Subgroups — #33104397

Order 18688.

1062 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×Dic₇₃	C₂	1	2	yes	maximal
2	#22612841	C₂	1	2	yes	maximal, complemented
3	#22610665	C₂	1	2	yes	maximal, complemented
4	#22610194	C₂	1	4	yes	maximal, complemented
5	#22610095	C₂	1	1	yes	maximal, characteristic, complemented
6	#22609941	C₂	1	2	yes	maximal, complemented
7	C₈×C₄⋊Dic₇₃	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	D₈×Dic₇₃	—	2	8	no	—
11	C₂×#14556525	C₂²	1	2	yes	—
12	C₂×#14556523	C₂²	1	2	yes	—
13	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	C₂×#14556512	C₂²	1	4	yes	—
15	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
16	#17988486	C₂²	1	4	yes	—
17	#17988462	—	2	4	no	—
18	#17988453	C₂²	1	2	yes	—
19	#17988447	—	2	2	no	—
20	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
21	C₂×#14556498	—	2	2	no	—
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	D₄	1	2	yes	—
36	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
37	C₂×C₂²⋊Dic₇₃	—	2	4	no	—
38	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
39	#14556525	—	2	4	no	—
40	#14556523	—	4	2	no	—
41	C₄⋊#10596831	—	2	2	no	—
42	#14556512	—	2	8	no	—
43	C₈×Dic₇₃	—	2	4	no	—
44	#14556498	—	2	4	no	—
45	#14556498	—	2	2	no	—
46	#14556498	—	2	2	no	—
47	#14556498	D₄	1	1	yes	characteristic
48	#14556498	D₄	1	1	yes	characteristic
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	2	yes	—
69	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
70	C₂×#10596831	—	2	2	no	—
71	C₂×#10596831	—	2	2	no	—
72	C₂×#10596831	—	2	2	no	—
73	C₄⋊Dic₇₃	—	2	4	no	—
74	C₄⋊Dic₇₃	—	2	4	no	—
75	C₄⋊Dic₇₃	—	2	4	no	—
76	C₄⋊Dic₇₃	—	8	4	no	—
77	C₂²⋊Dic₇₃	—	4	8	no	—
78	C₄×Dic₇₃	—	2	2	no	—
79	C₄×Dic₇₃	—	2	2	no	—
80	C₂⁴×C₇₃	—	2	2	no	—
81	C₂×D₄×C₇₃	—	2	2	no	—
82	C₂×D₄×C₇₃	—	2	2	no	—
83	C₂×D₄×C₇₃	—	2	2	no	—
84	C₂×D₄×C₇₃	C₂²×C₄	1	4	yes	—
85	C₂×D₄×C₇₃	C₂×D₄	1	4	yes	—
86	C₂×D₄×C₇₃	C₂×Q₈	1	4	yes	—
87	C₂²×C₄×C₇₃	—	2	2	no	—
88	C₂²×C₄×C₇₃	—	2	8	no	—
89	C₂²×C₄×C₇₃	—	8	4	no	—
90	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
93	D₈×C₇₃	C₄⋊C₄	1	16	yes	complemented
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	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
100	C₄⋊C₄×C₇₃	—	2	2	no	—
101	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
102	C₂²⋊C₄×C₇₃	—	2	8	no	—
103	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₇₃	#41	1	2	yes	—
105	C₂³×C₇₃	—	2	4	no	—
106	C₂³×C₇₃	—	2	4	no	—
107	C₂³×C₇₃	—	2	4	no	—
108	C₂³×C₇₃	—	4	4	no	—
109	C₂³×C₇₃	—	4	4	no	—
110	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
111	D₄×C₇₃	—	2	4	no	—
112	D₄×C₇₃	—	2	4	no	—
113	D₄×C₇₃	—	2	4	no	—
114	D₄×C₇₃	C₂×C₄⋊C₄	1	8	yes	—
115	C₂×C₄×C₇₃	—	2	2	no	—
116	C₂×C₄×C₇₃	—	2	2	no	—
117	C₂×C₄×C₇₃	—	2	4	no	—
118	C₂×C₄×C₇₃	—	4	4	no	—
119	C₂×C₄×C₇₃	—	8	8	no	—
120	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×Dic₇₃	—	2	2	no	—
129	C₂×Dic₇₃	—	2	2	no	—
130	C₂×Dic₇₃	—	2	2	no	—
131	C₂×Dic₇₃	—	2	2	no	—
132	C₂×Dic₇₃	—	2	2	no	—
133	C₂×Dic₇₃	—	4	4	no	—
134	C₂×Dic₇₃	—	4	4	no	—
135	C₂×Dic₇₃	—	4	4	no	—
136	C₂×Dic₇₃	—	8	16	no	—
137	C₂×Dic₇₃	#135	1	4	yes	—
138	C₈×C₇₃	—	2	2	no	—
139	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
140	#10596831	—	4	2	no	—
141	C₂²×C₇₃	—	2	8	no	—
