Subgroups — #34018030

Order 18688.

666 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626852	C₂	1	2	yes	maximal
2	#22625496	C₂	1	4	yes	maximal, complemented
3	#22625492	C₂	1	2	yes	maximal, complemented
4	#22625486	C₂	1	2	yes	maximal, complemented
5	#22625482	C₂	1	2	yes	maximal, complemented
6	#22625478	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#17989448	C₂	1	1	yes	maximal, characteristic, complemented
8	#22618353	C₂	1	1	yes	maximal, characteristic
9	#17989637	—	2	2	no	—
10	#17989621	—	2	2	no	—
11	#17989616	—	2	4	no	—
12	#17989609	C₂²	1	4	yes	—
13	#17989603	C₂²	1	2	yes	—
14	#17989591	C₂²	1	4	yes	—
15	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
16	#17989450	C₂²	1	2	yes	—
17	#17989450	C₂²	1	4	yes	—
18	#17989448	C₂²	1	2	yes	—
19	#17989448	C₂²	1	2	yes	—
20	#17989448	C₂²	1	1	yes	characteristic
21	#17989448	C₂²	1	1	yes	characteristic
22	#17989448	C₂²	1	4	yes	—
23	C₂×#14556599	C₄	1	1	yes	characteristic
24	C₂×#14556597	C₄	1	1	yes	characteristic
25	#17989055	C₂²	1	1	yes	characteristic
26	#17988544	C₄	1	2	yes	—
27	#17988543	C₄	1	1	yes	characteristic
28	#17988541	C₄	1	2	yes	—
29	#17988539	C₂²	1	2	yes	—
30	M₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₇₃	C₂²	1	2	yes	—
33	#405×C₇₃	C₄	1	1	yes	characteristic
34	C₂×#11924038	—	2	4	no	—
35	C₂²×#10596833	—	2	2	no	—
36	C₄⋊#10596833	—	2	2	no	—
37	C₄⋊#10596833	—	2	4	no	—
38	C₄⋊#10596833	C₂³	1	4	yes	—
39	C₄⋊#10596833	C₂³	1	2	yes	—
40	C₄⋊#10596833	C₂³	1	2	yes	—
41	#14556648	—	2	4	no	—
42	C₂²⋊#10596833	—	2	2	no	—
43	#14556646	—	2	4	no	—
44	C₄×#10596833	—	2	2	no	—
45	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
46	#14556599	—	2	4	no	—
47	#14556597	—	2	4	no	—
48	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
49	C₂×#11924011	C₂×C₄	1	2	yes	—
50	C₂×#11924011	C₂×C₄	1	2	yes	—
51	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
53	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
54	#14556556	D₄	1	1	yes	characteristic
55	#14556556	Q₈	1	1	yes	characteristic
56	#14556553	D₄	1	1	yes	characteristic
57	#14556553	Q₈	1	1	yes	characteristic
58	#14556551	D₄	1	2	yes	—
59	#14556551	Q₈	1	2	yes	—
60	#14556548	C₂×C₄	1	2	yes	—
61	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
62	C₄²×D₇₃	C₂³	1	2	yes	—
63	#14556502	C₂×C₄	1	2	yes	—
64	#14556501	C₂×C₄	1	2	yes	—
65	C₄×#10596831	C₂×C₄	1	2	yes	—
66	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
69	C₈×Dic₇₃	C₂³	1	2	yes	—
70	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
71	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
73	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
74	#97×C₇₃	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
76	#11924038	—	8	4	no	—
77	C₂×#10596833	—	2	4	no	—
