Subgroups — #508×#64408 #34030255

Order 18688.

690 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×#64408	C₂	1	2	yes	maximal, complemented
2	Q₃₂×#64408	C₂	1	2	yes	maximal, complemented
3	SD₃₂×#64408	C₂	1	4	yes	maximal, complemented
4	D₁₆×#64408	C₂	1	2	yes	maximal, complemented
5	#22626282	C₂	1	1	yes	maximal, characteristic, complemented
6	#22626238	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₆×#64408	C₂	1	1	yes	maximal, characteristic, complemented
8	#508×D₇₃	C₂	1	1	yes	maximal, characteristic
9	#41×#64408	—	2	2	no	—
10	C₂×#14556664	C₂²	1	1	yes	characteristic
11	Q₁₆×#64408	C₂²	1	4	yes	—
12	SD₁₆×#64408	—	2	4	no	—
13	D₈×#64408	C₂²	1	4	yes	—
14	#17989597	C₂²	1	2	yes	—
15	#17989577	—	2	2	no	—
16	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
17	#17989514	C₂²	1	2	yes	—
18	#17989513	C₂²	1	2	yes	—
19	#17989500	C₂²	1	4	yes	—
20	#17989499	C₂²	1	4	yes	—
21	C₁₆×#64408	C₂²	1	2	yes	—
22	C₁₆×#64408	C₂²	1	2	yes	—
23	#135×D₇₃	C₂²	1	2	yes	—
24	#17989020	C₄	1	1	yes	characteristic, complemented
25	#17989019	C₄	1	1	yes	characteristic, complemented
26	#17989018	C₄	1	2	yes	complemented
27	#17989017	C₄	1	2	yes	complemented
28	#17989016	C₄	1	1	yes	characteristic, complemented
29	Q₃₂×D₇₃	C₂²	1	1	yes	characteristic
30	SD₃₂×D₇₃	C₂²	1	2	yes	—
31	D₁₆×D₇₃	C₂²	1	1	yes	characteristic
32	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
33	#508×C₇₃	C₄	1	1	yes	characteristic, complemented
34	C₂×#11924036	D₄	1	1	yes	characteristic
35	Q₈×#64408	—	2	4	no	—
36	D₄×#64408	—	2	4	no	—
37	D₄×#64408	—	4	4	no	—
38	#14556670	—	2	2	no	—
39	#14556670	—	4	2	no	—
40	C₂×C₄×#64408	—	4	2	no	—
41	C₂×C₄×#64408	D₄	1	1	yes	characteristic
42	#14556664	—	2	1	no	—
43	#14556664	C₂³	1	2	yes	—
44	C₂×#11924031	—	2	1	no	—
45	#14556660	C₂³	1	4	yes	—
46	#14556659	—	2	2	no	—
47	#14556654	—	2	4	no	—
48	#14556653	—	2	4	no	—
49	C₈×#64408	D₄	1	2	yes	—
50	C₈×#64408	C₂³	1	2	yes	—
51	#41×D₇₃	—	2	2	no	—
52	#14556614	C₂×C₄	1	2	yes	complemented
53	#14556613	C₂×C₄	1	2	yes	complemented
54	#14556612	—	2	2	no	—
55	#14556611	—	2	2	no	—
56	#14556610	C₂×C₄	1	2	yes	complemented
57	Q₁₆×D₇₃	C₂³	1	2	yes	—
58	SD₁₆×D₇₃	—	2	2	no	—
59	D₈×D₇₃	C₂³	1	2	yes	—
60	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
61	Dic₅₈₄	C₂×C₄	1	1	yes	characteristic, complemented
62	#14556537	C₂×C₄	1	2	yes	complemented
63	#14556536	C₂×C₄	1	2	yes	complemented
64	#14556535	C₂×C₄	1	2	yes	complemented
65	#14556534	C₂×C₄	1	2	yes	complemented
66	D₁₁₆₈	C₂×C₄	1	1	yes	characteristic, complemented
