Subgroups — #36701127

Order 22784.

616 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26025637	C₂	1	2	yes	maximal
2	#26021753	C₂	1	2	yes	maximal
3	#26021593	C₂	1	2	yes	maximal
4	#26021588	C₂	1	2	yes	maximal
5	#26021188	C₂	1	1	yes	maximal, characteristic
6	#26020985	C₂	1	1	yes	maximal, characteristic
7	#26020843	C₂	1	2	yes	maximal
8	#26020825	C₂	1	2	yes	maximal
9	C₈₉×#1942	C₂	1	1	yes	maximal, characteristic
10	C₂×#15668003	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
12	C₂×#15667946	C₂²	1	1	yes	characteristic
13	C₂×#15667946	C₂²	1	1	yes	characteristic
14	C₂×#15667943	C₂²	1	2	yes	—
15	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
16	#19151532	—	2	2	no	—
17	#19151518	—	2	2	no	—
18	#19151497	—	2	2	no	—
19	#19151497	C₂²	1	2	yes	—
20	#19151496	C₂²	1	2	yes	—
21	#19151495	—	2	2	no	—
22	#19151494	C₂²	1	2	yes	—
23	C₄⋊C₄×Dic₈₉	C₂²	1	2	yes	—
24	#19151361	C₂²	1	2	yes	—
25	#19151360	C₂²	1	2	yes	—
26	#19151360	C₂²	1	2	yes	—
27	#19151360	C₂²	1	2	yes	—
28	#19151352	C₂²	1	2	yes	—
29	#19151352	C₂²	1	2	yes	—
30	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	#389×C₈₉	C₂²	1	2	yes	—
34	#340×C₈₉	C₂²	1	2	yes	—
35	C₂²×Dic₁₇₈	D₄	1	1	yes	characteristic
36	C₂²×#10623494	C₂³	1	2	yes	—
37	#15668003	—	4	2	no	—
38	#15668003	—	4	2	no	—
39	C₄×Dic₁₇₈	—	4	1	no	—
40	C₄×Dic₁₇₈	—	4	1	no	—
41	C₂×C₄⋊Dic₈₉	—	2	2	no	—
42	C₂×C₄⋊Dic₈₉	—	2	2	no	—
43	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
45	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
46	C₂×#13083553	—	2	2	no	—
47	C₂×#13083553	—	2	2	no	—
48	C₂×#13083553	—	2	2	no	—
49	C₂×#13083553	D₄	1	2	yes	—
50	C₂×C₄×Dic₈₉	—	2	2	no	—
51	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
52	#15667946	—	4	1	no	—
53	#15667946	Q₈	1	4	yes	complemented
54	#15667943	—	4	2	no	—
55	C₄⋊#10623494	—	4	1	no	—
56	#15667919	—	2	2	no	—
57	#15667919	—	2	2	no	—
58	#15667919	—	2	2	no	—
59	#15667919	—	2	2	no	—
60	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
63	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
65	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
66	#107×C₈₉	—	4	1	no	—
67	#107×C₈₉	Q₈	1	4	yes	complemented
68	C₄×C₈×C₈₉	—	4	1	no	—
69	#95×C₈₉	—	2	2	no	—
70	C₂×Dic₁₇₈	—	2	2	no	—
71	C₂×Dic₁₇₈	—	4	1	no	—
72	C₂×Dic₁₇₈	—	4	1	no	—
73	C₂²×Dic₈₉	—	2	2	no	—
74	C₂²×Dic₈₉	—	2	2	no	—
75	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
76	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
77	C₂×#10623494	—	2	2	no	—
78	C₂×#10623494	—	2	2	no	—
79	C₂×#10623494	—	2	2	no	—
80	C₄⋊Dic₈₉	—	2	2	no	—
81	C₄⋊Dic₈₉	—	2	2	no	—
82	C₄⋊Dic₈₉	—	2	2	no	—
83	C₄⋊Dic₈₉	—	4	1	no	—
84	C₄⋊Dic₈₉	—	4	4	no	—
85	C₄⋊Dic₈₉	—	8	2	no	—
86	C₄⋊Dic₈₉	C₂×Q₈	1	4	yes	—
87	#13083553	—	4	2	no	—
88	#13083553	—	4	4	no	—
89	#13083553	—	4	4	no	—
90	#13083553	—	8	2	no	—
91	C₄×Dic₈₉	—	2	2	no	—
92	C₄×Dic₈₉	—	2	2	no	—
93	C₄×Dic₈₉	—	8	2	no	—
94	C₂²×C₄×C₈₉	—	2	2	no	—
95	C₂²×C₄×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	C₂⁴	1	1	yes	characteristic
99	C₂×C₈×C₈₉	—	2	1	no	—
100	C₂×C₈×C₈₉	—	2	1	no	—
101	C₂×C₈×C₈₉	—	2	1	no	—
102	C₂×C₈×C₈₉	—	2	1	no	—
103	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
104	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
105	C₄⋊C₄×C₈₉	—	2	2	no	—
106	C₄⋊C₄×C₈₉	—	2	2	no	—
107	C₄⋊C₄×C₈₉	—	2	2	no	—
108	C₄⋊C₄×C₈₉	—	4	4	no	—
109	C₄⋊C₄×C₈₉	C₂×Q₈	1	4	yes	—
110	C₄²×C₈₉	—	2	1	no	—
111	C₄²×C₈₉	—	2	1	no	—
