Subgroups — #34030233

Order 18688.

662 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626896	C₂	1	2	yes	maximal, complemented
2	#22626320	C₂	1	2	yes	maximal, complemented
3	#22626319	C₂	1	4	yes	maximal, complemented
4	#22626318	C₂	1	2	yes	maximal, complemented
5	#22626270	C₂	1	1	yes	maximal, characteristic, complemented
6	#22626171	C₂	1	2	yes	maximal, complemented
7	C₄×#14556664	C₂	1	1	yes	maximal, characteristic, complemented
8	#443×D₇₃	C₂	1	1	yes	maximal, characteristic
9	#17989646	—	2	4	no	—
10	#17989646	C₂²	1	4	yes	complemented
11	#17989645	C₂²	1	4	yes	complemented
12	#17989643	—	2	2	no	—
13	#17989643	C₂²	1	2	yes	—
14	C₂×#14556664	—	2	2	no	—
15	C₂×#14556664	C₂²	1	1	yes	characteristic
16	C₂×#14556664	C₂²	1	1	yes	characteristic
17	#17989510	C₂²	1	2	yes	complemented
18	#17989509	C₂²	1	2	yes	complemented
19	#17989486	C₂²	1	4	yes	complemented
20	#17989485	C₂²	1	4	yes	complemented
21	#17989473	C₂²	1	2	yes	complemented
22	C₄×#11924031	C₂²	1	2	yes	complemented
23	#135×D₇₃	C₄	1	1	yes	characteristic
24	#131×D₇₃	C₂²	1	2	yes	—
25	#17988705	C₄	1	1	yes	characteristic
26	#17988704	C₂²	1	1	yes	characteristic
27	#17988703	C₂²	1	1	yes	characteristic
28	#17988702	C₄	1	2	yes	—
29	#17988701	C₂²	1	2	yes	—
30	#108×D₇₃	C₄	1	1	yes	characteristic
31	#104×D₇₃	C₄	1	2	yes	—
32	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
33	#443×C₇₃	C₄	1	1	yes	characteristic
34	#14556666	—	2	4	no	—
35	#14556666	—	2	2	no	—
36	#14556665	—	2	4	no	—
37	#14556665	C₂³	1	4	yes	—
38	#14556664	—	2	4	no	—
39	#14556664	—	2	2	no	—
40	#14556664	—	2	1	no	—
41	#14556664	—	2	1	no	—
42	#14556664	D₄	1	1	yes	characteristic
43	#14556664	D₄	1	1	yes	characteristic
44	#14556664	D₄	1	2	yes	—
45	C₂×#11924031	—	2	4	no	—
46	C₂×#11924031	C₂³	1	2	yes	—
47	C₂×#11924031	C₂³	1	2	yes	—
48	#41×D₇₃	C₂×C₄	1	2	yes	—
49	#14556614	C₈	1	1	yes	characteristic
50	#14556613	C₂×C₄	1	1	yes	characteristic
51	#14556612	C₈	1	2	yes	—
52	#14556611	C₂×C₄	1	2	yes	—
53	#14556610	C₈	1	1	yes	characteristic
54	Q₁₆×D₇₃	C₈	1	1	yes	characteristic
55	SD₁₆×D₇₃	C₈	1	2	yes	—
56	D₈×D₇₃	C₈	1	1	yes	characteristic
57	#14556599	—	2	2	no	—
58	#14556599	C₂³	1	2	yes	—
59	#14556598	C₂³	1	2	yes	—
60	#14556597	—	2	2	no	—
61	#14556597	C₂×C₄	1	2	yes	—
62	M₁₆×D₇₃	—	2	2	no	—
63	M₁₆×D₇₃	C₂×C₄	1	2	yes	—
64	C₂×C₈×D₇₃	—	2	2	no	—
65	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
67	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
68	#14556527	C₂×C₄	1	1	yes	characteristic
69	#14556526	C₂×C₄	1	2	yes	—
