Subgroups — #34019553

Order 18688.

626 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626833	C₂	1	2	yes	maximal
2	#22625570	C₂	1	2	yes	maximal, complemented
3	#22625554	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#10596833	C₂	1	2	yes	maximal, complemented
5	#22625492	C₂	1	1	yes	maximal, characteristic
6	#22625274	C₂	1	2	yes	maximal
7	C₈×#11924038	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
9	#17989637	—	2	4	no	—
10	#17989637	C₂²	1	4	yes	—
11	Q₈×#10596833	C₂²	1	4	yes	—
12	#17989609	—	2	2	no	—
13	#17989609	C₂²	1	2	yes	—
14	#17989606	—	2	2	no	—
15	#17989606	C₂²	1	1	yes	characteristic
16	C₄×#11924038	C₂²	1	1	yes	characteristic
17	#17989450	C₂²	1	2	yes	—
18	#17989448	C₂²	1	2	yes	—
19	#17989437	C₂²	1	4	yes	—
20	Q₈⋊#10596833	C₂²	1	4	yes	—
21	#17989429	C₂²	1	2	yes	—
22	C₈×#10596833	C₂²	1	2	yes	—
23	C₂×Q₁₆×D₇₃	C₄	1	1	yes	characteristic
24	C₄×Q₈×D₇₃	C₂²	1	2	yes	—
25	C₄×Dic₂₉₂	C₄	1	1	yes	characteristic
26	Q₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
27	#17988681	C₂²	1	1	yes	characteristic
28	C₄×#11924018	C₄	1	2	yes	—
29	#17988679	C₂²	1	2	yes	—
30	#107×D₇₃	C₄	1	1	yes	characteristic
31	#103×D₇₃	C₄	1	2	yes	—
32	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
33	C₄×Q₁₆×C₇₃	C₄	1	1	yes	characteristic
34	C₂×#11924038	—	2	1	no	—
35	C₄⋊#10596833	—	2	4	no	—
36	C₄⋊#10596833	—	2	4	no	—
37	C₄⋊#10596833	D₄	1	2	yes	—
38	C₄⋊#10596833	C₂³	1	4	yes	—
39	#14556648	—	2	2	no	—
40	#14556648	D₄	1	1	yes	characteristic
41	#14556648	D₄	1	1	yes	characteristic
42	#14556646	—	2	4	no	—
43	#14556646	—	2	2	no	—
44	#14556646	—	2	1	no	—
45	C₄×#10596833	—	2	4	no	—
46	C₄×#10596833	C₂³	1	2	yes	—
47	C₄×#10596833	C₂³	1	2	yes	—
48	C₂×Q₈×D₇₃	C₂×C₄	1	2	yes	—
49	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
50	C₂×#11924018	C₂×C₄	1	2	yes	—
51	Q₁₆×D₇₃	—	2	4	no	—
52	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
54	Q₈×Dic₇₃	C₂³	1	2	yes	—
55	C₄×Dic₁₄₆	—	2	2	no	—
56	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
57	#14556556	—	2	2	no	—
58	#14556556	C₂³	1	2	yes	—
59	C₄⋊C₄×D₇₃	—	2	2	no	—
60	C₄⋊C₄×D₇₃	C₂×C₄	1	2	yes	—
61	C₄²×D₇₃	—	2	2	no	—
62	C₄²×D₇₃	C₂³	1	1	yes	characteristic
63	#14556525	C₂×C₄	1	1	yes	characteristic
64	#14556524	C₂×C₄	1	2	yes	—
65	#14556516	C₂×C₄	1	2	yes	—
66	#14556515	C₂×C₄	1	2	yes	—
67	#14556514	C₂×C₄	1	2	yes	—
68	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
70	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
71	C₂×Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