67	#14556532	C₂×C₄	1	2	yes	complemented
68	C₂×#11924003	C₂×C₄	1	1	yes	characteristic, complemented
69	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic, complemented
70	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic, complemented
71	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
72	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
73	#135×C₇₃	C₂×C₄	1	2	yes	complemented
74	Q₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
75	SD₃₂×C₇₃	C₂×C₄	1	2	yes	complemented
76	D₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
77	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
78	C₂²×#64408	—	4	4	no	—
79	C₂²×#64408	D₈	1	2	yes	—
80	#11924036	—	2	1	no	—
81	#11924036	C₂×D₄	1	2	yes	—
82	C₂×#10596833	—	2	1	no	—
83	C₄⋊#64408	—	2	4	no	—
84	C₄⋊#64408	—	4	4	no	—
85	C₂²⋊#64408	—	4	2	no	—
86	C₂²⋊#64408	—	4	4	no	—
87	C₄×#64408	—	4	4	no	—
88	C₄×#64408	—	4	2	no	—
89	C₄×#64408	D₈	1	1	yes	characteristic
90	C₄×#64408	D₈	1	1	yes	characteristic
91	C₄×#64408	C₂×D₄	1	2	yes	—
92	#11924031	—	2	2	no	—
93	#11924029	—	2	2	no	—
94	#11924028	—	2	2	no	—
95	#11924028	—	4	2	no	—
96	#11924027	—	2	2	no	—
97	#11924027	—	4	2	no	—
98	Q₈×D₇₃	—	2	2	no	—
99	D₄×D₇₃	—	2	2	no	—
100	D₄×D₇₃	—	4	2	no	—
101	C₂×C₄×D₇₃	—	4	2	no	—
102	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
103	Dic₂₉₂	C₂²×C₄	1	2	yes	—
104	#11924018	—	2	2	no	—
105	#11924017	—	2	2	no	—
106	#11924016	—	2	2	no	—
107	#11924015	—	2	2	no	—
108	D₅₈₄	C₂²×C₄	1	2	yes	—
109	#11924013	—	2	2	no	—
110	C₂×#10596831	#41	1	1	yes	characteristic
111	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
113	C₈×D₇₃	#41	1	1	yes	characteristic
114	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
115	#11924003	C₂²×C₄	1	1	yes	characteristic
116	#11924003	C₂²×C₄	1	1	yes	characteristic
117	#41×C₇₃	—	2	2	no	—
118	Q₁₆×C₇₃	C₂²×C₄	1	2	yes	—
119	SD₁₆×C₇₃	—	2	2	no	—
120	D₈×C₇₃	C₂²×C₄	1	2	yes	—
121	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
122	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
123	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²×D₇₃	—	4	2	no	—
125	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
126	C₂×#64408	—	2	4	no	—
127	C₂×#64408	—	4	4	no	—
128	C₂×#64408	—	8	8	no	—
129	C₂×#64408	C₂×D₈	1	2	yes	—
130	C₂×#64408	C₂×D₈	1	2	yes	—
131	Q₈×C₇₃	—	2	2	no	—
132	D₄×C₇₃	—	2	2	no	—
133	D₄×C₇₃	—	4	2	no	—
134	C₂×C₄×C₇₃	—	4	2	no	—
135	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
136	C₂²⋊D₇₃	—	4	2	no	—
137	C₂²⋊D₇₃	—	4	2	no	—
138	C₂²⋊D₇₃	—	4	2	no	—
139	C₂×Dic₇₃	—	4	2	no	—
