Subgroups — #32995853

Order 18688.

362 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11924027	C₂	1	1	yes	maximal, characteristic
2	#22611728	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#22607911	C₂	1	1	yes	maximal, characteristic, complemented
5	#22607909	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊#11924003	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11924027	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₇₃	—	2	1	no	—
10	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
11	#17988711	C₄	1	1	yes	characteristic
12	D₄×#10596831	C₄	1	1	yes	characteristic
13	C₈×C₂²⋊D₇₃	—	2	1	no	—
14	#17988650	—	2	1	no	—
15	#17988639	C₄	1	1	yes	characteristic
16	#17988637	—	2	1	no	—
17	C₂×C₈×Dic₇₃	—	2	1	no	—
18	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
19	C₄⋊#11924003	C₂²	1	1	yes	characteristic
20	#17988405	—	2	1	no	—
21	#17988405	—	2	1	no	—
22	#17988366	—	2	1	no	—
23	C₄×#11924003	C₂²	1	1	yes	characteristic
24	C₈×D₄×C₇₃	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×#11924027	C₈	1	1	yes	characteristic
28	C₂²×#10596831	—	2	1	no	—
29	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
30	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
31	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
32	#14556556	C₈	1	1	yes	characteristic
33	D₄×Dic₇₃	C₈	1	1	yes	characteristic
34	C₄×C₂²⋊D₇₃	—	2	1	no	—
35	#14556551	—	2	1	no	—
36	#14556549	C₈	1	1	yes	characteristic
37	#14556547	—	2	1	no	—
38	C₂×C₄×Dic₇₃	—	2	1	no	—
39	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×#11924003	—	2	1	no	—
41	C₂×#11924003	—	2	1	no	—
42	C₁₆×D₇₃	—	4	2	no	—
43	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
44	#14556520	—	2	1	no	—
45	#14556520	—	2	1	no	—
46	C₂²⋊#10596831	—	2	1	no	—
47	#14556503	—	2	1	no	—
48	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
49	C₈×Dic₇₃	—	2	1	no	—
50	C₈×Dic₇₃	—	2	1	no	—
51	C₈×Dic₇₃	—	2	1	no	—
52	C₈×Dic₇₃	D₄	1	1	yes	characteristic
53	C₈×Dic₇₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₇₃	—	2	1	no	—
55	C₄×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₆×C₇₃	—	2	1	no	—
57	C₂×C₁₆×C₇₃	—	2	1	no	—
58	C₄⋊C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊C₈×C₇₃	—	2	1	no	—
60	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
61	#11924027	—	2	2	no	—
62	#11924027	C₁₆	1	4	yes	complemented
63	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
64	C₂×C₂²⋊D₇₃	—	2	1	no	—
65	C₂²×Dic₇₃	—	2	1	no	—
66	C₂²×Dic₇₃	—	2	1	no	—
67	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
70	C₂×#10596831	—	2	1	no	—
71	C₂×#10596831	—	4	2	no	—
72	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
74	C₈×D₇₃	—	2	1	no	—
75	C₈×D₇₃	—	2	1	no	—
76	C₈×D₇₃	—	4	1	no	—
77	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
78	#11924007	—	2	1	no	—
79	#11924007	—	2	1	no	—
80	C₂²⋊Dic₇₃	—	2	1	no	—
81	#11924005	—	2	1	no	—
82	C₄×Dic₇₃	—	2	1	no	—
83	C₄×Dic₇₃	—	2	1	no	—
84	C₄×Dic₇₃	—	2	1	no	—
85	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
87	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
88	#11924003	—	4	1	no	—
89	#11924003	—	4	1	no	—
90	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	—	2	1	no	—
92	C₂×C₈×C₇₃	—	2	1	no	—
93	C₂×C₈×C₇₃	—	4	1	no	—
94	C₂×C₈×C₇₃	—	4	1	no	—
95	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
97	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
98	C₂²⋊C₄×C₇₃	—	2	1	no	—
