Subgroups — #34030241

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 394 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	#22626323	C₂	1	1	yes	maximal, characteristic, complemented
3	#22626322	C₂	1	2	yes	maximal, complemented
4	#22626321	C₂	1	1	yes	maximal, characteristic, complemented
5	#22626320	C₂	1	1	yes	maximal, characteristic, complemented
6	#22626319	C₂	1	2	yes	maximal, complemented
7	#22626318	C₂	1	1	yes	maximal, characteristic, complemented
8	#22626273	C₂	1	1	yes	maximal, characteristic, complemented
9	#22626174	C₂	1	2	yes	maximal, complemented
10	#22626141	C₂	1	1	yes	maximal, characteristic, complemented
11	#22618831	C₂	1	1	yes	maximal, characteristic
12	#17989646	—	2	2	no	—
13	#17989646	—	2	2	no	—
14	#17989646	C₂²	1	2	yes	complemented
15	#17989646	C₂²	1	2	yes	complemented
16	#17989645	C₂²	1	2	yes	complemented
17	#17989645	C₂²	1	2	yes	complemented
18	C₂×#14556666	C₂²	1	1	yes	characteristic
19	#17989643	—	2	2	no	—
20	#17989643	C₂²	1	2	yes	—
21	C₂×#14556664	—	2	2	no	—
22	C₂×#14556664	C₂²	1	1	yes	characteristic
23	#17989510	C₂²	1	2	yes	complemented
24	#17989509	C₂²	1	1	yes	characteristic, complemented
25	#17989509	C₂²	1	1	yes	characteristic, complemented
26	#17989486	C₂²	1	2	yes	complemented
27	#17989486	C₂²	1	2	yes	complemented
28	#17989485	C₂²	1	2	yes	complemented
29	#17989485	C₂²	1	2	yes	complemented
30	#17989476	C₂²	1	1	yes	characteristic, complemented
31	#17989475	C₂²	1	1	yes	characteristic, complemented
32	#17989473	C₂²	1	1	yes	characteristic, complemented
33	C₄×#11924031	C₂²	1	1	yes	characteristic, complemented
34	#17989384	C₄	1	1	yes	characteristic
35	#131×D₇₃	C₂²	1	2	yes	—
36	#17988710	C₄	1	1	yes	characteristic
37	#17988707	C₄	1	2	yes	—
38	#17988704	C₂²	1	1	yes	characteristic
39	#17988703	C₂²	1	1	yes	characteristic
40	#17988701	C₂²	1	2	yes	—
41	#17988634	C₄	1	1	yes	characteristic
42	#17988602	C₄	1	2	yes	—
43	#17988544	C₂²	1	1	yes	characteristic
44	#444×C₇₃	C₄	1	1	yes	characteristic
45	#14556666	—	2	2	no	—
46	#14556666	—	2	2	no	—
47	#14556666	—	2	2	no	—
48	#14556666	—	2	2	no	—
49	#14556665	—	2	2	no	—
50	#14556665	—	2	2	no	—
51	#14556665	C₂³	1	2	yes	—
52	#14556665	C₂³	1	2	yes	—
53	#14556664	—	2	2	no	—
54	#14556664	—	2	2	no	—
55	#14556664	—	2	2	no	—
56	#14556664	D₄	1	2	yes	—
57	#14556664	D₄	1	2	yes	—
58	C₂×#11924031	—	2	2	no	—
59	C₂×#11924031	—	2	2	no	—
60	C₂×#11924031	C₂³	1	1	yes	characteristic
61	C₂×#11924031	C₂³	1	1	yes	characteristic
62	C₂×#11924031	C₂³	1	1	yes	characteristic
63	C₂×#11924031	C₂³	1	1	yes	characteristic