72	C₄×Q₈×C₇₃	C₂×C₄	1	2	yes	—
73	#107×C₇₃	C₂×C₄	1	1	yes	characteristic
74	#103×C₇₃	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
76	#11924038	—	2	4	no	—
77	C₂×#10596833	—	2	2	no	—
78	C₂×#10596833	—	2	4	no	—
79	C₂×#10596833	C₂×D₄	1	2	yes	—
80	C₂×#10596833	C₂×D₄	1	2	yes	—
81	C₂×Dic₁₄₆	—	2	2	no	—
82	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
83	Q₈×D₇₃	—	2	4	no	—
84	Q₈×D₇₃	—	4	2	no	—
85	C₂×C₄×D₇₃	—	2	2	no	—
86	C₂×C₄×D₇₃	—	2	2	no	—
87	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
90	Dic₂₉₂	—	2	2	no	—
91	#11924018	—	2	4	no	—
92	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
93	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
94	C₈×D₇₃	—	2	2	no	—
95	C₈×D₇₃	—	4	1	no	—
96	C₄⋊Dic₇₃	—	2	2	no	—
97	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
98	#11924007	—	2	2	no	—
99	#11924007	—	2	2	no	—
100	#11924007	—	2	2	no	—
101	C₄×Dic₇₃	—	2	2	no	—
102	C₄×Dic₇₃	—	2	2	no	—
103	C₄×Dic₇₃	—	2	2	no	—
104	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
105	C₄×Dic₇₃	#41	1	1	yes	characteristic
106	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
107	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
108	Q₁₆×C₇₃	M₁₆	1	4	yes	complemented
109	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₄⋊C₄×C₇₃	—	2	2	no	—
112	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
113	C₄²×C₇₃	—	2	2	no	—
114	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
115	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
116	Q₈×C₇₃	—	2	2	no	—
117	Q₈×C₇₃	C₂×M₁₆	1	4	yes	—
118	C₂×C₄×C₇₃	—	2	2	no	—
119	C₂×C₄×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×Dic₇₃	—	2	2	no	—
124	C₂×Dic₇₃	—	2	2	no	—
125	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
126	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₇₃	C₂×#41	1	1	yes	characteristic
128	C₄×D₇₃	—	2	2	no	—
129	C₄×D₇₃	—	2	1	no	—
130	C₄×D₇₃	—	2	1	no	—
131	C₄×D₇₃	—	2	2	no	—
132	C₄×D₇₃	—	4	4	no	—
133	C₄×D₇₃	—	4	4	no	—
134	Dic₁₄₆	—	2	2	no	—
135	Dic₁₄₆	—	2	2	no	—
136	Dic₁₄₆	—	4	4	no	—
137	#10596833	—	2	4	no	—
138	#10596833	—	2	2	no	—
139	#10596833	—	2	2	no	—
140	#10596833	—	4	4	no	—
141	C₈×C₇₃	—	2	1	no	—
142	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
143	#10596831	—	2	1	no	—
144	#10596831	—	2	1	no	—
145	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂×D₇₃	—	2	1	no	—
147	C₂×D₇₃	—	2	1	no	—
148	C₂×D₇₃	#443	1	2	yes	—
149	C₄×C₇₃	—	2	4	no	—
150	C₄×C₇₃	—	2	1	no	—
151	C₄×C₇₃	—	2	1	no	—
152	C₄×C₇₃	—	4	2	no	—
153	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
