Subgroups — #34034531

Order 18688.

782 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627173	C₂	1	2	yes	maximal, complemented
2	#22627062	C₂	1	4	yes	maximal, complemented
3	D₈×#10596842	C₂	1	4	yes	maximal, complemented
4	#22627055	C₂	1	1	yes	maximal, characteristic, complemented
5	#22627051	C₂	1	2	yes	maximal, complemented
6	#22627040	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₈×#64408	C₂	1	1	yes	maximal, characteristic
8	D₄×#10596842	—	2	4	no	—
9	D₄×#10596842	C₂²	1	8	yes	complemented
10	#17989700	C₂²	1	2	yes	—
11	#17989700	C₂²	1	1	yes	characteristic
12	#17989700	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊#10596842	—	2	4	no	—
14	#17989677	C₂²	1	2	yes	complemented
15	#17989676	C₂²	1	2	yes	complemented
16	D₄⋊#10596842	C₂²	1	8	yes	complemented
17	#17989671	C₂²	1	2	yes	complemented
18	C₈×#10596842	C₂²	1	2	yes	complemented
19	C₂×D₄×#64408	C₂²	1	2	yes	—
20	D₈×#64408	C₄	1	2	yes	—
21	D₈×#64408	C₄	1	2	yes	—
22	D₈×#64408	C₂²	1	2	yes	—
23	D₈×#64408	C₂²	1	2	yes	—
24	C₂×#14556660	C₄	1	1	yes	characteristic
25	C₂×#14556653	C₄	1	2	yes	—
26	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
27	C₂×D₈×D₇₃	C₄	1	1	yes	characteristic
28	C₂²×#10596842	—	2	8	no	—
29	C₄⋊#10596842	—	2	1	no	—
30	C₄⋊#10596842	D₄	1	2	yes	—
31	C₄⋊#10596842	C₂³	1	4	yes	—
32	C₂²⋊#10596842	—	2	8	no	—
33	C₂²⋊#10596842	—	2	2	no	—
34	C₂²⋊#10596842	—	2	1	no	—
35	C₂²⋊#10596842	D₄	1	2	yes	—
36	C₄×#10596842	—	2	2	no	—
37	C₄×#10596842	C₂³	1	2	yes	—
38	C₄×#10596842	C₂³	1	2	yes	—
39	C₂³×#64408	—	2	2	no	—
40	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
41	D₄×#64408	—	2	2	no	—
42	D₄×#64408	—	2	2	no	—
43	D₄×#64408	C₂×C₄	1	4	yes	—
44	D₄×#64408	C₂³	1	4	yes	—
45	C₂×C₄⋊#64408	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊#64408	—	2	2	no	—
47	C₂×C₄×#64408	C₂³	1	1	yes	characteristic
48	#14556660	C₂×C₄	1	2	yes	—
49	#14556660	C₂×C₄	1	2	yes	—
50	#14556653	C₂×C₄	1	4	yes	—
51	#14556653	C₂×C₄	1	4	yes	—
52	C₈×#64408	C₂×C₄	1	1	yes	characteristic
53	C₈×#64408	C₂×C₄	1	1	yes	characteristic
54	C₈×#64408	C₂³	1	1	yes	characteristic
55	C₈×#64408	C₂³	1	1	yes	characteristic
56	C₂×D₄×D₇₃	C₂×C₄	1	2	yes	—
57	C₂×D₅₈₄	C₈	1	1	yes	characteristic, complemented
58	C₂×#11924013	C₈	1	2	yes	complemented
59	D₈×D₇₃	C₈	1	2	yes	complemented
60	D₈×D₇₃	C₈	1	2	yes	complemented
61	D₈×D₇₃	C₂×C₄	1	2	yes	—
62	D₈×D₇₃	C₂×C₄	1	2	yes	—
63	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
