Subgroups — #34018073

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626609	C₂	1	1	yes	maximal, characteristic
2	#22626595	C₂	1	1	yes	maximal, characteristic
3	#22625507	C₂	1	4	yes	maximal, complemented
4	#22625503	C₂	1	4	yes	maximal, complemented
5	#22625483	C₂	1	4	yes	maximal, complemented
6	#22618561	C₂	1	1	yes	maximal, characteristic
7	#17989610	—	2	1	no	—
8	#17989609	C₂²	1	2	yes	—
9	#17989606	C₂²	1	2	yes	—
10	#17989606	C₂²	1	2	yes	—
11	#17989605	—	2	1	no	—
12	#17989605	—	2	1	no	—
13	#17989604	—	2	1	no	—
14	#17989603	C₂²	1	2	yes	—
15	#17989590	C₂²	1	2	yes	—
16	#17989588	C₂²	1	2	yes	—
17	#17989559	—	2	1	no	—
18	#17989555	—	2	1	no	—
19	C₄×#11924037	—	2	1	no	—
20	C₄×#11924036	—	2	1	no	—
21	#17989451	C₂²	1	4	yes	complemented
22	#17989450	C₂²	1	4	yes	complemented
23	#17989449	C₂²	1	4	yes	complemented
24	#17989448	C₂²	1	4	yes	complemented
25	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
26	#17988644	C₄	1	1	yes	characteristic
27	#17988642	C₄	1	2	yes	—
28	#17988639	C₂²	1	2	yes	—
29	#17988637	C₂²	1	2	yes	—
30	#17988609	C₄	1	2	yes	—
31	#17988549	C₄	1	2	yes	—
32	#17988539	C₂²	1	2	yes	—
33	#432×C₇₃	C₄	1	1	yes	characteristic
34	C₂×#11924037	—	2	2	no	—
35	C₂×#11924036	—	2	2	no	—
36	#14556658	—	2	2	no	—
37	#14556658	—	2	2	no	—
38	C₄⋊#10596833	C₂³	1	4	yes	—
39	C₄⋊#10596833	C₂³	1	4	yes	—
40	#14556648	—	2	2	no	—
41	#14556648	—	2	2	no	—
42	#14556648	—	2	2	no	—
43	C₂²⋊#10596833	—	2	2	no	—
44	#14556646	—	2	2	no	—
45	C₄×#10596833	—	2	2	no	—
46	C₄×#10596833	—	2	2	no	—
47	C₄×#10596833	—	2	2	no	—
48	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×D₇₃	Q₈	1	1	yes	characteristic
51	C₂×#11924011	C₂×C₄	1	4	yes	—
52	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
53	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
54	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
55	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×D₇₃	D₄	1	1	yes	characteristic
58	C₄²×D₇₃	D₄	1	1	yes	characteristic
59	C₄²×D₇₃	Q₈	1	1	yes	characteristic
60	C₄²×D₇₃	Q₈	1	1	yes	characteristic
61	C₄²×D₇₃	C₂³	1	1	yes	characteristic
62	C₄²×D₇₃	C₂³	1	1	yes	characteristic
63	C₄⋊#10596831	C₂×C₄	1	2	yes	—
64	#14556520	C₂×C₄	1	4	yes	—
65	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
66	#14556503	C₂×C₄	1	4	yes	—
67	#14556502	C₂×C₄	1	2	yes	—
68	C₈×Dic₇₃	C₂³	1	4	yes	—
69	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
71	C₂²⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
72	#97×C₇₃	C₂×C₄	1	2	yes	—
73	#11924037	—	4	4	no	—
74	#11924036	—	4	4	no	—
75	C₂×#10596833	—	2	4	no	—
76	C₂×#10596833	—	2	4	no	—