140	C₂×Dic₇₃	#135	1	1	yes	characteristic
141	D₂₉₂	—	2	2	no	—
142	D₂₉₂	—	4	2	no	—
143	C₄×D₇₃	—	4	2	no	—
144	C₄×D₇₃	—	4	2	no	—
145	C₄×D₇₃	—	4	2	no	—
146	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
147	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
148	C₄×D₇₃	#135	1	1	yes	characteristic
149	C₄×D₇₃	C₂²×D₄	1	1	yes	characteristic
150	Dic₁₄₆	—	2	2	no	—
151	Dic₁₄₆	—	4	2	no	—
152	#10596833	—	2	2	no	—
153	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
154	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
155	#10596831	C₄×D₄	1	1	yes	characteristic
156	#10596831	C₂×#41	1	1	yes	characteristic
157	C₂²×C₇₃	—	4	2	no	—
158	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
159	C₂×D₇₃	—	2	2	no	—
160	C₂×D₇₃	—	4	2	no	—
161	C₂×D₇₃	—	8	4	no	—
162	C₂×D₇₃	C₄×D₈	1	1	yes	characteristic
163	C₂×D₇₃	C₂²×D₈	1	1	yes	characteristic
164	#64408	—	2	2	no	—
165	#64408	—	8	4	no	—
166	#64408	#508	1	4	yes	direct product factor
167	C₄×C₇₃	—	4	2	no	—
168	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
169	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
170	Dic₇₃	—	4	2	no	—
171	Dic₇₃	C₄×D₈	1	1	yes	characteristic
172	Dic₇₃	C₂×#135	1	1	yes	characteristic
173	C₄×#508	—	73	1	no	maximal, Sylow
174	C₂×C₇₃	—	2	1	no	—
175	C₂×C₇₃	—	8	2	no	—
176	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
177	D₇₃	—	2	1	no	—
178	D₇₃	—	8	2	no	—
179	D₇₃	C₂×#508	1	2	yes	—
180	C₂×#508	—	73	1	no	—
181	C₄×#135	—	73	2	no	—
182	C₄×Q₃₂	—	73	2	no	—
183	C₄×SD₃₂	—	73	4	no	—
184	C₄×D₁₆	—	73	2	no	—
185	#2158	—	73	1	no	—
186	#2138	—	73	2	no	—
187	C₂×C₄×C₁₆	—	73	1	no	—
188	C₇₃	C₄×#508	1	1	yes	characteristic, complemented, Sylow
189	C₂×#135	—	73	2	no	—
190	C₄×#41	—	146	2	no	—
191	#508	—	73	2	no	—
192	#508	—	73	1	no	—
193	#508	—	73	2	no	—
194	#508	—	73	1	no	—
195	#508	—	73	1	no	—
196	#508	#64408	1	1	yes	characteristic, direct product factor
197	C₂×Q₃₂	—	73	1	no	—
198	C₂×SD₃₂	—	73	2	no	—
199	C₂×D₁₆	—	73	1	no	—
200	C₂²×C₁₆	—	73	1	no	—
201	C₂²×C₁₆	—	73	1	no	—
202	C₄×Q₁₆	—	73	4	no	—
203	C₄×SD₁₆	—	146	4	no	—
204	C₄×D₈	—	73	4	no	—
205	#427	—	73	2	no	—
206	#416	—	146	2	no	—
207	C₂×C₄×C₈	—	73	1	no	—
208	#367	—	73	2	no	—
209	#366	—	73	2	no	—
210	#358	—	73	4	no	—
211	#357	—	73	4	no	—
212	C₄×C₁₆	—	73	2	no	—
213	C₄×C₁₆	—	73	2	no	—
214	C₂×#41	—	146	2	no	—
215	#135	—	73	2	no	—
216	#135	—	73	2	no	—
217	#135	—	73	2	no	—
218	#135	—	146	2	no	—
219	#135	—	146	2	no	—
220	#135	C₂×#64408	1	2	yes	complemented
221	C₂×Q₁₆	—	73	2	no	—
222	C₂×SD₁₆	—	146	2	no	—
223	C₂×D₈	—	73	2	no	—
224	C₂²×C₈	—	73	1	no	—