64	C₂×D₈×C₇₃	C₈	1	1	yes	characteristic, complemented
65	C₂×#10596842	—	2	4	no	—
66	C₂×#10596842	—	2	2	no	—
67	C₂×#10596842	—	8	8	no	—
68	C₂×#10596842	C₂×D₄	1	4	yes	—
69	C₂²×#64408	—	2	2	no	—
70	C₂²×#64408	—	2	4	no	—
71	C₂²×#64408	—	2	4	no	—
72	C₂²×#64408	—	8	4	no	—
73	C₂²×#64408	C₂×D₄	1	1	yes	characteristic
74	C₂²×#64408	#41	1	1	yes	characteristic
75	#11924036	—	2	1	no	—
76	#11924036	C₂²×C₄	1	1	yes	characteristic
77	#11924036	C₂²×C₄	1	1	yes	characteristic
78	C₂×#10596833	—	2	1	no	—
79	C₄⋊#64408	—	2	2	no	—
80	C₄⋊#64408	C₂²×C₄	1	4	yes	—
81	C₂²⋊#64408	—	2	4	no	—
82	C₂²⋊#64408	—	2	4	no	—
83	C₄×#64408	C₂²×C₄	1	1	yes	characteristic
84	C₄×#64408	C₂×D₄	1	1	yes	characteristic
85	C₄×#64408	#41	1	1	yes	characteristic
86	C₄×#64408	C₂⁴	1	1	yes	characteristic
87	C₂³×D₇₃	—	2	2	no	—
88	C₂×D₂₉₂	C₂×C₈	1	2	yes	complemented
89	C₂×C₂²⋊D₇₃	—	2	2	no	—
90	D₄×D₇₃	—	2	2	no	—
91	D₄×D₇₃	—	2	2	no	—
92	D₄×D₇₃	C₂×C₈	1	4	yes	complemented
93	D₄×D₇₃	C₂²×C₄	1	4	yes	—
94	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
95	D₅₈₄	C₂×C₈	1	2	yes	complemented
96	D₅₈₄	C₂×C₈	1	2	yes	complemented
97	#11924013	C₂×C₈	1	4	yes	complemented
98	#11924013	C₂×C₈	1	4	yes	complemented
99	C₂×#10596831	C₂×C₈	1	1	yes	characteristic, complemented
100	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic, complemented
101	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic, complemented
102	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	complemented
105	D₈×C₇₃	C₂×C₈	1	2	yes	complemented
106	D₈×C₇₃	C₂×C₈	1	2	yes	complemented
107	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic, complemented
108	C₂³×C₇₃	—	2	2	no	—
109	C₂²×D₇₃	—	2	2	no	—
110	C₂²×D₇₃	—	2	4	no	—
111	C₂²×D₇₃	—	2	4	no	—
112	C₂²×D₇₃	—	8	2	no	—
113	C₂²×D₇₃	—	8	2	no	—
114	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
115	C₂×#64408	—	2	2	no	—
116	C₂×#64408	—	2	4	no	—
117	C₂×#64408	—	4	1	no	—
118	C₂×#64408	—	4	8	no	—
119	C₂×#64408	—	8	4	no	—
120	C₂×#64408	—	8	2	no	—
121	C₂×#64408	C₄×D₄	1	1	yes	characteristic
122	C₂×#64408	C₄×D₄	1	1	yes	characteristic
123	C₂×#64408	C₂²×D₄	1	1	yes	characteristic
124	C₂×#64408	C₂×#41	1	1	yes	characteristic
125	#10596842	—	2	4	no	—
126	#10596842	—	4	2	no	—
127	#10596842	—	4	4	no	—
128	D₄×C₇₃	—	2	2	no	—
129	D₄×C₇₃	C₂²×C₈	1	4	yes	—
130	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
131	C₂²⋊D₇₃	—	2	4	no	—
132	C₂²⋊D₇₃	—	2	4	no	—
133	C₂×Dic₇₃	#131	1	1	yes	characteristic
