Subgroups — #34018126

Order 18688.

618 conjugacy classes of subgroups.

The subgroup lattice has 388 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	D₇₃⋊#432	C₂	1	1	yes	maximal, characteristic
3	#22625505	C₂	1	4	yes	maximal, complemented
4	#22625503	C₂	1	4	yes	maximal, complemented
5	#22625495	C₂	1	4	yes	maximal, complemented
6	#22618563	C₂	1	1	yes	maximal, characteristic
7	#17989610	—	2	1	no	—
8	#17989610	—	2	1	no	—
9	#17989609	C₂²	1	2	yes	—
10	#17989609	C₂²	1	2	yes	—
11	#17989608	C₂²	1	2	yes	—
12	#17989607	—	2	1	no	—
13	#17989606	C₂²	1	2	yes	—
14	#17989605	—	2	1	no	—
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	#17989546	—	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	#17988643	C₄	1	2	yes	—
27	#17988643	C₄	1	2	yes	—
28	#17988639	C₂²	1	2	yes	—
29	#17988639	C₂²	1	2	yes	—
30	#17988611	C₄	1	1	yes	characteristic
31	#17988550	C₄	1	2	yes	—
32	M₁₆×Dic₇₃	C₂²	1	2	yes	—
33	#423×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	#14556646	—	2	2	no	—
46	#14556646	—	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×#11924012	C₂×C₄	1	2	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	C₄⋊#10596831	C₂×C₄	1	2	yes	—
65	#14556503	C₂×C₄	1	4	yes	—
66	#14556503	C₂×C₄	1	4	yes	—
67	#14556501	C₂×C₄	1	4	yes	—
68	C₈×Dic₇₃	C₂³	1	4	yes	—
69	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
72	C₄⋊C₈×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	—	4	2	no	—
82	C₂×C₄×D₇₃	—	4	1	no	—
83	C₂×C₄×D₇₃	—	4	1	no	—
84	C₂×C₄×D₇₃	—	4	2	no	—
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	C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
97	#11924012	—	4	2	no	—
98	C₂×#10596831	C₂²×C₄	1	4	yes	—
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	C₄⋊C₄	1	1	yes	characteristic
103	C₄×Dic₇₃	C₂²×C₄	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×D₄	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	C₂×Q₈	1	1	yes	characteristic
110	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
111	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	M₁₆×C₇₃	—	4	2	no	—
115	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
116	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
117	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
118	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
120	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×D₇₃	—	2	2	no	—
122	C₂²×D₇₃	—	4	1	no	—
123	C₂²×D₇₃	—	4	1	no	—
124	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	—	2	1	no	—
127	C₂×C₄×C₇₃	—	2	1	no	—
128	C₂×C₄×C₇₃	—	4	1	no	—
129	C₂×C₄×C₇₃	—	4	1	no	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×Dic₇₃	—	2	2	no	—
137	C₂×Dic₇₃	—	4	1	no	—
138	C₂×Dic₇₃	—	4	1	no	—
139	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
145	C₄×D₇₃	—	2	2	no	—
146	C₄×D₇₃	—	2	2	no	—
147	C₄×D₇₃	—	2	1	no	—
148	C₄×D₇₃	—	2	1	no	—
149	C₄×D₇₃	—	2	1	no	—
150	C₄×D₇₃	—	2	1	no	—
151	C₄×D₇₃	—	2	2	no	—
152	C₄×D₇₃	—	2	2	no	—
153	C₄×D₇₃	—	4	1	no	—
154	C₄×D₇₃	—	4	1	no	—
155	C₄×D₇₃	—	4	1	no	—
156	C₄×D₇₃	—	4	1	no	—
157	C₄×D₇₃	—	4	1	no	—
158	C₄×D₇₃	—	4	1	no	—
159	#10596833	—	4	4	no	—