64	#41×D₇₃	C₂×C₄	1	2	yes	—
65	#14556623	C₈	1	2	yes	—
66	#14556620	C₈	1	2	yes	—
67	#14556618	C₈	1	2	yes	—
68	#14556615	C₈	1	2	yes	—
69	#14556613	C₂×C₄	1	1	yes	characteristic
70	#14556611	C₂×C₄	1	2	yes	—
71	#14556599	—	2	2	no	—
72	#14556599	C₂³	1	2	yes	—
73	#14556598	C₂³	1	2	yes	—
74	#14556597	—	2	2	no	—
75	#14556597	C₂×C₄	1	2	yes	—
76	C₂×#11924011	C₂×C₄	1	1	yes	characteristic
77	M₁₆×D₇₃	—	2	2	no	—
78	M₁₆×D₇₃	C₂×C₄	1	2	yes	—
79	C₂×C₈×D₇₃	—	2	2	no	—
80	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
81	#14556548	C₂×C₄	1	1	yes	characteristic
82	#14556527	C₂×C₄	1	1	yes	characteristic
83	#14556526	C₂×C₄	1	2	yes	—
84	#14556519	C₂×C₄	1	2	yes	—
85	#14556518	C₂×C₄	1	2	yes	—
86	#14556517	C₂×C₄	1	2	yes	—
87	#14556501	C₂×C₄	1	1	yes	characteristic
88	C₄×#10596831	C₂³	1	1	yes	characteristic
89	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
90	#135×C₇₃	C₂×C₄	1	1	yes	characteristic
91	#131×C₇₃	C₂×C₄	1	2	yes	—
92	#108×C₇₃	C₂×C₄	1	1	yes	characteristic
93	#104×C₇₃	C₂×C₄	1	2	yes	—
94	#97×C₇₃	C₂×C₄	1	1	yes	characteristic
95	#11924031	—	2	2	no	—
96	#11924031	—	2	2	no	—
97	#11924031	—	2	1	no	—
98	#11924031	—	2	1	no	—
99	#11924031	C₂×D₄	1	1	yes	characteristic
100	#11924031	C₂×D₄	1	2	yes	—
101	#11924031	C₂×D₄	1	1	yes	characteristic
102	#11924029	C₂×C₈	1	2	yes	—
103	#11924028	—	2	2	no	—
104	#11924028	C₂²×C₄	1	2	yes	—
105	#11924027	—	2	2	no	—
106	#11924027	C₂×C₈	1	2	yes	—
107	Q₈×D₇₃	C₂×C₈	1	2	yes	—
108	D₄×D₇₃	—	2	2	no	—
109	D₄×D₇₃	C₂×C₈	1	2	yes	—
110	C₂×C₄×D₇₃	—	2	2	no	—
111	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
112	Dic₂₉₂	C₂×C₈	1	1	yes	characteristic
113	#11924018	C₂×C₈	1	2	yes	—
114	#11924017	C₂×C₈	1	2	yes	—
115	#11924016	C₂×C₈	1	2	yes	—
116	#11924015	C₂×C₈	1	2	yes	—
117	D₅₈₄	C₂×C₈	1	1	yes	characteristic
118	#11924013	C₂×C₈	1	2	yes	—
119	#11924012	—	2	2	no	—
120	#11924012	C₂²×C₄	1	2	yes	—
121	#11924011	—	2	2	no	—
122	#11924011	—	2	2	no	—
123	#11924011	—	2	2	no	—
124	#11924011	C₂×C₈	1	2	yes	—
125	#11924011	C₂×C₈	1	2	yes	—
126	C₂×#10596831	—	2	2	no	—
127	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
128	C₂×#10596831	C₂⁴	1	1	yes	characteristic
129	C₈×D₇₃	—	2	2	no	—
130	C₈×D₇₃	—	2	2	no	—
131	C₈×D₇₃	—	2	2	no	—
132	C₈×D₇₃	C₂×D₄	1	2	yes	—
133	C₈×D₇₃	#41	1	2	yes	—
134	#11924007	—	2	1	no	—
135	#11924005	—	2	1	no	—
136	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
137	#41×C₇₃	C₂²×C₄	1	2	yes	—
138	Q₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
139	SD₁₆×C₇₃	C₂×C₈	1	2	yes	—
140	D₈×C₇₃	C₂×C₈	1	1	yes	characteristic
141	M₁₆×C₇₃	—	2	2	no	—
142	M₁₆×C₇₃	C₂²×C₄	1	2	yes	—
