Subgroups — #36538302

Order 22784.

606 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030378	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
3	#26019679	C₂	1	4	yes	maximal, complemented
4	#26019643	C₂	1	2	yes	maximal, complemented
5	#26019619	C₂	1	2	yes	maximal, complemented
6	#26019015	C₂	1	1	yes	maximal, characteristic
7	#26018952	C₂	1	2	yes	maximal
8	C₈₉×#3170	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
10	#19151926	—	2	2	no	—
11	#19151880	C₂²	1	4	yes	—
12	#19151805	C₂²	1	4	yes	—
13	#19151753	—	2	2	no	—
14	C₈×Dic₁₇₈	—	2	1	no	—
15	C₈×Dic₁₇₈	—	2	1	no	—
16	C₈×Dic₁₇₈	C₂²	1	2	yes	—
17	C₈×Dic₁₇₈	C₂²	1	2	yes	—
18	#19151703	—	2	4	no	—
19	#19151655	C₂²	1	2	yes	—
20	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
21	C₂×#15667941	—	2	1	no	—
22	C₂×C₈×Dic₈₉	—	2	1	no	—
23	#19151351	C₂²	1	2	yes	—
24	#19151349	C₂²	1	2	yes	—
25	#19151344	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×#13083564	—	2	4	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	#15667943	—	2	4	no	—
47	C₄⋊#10623494	—	2	1	no	—
48	C₄⋊#10623494	C₂³	1	2	yes	—
49	C₄⋊#10623494	C₂³	1	4	yes	—
50	#15667941	—	2	2	no	—
51	#15667941	—	2	2	no	—
52	#15667936	—	2	4	no	—
53	#15667935	—	2	4	no	—
54	C₂²⋊#10623494	—	2	2	no	—
55	#15667923	—	2	2	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	4	no	—
81	C₂×Dic₁₇₈	C₂×D₄	1	2	yes	—
82	C₂²×Dic₈₉	—	2	1	no	—
83	C₂²×Dic₈₉	—	2	1	no	—
84	#13083564	—	4	8	no	—
85	C₂×#10623494	—	2	2	no	—
86	C₂×#10623494	—	2	4	no	—
87	C₂×#10623494	—	4	2	no	—
88	C₄⋊Dic₈₉	—	2	1	no	—
89	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
90	#13083553	—	2	2	no	—
91	#13083553	—	2	2	no	—
92	C₄×Dic₈₉	—	2	2	no	—
93	C₄×Dic₈₉	—	2	2	no	—
94	C₂×Q₈×C₈₉	—	2	2	no	—
95	C₂×Q₈×C₈₉	—	2	2	no	—
96	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
99	Q₁₆×C₈₉	—	4	2	no	—
100	Q₁₆×C₈₉	—	4	2	no	—
101	C₂×C₈×C₈₉	—	2	2	no	—
102	C₂×C₈×C₈₉	—	4	1	no	—
103	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₈₉	—	2	2	no	—
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	—	4	2	no	—
112	C₄⋊C₄×C₈₉	—	4	2	no	—
113	C₂²⋊C₄×C₈₉	—	4	2	no	—
114	C₄²×C₈₉	—	4	2	no	—
115	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₈₉	#41	1	1	yes	characteristic
118	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
119	C₂³×C₈₉	#121	1	1	yes	characteristic
120	Q₈×C₈₉	—	4	4	no	—
121	Q₈×C₈₉	—	4	2	no	—
122	Q₈×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	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	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₈₉	—	2	4	no	—
136	C₂×Dic₈₉	—	2	2	no	—
137	C₂×Dic₈₉	—	2	2	no	—
138	C₂×Dic₈₉	—	4	2	no	—
139	Dic₁₇₈	—	2	4	no	—
140	Dic₁₇₈	—	2	2	no	—
141	Dic₁₇₈	#135	1	4	yes	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	—	2	1	no	—
144	C₈×C₈₉	—	2	1	no	—
145	C₈×C₈₉	#137	1	2	yes	—
146	#10623494	—	4	4	no	—
147	#10623494	—	4	2	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	#473	1	2	yes	—
151	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	8	2	no	—
157	C₄×C₈₉	—	8	2	no	—
158	C₄×C₈₉	C₂×#135	1	2	yes	—
159	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
160	C₄×C₈₉	#575	1	1	yes	characteristic
161	Dic₈₉	—	2	4	no	—
162	Dic₈₉	—	4	2	no	—
163	#19265	—	89	1	no	maximal, Sylow
164	C₂×C₈₉	—	2	2	no	—
165	C₂×C₈₉	#3037	1	1	yes	characteristic
166	C₂×C₈₉	#3050	1	1	yes	characteristic
167	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
168	#3170	D₈₉	1	1	yes	characteristic
169	#3113	—	89	2	no	—
170	C₂×C₈×Q₈	—	89	1	no	—
171	#1680	—	89	4	no	—
172	#1674	—	89	2	no	—
173	#1670	—	89	2	no	—
174	#1568	—	89	1	no	—
175	#1552	—	89	2	no	—
176	C₈₉	#19265	1	1	yes	characteristic, complemented, Sylow
177	#533	C₂×D₈₉	1	2	yes	—
178	C₂×C₄×Q₈	—	89	1	no	—
179	#500	C₂×D₈₉	1	1	yes	characteristic
180	#494	C₂×D₈₉	1	1	yes	characteristic
181	#487	C₂×D₈₉	1	2	yes	—
182	#483	—	178	2	no	—
183	#475	—	89	4	no	—
184	#467	—	2	1	no	—
185	#467	—	2	1	no	—
186	#462	—	89	4	no	—
187	#451	—	178	2	no	—