70	#14556519	C₂×C₄	1	2	yes	—
71	#14556518	C₂×C₄	1	2	yes	—
72	#14556517	C₂×C₄	1	2	yes	—
73	C₄×#10596831	C₂³	1	1	yes	characteristic
74	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
75	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
76	#135×C₇₃	C₂×C₄	1	1	yes	characteristic
77	#131×C₇₃	C₂×C₄	1	2	yes	—
78	#108×C₇₃	C₂×C₄	1	1	yes	characteristic
79	#104×C₇₃	C₂×C₄	1	2	yes	—
80	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
81	#11924031	—	2	4	no	—
82	#11924031	—	2	2	no	—
83	#11924031	C₂×D₄	1	2	yes	—
84	#11924031	C₂×D₄	1	2	yes	—
85	#11924029	C₂×C₈	1	2	yes	—
86	#11924028	—	2	2	no	—
87	#11924028	C₂²×C₄	1	2	yes	—
88	#11924027	—	2	2	no	—
89	#11924027	C₂×C₈	1	2	yes	—
90	Q₈×D₇₃	C₂×C₈	1	2	yes	—
91	D₄×D₇₃	—	2	2	no	—
92	D₄×D₇₃	C₂×C₈	1	2	yes	—
93	C₂×C₄×D₇₃	—	2	1	no	—
94	C₂×C₄×D₇₃	—	2	2	no	—
95	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
96	Dic₂₉₂	C₂×C₈	1	1	yes	characteristic
97	#11924018	C₂×C₈	1	2	yes	—
98	#11924017	C₂×C₈	1	2	yes	—
99	#11924016	C₂×C₈	1	2	yes	—
100	#11924015	C₂×C₈	1	2	yes	—
101	D₅₈₄	C₂×C₈	1	1	yes	characteristic
102	#11924013	C₂×C₈	1	2	yes	—
103	#11924012	—	2	2	no	—
104	#11924012	C₂²×C₄	1	2	yes	—
105	#11924011	—	2	2	no	—
106	#11924011	—	2	2	no	—
107	#11924011	—	2	2	no	—
108	C₂×#10596831	—	2	2	no	—
109	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
110	C₂×#10596831	C₂⁴	1	1	yes	characteristic
111	C₈×D₇₃	—	2	2	no	—
112	C₈×D₇₃	—	2	2	no	—
113	C₈×D₇₃	—	2	2	no	—
114	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
115	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
116	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
117	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
118	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
119	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
120	C₈×D₇₃	#41	1	1	yes	characteristic
121	C₈×D₇₃	#41	1	1	yes	characteristic
122	C₄×Dic₇₃	—	2	1	no	—
123	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
124	#41×C₇₃	C₂²×C₄	1	2	yes	—
125	Q₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
126	SD₁₆×C₇₃	C₂×C₈	1	2	yes	—
127	D₈×C₇₃	C₂×C₈	1	1	yes	characteristic
128	M₁₆×C₇₃	—	2	2	no	—
129	M₁₆×C₇₃	C₂²×C₄	1	2	yes	—
130	C₂×C₈×C₇₃	—	2	2	no	—
131	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
134	C₂²×D₇₃	—	2	2	no	—
135	C₂²×D₇₃	#131	1	1	yes	characteristic
136	Q₈×C₇₃	C₂²×C₈	1	2	yes	—
137	D₄×C₇₃	—	2	2	no	—
138	D₄×C₇₃	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₇₃	—	2	1	no	—
140	C₂×C₄×C₇₃	—	2	2	no	—
141	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