143	C₂×C₈×C₇₃	—	2	2	no	—
144	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
146	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
147	C₂²×D₇₃	—	2	2	no	—
148	C₂²×D₇₃	#131	1	1	yes	characteristic
149	Q₈×C₇₃	C₂²×C₈	1	2	yes	—
150	D₄×C₇₃	—	2	2	no	—
151	D₄×C₇₃	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₇₃	—	2	2	no	—
153	C₂×C₄×C₇₃	—	2	1	no	—
154	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
155	C₂²⋊D₇₃	—	2	2	no	—
156	C₂²⋊D₇₃	—	2	2	no	—
157	C₂²⋊D₇₃	—	2	2	no	—
158	C₂×Dic₇₃	—	2	2	no	—
159	C₂×Dic₇₃	—	2	1	no	—
160	C₂×Dic₇₃	#131	1	1	yes	characteristic
161	D₂₉₂	—	2	2	no	—
162	D₂₉₂	C₂²×C₈	1	2	yes	—
163	C₄×D₇₃	—	2	2	no	—
164	C₄×D₇₃	—	2	2	no	—
165	C₄×D₇₃	—	2	2	no	—
166	C₄×D₇₃	C₄×D₄	1	2	yes	—
167	C₄×D₇₃	C₂²×C₈	1	2	yes	—
168	Dic₁₄₆	—	2	2	no	—
169	Dic₁₄₆	C₂²×C₈	1	2	yes	—
170	C₈×C₇₃	—	2	2	no	—
171	C₈×C₇₃	C₄×D₄	1	2	yes	—
172	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
173	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
174	#10596831	—	2	2	no	—
175	#10596831	C₂²×C₈	1	2	yes	—
176	#10596831	C₂²×D₄	1	1	yes	characteristic
177	#10596831	C₂×#41	1	1	yes	characteristic
178	C₂²×C₇₃	—	2	2	no	—
179	C₂²×C₇₃	C₂×#131	1	1	yes	characteristic
180	C₂×D₇₃	—	2	2	no	—
181	C₂×D₇₃	—	4	1	no	—
182	C₂×D₇₃	—	4	4	no	—
183	C₂×D₇₃	C₈×D₄	1	2	yes	—
184	C₄×C₇₃	—	2	2	no	—
185	C₄×C₇₃	—	4	1	no	—
186	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
188	Dic₇₃	—	2	2	no	—
189	Dic₇₃	—	4	1	no	—
190	Dic₇₃	C₈×D₄	1	2	yes	—
191	#33441	—	73	1	no	maximal, Sylow
192	C₂×C₇₃	—	2	1	no	—
193	C₂×C₇₃	—	4	2	no	—
194	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
195	D₇₃	—	2	2	no	—
196	D₇₃	—	4	2	no	—
197	C₂×#504	—	73	2	no	—
198	#2955	—	73	1	no	—
199	#2171	—	73	1	no	—
200	#2171	—	73	2	no	—
201	#2171	—	73	1	no	—
202	#2170	—	73	1	no	—
203	#2170	—	73	2	no	—
204	#2170	—	73	1	no	—
205	#2153	—	73	1	no	—
206	#2117	—	73	2	no	—
207	#2108	—	73	1	no	—
208	C₇₃	#33441	1	1	yes	characteristic, complemented, Sylow
209	#575	—	73	1	no	—
210	C₂×#131	—	73	2	no	—
211	#504	—	73	2	no	—
212	#504	—	73	2	no	—
213	#504	—	73	2	no	—
214	#504	—	73	2	no	—
215	#504	—	146	2	no	—
216	#504	—	146	2	no	—
217	C₂×M₃₂	—	73	1	no	—
218	C₂×M₃₂	—	73	2	no	—
219	C₂×M₃₂	—	146	2	no	—
220	C₂²×C₁₆	—	73	1	no	—
221	C₂²×C₁₆	—	146	2	no	—
222	#444	—	73	2	no	—
223	#444	—	73	1	no	—
224	#444	#64408	1	1	yes	characteristic
225	#443	—	73	1	no	—
226	#443	—	73	1	no	—
227	#443	—	73	2	no	—
228	#430	—	73	1	no	—
229	#421	—	73	2	no	—
230	#405	—	73	1	no	—
231	#364	—	73	1	no	—
232	#364	—	73	1	no	—
233	#364	—	73	2	no	—
234	#350	—	73	2	no	—