160	#10596833	—	4	4	no	—
161	C₈×C₇₃	—	2	4	no	—
162	#10596831	—	2	4	no	—
163	C₂²×C₇₃	—	2	1	no	—
164	C₂²×C₇₃	—	2	1	no	—
165	C₂²×C₇₃	—	2	1	no	—
166	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
167	C₂×D₇₃	—	2	1	no	—
168	C₂×D₇₃	—	2	1	no	—
169	C₂×D₇₃	—	2	1	no	—
170	C₂×D₇₃	—	2	1	no	—
171	C₂×D₇₃	—	2	1	no	—
172	C₂×D₇₃	—	4	4	no	—
173	C₂×D₇₃	#430	1	2	yes	—
174	C₄×C₇₃	—	2	1	no	—
175	C₄×C₇₃	—	2	1	no	—
176	C₄×C₇₃	—	2	1	no	—
177	C₄×C₇₃	—	2	1	no	—
178	C₄×C₇₃	—	2	1	no	—
179	C₄×C₇₃	#568	1	1	yes	characteristic
180	C₄×C₇₃	#568	1	1	yes	characteristic
181	Dic₇₃	—	2	1	no	—
182	Dic₇₃	—	2	1	no	—
183	Dic₇₃	—	2	1	no	—
184	Dic₇₃	—	2	1	no	—
185	Dic₇₃	—	2	1	no	—
186	Dic₇₃	#575	1	2	yes	—
187	#17821	—	73	1	no	maximal, Sylow
188	C₂×C₇₃	—	4	1	no	—
189	C₂×C₇₃	#2905	1	1	yes	characteristic
190	C₂×C₇₃	#2911	1	1	yes	characteristic
191	C₂×C₇₃	C₂×#430	1	1	yes	characteristic
192	D₇₃	—	2	2	no	—
193	D₇₃	—	4	1	no	—
194	#2920	—	73	1	no	—
195	#2920	—	73	1	no	—
196	C₂×#432	—	73	1	no	—
197	#1573	—	73	4	no	—
198	#1572	—	73	4	no	—
199	#1569	—	73	4	no	—
200	C₇₃	#17821	1	1	yes	characteristic, complemented, Sylow
201	C₂²×C₄²	—	73	1	no	—
202	#432	—	73	2	no	—
203	#432	—	73	2	no	—
204	#432	—	73	2	no	—
205	#432	—	73	2	no	—
206	#432	—	73	2	no	—
207	#432	—	146	1	no	—
208	#432	—	146	1	no	—
209	#432	—	146	1	no	—
210	#431	—	73	2	no	—
211	#431	—	73	2	no	—
212	#431	—	73	2	no	—
213	#431	—	146	1	no	—
214	#423	—	73	1	no	—
215	#423	—	73	2	no	—
216	#423	#64408	1	1	yes	characteristic
217	C₂×C₄⋊C₈	—	73	2	no	—
218	#407	—	73	2	no	—
219	#407	—	146	1	no	—
220	C₂×C₂²⋊C₈	—	146	1	no	—
221	C₄×M₁₆	—	73	2	no	—
222	C₄×M₁₆	—	146	1	no	—
223	C₂×#97	—	146	1	no	—
224	#335	—	73	4	no	—
225	#335	—	73	4	no	—
226	#334	—	73	4	no	—
227	#334	—	73	4	no	—
228	C₂³×C₄	—	73	1	no	—
229	C₂³×C₄	—	73	1	no	—
230	C₂³×C₄	—	73	1	no	—
231	C₂×M₁₆	—	73	2	no	—
232	C₂×M₁₆	—	146	2	no	—
233	C₂×M₁₆	C₂×#64408	1	2	yes	—
234	C₂²×C₈	—	146	2	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	—	73	1	no	—
240	C₂×C₄²	—	73	1	no	—
241	C₂×C₄²	—	73	1	no	—
242	C₂×C₄²	—	73	1	no	—
243	C₂×C₄²	—	73	1	no	—
244	C₂×C₄²	—	73	1	no	—
245	C₂×C₄²	—	73	1	no	—
246	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
247	C₄⋊C₈	—	73	4	no	—
248	C₄⋊C₈	—	73	4	no	—
249	C₄⋊C₈	—	73	2	no	—
250	C₄⋊C₈	—	73	2	no	—
251	C₄⋊C₈	—	146	2	no	—
252	C₄⋊C₈	—	146	2	no	—
253	C₄⋊C₈	C₂×#64408	1	2	yes	—
254	C₄⋊C₈	C₂×#64408	1	2	yes	—
255	C₂²⋊C₈	—	73	4	no	—
256	C₂²⋊C₈	—	73	4	no	—
257	C₂²⋊C₈	—	146	2	no	—
258	C₂²⋊C₈	—	146	2	no	—
259	C₂²⋊C₈	—	146	2	no	—
260	C₂²⋊C₈	—	146	2	no	—
261	#97	—	73	4	no	—
262	#97	—	146	2	no	—
263	#97	—	146	2	no	—
264	#97	—	146	2	no	—
265	C₄×C₈	—	73	4	no	—
266	C₄×C₈	—	146	2	no	—
267	C₂⁴	—	73	1	no	—
268	C₂²×C₄	—	292	1	no	—
269	C₂²×C₄	—	292	1	no	—
270	C₂²×C₄	—	292	2	no	—
271	C₂²×C₄	—	292	2	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	—	73	1	no	—
280	C₂²×C₄	—	73	1	no	—
281	C₂²×C₄	—	73	1	no	—
282	C₂²×C₄	—	73	1	no	—
283	C₂²×C₄	—	73	1	no	—
284	C₂²×C₄	—	73	1	no	—
285	C₂²×C₄	—	73	1	no	—
286	C₂²×C₄	—	73	1	no	—
287	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