142	C₂²⋊D₇₃	—	2	2	no	—
143	C₂²⋊D₇₃	—	2	2	no	—
144	C₂²⋊D₇₃	—	2	2	no	—
145	C₂×Dic₇₃	—	2	1	no	—
146	C₂×Dic₇₃	—	2	2	no	—
147	C₂×Dic₇₃	#131	1	1	yes	characteristic
148	D₂₉₂	—	2	2	no	—
149	D₂₉₂	C₂²×C₈	1	2	yes	—
150	C₄×D₇₃	—	2	4	no	—
151	C₄×D₇₃	—	2	2	no	—
152	C₄×D₇₃	—	2	2	no	—
153	C₄×D₇₃	—	2	2	no	—
154	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
155	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
156	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
157	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
158	Dic₁₄₆	—	2	2	no	—
159	Dic₁₄₆	C₂²×C₈	1	2	yes	—
160	C₈×C₇₃	—	2	2	no	—
161	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
162	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
163	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
164	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
165	#10596831	—	2	2	no	—
166	#10596831	C₂²×C₈	1	1	yes	characteristic
167	#10596831	C₂²×C₈	1	1	yes	characteristic
168	#10596831	C₂²×D₄	1	1	yes	characteristic
169	#10596831	C₂×#41	1	1	yes	characteristic
170	C₂²×C₇₃	—	2	2	no	—
171	C₂²×C₇₃	C₂×#131	1	1	yes	characteristic
172	C₂×D₇₃	—	2	2	no	—
173	C₂×D₇₃	—	2	2	no	—
174	C₂×D₇₃	—	4	4	no	—
175	C₂×D₇₃	C₈×D₄	1	1	yes	characteristic
176	C₂×D₇₃	C₈×D₄	1	1	yes	characteristic
177	C₄×C₇₃	—	2	2	no	—
178	C₄×C₇₃	—	2	2	no	—
179	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
180	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
181	Dic₇₃	—	2	2	no	—
182	Dic₇₃	—	2	2	no	—
183	Dic₇₃	C₈×D₄	1	1	yes	characteristic
184	Dic₇₃	C₈×D₄	1	1	yes	characteristic
185	C₂×#2170	—	73	1	no	maximal, Sylow
186	C₂×C₇₃	—	2	1	no	—
187	C₂×C₇₃	—	4	2	no	—
188	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
189	D₇₃	—	2	1	no	—
190	D₇₃	—	4	2	no	—
191	D₇₃	#2170	1	2	yes	complemented
192	C₂×#504	—	73	2	no	—
193	C₂×#443	—	73	1	no	—
194	#2170	—	73	2	no	—
195	#2170	—	73	4	no	—
196	#2170	—	73	2	no	—
197	C₂×#364	—	73	1	no	—
198	C₂×#350	—	73	2	no	—
199	C₂×C₄×C₁₆	—	73	1	no	—
200	C₇₃	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
201	C₂×#135	—	73	1	no	—
202	C₂×#131	—	73	2	no	—
203	#504	—	73	4	no	—
204	#504	—	73	4	no	—
205	#504	—	146	4	no	—
206	C₂×M₃₂	—	73	2	no	—
207	C₂×M₃₂	—	146	2	no	—
208	C₂²×C₁₆	—	73	1	no	—
209	C₂²×C₁₆	—	73	1	no	—
210	C₂²×C₁₆	—	146	2	no	—
211	#443	—	73	1	no	—
212	#443	—	73	2	no	—
213	#443	—	73	1	no	—
214	#443	—	73	2	no	—
215	#443	—	73	1	no	—
216	#443	#64408	1	1	yes	characteristic
217	C₂×#108	—	73	1	no	—
218	C₂×#104	—	73	2	no	—
219	C₂×C₄×C₈	—	73	1	no	—