235	#350	—	73	2	no	—
236	#350	—	73	2	no	—
237	#350	—	73	2	no	—
238	#346	—	73	1	no	—
239	#346	—	73	1	no	—
240	C₄×C₁₆	—	73	1	no	—
241	C₄×C₁₆	—	73	1	no	—
242	C₂×#41	—	73	2	no	—
243	#137	—	73	2	no	—
244	#137	—	73	2	no	—
245	AGL(1,ℤ/8)	—	73	2	no	—
246	AGL(1,ℤ/8)	—	73	2	no	—
247	#135	—	73	2	no	—
248	#135	—	73	1	no	—
249	#135	C₂×#64408	1	1	yes	characteristic
250	#131	—	73	2	no	—
251	#131	—	73	2	no	—
252	#131	—	73	2	no	—
253	#131	—	146	2	no	—
254	#131	—	146	2	no	—
255	#131	C₂×#64408	1	2	yes	—
256	C₂×M₁₆	—	73	1	no	—
257	C₂×M₁₆	—	73	2	no	—
258	C₂×M₁₆	—	146	2	no	—
259	C₂²×C₈	—	73	1	no	—
260	C₂²×C₈	—	146	2	no	—
261	#117	—	73	1	no	—
262	M₃₂	—	73	2	no	—
263	M₃₂	—	73	2	no	—
264	M₃₂	—	146	2	no	—
265	M₃₂	—	146	2	no	—
266	M₃₂	—	146	2	no	—
267	M₃₂	—	146	2	no	—
268	M₃₂	—	146	2	no	—
269	M₃₂	—	146	2	no	—
270	C₂×C₁₆	—	73	1	no	—
271	C₂×C₁₆	—	73	1	no	—
272	C₂×C₁₆	—	73	2	no	—
273	C₂×C₁₆	—	73	2	no	—
274	C₂×C₁₆	—	73	1	no	—
275	C₂×C₁₆	—	73	1	no	—
276	C₂×C₁₆	—	146	2	no	—
277	C₂×C₁₆	—	146	2	no	—
278	C₂×C₁₆	—	146	2	no	—
279	C₂×C₁₆	—	146	2	no	—
280	C₂×C₁₆	—	146	2	no	—
281	#108	—	73	2	no	—
282	#108	—	73	1	no	—
283	#108	C₂×#64408	1	1	yes	characteristic
284	#104	—	73	2	no	—
285	#104	—	73	2	no	—
286	#104	—	73	2	no	—
287	#104	C₂×#64408	1	2	yes	—
288	#97	—	73	1	no	—
289	#97	C₂×#64408	1	1	yes	characteristic
290	C₄×C₈	—	73	1	no	—
291	C₄×C₈	—	73	1	no	—
292	#41	—	73	2	no	—
293	#41	—	73	2	no	—
294	#41	—	73	2	no	—
295	#41	—	146	2	no	—
296	#41	—	146	2	no	—
297	#41	C₂²×#64408	1	2	yes	—
298	C₂×Q₈	—	73	2	no	—
299	C₂×D₄	—	73	2	no	—
300	C₂×D₄	—	146	2	no	—
301	C₂²×C₄	—	73	1	no	—
302	C₂²×C₄	—	146	2	no	—
303	Q₁₆	—	73	2	no	—
304	Q₁₆	—	73	1	no	—
305	Q₁₆	#11924036	1	1	yes	characteristic
306	SD₁₆	—	73	2	no	—
307	SD₁₆	—	73	2	no	—
308	SD₁₆	—	73	2	no	—
309	SD₁₆	#11924036	1	2	yes	—
310	D₈	—	73	2	no	—
311	D₈	—	73	1	no	—
312	D₈	#11924036	1	1	yes	characteristic
313	M₁₆	—	2	2	no	—
314	M₁₆	—	73	2	no	—
315	M₁₆	—	73	2	no	—
316	M₁₆	—	73	2	no	—
317	M₁₆	—	146	2	no	—
318	M₁₆	—	146	2	no	—
319	M₁₆	—	146	2	no	—
320	M₁₆	—	146	2	no	—
321	M₁₆	C₂²×#64408	1	2	yes	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	73	1	no	—
324	C₂×C₈	—	73	1	no	—
325	C₂×C₈	—	73	2	no	—
326	C₂×C₈	—	73	2	no	—
327	C₂×C₈	—	146	2	no	—
328	C₂×C₈	—	146	2	no	—
329	C₂×C₈	—	146	2	no	—
330	C₂×C₈	—	146	2	no	—
331	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
332	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
333	C₄⋊C₄	—	146	1	no	—
334	C₂²⋊C₄	—	146	1	no	—
335	C₄²	—	73	1	no	—
336	C₄²	C₂²×#64408	1	1	yes	characteristic
337	C₁₆	—	73	1	no	—
338	C₁₆	—	73	2	no	—