134	D₂₉₂	—	2	2	no	—
135	D₂₉₂	C₂²×C₈	1	4	yes	—
136	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
137	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
138	C₄×D₇₃	#131	1	1	yes	characteristic
139	C₄×D₇₃	C₂³×C₄	1	1	yes	characteristic
140	#10596833	—	2	1	no	—
141	#10596833	—	2	1	no	—
142	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
143	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
144	#10596831	C₂²×C₈	1	1	yes	characteristic
145	#10596831	C₂²×C₈	1	1	yes	characteristic
146	C₂²×C₇₃	—	2	4	no	—
147	C₂²×C₇₃	—	8	2	no	—
148	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
149	C₂×D₇₃	—	2	1	no	—
150	C₂×D₇₃	—	2	1	no	—
151	C₂×D₇₃	—	2	4	no	—
152	C₂×D₇₃	—	4	4	no	—
153	C₂×D₇₃	—	4	4	no	—
154	C₂×D₇₃	—	8	4	no	—
155	C₂×D₇₃	—	8	2	no	—
156	C₂×D₇₃	C₈×D₄	1	1	yes	characteristic
157	C₂×D₇₃	C₂×C₄×D₄	1	1	yes	characteristic
158	#64408	—	2	1	no	—
159	#64408	—	2	1	no	—
160	#64408	—	2	1	no	—
161	#64408	—	4	4	no	—
162	#64408	#576	1	2	yes	—
163	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
164	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
165	Dic₇₃	C₈×D₄	1	1	yes	characteristic
166	Dic₇₃	C₂×#131	1	1	yes	characteristic
167	#17854	—	73	1	no	maximal, Sylow
168	C₂×C₇₃	—	2	1	no	—
169	C₂×C₇₃	—	4	4	no	—
170	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
171	D₇₃	—	2	1	no	—
172	D₇₃	—	4	4	no	—
173	D₇₃	#2948	1	1	yes	characteristic
174	D₇₃	#2957	1	1	yes	characteristic
175	#2972	—	73	2	no	—
176	C₂×C₄×D₈	—	73	1	no	—
177	#1580	—	73	4	no	—
178	C₈×D₈	—	73	4	no	—
179	#1564	—	73	1	no	—
180	#1478	—	73	2	no	—
181	#1450	—	73	1	no	—
182	C₇₃	#17854	1	1	yes	characteristic, complemented, Sylow
183	C₂²×D₈	—	73	1	no	—
184	C₂×C₄×D₄	—	73	2	no	—
185	C₄×D₈	—	73	2	no	—
186	C₄×D₈	—	73	2	no	—
187	C₄×D₈	—	73	2	no	—
188	C₄×D₈	—	73	2	no	—
189	C₈×D₄	—	73	8	no	—
190	C₈×D₄	—	146	4	no	—
191	#431	—	73	1	no	—
192	#431	—	73	1	no	—
193	#431	—	73	2	no	—
194	C₂×#107	—	73	1	no	—
195	C₂×#102	—	73	2	no	—
196	C₂×C₂²⋊C₈	—	146	4	no	—
197	C₂×C₄×C₈	—	73	1	no	—
198	#335	—	73	2	no	—
199	#334	—	73	2	no	—
200	D₄⋊C₈	—	73	8	no	—
201	#322	—	73	2	no	—
202	C₈²	—	73	2	no	—
203	C₂²×D₄	—	73	2	no	—
204	C₂³×C₄	—	146	2	no	—
205	C₂×D₈	—	73	2	no	—
206	C₂×D₈	—	73	2	no	—
207	C₂×D₈	—	73	2	no	—
208	C₂×D₈	—	73	2	no	—
209	C₂×D₈	—	73	2	no	—
210	C₂×D₈	—	73	1	no	—
211	C₂×D₈	#10596842	1	1	yes	characteristic, complemented
212	C₂²×C₈	—	73	1	no	—
213	C₂²×C₈	—	73	1	no	—
214	C₂²×C₈	—	146	8	no	—