77	C₂³×D₇₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₄×D₇₃	—	2	4	no	—
82	C₂×C₄×D₇₃	—	4	2	no	—
83	C₂×C₄×D₇₃	—	4	2	no	—
84	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
96	#11924011	—	2	8	no	—
97	C₂×#10596831	C₂²×C₄	1	4	yes	—
98	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
104	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
105	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
106	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
107	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
108	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
109	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
110	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
114	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
115	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
116	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
118	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×D₇₃	—	2	2	no	—
120	C₂²×D₇₃	—	4	1	no	—
121	C₂²×D₇₃	—	4	1	no	—
122	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	—	2	2	no	—
125	C₂×C₄×C₇₃	—	4	1	no	—
126	C₂×C₄×C₇₃	—	4	1	no	—
127	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×Dic₇₃	—	2	2	no	—
134	C₂×Dic₇₃	—	2	2	no	—
135	C₂×Dic₇₃	—	2	2	no	—
136	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
142	C₄×D₇₃	—	2	2	no	—
143	C₄×D₇₃	—	2	2	no	—
144	C₄×D₇₃	—	2	2	no	—
145	C₄×D₇₃	—	2	2	no	—
146	C₄×D₇₃	—	2	2	no	—
147	C₄×D₇₃	—	2	2	no	—
148	C₄×D₇₃	—	2	2	no	—
149	C₄×D₇₃	—	4	1	no	—
150	C₄×D₇₃	—	4	1	no	—
151	C₄×D₇₃	—	4	1	no	—
152	C₄×D₇₃	—	4	1	no	—
153	C₄×D₇₃	#131	1	4	yes	—
154	#10596833	—	4	4	no	—
155	#10596833	—	4	4	no	—
156	C₈×C₇₃	—	2	4	no	—
157	#10596831	—	2	4	no	—
158	C₂²×C₇₃	—	2	1	no	—
159	C₂²×C₇₃	—	2	1	no	—
160	C₂²×C₇₃	—	2	1	no	—
161	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
162	C₂×D₇₃	—	2	1	no	—
163	C₂×D₇₃	—	2	1	no	—
164	C₂×D₇₃	—	2	1	no	—
165	C₂×D₇₃	—	2	1	no	—
166	C₂×D₇₃	—	2	1	no	—
167	C₂×D₇₃	—	4	4	no	—
168	C₂×D₇₃	#424	1	2	yes	—
169	C₄×C₇₃	—	2	1	no	—
170	C₄×C₇₃	—	2	1	no	—
171	C₄×C₇₃	—	2	1	no	—
172	C₄×C₇₃	—	2	1	no	—
173	C₄×C₇₃	—	2	1	no	—
174	C₄×C₇₃	C₂×#131	1	2	yes	—
175	Dic₇₃	—	2	1	no	—
176	Dic₇₃	—	2	1	no	—
177	Dic₇₃	—	2	1	no	—
178	Dic₇₃	#424	1	2	yes	—
179	Dic₇₃	#430	1	2	yes	—
180	Dic₇₃	#575	1	2	yes	—
181	#17808	—	73	1	no	maximal, Sylow
182	C₂×C₇₃	—	4	1	no	—
183	C₂×C₇₃	C₂×#424	1	1	yes	characteristic
184	C₂×C₇₃	#2911	1	1	yes	characteristic
185	C₂×C₇₃	C₂×#430	1	1	yes	characteristic
186	D₇₃	—	2	2	no	—
187	D₇₃	—	4	1	no	—
188	#2920	—	73	1	no	—
189	#2920	—	73	1	no	—
190	C₂×#431	—	73	1	no	—
191	#1574	—	73	4	no	—
192	#1572	—	73	4	no	—
193	#1565	—	73	4	no	—