225	C₂²×C₈	—	73	1	no	—
226	C₄×Q₈	—	146	4	no	—
227	C₄×D₄	—	292	4	no	—
228	C₄×D₄	—	146	4	no	—
229	#117	—	292	2	no	—
230	#117	—	146	2	no	—
231	C₂×C₄²	—	292	2	no	—
232	C₂×C₄²	—	73	1	no	—
233	Q₃₂	—	73	2	no	—
234	Q₃₂	—	73	1	no	—
235	Q₃₂	C₂×#64408	1	1	yes	characteristic, complemented
236	SD₃₂	—	73	2	no	—
237	SD₃₂	—	73	2	no	—
238	SD₃₂	—	73	2	no	—
239	SD₃₂	C₂×#64408	1	2	yes	complemented
240	D₁₆	—	73	2	no	—
241	D₁₆	—	73	1	no	—
242	D₁₆	C₂×#64408	1	1	yes	characteristic, complemented
243	C₂×C₁₆	—	73	1	no	—
244	C₂×C₁₆	—	73	1	no	—
245	C₂×C₁₆	—	73	1	no	—
246	C₂×C₁₆	—	73	1	no	—
247	C₂×C₁₆	—	73	1	no	—
248	C₂×C₁₆	—	73	2	no	—
249	C₂×C₁₆	—	146	1	no	—
250	C₂×C₁₆	—	146	1	no	—
251	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic, complemented
252	#107	—	73	4	no	—
253	#106	—	146	2	no	—
254	#103	—	146	4	no	—
255	#102	—	146	4	no	—
256	C₄×C₈	—	73	2	no	—
257	C₄×C₈	—	73	2	no	—
258	#41	—	2	2	no	—
259	#41	—	292	2	no	—
260	#41	—	292	2	no	—
261	#41	—	146	2	no	—
262	#41	—	146	2	no	—
263	#41	—	146	2	no	—
264	C₂×Q₈	—	146	2	no	—
265	C₂×D₄	—	292	2	no	—
266	C₂×D₄	—	146	2	no	—
267	C₂²×C₄	—	292	2	no	—
268	C₂²×C₄	—	292	4	no	—
269	C₂²×C₄	—	73	1	no	—
270	C₂²×C₄	—	73	2	no	—
271	Q₁₆	—	73	2	no	—
272	Q₁₆	—	146	2	no	—
273	Q₁₆	C₂²×#64408	1	2	yes	—
274	SD₁₆	—	2	2	no	—
275	SD₁₆	—	146	2	no	—
276	SD₁₆	—	146	2	no	—
277	SD₁₆	—	146	2	no	—
278	D₈	—	73	2	no	—
279	D₈	—	146	2	no	—
280	D₈	C₂²×#64408	1	2	yes	—
281	C₂×C₈	—	73	1	no	—
282	C₂×C₈	—	73	1	no	—
283	C₂×C₈	—	73	1	no	—
284	C₂×C₈	—	73	1	no	—
285	C₂×C₈	—	73	1	no	—
286	C₂×C₈	—	73	2	no	—
287	C₂×C₈	—	146	1	no	—
288	C₂×C₈	—	146	1	no	—
289	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
290	C₄⋊C₄	—	292	4	no	—
291	C₄⋊C₄	—	146	4	no	—
292	C₂²⋊C₄	—	292	2	no	—
293	C₂²⋊C₄	—	292	4	no	—
294	C₄²	—	292	2	no	—
295	C₄²	—	292	4	no	—
296	C₄²	—	73	1	no	—
297	C₄²	—	73	1	no	—
298	C₄²	—	73	2	no	—
299	C₁₆	—	73	1	no	—
300	C₁₆	—	73	1	no	—
301	C₁₆	—	146	2	no	—
302	C₁₆	C₂²×#64408	1	1	yes	characteristic
303	C₁₆	C₂²×#64408	1	1	yes	characteristic
304	C₂³	—	292	2	no	—
305	C₂³	—	73	1	no	—
306	Q₈	—	2	2	no	—
307	Q₈	—	292	2	no	—
308	Q₈	—	146	2	no	—
309	D₄	—	2	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	292	2	no	—
312	D₄	—	292	2	no	—
313	D₄	—	292	2	no	—
314	D₄	—	292	2	no	—
315	D₄	—	146	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	292	2	no	—
318	C₂×C₄	—	292	2	no	—
319	C₂×C₄	—	292	2	no	—
320	C₂×C₄	—	292	2	no	—
321	C₂×C₄	—	292	4	no	—
322	C₂×C₄	—	584	8	no	—
323	C₂×C₄	—	73	2	no	—
324	C₂×C₄	—	73	1	no	—
325	C₂×C₄	—	73	1	no	—