288	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
289	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
290	M₁₆	—	4	2	no	—
291	M₁₆	—	292	4	no	—
292	M₁₆	—	292	2	no	—
293	C₂×C₈	—	292	4	no	—
294	C₂×C₈	—	73	4	no	—
295	C₂×C₈	—	146	4	no	—
296	C₂×C₈	—	146	4	no	—
297	C₂×C₈	C₂²×#64408	1	4	yes	—
298	C₄²	—	73	1	no	—
299	C₄²	—	73	1	no	—
300	C₄²	—	73	1	no	—
301	C₄²	—	73	1	no	—
302	C₄²	—	73	1	no	—
303	C₄²	—	73	1	no	—
304	C₄²	—	73	1	no	—
305	C₄²	—	73	1	no	—
306	C₄²	—	73	1	no	—
307	C₄²	—	73	1	no	—
308	C₄²	—	73	1	no	—
309	C₄²	—	73	1	no	—
310	C₄²	C₄⋊#64408	1	1	yes	characteristic
311	C₄²	C₄⋊#64408	1	1	yes	characteristic
312	C₄²	C₂²×#64408	1	1	yes	characteristic
313	C₄²	C₂²×#64408	1	1	yes	characteristic
314	C₂³	—	292	1	no	—
315	C₂³	—	292	1	no	—
316	C₂³	—	73	1	no	—
317	C₂³	—	73	1	no	—
318	C₂³	—	146	2	no	—
319	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	292	1	no	—
325	C₂×C₄	—	292	1	no	—
326	C₂×C₄	—	292	1	no	—
327	C₂×C₄	—	292	1	no	—
328	C₂×C₄	—	292	1	no	—
329	C₂×C₄	—	292	1	no	—
330	C₂×C₄	—	292	1	no	—
331	C₂×C₄	—	292	1	no	—
332	C₂×C₄	—	73	1	no	—
333	C₂×C₄	—	73	1	no	—
334	C₂×C₄	—	73	1	no	—
335	C₂×C₄	—	73	1	no	—
336	C₂×C₄	—	73	1	no	—
337	C₂×C₄	—	73	1	no	—
338	C₂×C₄	—	146	2	no	—
339	C₂×C₄	—	146	2	no	—
340	C₂×C₄	—	146	2	no	—
341	C₂×C₄	—	146	2	no	—
342	C₂×C₄	—	146	2	no	—
343	C₂×C₄	—	146	1	no	—
344	C₂×C₄	—	146	1	no	—
345	C₂×C₄	—	146	1	no	—
346	C₂×C₄	—	146	1	no	—
347	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
348	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
349	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
350	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
351	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
352	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
353	C₈	—	2	4	no	—
354	C₈	—	292	4	no	—
355	C₈	—	292	4	no	—
356	C₈	—	146	4	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	292	4	no	—
361	C₂²	—	73	2	no	—
362	C₂²	—	146	1	no	—
363	C₂²	—	146	1	no	—
364	C₂²	—	146	1	no	—
365	C₂²	—	146	1	no	—
366	C₂²	—	146	1	no	—
367	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	73	2	no	—
374	C₄	—	146	1	no	—
375	C₄	—	146	1	no	—
376	C₄	—	146	1	no	—
377	C₄	—	146	1	no	—
378	C₄	—	146	1	no	—
379	C₄	#17989665	1	1	yes	characteristic
380	C₄	#17989667	1	1	yes	characteristic
381	C₂	—	4	1	no	—
382	C₂	—	292	1	no	—
383	C₂	—	146	2	no	—
384	C₂	#22626575	1	1	yes	characteristic
385	C₂	#22626584	1	1	yes	characteristic
386	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	#423×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₇₃	C₄⋊C₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	#423	#64408
hypocenter	C₇₃	#17821
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
3. #22625505	C₂	semidirect
4. #22625503	C₂	semidirect
5. #22625495	C₂	semidirect
21. #17989451	C₂²	semidirect
22. #17989450	C₂²	semidirect
23. #17989449	C₂²	semidirect
24. #17989448	C₂²	semidirect
200. C₇₃	#17821	semidirect

Maximal subgroups

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

subgroup	index
#22626609	2
D₇₃⋊#432	2
#22625505	2
#22625503	2
#22625495	2
#22618563	2
#17821	73