194	C₇₃	#17808	1	1	yes	characteristic, complemented, Sylow
195	C₂²×C₄²	—	73	1	no	—
196	#432	—	73	1	no	—
197	#432	—	73	2	no	—
198	#432	—	73	2	no	—
199	#432	—	146	1	no	—
200	#432	#64408	1	1	yes	characteristic
201	#431	—	73	2	no	—
202	#431	—	73	2	no	—
203	#431	—	73	2	no	—
204	#431	—	73	2	no	—
205	#431	—	73	2	no	—
206	#431	—	146	1	no	—
207	#431	—	146	1	no	—
208	#431	—	146	1	no	—
209	#423	—	73	2	no	—
210	#423	—	73	2	no	—
211	C₂×C₄⋊C₈	—	73	2	no	—
212	#407	—	73	2	no	—
213	#407	—	146	1	no	—
214	C₂×C₂²⋊C₈	—	146	1	no	—
215	C₄×M₁₆	—	73	2	no	—
216	C₄×M₁₆	—	146	1	no	—
217	C₂×C₄×C₈	—	146	1	no	—
218	#335	—	73	4	no	—
219	#335	—	73	4	no	—
220	#334	—	73	4	no	—
221	#334	—	73	4	no	—
222	C₂³×C₄	—	73	1	no	—
223	C₂³×C₄	—	73	1	no	—
224	C₂³×C₄	—	73	1	no	—
225	C₂×M₁₆	—	73	4	no	—
226	C₂×M₁₆	—	146	2	no	—
227	C₂²×C₈	—	146	2	no	—
228	C₂×C₄²	—	73	1	no	—
229	C₂×C₄²	—	73	1	no	—
230	C₂×C₄²	—	73	1	no	—
231	C₂×C₄²	—	73	1	no	—
232	C₂×C₄²	—	73	1	no	—
233	C₂×C₄²	—	73	1	no	—
234	C₂×C₄²	—	73	1	no	—
235	C₂×C₄²	—	73	1	no	—
236	C₂×C₄²	—	73	1	no	—
237	C₂×C₄²	—	73	1	no	—
238	C₂×C₄²	—	73	1	no	—
239	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
240	C₄⋊C₈	—	73	4	no	—
241	C₄⋊C₈	—	73	4	no	—
242	C₄⋊C₈	—	73	4	no	—
243	C₄⋊C₈	—	73	2	no	—
244	C₄⋊C₈	—	146	2	no	—
245	C₄⋊C₈	—	146	2	no	—
246	C₄⋊C₈	C₂×#64408	1	2	yes	—
247	C₂²⋊C₈	—	73	2	no	—
248	C₂²⋊C₈	—	73	4	no	—
249	C₂²⋊C₈	—	146	2	no	—
250	C₂²⋊C₈	—	146	2	no	—
251	C₂²⋊C₈	—	146	2	no	—
252	C₂²⋊C₈	—	146	2	no	—
253	C₂²⋊C₈	C₂×#64408	1	2	yes	—
254	#97	—	73	2	no	—
255	#97	—	146	2	no	—
256	#97	C₂×#64408	1	2	yes	—
257	C₄×C₈	—	73	4	no	—
258	C₄×C₈	—	146	2	no	—
259	C₄×C₈	—	146	2	no	—
260	C₄×C₈	—	146	2	no	—
261	C₂⁴	—	73	1	no	—
262	C₂²×C₄	—	292	2	no	—
263	C₂²×C₄	—	292	2	no	—
264	C₂²×C₄	—	73	1	no	—
265	C₂²×C₄	—	73	1	no	—
266	C₂²×C₄	—	73	1	no	—
267	C₂²×C₄	—	73	1	no	—
268	C₂²×C₄	—	73	1	no	—
269	C₂²×C₄	—	73	1	no	—
270	C₂²×C₄	—	73	1	no	—
271	C₂²×C₄	—	73	1	no	—
272	C₂²×C₄	—	73	1	no	—
273	C₂²×C₄	—	73	1	no	—
274	C₂²×C₄	—	73	1	no	—
275	C₂²×C₄	—	73	1	no	—
276	C₂²×C₄	—	73	1	no	—
277	C₂²×C₄	—	73	1	no	—
278	C₂²×C₄	—	73	1	no	—
279	C₂²×C₄	—	146	4	no	—
280	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
281	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
282	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
283	M₁₆	—	292	4	no	—
284	M₁₆	—	146	8	no	—
285	C₂×C₈	—	292	4	no	—
286	C₂×C₈	—	73	4	no	—
287	C₂×C₈	—	146	4	no	—
288	C₂×C₈	—	146	4	no	—
289	C₂×C₈	C₂²×#64408	1	4	yes	—
290	C₄²	—	73	1	no	—
291	C₄²	—	73	1	no	—
292	C₄²	—	73	1	no	—
293	C₄²	—	73	1	no	—
294	C₄²	—	73	1	no	—
295	C₄²	—	73	1	no	—
296	C₄²	—	73	1	no	—
297	C₄²	—	73	1	no	—