112	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	4	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	#135	1	2	yes	—
125	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
130	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
131	C₂×Dic₈₉	—	2	2	no	—
132	C₂×Dic₈₉	—	2	2	no	—
133	C₂×Dic₈₉	—	2	4	no	—
134	C₂×Dic₈₉	—	2	2	no	—
135	C₂×Dic₈₉	—	2	2	no	—
136	C₂×Dic₈₉	—	2	2	no	—
137	C₂×Dic₈₉	—	2	2	no	—
138	C₂×Dic₈₉	—	4	2	no	—
139	C₂×Dic₈₉	—	4	2	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	C₂×Dic₈₉	—	4	2	no	—
142	C₂×Dic₈₉	—	4	2	no	—
143	Dic₁₇₈	—	4	4	no	—
144	C₈×C₈₉	—	2	2	no	—
145	C₈×C₈₉	—	2	2	no	—
146	#10623494	—	4	2	no	—
147	C₂²×C₈₉	#534	1	1	yes	characteristic
148	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
149	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
150	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
151	C₂²×C₈₉	#575	1	1	yes	characteristic
152	C₂²×C₈₉	#576	1	1	yes	characteristic
153	C₂²×C₈₉	#578	1	1	yes	characteristic
154	C₄×C₈₉	—	2	2	no	—
155	C₄×C₈₉	—	4	4	no	—
156	C₄×C₈₉	—	4	1	no	—
157	C₄×C₈₉	—	8	2	no	—
158	C₄×C₈₉	#544	1	1	yes	characteristic
159	C₄×C₈₉	#544	1	1	yes	characteristic
160	C₄×C₈₉	Q₈²	1	1	yes	characteristic
161	C₄×C₈₉	#563	1	1	yes	characteristic
162	Dic₈₉	—	4	4	no	—
163	Dic₈₉	—	4	2	no	—
164	Dic₈₉	—	4	2	no	—
165	Dic₈₉	—	8	2	no	—
166	#32686	—	89	1	no	maximal, Sylow
167	C₂×C₈₉	#2734	1	1	yes	characteristic
168	C₂×C₈₉	#3191	1	1	yes	characteristic
169	C₂×C₈₉	#3203	1	1	yes	characteristic
170	C₂×C₈₉	#3380	1	1	yes	characteristic
171	C₂×C₈₉	#3383	1	1	yes	characteristic
172	C₂×C₈₉	#3389	1	1	yes	characteristic
173	C₂×C₈₉	#3390	1	1	yes	characteristic
174	#2584	—	89	2	no	—
175	#2088	—	89	2	no	—
176	#2059	—	89	2	no	—
177	#2059	—	89	2	no	—
178	#1984	—	89	1	no	—
179	#1942	—	89	1	no	—
180	#1942	D₈₉	1	1	yes	characteristic
181	#1921	—	89	2	no	—
182	#1918	—	89	2	no	—
183	C₈₉	#32686	1	1	yes	characteristic, complemented, Sylow
184	C₂×#125	—	89	2	no	—
185	C₂×C₄×Q₈	—	89	1	no	—
186	C₂×#107	—	89	1	no	—
187	C₂×#107	—	89	1	no	—
188	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
189	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
190	C₂×#106	—	89	2	no	—
191	C₂×C₄⋊C₈	—	89	1	no	—
192	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
193	#398	—	178	2	no	—
194	#395	—	178	2	no	—
195	#389	—	89	2	no	—
196	#389	—	89	2	no	—
197	#389	—	89	2	no	—
198	#389	—	178	2	no	—
199	#389	—	178	2	no	—
200	#389	C₂×D₈₉	1	2	yes	—
201	C₄×C₄⋊C₄	—	89	2	no	—
202	#340	—	89	2	no	—
203	#340	—	89	2	no	—
204	#340	—	89	2	no	—
205	#340	—	89	2	no	—
206	#340	C₂×D₈₉	1	2	yes	—
207	#336	—	89	2	no	—
208	#336	—	89	2	no	—
209	C₂²×Q₈	—	89	1	no	—
210	C₂²×C₈	—	89	2	no	—
211	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
213	#125	—	356	2	no	—
214	#125	—	356	2	no	—
215	C₄×Q₈	—	356	1	no	—
216	C₄×Q₈	—	356	1	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	89	2	no	—
219	C₂×C₄⋊C₄	—	89	1	no	—
220	C₂×C₄⋊C₄	—	89	2	no	—
221	C₂×C₄⋊C₄	—	89	2	no	—
222	C₂×C₄⋊C₄	—	178	2	no	—
223	C₂×C₄⋊C₄	—	178	2	no	—
224	C₂×C₄⋊C₄	—	178	2	no	—
225	C₂×C₄⋊C₄	—	178	2	no	—
226	C₂×C₄⋊C₄	—	178	2	no	—
227	C₂×C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
228	C₂×C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
229	C₂×C₄²	—	89	2	no	—
230	C₂×C₄²	—	178	2	no	—
231	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
232	#107	—	4	1	no	—
233	#107	—	89	4	no	—
234	#107	—	356	1	no	—