99	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₇₃	—	4	1	no	—
101	C₁₆×C₇₃	—	4	1	no	—
102	C₂³×C₇₃	—	2	1	no	—
103	C₂²×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
104	D₄×C₇₃	—	2	1	no	—
105	D₄×C₇₃	C₂×C₁₆	1	2	yes	complemented
106	C₂×C₄×C₇₃	—	2	1	no	—
107	C₂×C₄×C₇₃	—	4	1	no	—
108	C₂×C₄×C₇₃	—	4	1	no	—
109	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
112	C₂²⋊D₇₃	—	2	4	no	—
113	C₂×Dic₇₃	—	2	1	no	—
114	C₂×Dic₇₃	—	2	4	no	—
115	C₂×Dic₇₃	—	2	1	no	—
116	C₂×Dic₇₃	—	4	2	no	—
117	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
119	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
120	C₄×D₇₃	—	2	1	no	—
121	C₄×D₇₃	—	2	1	no	—
122	C₄×D₇₃	—	4	1	no	—
123	C₄×D₇₃	C₂×C₁₆	1	2	yes	—
124	C₄×D₇₃	M₃₂	1	2	yes	—
125	Dic₁₄₆	—	2	1	no	—
126	Dic₁₄₆	C₂×C₁₆	1	2	yes	complemented
127	C₈×C₇₃	—	2	1	no	—
128	C₈×C₇₃	—	4	1	no	—
129	C₈×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₈×C₇₃	#137	1	1	yes	characteristic
131	#10596831	—	2	1	no	—
132	#10596831	—	2	1	no	—
133	#10596831	—	4	1	no	—
134	C₂²×C₇₃	—	2	2	no	—
135	C₂²×C₇₃	—	4	1	no	—
136	C₂²×C₇₃	—	4	1	no	—
137	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₂×D₇₃	—	2	1	no	—
139	C₂×D₇₃	—	2	1	no	—
140	C₂×D₇₃	C₂²⋊C₁₆	1	2	yes	—
141	C₄×C₇₃	—	2	1	no	—
142	C₄×C₇₃	—	4	1	no	—
143	C₄×C₇₃	#417	1	1	yes	characteristic
144	C₄×C₇₃	#421	1	1	yes	characteristic
145	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
146	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
147	Dic₇₃	—	2	2	no	—
148	Dic₇₃	—	2	1	no	—
149	Dic₇₃	—	2	1	no	—
150	Dic₇₃	—	4	1	no	—
151	Dic₇₃	C₂²⋊C₁₆	1	2	yes	—
152	#11621	—	73	1	no	maximal, Sylow
153	C₂×C₇₃	—	4	2	no	—
154	C₂×C₇₃	#1477	1	1	yes	characteristic
155	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
156	C₂×C₇₃	#2117	1	1	yes	characteristic
157	D₇₃	—	2	2	no	—
158	C₈×#41	—	73	1	no	—
159	#2162	—	73	1	no	—
160	C₂×C₄⋊C₁₆	—	73	1	no	—
161	Q₈⋊C₁₆	—	73	1	no	—
162	Q₈⋊C₁₆	—	73	1	no	—
163	D₄⋊C₁₆	—	73	1	no	—
164	D₄⋊C₁₆	D₇₃	1	1	yes	characteristic, complemented
165	C₇₃	#11621	1	1	yes	characteristic, complemented, Sylow
166	C₄×#41	—	73	1	no	—
167	C₂²×C₁₆	—	146	1	no	—
168	C₈×Q₈	—	73	1	no	—
169	C₈×Q₈	—	73	1	no	—
170	C₈×D₄	—	73	1	no	—
171	C₈×D₄	—	146	1	no	—
172	C₈×D₄	—	146	1	no	—
173	C₈×D₄	C₂×D₇₃	1	1	yes	characteristic
174	#431	—	73	1	no	—
175	#431	—	146	1	no	—
176	C₂×C₄×C₈	—	73	1	no	—
177	C₂×C₄×C₈	—	146	1	no	—
178	C₄⋊C₁₆	—	73	1	no	—
179	C₄⋊C₁₆	—	146	1	no	—
180	C₄⋊C₁₆	—	146	1	no	—
181	C₄⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
182	C₂²⋊C₁₆	—	146	1	no	—
183	C₄×C₁₆	—	73	1	no	—
184	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
185	C₂×#41	—	73	1	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	73	1	no	—
188	C₂²×C₈	—	73	1	no	—
189	C₂²×C₈	—	146	1	no	—
190	C₄×Q₈	—	73	1	no	—
191	C₄×Q₈	—	73	1	no	—
192	C₄×D₄	—	73	1	no	—
193	C₄×D₄	—	146	1	no	—
194	C₄×D₄	—	146	1	no	—
195	C₄×D₄	C₄×D₇₃	1	1	yes	characteristic
196	#117	—	73	1	no	—
197	#117	—	146	1	no	—
198	C₂×C₄²	—	73	1	no	—
199	C₂×C₄²	—	146	1	no	—
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	—	2	1	no	—
202	C₂×C₁₆	—	292	2	no	—
203	C₂×C₁₆	—	146	1	no	—
204	C₂×C₁₆	—	146	1	no	—
205	C₄⋊C₈	—	73	1	no	—
206	C₄⋊C₈	—	146	1	no	—
207	C₄⋊C₈	—	146	1	no	—
208	C₄⋊C₈	C₄×D₇₃	1	1	yes	characteristic
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	146	1	no	—
211	C₂²⋊C₈	—	146	1	no	—
212	C₄×C₈	—	73	1	no	—
213	C₄×C₈	—	73	1	no	—
214	C₄×C₈	—	73	1	no	—
215	C₄×C₈	—	146	1	no	—