215	C₄×D₄	—	73	4	no	—
216	C₄×D₄	—	73	4	no	—
217	C₄×D₄	—	146	2	no	—
218	C₄×D₄	—	146	2	no	—
219	C₂×C₄⋊C₄	—	73	2	no	—
220	C₂×C₂²⋊C₄	—	146	2	no	—
221	C₂×C₄²	—	73	1	no	—
222	#107	—	73	2	no	—
223	#107	—	73	2	no	—
224	C₄⋊C₈	—	73	2	no	—
225	C₄⋊C₈	—	73	4	no	—
226	C₄⋊C₈	—	146	1	no	—
227	#102	—	73	4	no	—
228	#102	—	73	4	no	—
229	C₂²⋊C₈	—	73	2	no	—
230	C₂²⋊C₈	—	146	8	no	—
231	C₂²⋊C₈	—	146	2	no	—
232	C₂²⋊C₈	—	146	1	no	—
233	C₄×C₈	—	73	2	no	—
234	C₄×C₈	—	73	1	no	—
235	C₄×C₈	—	73	1	no	—
236	C₄×C₈	—	73	1	no	—
237	C₄×C₈	—	73	1	no	—
238	C₄×C₈	—	73	2	no	—
239	C₄×C₈	—	146	2	no	—
240	C₂⁴	—	146	2	no	—
241	C₂×D₄	—	73	4	no	—
242	C₂×D₄	—	73	4	no	—
243	C₂×D₄	—	73	2	no	—
244	C₂×D₄	—	146	2	no	—
245	C₂×D₄	—	146	2	no	—
246	C₂×D₄	—	146	2	no	—
247	C₂×D₄	C₂×#10596842	1	2	yes	complemented
248	C₂²×C₄	—	584	4	no	—
249	C₂²×C₄	—	73	1	no	—
250	C₂²×C₄	—	73	1	no	—
251	C₂²×C₄	—	73	1	no	—
252	C₂²×C₄	—	146	4	no	—
253	C₂²×C₄	—	146	4	no	—
254	C₂²×C₄	—	146	2	no	—
255	D₈	—	73	4	no	—
256	D₈	—	73	4	no	—
257	D₈	—	73	2	no	—
258	D₈	—	73	2	no	—
259	D₈	C₂×#10596842	1	2	yes	complemented
260	D₈	C₂×#10596842	1	2	yes	complemented
261	C₂×C₈	—	584	8	no	—
262	C₂×C₈	—	73	1	no	—
263	C₂×C₈	—	73	1	no	—
264	C₂×C₈	—	73	1	no	—
265	C₂×C₈	—	73	1	no	—
266	C₂×C₈	—	73	1	no	—
267	C₂×C₈	—	73	1	no	—
268	C₂×C₈	—	73	1	no	—
269	C₂×C₈	—	73	4	no	—
270	C₂×C₈	—	146	2	no	—
271	C₂×C₈	—	146	4	no	—
272	C₂×C₈	—	146	1	no	—
273	C₂×C₈	—	146	1	no	—
274	C₂×C₈	C₂×#10596842	1	1	yes	characteristic, complemented
275	C₄⋊C₄	—	73	4	no	—
276	C₄⋊C₄	—	146	2	no	—
277	C₂²⋊C₄	—	146	4	no	—
278	C₂²⋊C₄	—	146	4	no	—
279	C₄²	—	73	1	no	—
280	C₄²	—	73	1	no	—
281	C₄²	—	73	1	no	—
282	C₄²	—	73	1	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	584	2	no	—
285	C₂³	—	584	2	no	—
286	C₂³	—	73	1	no	—
287	C₂³	—	146	4	no	—
288	C₂³	—	146	4	no	—
289	C₂³	—	146	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	73	4	no	—
292	D₄	—	146	4	no	—
293	D₄	—	146	4	no	—
294	D₄	—	146	2	no	—
295	D₄	C₂²×#10596842	1	4	yes	—
296	C₂×C₄	—	292	8	no	—
297	C₂×C₄	—	292	1	no	—
298	C₂×C₄	—	584	2	no	—
299	C₂×C₄	—	584	4	no	—
300	C₂×C₄	—	73	1	no	—
301	C₂×C₄	—	73	1	no	—
302	C₂×C₄	—	73	1	no	—
303	C₂×C₄	—	73	1	no	—
304	C₂×C₄	—	73	1	no	—
305	C₂×C₄	—	73	1	no	—
306	C₂×C₄	—	73	1	no	—
307	C₂×C₄	—	73	1	no	—
308	C₂×C₄	—	73	1	no	—
309	C₂×C₄	—	146	4	no	—
310	C₂×C₄	—	146	2	no	—
311	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
312	C₈	—	292	4	no	—
313	C₈	—	292	2	no	—
314	C₈	—	73	1	no	—