220	#364	—	73	2	no	—
221	#364	—	73	2	no	—
222	#350	—	73	4	no	—
223	#350	—	73	4	no	—
224	C₄×C₁₆	—	73	2	no	—
225	C₄×C₁₆	—	73	2	no	—
226	C₂×#41	—	73	2	no	—
227	#135	—	73	2	no	—
228	#135	—	73	1	no	—
229	#135	—	73	2	no	—
230	#135	—	73	1	no	—
231	#135	—	73	1	no	—
232	#135	C₂×#64408	1	1	yes	characteristic
233	C₂×Q₁₆	—	73	1	no	—
234	C₂×SD₁₆	—	73	2	no	—
235	C₂×D₈	—	73	1	no	—
236	#131	—	73	2	no	—
237	#131	—	73	2	no	—
238	#131	—	73	2	no	—
239	#131	—	146	2	no	—
240	#131	—	146	2	no	—
241	#131	C₂×#64408	1	2	yes	—
242	C₂×M₁₆	—	73	2	no	—
243	C₂×M₁₆	—	146	2	no	—
244	C₂²×C₈	—	73	1	no	—
245	C₂²×C₈	—	73	1	no	—
246	C₂²×C₈	—	146	2	no	—
247	C₂×C₄²	—	73	1	no	—
248	M₃₂	—	73	4	no	—
249	M₃₂	—	146	4	no	—
250	M₃₂	—	146	4	no	—
251	M₃₂	—	146	2	no	—
252	C₂×C₁₆	—	73	2	no	—
253	C₂×C₁₆	—	73	1	no	—
254	C₂×C₁₆	—	73	1	no	—
255	C₂×C₁₆	—	73	2	no	—
256	C₂×C₁₆	—	73	2	no	—
257	C₂×C₁₆	—	146	4	no	—
258	C₂×C₁₆	—	146	4	no	—
259	C₂×C₁₆	—	146	1	no	—
260	C₂×C₁₆	—	146	1	no	—
261	C₂×C₁₆	—	146	2	no	—
262	#108	—	73	2	no	—
263	#108	—	73	1	no	—
264	#108	C₂×#64408	1	1	yes	characteristic
265	#104	—	73	2	no	—
266	#104	—	73	2	no	—
267	#104	—	73	2	no	—
268	#104	C₂×#64408	1	2	yes	—
269	C₄×C₈	—	73	1	no	—
270	C₄×C₈	—	73	1	no	—
271	C₄×C₈	—	73	1	no	—
272	C₄×C₈	C₂×#64408	1	1	yes	characteristic
273	#41	—	73	2	no	—
274	#41	—	73	2	no	—
275	#41	—	73	2	no	—
276	#41	—	146	2	no	—
277	#41	—	146	2	no	—
278	#41	C₂²×#64408	1	2	yes	—
279	C₂×Q₈	—	73	2	no	—
280	C₂×D₄	—	73	2	no	—
281	C₂×D₄	—	146	2	no	—
282	C₂²×C₄	—	73	1	no	—
283	C₂²×C₄	—	146	2	no	—
284	C₂²×C₄	—	146	1	no	—
285	Q₁₆	—	73	2	no	—
286	Q₁₆	—	73	1	no	—
287	Q₁₆	#11924036	1	1	yes	characteristic
288	SD₁₆	—	73	2	no	—
289	SD₁₆	—	73	2	no	—
290	SD₁₆	—	73	2	no	—
291	SD₁₆	#11924036	1	2	yes	—
292	D₈	—	73	2	no	—
293	D₈	—	73	1	no	—
294	D₈	#11924036	1	1	yes	characteristic
295	M₁₆	—	2	2	no	—
296	M₁₆	—	73	2	no	—
297	M₁₆	—	146	2	no	—
298	M₁₆	—	146	2	no	—
299	M₁₆	—	146	2	no	—
300	M₁₆	—	146	2	no	—
301	M₁₆	C₂²×#64408	1	2	yes	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	73	1	no	—
304	C₂×C₈	—	73	1	no	—
305	C₂×C₈	—	73	1	no	—
306	C₂×C₈	—	73	1	no	—
307	C₂×C₈	—	73	1	no	—
308	C₂×C₈	—	73	1	no	—
309	C₂×C₈	—	73	1	no	—
310	C₂×C₈	—	73	1	no	—
311	C₂×C₈	—	73	1	no	—
312	C₂×C₈	—	73	1	no	—
313	C₂×C₈	—	146	2	no	—
314	C₂×C₈	—	146	2	no	—
315	C₂×C₈	—	146	2	no	—
316	C₂×C₈	—	146	2	no	—
317	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
318	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
319	C₄²	—	73	1	no	—
320	C₄²	—	146	1	no	—
