Subgroups — #36709348

Order 22784.

654 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022430	C₂	1	2	yes	maximal, complemented
3	#26022386	C₂	1	2	yes	maximal, complemented
4	#26022284	C₂	1	2	yes	maximal, complemented
5	#26022163	C₂	1	2	yes	maximal, complemented
6	C₄×#15667953	C₂	1	4	yes	maximal, complemented
7	#26022051	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#3169	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₃₅₆	—	2	1	no	—
11	C₂×#15667953	C₂²	1	4	yes	—
12	C₄×D₇₁₂	—	2	1	no	—
13	C₄×D₇₁₂	—	2	1	no	—
14	C₄×D₇₁₂	C₂²	1	2	yes	—
15	C₄×D₇₁₂	C₂²	1	2	yes	—
16	C₂×#15667946	C₂²	1	1	yes	characteristic
17	C₂×#15667934	—	2	1	no	—
18	#19151423	C₂²	1	4	yes	complemented
19	C₄⋊#13083549	C₂²	1	2	yes	complemented
20	#19151399	C₂²	1	4	yes	complemented
21	#19151398	C₂²	1	4	yes	—
22	C₂²⋊#13083549	C₂²	1	2	yes	—
23	C₄×#13083549	C₂²	1	2	yes	complemented
24	#532×C₈₉	—	2	1	no	—
25	#500×C₈₉	C₂²	1	1	yes	characteristic, complemented
26	#493×C₈₉	C₂²	1	1	yes	characteristic, complemented
27	#486×C₈₉	—	2	1	no	—
28	#466×C₈₉	—	2	1	no	—
29	#466×C₈₉	C₂²	1	2	yes	—
30	C₄×D₈×C₈₉	—	2	1	no	—
31	C₄×D₈×C₈₉	C₂²	1	2	yes	complemented
32	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	C₂²×D₃₅₆	—	2	1	no	—
34	C₂²×C₄×D₈₉	—	4	1	no	—
35	C₂×D₇₁₂	—	2	4	no	—
36	C₂×D₇₁₂	—	2	1	no	—
37	C₂×D₇₁₂	C₂³	1	2	yes	—
38	C₄×D₃₅₆	—	2	2	no	—
39	C₄×D₃₅₆	—	2	2	no	—
40	C₄×D₃₅₆	—	4	1	no	—
41	C₄×D₃₅₆	—	4	1	no	—
42	C₂×C₄⋊Dic₈₉	—	2	1	no	—
43	C₂×#13083551	—	4	1	no	—
44	#15667953	—	2	8	no	—
45	C₂×#13083549	C₂³	1	4	yes	—
46	#15667946	—	2	1	no	—
47	#15667946	C₂³	1	2	yes	—
48	#15667934	—	2	2	no	—
49	#15667934	—	2	2	no	—
50	C₂×#41×C₈₉	—	4	1	no	—
51	C₂×D₈×C₈₉	—	2	1	no	—
52	C₂×D₈×C₈₉	C₂³	1	2	yes	—
53	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₈₉	—	2	1	no	—
56	#127×C₈₉	—	2	1	no	—
57	#124×C₈₉	—	4	1	no	—
58	#121×C₈₉	—	2	2	no	—
59	#121×C₈₉	—	4	1	no	—
60	C₄×D₄×C₈₉	—	2	2	no	—
61	C₄×D₄×C₈₉	—	4	1	no	—
62	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
63	#107×C₈₉	—	2	1	no	—
64	#107×C₈₉	C₂³	1	2	yes	—
65	#102×C₈₉	—	2	2	no	—
66	#102×C₈₉	—	2	2	no	—
67	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
68	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
69	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂³×D₈₉	—	4	1	no	—
72	C₂×D₃₅₆	—	2	4	no	—
73	C₂×D₃₅₆	—	2	2	no	—
74	C₂×D₃₅₆	—	4	2	no	—
75	C₂×D₃₅₆	—	4	1	no	—
76	C₂²×Dic₈₉	—	4	1	no	—
77	C₂×C₄×D₈₉	—	4	2	no	—
78	C₂×C₄×D₈₉	—	4	2	no	—
79	C₂×C₄×D₈₉	—	8	4	no	—
80	D₇₁₂	—	2	4	no	—
81	D₇₁₂	—	2	2	no	—
82	D₇₁₂	#41	1	4	yes	complemented
83	C₄⋊Dic₈₉	—	2	2	no	—
84	C₄⋊Dic₈₉	—	4	1	no	—
85	#13083551	—	4	2	no	—
86	#13083551	—	4	2	no	—
87	#13083549	—	2	4	no	—
88	#41×C₈₉	—	8	4	no	—
89	C₂×Q₈×C₈₉	—	4	1	no	—
90	C₂×D₄×C₈₉	—	2	2	no	—
91	C₂×D₄×C₈₉	—	4	2	no	—
92	C₂×D₄×C₈₉	—	4	1	no	—
93	C₂×D₄×C₈₉	—	4	1	no	—
94	C₂²×C₄×C₈₉	—	4	2	no	—
95	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	D₈×C₈₉	—	2	4	no	—
99	D₈×C₈₉	—	4	2	no	—
100	C₂×C₈×C₈₉	—	2	2	no	—
101	C₂×C₈×C₈₉	—	4	1	no	—
102	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₈₉	—	2	2	no	—
107	C₄⋊C₄×C₈₉	—	4	1	no	—
108	C₂²⋊C₄×C₈₉	—	4	2	no	—
109	C₂²⋊C₄×C₈₉	—	4	2	no	—
110	C₄²×C₈₉	D₈	1	1	yes	characteristic
111	C₄²×C₈₉	D₈	1	1	yes	characteristic
112	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂³×C₈₉	—	4	2	no	—
115	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
116	C₂²×D₈₉	—	4	4	no	—