78	C₂×#10596833	—	2	2	no	—
79	C₂×#10596833	—	2	2	no	—
80	C₂×#10596833	—	8	2	no	—
81	#11924028	—	2	2	no	—
82	#11924028	—	2	2	no	—
83	C₂×Dic₁₄₆	C₄⋊C₄	1	1	yes	characteristic
84	C₂×D₂₉₂	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₂²⋊D₇₃	C₄⋊C₄	1	2	yes	—
86	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
90	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
91	#11924012	—	2	2	no	—
92	#11924011	—	2	4	no	—
93	#11924011	—	2	4	no	—
94	C₂×#10596831	—	2	2	no	—
95	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
96	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
97	C₂×#10596831	C₂²×C₄	1	2	yes	—
98	C₈×D₇₃	—	2	4	no	—
99	C₈×D₇₃	—	2	4	no	—
100	#11924007	C₄⋊C₄	1	2	yes	—
101	#11924007	C₄⋊C₄	1	2	yes	—
102	#11924005	C₄⋊C₄	1	2	yes	—
103	#11924005	C₄⋊C₄	1	2	yes	—
104	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
106	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
107	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
108	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
109	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
110	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	M₁₆×C₇₃	M₁₆	1	4	yes	complemented
112	C₂×C₈×C₇₃	M₁₆	1	4	yes	complemented
113	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
116	C₄⋊C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
117	C₄⋊C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
118	C₂²⋊C₄×C₇₃	C₄⋊C₄	1	2	yes	—
119	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
120	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
128	C₂²⋊D₇₃	—	2	4	no	—
129	C₂×Dic₇₃	—	2	2	no	—
130	C₂×Dic₇₃	—	4	1	no	—
131	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
136	D₂₉₂	—	2	2	no	—
137	C₄×D₇₃	—	2	2	no	—
138	C₄×D₇₃	—	2	2	no	—
139	C₄×D₇₃	—	2	4	no	—
140	C₄×D₇₃	—	4	2	no	—
141	Dic₁₄₆	—	2	2	no	—
142	#10596833	—	4	2	no	—
143	#10596833	—	4	4	no	—
144	#10596833	—	4	2	no	—
145	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
146	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
147	C₈×C₇₃	C₂×M₁₆	1	4	yes	—
148	#10596831	—	2	1	no	—
149	#10596831	—	2	2	no	—
150	#10596831	—	2	1	no	—
151	C₂²×C₇₃	—	2	1	no	—
152	C₂²×C₇₃	—	2	1	no	—
153	C₂²×C₇₃	#423	1	2	yes	—
154	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₂×D₇₃	—	2	2	no	—
156	C₂×D₇₃	—	2	2	no	—
157	C₂×D₇₃	—	2	2	no	—
158	C₄×C₇₃	—	2	2	no	—
159	C₄×C₇₃	—	2	2	no	—
160	C₄×C₇₃	#423	1	2	yes	—
161	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
162	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
163	Dic₇₃	—	2	2	no	—
164	Dic₇₃	—	2	1	no	—
165	Dic₇₃	—	2	1	no	—
166	Dic₇₃	—	2	1	no	—
167	Dic₇₃	#577	1	2	yes	—
168	#17798	—	73	1	no	maximal, Sylow
169	C₂×C₇₃	—	2	2	no	—
170	C₂×C₇₃	C₂×#423	1	1	yes	characteristic
