107	C₂×Dic₇₃	semidirect
230. #102	C₂×Dic₇₃	semidirect
232. C₄×C₈	C₂×Dic₇₃	semidirect
254. D₈	C₄⋊Dic₇₃	semidirect
289. D₄	C₂×C₄⋊Dic₇₃	semidirect
313. C₈	C₂×C₄⋊Dic₇₃	semidirect
330. C₄	#17988699	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988699	2
D₄×C₄⋊Dic₇₃	2
#22610670	2
#22610194	2
#22610088	2
#22609931	2
#22609798	2
C₂×C₄×D₈×C₇₃	2
#22670	73