117	C₂²×D₈₉	—	4	2	no	—
118	C₂²×D₈₉	—	8	2	no	—
119	C₂²×D₈₉	—	8	2	no	—
120	Q₈×C₈₉	—	8	2	no	—
121	D₄×C₈₉	—	4	2	no	—
122	D₄×C₈₉	—	4	2	no	—
123	D₄×C₈₉	—	8	2	no	—
124	D₄×C₈₉	—	8	4	no	—
125	D₄×C₈₉	—	8	2	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	8	4	no	—
131	C₂×C₄×C₈₉	D₁₆	1	4	yes	complemented
132	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₈₉	—	4	2	no	—
139	C₂×Dic₈₉	—	8	2	no	—
140	D₃₅₆	—	2	4	no	—
141	D₃₅₆	—	4	4	no	—
142	D₃₅₆	—	4	2	no	—
143	C₄×D₈₉	—	8	4	no	—
144	C₄×D₈₉	—	8	4	no	—
145	C₈×C₈₉	—	2	1	no	—
146	C₈×C₈₉	—	2	1	no	—
147	C₈×C₈₉	—	2	1	no	—
148	C₈×C₈₉	C₂×#41	1	2	yes	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	—	8	2	no	—
152	C₂²×C₈₉	—	8	2	no	—
153	C₂²×C₈₉	—	8	2	no	—
154	C₂²×C₈₉	C₂×D₁₆	1	2	yes	—
155	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₈₉	—	4	4	no	—
157	C₂×D₈₉	—	8	8	no	—
158	C₂×D₈₉	—	8	2	no	—
159	C₂×D₈₉	—	8	2	no	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	8	2	no	—
164	C₄×C₈₉	C₂×D₁₆	1	2	yes	—
165	C₄×C₈₉	C₂×D₁₆	1	2	yes	—
166	C₄×C₈₉	#525	1	1	yes	characteristic
167	C₄×C₈₉	#575	1	1	yes	characteristic
168	Dic₈₉	—	8	2	no	—
169	#33660	—	89	1	no	maximal, Sylow
170	C₂×C₈₉	—	2	2	no	—
171	C₂×C₈₉	—	16	2	no	—
172	C₂×C₈₉	#3100	1	1	yes	characteristic
173	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
174	C₂×C₈₉	#3414	1	1	yes	characteristic
175	D₈₉	—	8	4	no	—
176	#3169	D₈₉	1	1	yes	characteristic, complemented
177	C₂×C₄×D₈	—	89	1	no	—
178	#2232	—	89	2	no	—
179	#2224	—	89	2	no	—
180	#2206	—	89	2	no	—
181	#2184	—	89	2	no	—
182	C₄×D₁₆	—	89	4	no	—
183	#2162	—	89	1	no	—
184	C₈₉	#33660	1	1	yes	characteristic, complemented, Sylow
185	C₂²×D₈	—	89	1	no	—
186	#532	—	2	1	no	—
187	C₂×C₄×D₄	—	178	1	no	—
188	C₂×D₁₆	—	89	4	no	—
189	#500	C₂×D₈₉	1	1	yes	characteristic, complemented
190	#493	C₂×D₈₉	1	1	yes	characteristic, complemented
191	#486	—	2	1	no	—
192	#466	—	2	1	no	—
193	#466	C₂×D₈₉	1	2	yes	—
194	C₄×D₈	—	2	1	no	—
195	C₄×D₈	—	89	2	no	—
196	C₄×D₈	—	89	2	no	—
197	C₄×D₈	—	178	1	no	—
198	C₄×D₈	—	178	1	no	—
199	C₄×D₈	C₂×D₈₉	1	2	yes	complemented
200	C₂×#107	—	89	1	no	—
201	C₂×#102	—	178	1	no	—
202	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
203	#366	—	89	4	no	—
204	C₄⋊C₁₆	—	89	2	no	—
205	#357	—	89	4	no	—
206	#357	—	89	4	no	—
207	C₂²⋊C₁₆	—	89	2	no	—
208	C₄×C₁₆	—	89	2	no	—
209	C₂×#41	—	4	1	no	—
210	C₂²×D₄	—	178	1	no	—
211	C₂³×C₄	—	356	1	no	—
212	C₂×D₈	—	2	1	no	—
213	C₂×D₈	—	89	2	no	—
214	C₂×D₈	—	178	4	no	—
215	C₂×D₈	—	178	1	no	—
216	C₂×D₈	C₂²×D₈₉	1	2	yes	—
217	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
219	C₄⋊Q₈	—	2	1	no	—
220	#127	—	2	1	no	—
221	#124	—	4	1	no	—
222	#121	—	2	2	no	—
223	#121	—	4	1	no	—
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	—	4	1	no	—
226	C₄×D₄	—	356	1	no	—
227	C₄×D₄	—	356	1	no	—
228	C₄×D₄	—	178	2	no	—
229	C₄×D₄	—	178	2	no	—
230	C₂×C₄⋊C₄	—	178	1	no	—
231	C₂×C₂²⋊C₄	—	356	1	no	—
232	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
233	D₁₆	—	178	8	no	—
234	C₂×C₁₆	—	89	4	no	—
235	#107	—	2	1	no	—
236	#107	—	89	2	no	—
237	#107	—	178	1	no	—
238	#107	C₂²×D₈₉	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	2	2	no	—
241	#102	—	178	2	no	—
242	#102	—	178	2	no	—
243	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
244	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
247	C₂⁴	—	356	1	no	—
248	#41	—	8	4	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	356	2	no	—
