Subgroups — #36538275

Order 22784.

602 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030053	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
3	#26019707	C₂	1	4	yes	maximal, complemented
4	#26019653	C₂	1	2	yes	maximal, complemented
5	#26019619	C₂	1	2	yes	maximal, complemented
6	#26019011	C₂	1	1	yes	maximal, characteristic
7	#26018942	C₂	1	2	yes	maximal
8	C₈₉×#3170	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
10	#19151902	—	2	4	no	—
11	#19151890	C₂²	1	4	yes	—
12	#19151880	C₂²	1	4	yes	—
13	C₈×Dic₁₇₈	—	2	1	no	—
14	C₈×Dic₁₇₈	—	2	1	no	—
15	C₈×Dic₁₇₈	C₂²	1	2	yes	—
16	C₈×Dic₁₇₈	C₂²	1	2	yes	—
17	#19151641	—	2	2	no	—
18	#19151622	C₂²	1	2	yes	—
19	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
20	C₂×#15667941	—	2	1	no	—
21	#19151606	—	2	2	no	—
22	C₂×C₈×Dic₈₉	—	2	1	no	—
23	#19151351	C₂²	1	2	yes	—
24	#19151351	C₂²	1	2	yes	—
25	#19151347	C₂²	1	4	yes	—
26	#19151344	C₂²	1	4	yes	—
27	#533×C₈₉	C₂²	1	2	yes	—
28	#500×C₈₉	C₂²	1	1	yes	characteristic
29	#494×C₈₉	C₂²	1	1	yes	characteristic
30	#487×C₈₉	C₂²	1	2	yes	—
31	#467×C₈₉	—	2	1	no	—
32	#467×C₈₉	—	2	1	no	—
33	C₄×Q₁₆×C₈₉	—	2	1	no	—
34	C₄×Q₁₆×C₈₉	—	2	1	no	—
35	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₂²×Dic₁₇₈	D₄	1	1	yes	characteristic
37	C₂×#13083558	—	2	2	no	—
38	C₂²×#10623494	—	2	1	no	—
39	C₄×Dic₁₇₈	—	2	1	no	—
40	C₄×Dic₁₇₈	—	2	1	no	—
41	C₄×Dic₁₇₈	D₄	1	2	yes	—
42	C₄×Dic₁₇₈	C₂³	1	2	yes	—
43	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
44	C₂×#13083553	—	2	1	no	—
45	C₂×C₄×Dic₈₉	—	2	1	no	—
46	#15667945	—	2	4	no	—
47	#15667943	—	2	4	no	—
48	C₄⋊#10623494	—	2	1	no	—
49	C₄⋊#10623494	—	2	2	no	—
50	C₄⋊#10623494	C₂³	1	2	yes	—
51	C₄⋊#10623494	C₂³	1	4	yes	—
52	#15667941	—	2	2	no	—
53	#15667941	—	2	2	no	—
54	#15667935	—	2	4	no	—
55	#15667935	—	2	4	no	—
56	C₈×Dic₈₉	—	2	2	no	—
57	C₈×Dic₈₉	—	2	2	no	—
58	C₂×Q₁₆×C₈₉	—	2	1	no	—
59	C₂×Q₁₆×C₈₉	—	2	1	no	—
60	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
64	#125×C₈₉	—	2	2	no	—
65	C₂²⋊Q₈×C₈₉	—	2	2	no	—
66	C₂²⋊Q₈×C₈₉	—	2	2	no	—
67	C₄×Q₈×C₈₉	—	2	2	no	—
68	C₄×Q₈×C₈₉	—	2	2	no	—
69	#117×C₈₉	—	2	2	no	—
70	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
71	#107×C₈₉	—	2	1	no	—
72	#107×C₈₉	—	2	1	no	—
73	#103×C₈₉	—	2	2	no	—
74	#103×C₈₉	—	2	2	no	—
75	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
76	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	2	1	no	—
80	C₂×Dic₁₇₈	—	2	2	no	—
81	C₂×Dic₁₇₈	Q₁₆	1	4	yes	complemented
82	C₂×Dic₁₇₈	C₂×D₄	1	2	yes	—
83	C₂²×Dic₈₉	—	2	1	no	—
84	C₂²×Dic₈₉	—	2	1	no	—
85	#13083558	—	4	4	no	—
86	C₂×#10623494	—	2	2	no	—
87	C₂×#10623494	—	2	4	no	—
88	C₂×#10623494	—	4	2	no	—
89	C₄⋊Dic₈₉	—	2	1	no	—
90	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
91	#13083553	—	2	2	no	—
92	#13083553	—	2	2	no	—
93	C₄×Dic₈₉	—	2	2	no	—
94	C₄×Dic₈₉	—	2	2	no	—
95	C₂×Q₈×C₈₉	—	2	2	no	—
96	C₂×Q₈×C₈₉	—	2	2	no	—
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
100	Q₁₆×C₈₉	—	4	2	no	—
101	Q₁₆×C₈₉	—	4	2	no	—
102	C₂×C₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	—	4	1	no	—
104	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	—	2	2	no	—
109	C₄⋊C₄×C₈₉	—	2	2	no	—
110	C₄⋊C₄×C₈₉	—	2	2	no	—
111	C₄⋊C₄×C₈₉	—	2	2	no	—
112	C₄⋊C₄×C₈₉	—	4	2	no	—
113	C₄⋊C₄×C₈₉	—	4	2	no	—
114	C₂²⋊C₄×C₈₉	—	4	2	no	—
115	C₄²×C₈₉	—	4	2	no	—