321	C₄²	C₂²×#64408	1	1	yes	characteristic
322	C₁₆	—	73	2	no	—
323	C₁₆	—	73	2	no	—
324	C₁₆	—	146	2	no	—
325	C₁₆	—	146	4	no	—
326	C₂³	—	73	1	no	—
327	C₂³	—	146	2	no	—
328	Q₈	—	73	2	no	—
329	Q₈	—	146	2	no	—
330	Q₈	C₂×#11924036	1	2	yes	—
331	D₄	—	2	2	no	—
332	D₄	—	73	2	no	—
333	D₄	—	146	2	no	—
334	D₄	—	146	2	no	—
335	D₄	—	146	2	no	—
336	D₄	—	146	2	no	—
337	D₄	C₂×#11924036	1	2	yes	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	73	1	no	—
341	C₂×C₄	—	73	1	no	—
342	C₂×C₄	—	73	1	no	—
343	C₂×C₄	—	73	1	no	—
344	C₂×C₄	—	73	1	no	—
345	C₂×C₄	—	146	2	no	—
346	C₂×C₄	—	146	2	no	—
347	C₂×C₄	—	146	4	no	—
348	C₂×C₄	—	146	2	no	—
349	C₂×C₄	—	146	1	no	—
350	C₂×C₄	—	146	2	no	—
351	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
352	C₈	—	2	2	no	—
353	C₈	—	73	1	no	—
354	C₈	—	73	1	no	—
355	C₈	—	73	1	no	—
356	C₈	—	73	1	no	—
357	C₈	—	146	2	no	—
358	C₈	D₄×#64408	1	1	yes	characteristic
359	C₈	D₄×#64408	1	1	yes	characteristic
360	C₈	C₂×#11924036	1	1	yes	characteristic
361	C₈	C₂×#11924036	1	1	yes	characteristic
362	C₂²	—	2	2	no	—
363	C₂²	—	292	4	no	—
364	C₂²	—	73	1	no	—
365	C₂²	—	73	1	no	—
366	C₂²	—	146	2	no	—
367	C₂²	—	146	2	no	—
368	C₂²	D₇₃⋊#131	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	2	2	no	—
371	C₄	—	73	1	no	—
372	C₄	—	73	1	no	—
373	C₄	—	146	2	no	—
374	C₄	—	146	2	no	—
375	C₄	C₂×D₄×#64408	1	1	yes	characteristic
376	C₄	C₂²×#11924036	1	1	yes	characteristic
377	C₂	—	2	1	no	—
378	C₂	—	4	2	no	—
379	C₂	—	292	2	no	—
380	C₂	—	73	2	no	—
381	C₂	—	146	1	no	—
382	C₂	D₄×#11924036	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×#64408
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	#443×C₇₃	C₄
socle	C₂×C₇₃	C₂×C₈×D₄
prime radical	#135×D₇₃	C₄
cosocle	C₂×#10596831	C₂⁴
hypercenter	#443	#64408
hypocenter	C₇₃	C₂×#2170
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #22626896	C₂	semidirect
2. #22626320	C₂	semidirect
3. #22626319	C₂	semidirect
4. #22626318	C₂	semidirect
5. #22626270	C₂	semidirect
6. #22626171	C₂	semidirect
7. C₄×#14556664	C₂	semidirect
10. #17989646	C₂²	semidirect
11. #17989645	C₂²	semidirect
17. #17989510	C₂²	semidirect
18. #17989509	C₂²	semidirect
19. #17989486	C₂²	semidirect
20. #17989485	C₂²	semidirect
21. #17989473	C₂²	semidirect
22. C₄×#11924031	C₂²	semidirect
191. D₇₃	#2170	semidirect
200. C₇₃	C₂×#2170	semidirect

Maximal subgroups

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

subgroup	index
#22626896	2
#22626320	2
#22626319	2
#22626318	2
#22626270	2
#22626171	2
C₄×#14556664	2
#443×D₇₃	2
C₂×#2170	73