298	C₄²	—	73	1	no	—
299	C₄²	—	73	1	no	—
300	C₄²	—	73	1	no	—
301	C₄²	—	73	1	no	—
302	C₄²	C₄⋊#64408	1	1	yes	characteristic
303	C₄²	C₄⋊#64408	1	1	yes	characteristic
304	C₄²	C₂²×#64408	1	1	yes	characteristic
305	C₄²	C₂²×#64408	1	1	yes	characteristic
306	C₂³	—	292	1	no	—
307	C₂³	—	292	1	no	—
308	C₂³	—	73	1	no	—
309	C₂³	—	73	1	no	—
310	C₂³	—	146	2	no	—
311	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	292	1	no	—
316	C₂×C₄	—	292	1	no	—
317	C₂×C₄	—	292	1	no	—
318	C₂×C₄	—	292	1	no	—
319	C₂×C₄	—	73	1	no	—
320	C₂×C₄	—	73	1	no	—
321	C₂×C₄	—	73	1	no	—
322	C₂×C₄	—	73	4	no	—
323	C₂×C₄	—	73	1	no	—
324	C₂×C₄	—	73	1	no	—
325	C₂×C₄	—	73	1	no	—
326	C₂×C₄	—	146	2	no	—
327	C₂×C₄	—	146	2	no	—
328	C₂×C₄	—	146	2	no	—
329	C₂×C₄	—	146	2	no	—
330	C₂×C₄	—	146	2	no	—
331	C₂×C₄	—	146	2	no	—
332	C₂×C₄	—	146	2	no	—
333	C₂×C₄	—	146	2	no	—
334	C₂×C₄	—	146	2	no	—
335	C₂×C₄	—	146	2	no	—
336	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
337	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
338	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
339	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
340	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
341	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
342	C₈	—	2	4	no	—
343	C₈	—	292	4	no	—
344	C₈	—	292	4	no	—
345	C₈	—	146	4	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	292	4	no	—
350	C₂²	—	73	2	no	—
351	C₂²	—	146	1	no	—
352	C₂²	—	146	1	no	—
353	C₂²	—	146	1	no	—
354	C₂²	—	146	1	no	—
355	C₂²	—	146	1	no	—
356	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	73	2	no	—
363	C₄	—	73	2	no	—
364	C₄	—	73	2	no	—
365	C₄	—	146	1	no	—
366	C₄	—	146	1	no	—
367	C₄	—	146	1	no	—
368	C₄	D₇₃⋊#131	1	2	yes	—
369	C₂	—	4	1	no	—
370	C₂	—	292	1	no	—
371	C₂	—	146	2	no	—
372	C₂	D₇₃⋊#424	1	1	yes	characteristic
373	C₂	#22626584	1	1	yes	characteristic
374	C₂	D₇₃⋊#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	#432×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₇₃	C₄⋊C₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	#432	#64408
hypocenter	C₇₃	#17808
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

Complements

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

N	H ≅ G/N	type
3. #22625507	C₂	semidirect
4. #22625503	C₂	semidirect
5. #22625483	C₂	semidirect
21. #17989451	C₂²	semidirect
22. #17989450	C₂²	semidirect
23. #17989449	C₂²	semidirect
24. #17989448	C₂²	semidirect
194. C₇₃	#17808	semidirect

Maximal subgroups

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

subgroup	index
#22626609	2
#22626595	2
#22625507	2
#22625503	2
#22625483	2
#22618561	2
#17808	73