116	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₈₉	#41	1	1	yes	characteristic
119	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
120	C₂³×C₈₉	#121	1	1	yes	characteristic
121	Q₈×C₈₉	—	4	2	no	—
122	Q₈×C₈₉	—	4	2	no	—
123	Q₈×C₈₉	—	8	2	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	4	1	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	4	2	no	—
129	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
130	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
133	C₂×C₄×C₈₉	#135	1	2	yes	—
134	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
137	C₂×Dic₈₉	—	2	2	no	—
138	C₂×Dic₈₉	—	2	4	no	—
139	C₂×Dic₈₉	—	2	2	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	Dic₁₇₈	—	2	2	no	—
142	Dic₁₇₈	—	2	4	no	—
143	Dic₁₇₈	C₂×Q₁₆	1	4	yes	—
144	C₈×C₈₉	—	2	1	no	—
145	C₈×C₈₉	—	2	1	no	—
146	C₈×C₈₉	—	2	1	no	—
147	C₈×C₈₉	#137	1	2	yes	—
148	#10623494	—	4	4	no	—
149	#10623494	—	4	2	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	—	2	1	no	—
152	C₂²×C₈₉	#467	1	2	yes	—
153	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	8	2	no	—
158	C₄×C₈₉	—	8	2	no	—
159	C₄×C₈₉	#467	1	2	yes	—
160	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
161	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
162	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
163	C₄×C₈₉	#575	1	1	yes	characteristic
164	Dic₈₉	—	2	4	no	—
165	Dic₈₉	—	4	2	no	—
166	#19262	—	89	1	no	maximal, Sylow
167	C₂×C₈₉	—	2	2	no	—
168	C₂×C₈₉	#3037	1	1	yes	characteristic
169	C₂×C₈₉	C₂×#467	1	1	yes	characteristic
170	C₂×C₈₉	#3067	1	1	yes	characteristic
171	#3170	D₈₉	1	1	yes	characteristic
172	#3078	—	89	2	no	—
173	C₂×C₈×Q₈	—	89	1	no	—
174	#1684	—	89	4	no	—
175	#1676	—	89	2	no	—
176	#1670	—	89	2	no	—
177	#1567	—	89	1	no	—
178	#1550	—	89	2	no	—
179	C₈₉	#19262	1	1	yes	characteristic, complemented, Sylow
180	#533	C₂×D₈₉	1	2	yes	—
181	C₂×C₄×Q₈	—	89	1	no	—
182	#500	C₂×D₈₉	1	1	yes	characteristic
183	#494	C₂×D₈₉	1	1	yes	characteristic
184	#487	C₂×D₈₉	1	2	yes	—
185	#479	—	178	4	no	—
186	#477	—	89	4	no	—
187	#475	—	89	4	no	—
188	#467	—	2	1	no	—
189	#467	—	2	1	no	—
190	C₈×Q₈	—	89	2	no	—
191	C₈×Q₈	—	89	2	no	—
192	C₈×Q₈	—	178	1	no	—
193	C₈×Q₈	—	178	1	no	—
194	C₄×Q₁₆	—	2	1	no	—
195	C₄×Q₁₆	—	2	1	no	—
196	#428	—	178	2	no	—
197	#423	—	89	2	no	—
198	C₂×C₄⋊C₈	—	89	1	no	—
199	C₂×C₄⋊C₈	—	178	1	no	—
200	#419	—	178	2	no	—
201	C₂×C₄×C₈	—	178	1	no	—
202	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
203	#335	—	89	2	no	—
204	#335	—	89	2	no	—
205	#333	—	89	4	no	—
206	#332	—	89	4	no	—
207	C₂²×Q₈	—	89	1	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	—	2	1	no	—
210	C₂×M₁₆	—	178	2	no	—
211	C₂²×C₈	—	178	1	no	—
212	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
213	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
214	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
215	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
216	#125	—	2	2	no	—
217	C₂²⋊Q₈	—	2	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	—	89	2	no	—
222	C₄×Q₈	—	89	2	no	—
223	C₄×Q₈	—	178	1	no	—
224	C₄×Q₈	—	178	1	no	—
225	#117	—	2	2	no	—
226	C₂×C₄⋊C₄	—	89	1	no	—
227	C₂×C₄⋊C₄	—	178	1	no	—
228	C₂×C₄²	—	178	1	no	—
229	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
230	#107	—	2	1	no	—
231	#107	—	2	1	no	—