216	C₄×C₈	—	146	1	no	—
217	C₄×C₈	—	146	1	no	—
218	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
219	#41	—	73	4	no	—
220	#41	—	146	2	no	—
221	C₂×Q₈	—	73	1	no	—
222	C₂×D₄	—	146	1	no	—
223	C₂×D₄	C₈×D₇₃	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	73	1	no	—
226	C₂²×C₄	—	73	1	no	—
227	C₂²×C₄	—	73	1	no	—
228	C₂²×C₄	—	146	1	no	—
229	C₂²×C₄	—	146	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	292	1	no	—
234	C₂×C₈	—	292	2	no	—
235	C₂×C₈	—	73	1	no	—
236	C₂×C₈	—	73	1	no	—
237	C₂×C₈	—	146	1	no	—
238	C₂×C₈	—	146	1	no	—
239	C₂×C₈	—	146	1	no	—
240	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
241	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
242	C₄⋊C₄	—	73	1	no	—
243	C₄⋊C₄	—	146	1	no	—
244	C₄⋊C₄	—	146	1	no	—
245	C₄⋊C₄	C₈×D₇₃	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	146	1	no	—
248	C₂²⋊C₄	—	146	1	no	—
249	C₄²	—	73	1	no	—
250	C₄²	—	73	1	no	—
251	C₄²	—	73	1	no	—
252	C₄²	—	146	1	no	—
253	C₄²	—	146	1	no	—
254	C₄²	—	146	1	no	—
255	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
256	C₁₆	—	4	1	no	—
257	C₁₆	—	4	1	no	—
258	C₁₆	—	292	1	no	—
259	C₁₆	—	292	1	no	—
260	C₂³	—	2	1	no	—
261	C₂³	—	73	1	no	—
262	Q₈	—	73	2	no	—
263	Q₈	—	146	1	no	—
264	D₄	—	2	1	no	—
265	D₄	—	146	4	no	—
266	D₄	C₁₆×D₇₃	1	2	yes	complemented
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	292	1	no	—
271	C₂×C₄	—	292	2	no	—
272	C₂×C₄	—	73	1	no	—
273	C₂×C₄	—	73	2	no	—
274	C₂×C₄	—	73	2	no	—
275	C₂×C₄	—	73	1	no	—
276	C₂×C₄	—	73	1	no	—
277	C₂×C₄	—	146	1	no	—
278	C₂×C₄	—	146	4	no	—
279	C₂×C₄	—	146	1	no	—
280	C₂×C₄	—	146	1	no	—
281	C₂×C₄	—	146	1	no	—
282	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
283	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
284	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	4	1	no	—
287	C₈	—	292	1	no	—
288	C₈	—	146	1	no	—
289	C₈	—	146	1	no	—
290	C₈	#14556615	1	1	yes	characteristic
291	C₈	#14556620	1	1	yes	characteristic
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	73	2	no	—
296	C₂²	—	146	1	no	—
297	C₂²	—	146	1	no	—
298	C₂²	C₂²⋊C₈×D₇₃	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	292	1	no	—
302	C₄	—	73	2	no	—
303	C₄	—	146	2	no	—
304	C₄	—	146	1	no	—
305	C₄	—	146	1	no	—
306	C₄	#17988585	1	1	yes	characteristic
307	C₄	#17988602	1	1	yes	characteristic
308	C₄	C₂×C₁₆×D₇₃	1	1	yes	characteristic
309	C₄	M₃₂×D₇₃	1	1	yes	characteristic
310	C₂	—	4	2	no	—
311	C₂	—	146	2	no	—
312	C₂	#22608315	1	1	yes	characteristic
313	C₂	C₂²⋊C₁₆×D₇₃	1	1	yes	characteristic
314	C₂	#22611557	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₃
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×D₄⋊C₁₆	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	C₂×#11924027	C₈
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	D₄⋊C₁₆	D₇₃
hypocenter	C₇₃	#11621
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ D₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #22611728	C₂	semidirect
3. C₄⋊C₁₆×D₇₃	C₂	semidirect
4. #22607911	C₂	semidirect
5. #22607909	C₂	semidirect
6. D₄⋊#11924003	C₂	semidirect
7. C₇₃×D₄⋊C₁₆	C₂	semidirect
62. #11924027	C₁₆	semidirect
105. D₄×C₇₃	C₂×C₁₆	semidirect
126. Dic₁₄₆	C₂×C₁₆	semidirect
164. D₄⋊C₁₆	D₇₃	semidirect
165. C₇₃	#11621	semidirect
266. D₄	C₁₆×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11924027	2
#22611728	2
C₄⋊C₁₆×D₇₃	2
#22607911	2
#22607909	2
D₄⋊#11924003	2
C₇₃×D₄⋊C₁₆	2
#11621	73