235	#107	Dic₁₇₈	1	4	yes	complemented
236	#106	—	356	2	no	—
237	C₄⋊C₈	—	356	1	no	—
238	C₄×C₈	—	4	1	no	—
239	#95	—	2	2	no	—
240	#95	—	178	2	no	—
241	#95	—	178	2	no	—
242	#95	—	178	2	no	—
243	#95	—	178	2	no	—
244	C₂×Q₈	—	356	1	no	—
245	C₂×Q₈	—	356	1	no	—
246	C₂×Q₈	—	178	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	89	2	no	—
250	C₂²×C₄	—	89	2	no	—
251	C₂²×C₄	—	178	2	no	—
252	C₂²×C₄	—	178	2	no	—
253	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
255	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	178	2	no	—
261	C₂×C₈	—	178	2	no	—
262	C₂×C₈	—	178	2	no	—
263	C₂×C₈	Q₈×D₈₉	1	2	yes	—
264	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₄⋊C₄	—	89	4	no	—
270	C₄⋊C₄	—	356	2	no	—
271	C₄⋊C₄	—	356	1	no	—
272	C₄⋊C₄	—	356	4	no	—
273	C₄⋊C₄	—	356	4	no	—
274	C₄⋊C₄	—	356	4	no	—
275	C₄⋊C₄	—	178	2	no	—
276	C₄⋊C₄	—	178	2	no	—
277	C₄⋊C₄	—	178	2	no	—
278	C₄⋊C₄	—	712	2	no	—
279	C₄⋊C₄	—	712	2	no	—
280	C₄⋊C₄	C₂×Dic₁₇₈	1	4	yes	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	178	2	no	—
284	C₄²	—	178	2	no	—
285	C₄²	—	712	2	no	—
286	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
287	Q₈	—	356	4	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	356	2	no	—
300	C₂×C₄	—	356	2	no	—
301	C₂×C₄	—	356	2	no	—
302	C₂×C₄	—	356	2	no	—
303	C₂×C₄	—	356	2	no	—
304	C₂×C₄	—	178	2	no	—
305	C₂×C₄	—	178	2	no	—
306	C₂×C₄	—	178	2	no	—
307	C₂×C₄	—	178	2	no	—
308	C₂×C₄	—	178	2	no	—
309	C₂×C₄	—	178	4	no	—
310	C₂×C₄	—	178	2	no	—
311	C₂×C₄	#15668031	1	2	yes	—
312	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
313	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
314	C₂×C₄	#15668057	1	1	yes	characteristic
315	C₂×C₄	#15668058	1	1	yes	characteristic
316	C₂×C₄	#15668059	1	1	yes	characteristic
317	C₂×C₄	#15668060	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	2	2	no	—
320	C₈	—	356	2	no	—
321	C₂²	#19152156	1	1	yes	characteristic
322	C₂²	#19152263	1	1	yes	characteristic
323	C₂²	D₄×Dic₁₇₈	1	1	yes	characteristic
324	C₂²	C₂×#15668031	1	1	yes	characteristic
325	C₂²	#19152395	1	1	yes	characteristic
326	C₂²	#19152400	1	1	yes	characteristic
327	C₂²	#19152411	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	4	no	—
331	C₄	—	8	2	no	—
332	C₄	—	356	2	no	—
333	C₄	—	356	4	no	—
334	C₄	—	356	2	no	—
335	C₄	—	712	2	no	—
336	C₄	#19152227	1	1	yes	characteristic
337	C₄	#19152229	1	1	yes	characteristic
338	C₄	Q₈×Dic₁₇₈	1	1	yes	characteristic
339	C₄	#19152352	1	1	yes	characteristic
340	C₂	#26027283	1	1	yes	characteristic
341	C₂	#26031039	1	1	yes	characteristic
342	C₂	#26031155	1	1	yes	characteristic
343	C₂	#26033117	1	1	yes	characteristic
344	C₂	#26033145	1	1	yes	characteristic
345	C₂	#26033213	1	1	yes	characteristic
346	C₂	#26033217	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₈₉
commutator, cosocle	C₂²×C₄×C₈₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1942	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
hypercenter	#1942	D₈₉
hypocenter	C₈₉	#32686
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1942

Complements

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

N	H ≅ G/N	type
53. #15667946	Q₈	semidirect
67. #107×C₈₉	Q₈	semidirect
183. C₈₉	#32686	semidirect
235. #107	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26025637	2
#26021753	2
#26021593	2
#26021588	2
#26021188	2
#26020985	2
#26020843	2
#26020825	2
C₈₉×#1942	2
#32686	89