326	C₂×C₄	—	73	1	no	—
327	C₂×C₄	—	73	1	no	—
328	C₂×C₄	—	73	1	no	—
329	C₂×C₄	—	73	2	no	—
330	C₂×C₄	—	146	4	no	—
331	C₂×C₄	D₄×#64408	1	1	yes	characteristic
332	C₈	—	73	1	no	—
333	C₈	—	73	1	no	—
334	C₈	—	146	2	no	—
335	C₈	D₄×#64408	1	1	yes	characteristic
336	C₈	C₂³×#64408	1	1	yes	characteristic
337	C₂²	—	4	2	no	—
338	C₂²	—	292	2	no	—
339	C₂²	—	584	4	no	—
340	C₂²	—	73	1	no	—
341	C₂²	—	73	1	no	—
342	C₂²	—	146	2	no	—
343	C₂²	D₈×#64408	1	1	yes	characteristic
344	C₄	—	4	2	no	—
345	C₄	—	292	2	no	—
346	C₄	—	584	4	no	—
347	C₄	—	73	4	no	—
348	C₄	—	73	1	no	—
349	C₄	—	73	1	no	—
350	C₄	—	146	2	no	—
351	C₄	D₈×#64408	1	1	yes	characteristic
352	C₄	C₂×D₄×#64408	1	1	yes	characteristic
353	C₂	—	2	1	no	—
354	C₂	—	8	2	no	—
355	C₂	—	584	2	no	—
356	C₂	—	73	2	no	—
357	C₂	—	146	1	no	—
358	C₂	C₂×D₈×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×#64408
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₈	C₂³×#64408
Fitting	#508×C₇₃	C₄
socle	C₂×C₇₃	C₂×C₄×D₈
prime radical	#508×D₇₃	C₂
cosocle	C₈×D₇₃	C₂⁴
hypercenter	#508	#64408
hypocenter	C₇₃	C₄×#508
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #135×#64408	C₂	semidirect
2. Q₃₂×#64408	C₂	semidirect
3. SD₃₂×#64408	C₂	semidirect
4. D₁₆×#64408	C₂	semidirect
5. #22626282	C₂	semidirect
6. #22626238	C₂	semidirect
7. C₂×C₁₆×#64408	C₂	semidirect
24. #17989020	C₄	semidirect
25. #17989019	C₄	semidirect
26. #17989018	C₄	semidirect
27. #17989017	C₄	semidirect
28. #17989016	C₄	semidirect
33. #508×C₇₃	C₄	semidirect
52. #14556614	C₂×C₄	semidirect
53. #14556613	C₂×C₄	semidirect
56. #14556610	C₂×C₄	semidirect
61. Dic₅₈₄	C₂×C₄	semidirect
62. #14556537	C₂×C₄	semidirect
63. #14556536	C₂×C₄	semidirect
64. #14556535	C₂×C₄	semidirect
65. #14556534	C₂×C₄	semidirect
66. D₁₁₆₈	C₂×C₄	semidirect
67. #14556532	C₂×C₄	semidirect
68. C₂×#11924003	C₂×C₄	semidirect
69. C₁₆×D₇₃	C₂×C₄	semidirect
70. C₁₆×D₇₃	C₂×C₄	semidirect
73. #135×C₇₃	C₂×C₄	semidirect
74. Q₃₂×C₇₃	C₂×C₄	semidirect
75. SD₃₂×C₇₃	C₂×C₄	semidirect
76. D₁₆×C₇₃	C₂×C₄	semidirect
77. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
166. #64408	#508	direct (N × H)
188. C₇₃	C₄×#508	semidirect
196. #508	#64408	direct (N × H)
220. #135	C₂×#64408	semidirect
235. Q₃₂	C₂×#64408	semidirect
239. SD₃₂	C₂×#64408	semidirect
242. D₁₆	C₂×#64408	semidirect
251. C₂×C₁₆	C₂×#64408	semidirect

Maximal subgroups

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

subgroup	index
#135×#64408	2
Q₃₂×#64408	2
SD₃₂×#64408	2
D₁₆×#64408	2
#22626282	2
#22626238	2
C₂×C₁₆×#64408	2
#508×D₇₃	2
C₄×#508	73