171	C₂×C₇₃	#2912	1	1	yes	characteristic
172	C₂×C₇₃	#2916	1	1	yes	characteristic
173	D₇₃	—	4	2	no	—
174	#2989	—	73	2	no	—
175	C₄×#131	—	73	1	no	—
176	#1569	—	73	4	no	—
177	#1568	—	73	2	no	—
178	#1566	—	73	2	no	—
179	#1565	—	73	2	no	—
180	#1564	—	73	1	no	—
181	C₂×#334	—	73	1	no	—
182	C₇₃	#17798	1	1	yes	characteristic, complemented, Sylow
183	C₂×#131	—	73	1	no	—
184	C₂×#131	—	73	1	no	—
185	C₄×#41	—	73	1	no	—
186	#446	—	146	2	no	—
187	#436	—	146	2	no	—
188	#435	—	146	4	no	—
189	#432	—	73	4	no	—
190	#431	—	73	2	no	—
191	#423	—	73	4	no	—
192	C₂×C₄⋊C₈	—	73	2	no	—
193	#405	—	73	1	no	—
194	#405	—	73	2	no	—
195	#405	—	73	2	no	—
196	#405	#64408	1	1	yes	characteristic
197	C₄×M₁₆	—	73	1	no	—
198	C₄×M₁₆	—	73	2	no	—
199	C₂×C₄×C₈	—	73	1	no	—
200	C₂×C₄×C₈	—	73	2	no	—
201	#335	—	73	2	no	—
202	#335	—	73	4	no	—
203	#334	—	73	2	no	—
204	#334	—	73	2	no	—
205	#334	—	73	1	no	—
206	#334	—	73	1	no	—
207	#334	—	73	4	no	—
208	C₂×#41	—	73	1	no	—
209	#131	—	146	4	no	—
210	#131	—	146	4	no	—
211	C₂×M₁₆	—	73	1	no	—
212	C₂×M₁₆	—	73	2	no	—
213	C₂×M₁₆	—	73	2	no	—
214	C₂×M₁₆	—	146	4	no	—
215	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
216	C₂²×C₈	—	73	1	no	—
217	C₂²×C₈	—	73	2	no	—
218	C₂²×C₈	—	73	2	no	—
219	C₂²×C₈	—	146	2	no	—
220	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
221	C₄×Q₈	—	73	1	no	—
222	C₄×Q₈	—	73	1	no	—
223	C₄×D₄	—	73	1	no	—
224	C₄×D₄	—	73	1	no	—
225	C₄×D₄	—	73	2	no	—
226	C₄×D₄	—	73	2	no	—
227	#117	—	73	2	no	—
228	#117	C₂×#64408	1	1	yes	characteristic
229	C₂×C₄²	—	73	1	no	—
230	C₂×C₄²	—	73	2	no	—
231	C₄⋊C₈	—	73	4	no	—
232	C₄⋊C₈	—	73	2	no	—
233	C₄⋊C₈	—	73	2	no	—
234	C₄⋊C₈	—	146	2	no	—
235	C₄⋊C₈	—	146	4	no	—
236	C₂²⋊C₈	—	146	2	no	—
237	C₂²⋊C₈	—	146	4	no	—
238	#97	—	73	2	no	—
239	#97	—	73	2	no	—
240	#97	—	146	4	no	—
241	#97	C₂×#64408	1	2	yes	—
242	C₄×C₈	—	73	1	no	—
243	C₄×C₈	—	73	1	no	—
244	C₄×C₈	—	73	2	no	—
245	C₄×C₈	—	73	1	no	—
246	C₄×C₈	—	73	1	no	—
247	C₄×C₈	—	73	2	no	—
248	C₄×C₈	—	146	2	no	—
249	C₄×C₈	C₂×#64408	1	2	yes	—
250	#41	—	146	2	no	—
251	#41	—	146	2	no	—
252	C₂×Q₈	—	73	1	no	—
253	C₂×D₄	—	73	2	no	—
254	C₂×D₄	—	73	1	no	—
255	C₂²×C₄	—	73	1	no	—
256	C₂²×C₄	—	73	2	no	—
257	C₂²×C₄	—	73	1	no	—
258	C₂²×C₄	—	73	2	no	—
259	C₂²×C₄	—	73	2	no	—
260	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
261	M₁₆	—	584	4	no	—
262	M₁₆	—	146	2	no	—
263	M₁₆	—	146	4	no	—
264	M₁₆	—	146	4	no	—
265	M₁₆	#11924038	1	4	yes	complemented
266	C₂×C₈	—	584	2	no	—
267	C₂×C₈	—	73	1	no	—
268	C₂×C₈	—	73	1	no	—
269	C₂×C₈	—	73	2	no	—
270	C₂×C₈	—	146	2	no	—
271	C₂×C₈	—	146	2	no	—
272	C₂×C₈	—	146	4	no	—
273	C₂×C₈	—	146	2	no	—
274	C₂×C₈	—	146	4	no	—
275	C₂×C₈	—	146	4	no	—
276	C₂×C₈	#11924038	1	4	yes	complemented
277	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