188	C₈×Q₈	—	89	2	no	—
189	C₈×Q₈	—	89	2	no	—
190	C₈×Q₈	—	178	1	no	—
191	C₈×Q₈	—	178	1	no	—
192	#441	—	178	4	no	—
193	C₄×Q₁₆	—	2	1	no	—
194	C₄×Q₁₆	—	2	1	no	—
195	#432	—	89	2	no	—
196	C₂×C₄⋊C₈	—	89	1	no	—
197	C₂×C₄⋊C₈	—	178	1	no	—
198	C₂×C₄×C₈	—	178	1	no	—
199	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
200	#335	—	89	2	no	—
201	#334	—	89	2	no	—
202	#332	—	89	4	no	—
203	#332	—	89	4	no	—
204	C₂²×Q₈	—	89	1	no	—
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	—	178	4	no	—
208	C₂²×C₈	—	178	1	no	—
209	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
210	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
211	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
212	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
213	#125	—	2	2	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	89	2	no	—
219	C₄×Q₈	—	89	2	no	—
220	C₄×Q₈	—	178	1	no	—
221	C₄×Q₈	—	178	1	no	—
222	#117	—	2	2	no	—
223	C₂×C₄⋊C₄	—	89	1	no	—
224	C₂×C₄⋊C₄	—	178	1	no	—
225	C₂×C₄²	—	178	1	no	—
226	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
227	#107	—	2	1	no	—
228	#107	—	2	1	no	—
229	#106	—	178	4	no	—
230	C₄⋊C₈	—	89	2	no	—
231	C₄⋊C₈	—	89	4	no	—
232	C₄⋊C₈	—	178	2	no	—
233	C₄⋊C₈	—	178	2	no	—
234	C₄⋊C₈	—	178	1	no	—
235	#103	—	2	2	no	—
236	#103	—	2	2	no	—
237	#103	—	178	4	no	—
238	#103	—	178	4	no	—
239	C₂²⋊C₈	—	178	2	no	—
240	#97	—	178	2	no	—
241	C₄×C₈	—	178	2	no	—
242	C₄×C₈	—	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	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	89	2	no	—
250	C₂×Q₈	—	178	4	no	—
251	C₂×Q₈	—	178	1	no	—
252	C₂²×C₄	—	178	1	no	—
253	C₂²×C₄	—	178	1	no	—
254	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
256	C₂²×C₄	#13083573	1	1	yes	characteristic
257	Q₁₆	—	4	2	no	—
258	Q₁₆	—	4	2	no	—
259	Q₁₆	—	356	8	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	356	2	no	—
263	C₂×C₈	—	178	4	no	—
264	C₂×C₈	—	178	2	no	—
265	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
266	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
267	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
268	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	89	2	no	—
276	C₄⋊C₄	—	178	1	no	—
277	C₄⋊C₄	—	178	2	no	—
278	C₄⋊C₄	—	178	2	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₄²	—	4	2	no	—
281	C₄²	—	178	2	no	—
282	C₄²	—	178	2	no	—
283	C₄²	D₄×D₈₉	1	1	yes	characteristic
284	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
285	C₄²	#13083573	1	1	yes	characteristic
286	C₄²	C₂³×D₈₉	1	1	yes	characteristic
287	C₂³	#15667983	1	1	yes	characteristic
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	4	no	—
291	Q₈	—	89	4	no	—
292	Q₈	—	178	4	no	—
293	Q₈	—	178	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	356	2	no	—
301	C₂×C₄	—	178	2	no	—
302	C₂×C₄	—	178	4	no	—
303	C₂×C₄	—	178	2	no	—
304	C₂×C₄	#15667983	1	1	yes	characteristic
305	C₂×C₄	#15667983	1	1	yes	characteristic
306	C₂×C₄	#15667983	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
309	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	356	2	no	—
314	C₈	—	356	4	no	—
315	C₈	#15668042	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	#19151870	1	2	yes	—
319	C₂²	C₂×#15667983	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	356	2	no	—
327	C₄	—	178	4	no	—
328	C₄	#135×D₈₉	1	2	yes	—
329	C₄	C₂×#15668042	1	1	yes	characteristic
330	C₄	#19152396	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	#26029720	1	1	yes	characteristic
333	C₂	#26029829	1	1	yes	characteristic
334	C₂	C₂×#19151870	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	#19265
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. #26019679	C₂	semidirect
4. #26019643	C₂	semidirect
5. #26019619	C₂	semidirect
176. C₈₉	#19265	semidirect

Maximal subgroups

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

subgroup	index
#26030378	2
C₂×C₈×Dic₁₇₈	2
#26019679	2
#26019643	2
#26019619	2
#26019015	2
#26018952	2
C₈₉×#3170	2
#19265	89