339	C₁₆	—	73	1	no	—
340	C₁₆	—	146	1	no	—
341	C₁₆	—	146	1	no	—
342	C₁₆	—	146	2	no	—
343	C₁₆	—	146	2	no	—
344	C₂³	—	73	1	no	—
345	C₂³	—	146	2	no	—
346	Q₈	—	73	2	no	—
347	Q₈	—	146	2	no	—
348	Q₈	C₂×#11924036	1	2	yes	—
349	D₄	—	2	2	no	—
350	D₄	—	73	2	no	—
351	D₄	—	146	2	no	—
352	D₄	—	146	2	no	—
353	D₄	—	146	2	no	—
354	D₄	—	146	2	no	—
355	D₄	C₂×#11924036	1	2	yes	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	73	1	no	—
359	C₂×C₄	—	73	2	no	—
360	C₂×C₄	—	73	2	no	—
361	C₂×C₄	—	146	2	no	—
362	C₂×C₄	—	146	2	no	—
363	C₂×C₄	—	146	1	no	—
364	C₂×C₄	—	146	2	no	—
365	C₂×C₄	—	146	2	no	—
366	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	73	1	no	—
369	C₈	—	73	1	no	—
370	C₈	—	73	2	no	—
371	C₈	—	146	2	no	—
372	C₈	D₄×#64408	1	2	yes	—
373	C₈	C₂×#11924036	1	1	yes	characteristic
374	C₈	C₂×#11924036	1	1	yes	characteristic
375	C₂²	—	2	2	no	—
376	C₂²	—	292	1	no	—
377	C₂²	—	292	4	no	—
378	C₂²	—	73	2	no	—
379	C₂²	—	146	2	no	—
380	C₂²	D₇₃⋊#131	1	1	yes	characteristic
381	C₄	—	2	2	no	—
382	C₄	—	4	1	no	—
383	C₄	—	292	1	no	—
384	C₄	—	73	2	no	—
385	C₄	—	146	2	no	—
386	C₄	C₂×D₄×#64408	1	1	yes	characteristic
387	C₄	C₂²×#11924036	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	4	2	no	—
390	C₂	—	292	2	no	—
391	C₂	—	146	2	no	—
392	C₂	D₄×#11924036	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×#64408
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	#444×C₇₃	C₄
socle	C₂×C₇₃	C₂×C₈×D₄
prime radical	#17989384	C₄
cosocle	C₂×#10596831	C₂⁴
hypercenter	#444	#64408
hypocenter	C₇₃	#33441
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
1. #22626896	C₂	semidirect
2. #22626323	C₂	semidirect
3. #22626322	C₂	semidirect
4. #22626321	C₂	semidirect
5. #22626320	C₂	semidirect
6. #22626319	C₂	semidirect
7. #22626318	C₂	semidirect
8. #22626273	C₂	semidirect
9. #22626174	C₂	semidirect
10. #22626141	C₂	semidirect
14. #17989646	C₂²	semidirect
15. #17989646	C₂²	semidirect
16. #17989645	C₂²	semidirect
17. #17989645	C₂²	semidirect
23. #17989510	C₂²	semidirect
24. #17989509	C₂²	semidirect
25. #17989509	C₂²	semidirect
26. #17989486	C₂²	semidirect
27. #17989486	C₂²	semidirect
28. #17989485	C₂²	semidirect
29. #17989485	C₂²	semidirect
30. #17989476	C₂²	semidirect
31. #17989475	C₂²	semidirect
32. #17989473	C₂²	semidirect
33. C₄×#11924031	C₂²	semidirect
208. C₇₃	#33441	semidirect

Maximal subgroups

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

subgroup	index
#22626896	2
#22626323	2
#22626322	2
#22626321	2
#22626320	2
#22626319	2
#22626318	2
#22626273	2
#22626174	2
#22626141	2
#22618831	2
#33441	73