142	C₂²×C₇₃	—	4	8	no	—
143	C₂²×C₇₃	—	4	4	no	—
144	C₂²×C₇₃	—	4	4	no	—
145	C₂²×C₇₃	—	4	8	no	—
146	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂²×C₇₃	D₄²	1	1	yes	characteristic
148	C₂²×C₇₃	D₄×Q₈	1	1	yes	characteristic
149	C₂²×C₇₃	C₂×#135	1	2	yes	—
150	C₂²×C₇₃	#576	1	2	yes	—
151	C₄×C₇₃	—	2	2	no	—
152	C₄×C₇₃	—	2	2	no	—
153	C₄×C₇₃	—	4	4	no	—
154	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₇₃	#520	1	1	yes	characteristic
156	C₄×C₇₃	#547	1	1	yes	characteristic
157	C₄×C₇₃	#554	1	1	yes	characteristic
158	Dic₇₃	—	2	4	no	—
159	Dic₇₃	—	4	2	no	—
160	Dic₇₃	—	8	4	no	—
161	#22682	—	73	1	no	maximal, Sylow
162	C₂×C₇₃	—	4	8	no	—
163	C₂×C₇₃	D₄×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₇₃	C₄×#135	1	1	yes	characteristic
165	C₂×C₇₃	#2948	1	1	yes	characteristic
166	C₂×C₇₃	#3280	1	2	yes	—
167	C₂×C₇₃	#3380	1	2	yes	—
168	C₂×C₄×D₈	—	73	2	no	—
169	C₂×C₄×D₈	D₇₃	1	1	yes	characteristic
170	#2349	—	73	2	no	—
171	#1942	—	73	2	no	—
172	#1864	—	73	4	no	—
173	#1846	—	73	1	no	—
174	#1804	—	73	2	no	—
175	C₈×C₄⋊C₄	—	73	1	no	—
176	C₇₃	#22682	1	1	yes	characteristic, complemented, Sylow
177	C₂²×D₈	C₂×D₇₃	1	1	yes	characteristic
178	C₂×C₄×D₄	—	73	4	no	—
179	C₂×C₄×D₄	C₂×D₇₃	1	2	yes	—
180	C₄×D₈	—	146	8	no	—
181	C₄×D₈	Dic₇₃	1	8	yes	complemented
182	C₂×#107	—	73	2	no	—
183	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
184	C₂×#106	—	73	2	no	—
185	C₂×C₄⋊C₈	—	73	1	no	—
186	C₂×#102	—	73	4	no	—
187	C₂×#102	C₂×D₇₃	1	2	yes	—
188	C₂×C₄×C₈	—	73	2	no	—
189	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
190	#389	—	73	4	no	—
191	#385	—	146	4	no	—
192	#382	—	73	2	no	—
193	#380	—	146	2	no	—
194	C₄×C₄⋊C₄	—	73	1	no	—
195	C₂×#95	—	146	2	no	—
196	#340	—	73	4	no	—
197	#336	—	73	2	no	—
198	C₂²×D₄	C₂²×D₇₃	1	2	yes	—
199	C₂³×C₄	—	2	2	no	—
200	C₂³×C₄	—	146	4	no	—
201	C₂×D₈	Dic₁₄₆	1	4	yes	—
202	C₂×D₈	D₂₉₂	1	4	yes	—
203	C₂×D₈	C₂×Dic₇₃	1	4	yes	—
204	C₂²×C₈	—	73	2	no	—
205	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
207	C₄×D₄	—	2	4	no	—
208	C₄×D₄	—	292	4	no	—
209	C₄×D₄	—	146	8	no	—
210	C₄×D₄	C₂×Dic₇₃	1	8	yes	complemented
211	C₂×C₄⋊C₄	—	73	2	no	—
212	C₂×C₄⋊C₄	—	73	4	no	—
213	C₂×C₄⋊C₄	—	146	4	no	—
214	C₂×C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
215	C₂×C₂²⋊C₄	—	2	2	no	—
216	C₂×C₂²⋊C₄	—	146	4	no	—
217	C₂×C₄²	—	73	2	no	—
218	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
219	#107	—	146	4	no	—
220	#107	C₂×Dic₇₃	1	4	yes	complemented
221	#106	—	292	2	no	—
222	C₄⋊C₈	—	146	2	no	—
223	#102	—	146	8	no	—
224	#102	C₂×Dic₇₃	1	8	yes	complemented
225	C₄×C₈	—	146	4	no	—
226	C₄×C₈	C₂×Dic₇₃	1	4	yes	complemented
227	#95	—	73	1	no	—
228	#95	—	73	1	no	—
229	#95	—	146	4	no	—
230	#95	—	146	2	no	—
231	#95	—	146	2	no	—
232	C₂⁴	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	C₂²×Dic₇₃	1	4	yes	—
237	C₂×D₄	C₂×D₂₉₂	1	4	yes	—
238	C₂×D₄	C₂×Dic₁₄₆	1	4	yes	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	8	no	—
241	C₂²×C₄	—	8	4	no	—
242	C₂²×C₄	—	292	8	no	—
243	C₂²×C₄	—	292	8	no	—
244	C₂²×C₄	—	584	4	no	—
245	C₂²×C₄	—	73	2	no	—
246	C₂²×C₄	—	73	2	no	—
247	C₂²×C₄	—	146	4	no	—
248	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
249	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