232	#107	—	178	4	no	—
233	#106	—	178	4	no	—
234	C₄⋊C₈	—	89	2	no	—
235	C₄⋊C₈	—	89	4	no	—
236	C₄⋊C₈	—	178	2	no	—
237	C₄⋊C₈	—	178	2	no	—
238	C₄⋊C₈	—	178	2	no	—
239	C₄⋊C₈	—	178	1	no	—
240	#103	—	2	2	no	—
241	#103	—	2	2	no	—
242	#103	—	178	4	no	—
243	#103	—	178	4	no	—
244	C₄×C₈	—	178	2	no	—
245	C₄×C₈	—	178	2	no	—
246	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
247	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	89	4	no	—
253	C₂×Q₈	—	89	2	no	—
254	C₂×Q₈	—	178	1	no	—
255	C₂×Q₈	—	178	2	no	—
256	C₂²×C₄	—	178	1	no	—
257	C₂²×C₄	—	178	1	no	—
258	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
260	C₂²×C₄	#13083573	1	1	yes	characteristic
261	Q₁₆	—	4	2	no	—
262	Q₁₆	—	4	2	no	—
263	M₁₆	—	356	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	356	2	no	—
267	C₂×C₈	—	178	4	no	—
268	C₂×C₈	—	178	2	no	—
269	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
270	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
271	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	89	2	no	—
280	C₄⋊C₄	—	178	1	no	—
281	C₄⋊C₄	—	178	2	no	—
282	C₄⋊C₄	—	178	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₄²	—	4	2	no	—
285	C₄²	—	178	2	no	—
286	C₄²	—	178	2	no	—
287	C₄²	D₄×D₈₉	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
289	C₄²	#13083573	1	1	yes	characteristic
290	C₄²	C₂³×D₈₉	1	1	yes	characteristic
291	C₂³	#15667983	1	1	yes	characteristic
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	8	2	no	—
295	Q₈	—	89	4	no	—
296	Q₈	—	178	4	no	—
297	Q₈	—	178	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	356	2	no	—
304	C₂×C₄	—	178	2	no	—
305	C₂×C₄	—	178	2	no	—
306	C₂×C₄	—	178	4	no	—
307	C₂×C₄	#15667983	1	1	yes	characteristic
308	C₂×C₄	#15667983	1	1	yes	characteristic
309	C₂×C₄	#15667983	1	1	yes	characteristic
310	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
311	C₂×C₄	#15668035	1	2	yes	—
312	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
313	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
314	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	356	2	no	—
319	C₈	—	356	4	no	—
320	C₈	#15668042	1	2	yes	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	#19151836	1	2	yes	—
324	C₂²	C₂×#15667983	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	—	356	2	no	—
331	C₄	—	178	4	no	—
332	C₄	#19151836	1	2	yes	—
333	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
334	C₄	C₂×#15668035	1	1	yes	characteristic
335	C₄	C₂×#15668042	1	1	yes	characteristic
336	C₄	#19152396	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	#26029720	1	1	yes	characteristic
339	C₂	C₂×#19151836	1	1	yes	characteristic
340	C₂	#26029968	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667983
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#3170	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂³×C₈₉	#121
hypercenter	#3170	D₈₉
hypocenter	C₈₉	#19262
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #26019707	C₂	semidirect
4. #26019653	C₂	semidirect
5. #26019619	C₂	semidirect
81. C₂×Dic₁₇₈	Q₁₆	semidirect
179. C₈₉	#19262	semidirect

Maximal subgroups

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

subgroup	index
#26030053	2
C₂×C₈×Dic₁₇₈	2
#26019707	2
#26019653	2
#26019619	2
#26019011	2
#26018942	2
C₈₉×#3170	2
#19262	89