154	C₄×C₇₃	#568	1	1	yes	characteristic
155	Dic₇₃	—	2	1	no	—
156	Dic₇₃	—	2	1	no	—
157	Dic₇₃	—	4	2	no	—
158	Dic₇₃	—	4	2	no	—
159	Dic₇₃	#578	1	2	yes	—
160	#18058	—	73	1	no	maximal, Sylow
161	C₂×C₇₃	D₄×M₁₆	1	1	yes	characteristic
162	C₂×C₇₃	#2950	1	1	yes	characteristic
163	C₂×C₇₃	C₂×#443	1	1	yes	characteristic
164	D₇₃	—	2	2	no	—
165	#2984	—	73	2	no	—
166	C₂×C₄×Q₁₆	—	73	1	no	—
167	#1591	—	73	2	no	—
168	#1587	—	73	4	no	—
169	C₈×Q₁₆	—	73	2	no	—
170	#1568	—	73	1	no	—
171	#1491	—	73	2	no	—
172	C₈×M₁₆	—	73	1	no	—
173	C₇₃	#18058	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₁₆	—	73	1	no	—
175	C₂×C₄×Q₈	—	73	2	no	—
176	#446	—	73	4	no	—
177	#446	—	146	4	no	—
178	C₈×Q₈	—	73	4	no	—
179	C₄×Q₁₆	—	73	1	no	—
180	C₄×Q₁₆	—	73	2	no	—
181	C₄×Q₁₆	—	73	1	no	—
182	C₄×Q₁₆	—	73	2	no	—
183	C₄×Q₁₆	—	73	1	no	—
184	C₄×Q₁₆	#64408	1	1	yes	characteristic
185	#432	—	73	2	no	—
186	#432	—	146	2	no	—
187	#431	—	73	1	no	—
188	#431	—	146	2	no	—
189	C₂×#107	—	73	1	no	—
190	C₂×#103	—	73	2	no	—
191	C₄×M₁₆	—	73	1	no	—
192	C₂×C₄×C₈	—	73	1	no	—
193	#335	—	73	2	no	—
194	#334	—	73	2	no	—
195	Q₈⋊C₈	—	73	4	no	—
196	Q₈⋊C₈	—	73	4	no	—
197	#322	—	73	2	no	—
198	C₈²	—	73	2	no	—
199	C₂²×Q₈	—	73	2	no	—
200	C₂×Q₁₆	—	73	2	no	—
201	C₂×Q₁₆	—	73	1	no	—
202	C₂×Q₁₆	—	146	4	no	—
203	C₂×Q₁₆	C₂×#64408	1	1	yes	characteristic
204	C₂×M₁₆	—	146	1	no	—
205	C₂²×C₈	—	73	1	no	—
206	C₂²×C₈	—	73	1	no	—
207	C₄×Q₈	—	73	2	no	—
208	C₄×Q₈	—	73	2	no	—
209	C₄×Q₈	—	73	2	no	—
210	C₄×Q₈	—	146	2	no	—
211	C₄×Q₈	—	146	2	no	—
212	C₄×Q₈	C₂×#64408	1	2	yes	—
213	C₂×C₄⋊C₄	—	73	2	no	—
214	C₂×C₄⋊C₄	—	146	2	no	—
215	C₂×C₄²	—	73	1	no	—
216	C₂×C₄²	—	146	2	no	—
217	#107	—	73	1	no	—
218	#107	—	73	2	no	—
219	#107	C₂×#64408	1	1	yes	characteristic
220	C₄⋊C₈	—	73	2	no	—
221	C₄⋊C₈	—	73	4	no	—
222	C₄⋊C₈	—	146	4	no	—
223	C₄⋊C₈	—	146	4	no	—
224	#103	—	73	2	no	—
225	#103	—	73	2	no	—
226	#103	—	73	2	no	—
227	#103	C₂×#64408	1	2	yes	—
228	C₂²⋊C₈	—	73	1	no	—
229	C₂²⋊C₈	—	73	1	no	—
230	C₂²⋊C₈	—	146	2	no	—
231	#97	—	146	1	no	—
232	#97	—	146	2	no	—
233	#97	—	146	4	no	—
234	C₄×C₈	—	73	2	no	—
235	C₄×C₈	—	73	1	no	—
236	C₄×C₈	—	73	1	no	—
237	C₄×C₈	—	73	2	no	—
238	C₄×C₈	—	73	1	no	—
239	C₄×C₈	—	146	4	no	—
240	C₄×C₈	C₂×#64408	1	1	yes	characteristic
241	C₂×Q₈	—	292	2	no	—
242	C₂×Q₈	—	73	2	no	—
243	C₂×Q₈	—	146	2	no	—
244	C₂×Q₈	—	146	4	no	—
245	C₂×Q₈	C₂²×#64408	1	2	yes	—
246	C₂²×C₄	—	73	1	no	—
247	C₂²×C₄	—	73	1	no	—
248	C₂²×C₄	—	73	1	no	—
249	C₂²×C₄	—	146	2	no	—
250	C₂²×C₄	—	146	2	no	—
251	Q₁₆	—	146	4	no	—