255	C₂×D₄	—	356	1	no	—
256	C₂×D₄	—	178	4	no	—
257	C₂×D₄	—	178	2	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	356	1	no	—
260	C₂²×C₄	—	356	2	no	—
261	C₂²×C₄	—	356	2	no	—
262	C₂²×C₄	—	712	4	no	—
263	C₂²×C₄	#13083559	1	1	yes	characteristic
264	C₂²×C₄	#13083559	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
266	D₈	—	2	4	no	—
267	D₈	—	4	2	no	—
268	D₈	—	89	4	no	—
269	D₈	—	178	4	no	—
270	D₈	—	178	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
274	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
275	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
276	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	356	1	no	—
280	C₄⋊C₄	—	178	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	356	2	no	—
284	C₂²⋊C₄	—	356	2	no	—
285	C₄²	#13083559	1	1	yes	characteristic
286	C₄²	#13083559	1	1	yes	characteristic
287	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
289	C₁₆	—	178	4	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	356	4	no	—
292	C₂³	—	356	2	no	—
293	C₂³	—	712	2	no	—
294	C₂³	—	712	2	no	—
295	C₂³	C₂×#13083559	1	1	yes	characteristic
296	Q₈	—	8	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	8	2	no	—
300	D₄	—	8	4	no	—
301	D₄	—	8	2	no	—
302	D₄	—	356	4	no	—
303	D₄	—	356	2	no	—
304	D₄	—	178	4	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	4	no	—
310	C₂×C₄	—	356	2	no	—
311	C₂×C₄	—	712	2	no	—
312	C₂×C₄	—	712	4	no	—
313	C₂×C₄	—	712	4	no	—
314	C₂×C₄	#15667953	1	4	yes	complemented
315	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
316	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
317	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
318	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
319	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	#41×D₈₉	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	356	4	no	—
331	C₂²	—	712	8	no	—
332	C₂²	—	712	2	no	—
333	C₂²	—	712	2	no	—
334	C₂²	C₂×#15667953	1	2	yes	—
335	C₂²	C₂²×#13083559	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	8	2	no	—
340	C₄	—	712	2	no	—
341	C₄	C₂×#15667953	1	2	yes	—
342	C₄	C₂×#15667953	1	2	yes	—
343	C₄	#19152105	1	1	yes	characteristic
344	C₄	#19152396	1	1	yes	characteristic
345	C₂	—	2	2	no	—
346	C₂	—	16	2	no	—
347	C₂	—	712	4	no	—
348	C₂	#26030248	1	1	yes	characteristic
349	C₂	C₂²×#15667953	1	1	yes	characteristic
350	C₂	#26033386	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3169	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#3169	D₈₉
hypocenter	C₈₉	#33660
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×D₇₁₂	C₂	semidirect
2. #26022430	C₂	semidirect
3. #26022386	C₂	semidirect
4. #26022284	C₂	semidirect
5. #26022163	C₂	semidirect
6. C₄×#15667953	C₂	semidirect
7. #26022051	C₂	semidirect
8. C₈₉×#3169	C₂	semidirect
18. #19151423	C₂²	semidirect
19. C₄⋊#13083549	C₂²	semidirect
20. #19151399	C₂²	semidirect
23. C₄×#13083549	C₂²	semidirect
25. #500×C₈₉	C₂²	semidirect
26. #493×C₈₉	C₂²	semidirect
31. C₄×D₈×C₈₉	C₂²	semidirect
82. D₇₁₂	#41	semidirect
131. C₂×C₄×C₈₉	D₁₆	semidirect
176. #3169	D₈₉	semidirect
184. C₈₉	#33660	semidirect
189. #500	C₂×D₈₉	semidirect
190. #493	C₂×D₈₉	semidirect
199. C₄×D₈	C₂×D₈₉	semidirect
314. C₂×C₄	#15667953	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₇₁₂	2
#26022430	2
#26022386	2
#26022284	2
#26022163	2
C₄×#15667953	2
#26022051	2
C₈₉×#3169	2
#33660	89