250	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
251	D₈	C₄⋊Dic₇₃	1	16	yes	complemented
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	146	2	no	—
255	C₂×C₈	—	146	2	no	—
256	C₂×C₈	—	146	2	no	—
257	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
258	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
259	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
260	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	584	4	no	—
263	C₄⋊C₄	—	146	4	no	—
264	C₄⋊C₄	—	146	4	no	—
265	C₄⋊C₄	—	146	4	no	—
266	C₄⋊C₄	C₂²×Dic₇₃	1	4	yes	—
267	C₂²⋊C₄	—	2	8	no	—
268	C₂²⋊C₄	—	292	8	no	—
269	C₄²	—	146	2	no	—
270	C₄²	—	146	2	no	—
271	C₄²	C₂²×Dic₇₃	1	2	yes	—
272	C₄²	#11924027	1	2	yes	—
273	C₂³	—	2	4	no	—
274	C₂³	—	2	4	no	—
275	C₂³	—	2	4	no	—
276	C₂³	—	4	4	no	—
277	C₂³	—	4	4	no	—
278	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
279	D₄	—	2	4	no	—
280	D₄	—	2	4	no	—
281	D₄	—	2	4	no	—
282	D₄	C₂×C₄⋊Dic₇₃	1	8	yes	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	8	8	no	—
288	C₂×C₄	—	292	4	no	—
289	C₂×C₄	—	292	4	no	—
290	C₂×C₄	—	292	4	no	—
291	C₂×C₄	—	584	16	no	—
292	C₂×C₄	—	73	4	no	—
293	C₂×C₄	—	146	2	no	—
294	C₂×C₄	—	146	2	no	—
295	C₂×C₄	—	146	2	no	—
296	C₂×C₄	—	146	2	no	—
297	C₂×C₄	—	146	2	no	—
298	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
299	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
300	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
302	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
303	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
304	C₂×C₄	#14556637	1	1	yes	characteristic
305	C₂×C₄	#14556638	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	292	2	no	—
308	C₈	C₂×C₄⋊Dic₇₃	1	4	yes	—
309	C₂²	—	2	8	no	—
310	C₂²	—	4	8	no	—
311	C₂²	—	4	4	no	—
312	C₂²	—	4	4	no	—
313	C₂²	—	4	8	no	—
314	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
315	C₂²	D₄×D₂₉₂	1	1	yes	characteristic
316	C₂²	D₄×Dic₁₄₆	1	1	yes	characteristic
317	C₂²	#135×D₇₃	1	2	yes	—
318	C₂²	#17989388	1	2	yes	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	4	no	—
322	C₄	—	292	2	no	—
323	C₄	—	584	4	no	—
324	C₄	—	146	4	no	—
325	C₄	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
326	C₄	#17989070	1	1	yes	characteristic
327	C₄	#17989241	1	1	yes	characteristic
328	C₄	#17989294	1	1	yes	characteristic
329	C₂	—	4	8	no	—
330	C₂	D₄×C₄⋊Dic₇₃	1	1	yes	characteristic
331	C₂	#135×Dic₇₃	1	1	yes	characteristic
332	C₂	#22618783	1	1	yes	characteristic
333	C₂	#22621617	1	2	yes	—
334	C₂	#22622798	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	#22682
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. #22612841	C₂	semidirect
3. #22610665	C₂	semidirect
4. #22610194	C₂	semidirect
5. #22610095	C₂	semidirect
6. #22609941	C₂	semidirect
7. C₈×C₄⋊Dic₇₃	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
93. D₈×C₇₃	C₄⋊C₄	semidirect
176. C₇₃	#22682	semidirect
181. C₄×D₈	Dic₇₃	semidirect
210. C₄×D₄	C₂×Dic₇₃	semidirect
220. #107	C₂×Dic₇₃	semidirect
224. #102	C₂×Dic₇₃	semidirect
226. C₄×C₈	C₂×Dic₇₃	semidirect
251. D₈	C₄⋊Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈×Dic₇₃	2
#22612841	2
#22610665	2
#22610194	2
#22610095	2
#22609941	2
C₈×C₄⋊Dic₇₃	2
C₂×C₄×D₈×C₇₃	2
#22682	73