278	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
279	C₂×C₈	C₂²×#64408	1	2	yes	—
280	C₄⋊C₄	—	73	2	no	—
281	C₄⋊C₄	—	73	2	no	—
282	C₄⋊C₄	C₄⋊#64408	1	1	yes	characteristic
283	C₄⋊C₄	C₄⋊#64408	1	1	yes	characteristic
284	C₂²⋊C₄	—	73	2	no	—
285	C₂²⋊C₄	—	73	2	no	—
286	C₂²⋊C₄	C₄⋊#64408	1	2	yes	—
287	C₄²	—	73	1	no	—
288	C₄²	—	73	1	no	—
289	C₄²	—	73	1	no	—
290	C₄²	—	73	2	no	—
291	C₄²	—	73	1	no	—
292	C₄²	—	73	2	no	—
293	C₄²	C₂²×#64408	1	2	yes	—
294	C₂³	—	73	2	no	—
295	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
296	Q₈	—	146	2	no	—
297	D₄	—	146	4	no	—
298	D₄	—	146	2	no	—
299	C₂×C₄	—	292	2	no	—
300	C₂×C₄	—	292	1	no	—
301	C₂×C₄	—	73	1	no	—
302	C₂×C₄	—	73	1	no	—
303	C₂×C₄	—	73	1	no	—
304	C₂×C₄	—	73	1	no	—
305	C₂×C₄	—	73	2	no	—
306	C₂×C₄	—	146	4	no	—
307	C₂×C₄	—	146	2	no	—
308	C₂×C₄	—	146	2	no	—
309	C₂×C₄	—	146	2	no	—
310	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
311	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
312	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
313	C₂×C₄	C₂×#11924038	1	2	yes	—
314	C₂×C₄	C₂×#11924038	1	2	yes	—
315	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
316	C₈	—	292	4	no	—
317	C₈	—	292	2	no	—
318	C₈	—	292	2	no	—
319	C₈	—	146	1	no	—
320	C₈	—	146	1	no	—
321	C₈	—	146	2	no	—
322	C₈	C₂×#11924038	1	2	yes	—
323	C₈	C₂×#11924038	1	4	yes	—
324	C₈	C₂×#11924038	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	146	2	no	—
328	C₂²	—	146	2	no	—
329	C₂²	—	146	2	no	—
330	C₂²	#17989591	1	2	yes	—
331	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	73	2	no	—
335	C₄	—	146	2	no	—
336	C₄	—	146	1	no	—
337	C₄	—	146	1	no	—
338	C₄	—	146	1	no	—
339	C₄	#17989591	1	2	yes	—
340	C₄	C₂²×#11924038	1	1	yes	characteristic
341	C₄	C₂²×#11924038	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	292	2	no	—
344	C₂	C₂×#17989591	1	1	yes	characteristic
345	C₂	#22626585	1	1	yes	characteristic
346	C₂	#22626592	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	#405×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂×#11924028	C₂×C₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	#405	#64408
hypocenter	C₇₃	#17798
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #22625496	C₂	semidirect
3. #22625492	C₂	semidirect
4. #22625486	C₂	semidirect
5. #22625482	C₂	semidirect
6. #22625478	C₂	semidirect
7. C₂×#17989448	C₂	semidirect
111. M₁₆×C₇₃	M₁₆	semidirect
112. C₂×C₈×C₇₃	M₁₆	semidirect
182. C₇₃	#17798	semidirect
265. M₁₆	#11924038	semidirect
276. C₂×C₈	#11924038	semidirect

Maximal subgroups

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

subgroup	index
#22626852	2
#22625496	2
#22625492	2
#22625486	2
#22625482	2
#22625478	2
C₂×#17989448	2
#22618353	2
#17798	73