315	C₈	—	73	1	no	—
316	C₈	—	146	4	no	—
317	C₈	—	146	1	no	—
318	C₈	—	146	1	no	—
319	C₈	C₂²×#10596842	1	1	yes	characteristic
320	C₈	C₂²×#10596842	1	1	yes	characteristic
321	C₂²	—	2	4	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	292	4	no	—
324	C₂²	—	292	4	no	—
325	C₂²	—	584	2	no	—
326	C₂²	—	584	4	no	—
327	C₂²	—	73	1	no	—
328	C₂²	—	73	1	no	—
329	C₂²	—	146	1	no	—
330	C₂²	—	146	1	no	—
331	C₂²	—	146	4	no	—
332	C₂²	D₄×#10596842	1	1	yes	characteristic
333	C₄	—	292	4	no	—
334	C₄	—	73	2	no	—
335	C₄	—	73	1	no	—
336	C₄	—	73	1	no	—
337	C₄	—	146	1	no	—
338	C₄	—	146	1	no	—
339	C₄	—	146	1	no	—
340	C₄	D₄×#10596842	1	1	yes	characteristic
341	C₄	C₂³×#10596842	1	1	yes	characteristic
342	C₂	—	2	1	no	—
343	C₂	—	4	4	no	—
344	C₂	—	292	4	no	—
345	C₂	—	73	1	no	—
346	C₂	—	73	1	no	—
347	C₂	—	146	1	no	—
348	C₂	C₂×D₄×#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#10596842
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₄	C₂³×#10596842
Fitting	C₂×D₈×C₇₃	C₈
socle	C₂×C₇₃	C₂×C₈×D₄
prime radical	C₂×D₈×D₇₃	C₄
cosocle	C₄×#64408	C₂⁴
hypercenter	C₂×D₈	#10596842
hypocenter	C₇₃	#17854
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #22627173	C₂	semidirect
2. #22627062	C₂	semidirect
3. D₈×#10596842	C₂	semidirect
4. #22627055	C₂	semidirect
5. #22627051	C₂	semidirect
6. #22627040	C₂	semidirect
9. D₄×#10596842	C₂²	semidirect
14. #17989677	C₂²	semidirect
15. #17989676	C₂²	semidirect
16. D₄⋊#10596842	C₂²	semidirect
17. #17989671	C₂²	semidirect
18. C₈×#10596842	C₂²	semidirect
57. C₂×D₅₈₄	C₈	semidirect
58. C₂×#11924013	C₈	semidirect
59. D₈×D₇₃	C₈	semidirect
60. D₈×D₇₃	C₈	semidirect
64. C₂×D₈×C₇₃	C₈	semidirect
88. C₂×D₂₉₂	C₂×C₈	semidirect
92. D₄×D₇₃	C₂×C₈	semidirect
95. D₅₈₄	C₂×C₈	semidirect
96. D₅₈₄	C₂×C₈	semidirect
97. #11924013	C₂×C₈	semidirect
98. #11924013	C₂×C₈	semidirect
99. C₂×#10596831	C₂×C₈	semidirect
100. C₈×D₇₃	C₂×C₈	semidirect
101. C₈×D₇₃	C₂×C₈	semidirect
104. C₂×D₄×C₇₃	C₂×C₈	semidirect
105. D₈×C₇₃	C₂×C₈	semidirect
106. D₈×C₇₃	C₂×C₈	semidirect
107. C₂×C₈×C₇₃	C₂×C₈	semidirect
182. C₇₃	#17854	semidirect
211. C₂×D₈	#10596842	semidirect
247. C₂×D₄	C₂×#10596842	semidirect
259. D₈	C₂×#10596842	semidirect
260. D₈	C₂×#10596842	semidirect
274. C₂×C₈	C₂×#10596842	semidirect

Maximal subgroups

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

subgroup	index
#22627173	2
#22627062	2
D₈×#10596842	2
#22627055	2
#22627051	2
#22627040	2
C₂×D₈×#64408	2
#17854	73