252	Q₁₆	—	146	2	no	—
253	Q₁₆	#11924038	1	4	yes	complemented
254	M₁₆	—	146	4	no	—
255	C₂×C₈	—	292	1	no	—
256	C₂×C₈	—	73	2	no	—
257	C₂×C₈	—	73	2	no	—
258	C₂×C₈	—	73	1	no	—
259	C₂×C₈	—	73	1	no	—
260	C₂×C₈	—	146	4	no	—
261	C₂×C₈	—	146	2	no	—
262	C₂×C₈	—	146	2	no	—
263	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
264	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	73	2	no	—
267	C₄⋊C₄	—	146	2	no	—
268	C₄⋊C₄	—	146	2	no	—
269	C₄⋊C₄	—	146	2	no	—
270	C₄⋊C₄	—	146	2	no	—
271	C₄⋊C₄	C₂²×#64408	1	2	yes	—
272	C₄²	—	2	2	no	—
273	C₄²	—	73	1	no	—
274	C₄²	—	73	1	no	—
275	C₄²	—	73	1	no	—
276	C₄²	—	146	2	no	—
277	C₄²	—	146	2	no	—
278	C₄²	—	146	2	no	—
279	C₄²	C₂²×#64408	1	1	yes	characteristic
280	C₂³	—	73	1	no	—
281	Q₈	—	2	2	no	—
282	Q₈	—	292	4	no	—
283	Q₈	—	146	2	no	—
284	Q₈	—	146	2	no	—
285	Q₈	C₂×#11924038	1	4	yes	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	292	4	no	—
289	C₂×C₄	—	292	4	no	—
290	C₂×C₄	—	73	1	no	—
291	C₂×C₄	—	73	1	no	—
292	C₂×C₄	—	73	1	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	1	no	—
298	C₂×C₄	—	146	1	no	—
299	C₂×C₄	D₄×#64408	1	1	yes	characteristic
300	C₂×C₄	D₄×#64408	1	1	yes	characteristic
301	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	292	4	no	—
304	C₈	—	146	2	no	—
305	C₈	—	146	2	no	—
306	C₈	—	146	4	no	—
307	C₈	—	146	1	no	—
308	C₈	—	146	1	no	—
309	C₈	C₂×#11924038	1	2	yes	—
310	C₂²	—	73	2	no	—
311	C₂²	—	146	1	no	—
312	C₂²	—	146	1	no	—
313	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
314	C₄	—	2	4	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	292	2	no	—
319	C₄	—	292	2	no	—
320	C₄	—	73	2	no	—
321	C₄	—	146	1	no	—
322	C₄	—	146	1	no	—
323	C₄	C₂²×#11924038	1	1	yes	characteristic
324	C₄	#17989666	1	1	yes	characteristic
325	C₂	—	146	2	no	—
326	C₂	D₄×#11924038	1	1	yes	characteristic
327	C₂	#22626692	1	1	yes	characteristic
328	C₂	D₇₃⋊#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₄×Q₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂²×D₇₃	C₄×D₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₄×Q₁₆	#64408
hypocenter	C₇₃	#18058
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #22625570	C₂	semidirect
3. #22625554	C₂	semidirect
4. Q₁₆×#10596833	C₂	semidirect
108. Q₁₆×C₇₃	M₁₆	semidirect
173. C₇₃	#18058	semidirect
253. Q₁₆	#11924038	semidirect

Maximal subgroups

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

subgroup	index
#22626833	2
#22625570	2
#22625554	2
Q₁₆×#10596833	2
#22625492	2
#22625274	2
C₈×#11924038	2
C₄×Q₁₆×D₇₃	2
#18058	73