Subgroups — #36574322

Order 22784.

784 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	#26021187	C₂	1	2	yes	maximal
3	#26020806	C₂	1	4	yes	maximal
4	#26020368	C₂	1	2	yes	maximal
5	#26020117	C₂	1	2	yes	maximal
6	C₄⋊C₈×Dic₈₉	C₂	1	2	yes	maximal
7	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#1842	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₇₈	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₇₈	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
14	C₂×C₄×#10623494	C₂²	1	2	yes	—
15	C₂×C₈×Dic₈₉	C₂²	1	4	yes	—
16	C₄×#13083553	C₄	1	2	yes	—
17	C₄×#13083553	C₄	1	2	yes	—
18	C₄×#13083553	C₂²	1	2	yes	—
19	C₄×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
20	C₄×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
21	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
22	C₄²×Dic₈₉	C₂²	1	2	yes	—
23	#19151353	C₂²	1	2	yes	—
24	#19151352	C₂²	1	4	yes	—
25	#19151352	C₂²	1	4	yes	—
26	#19151352	C₂²	1	4	yes	—
27	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
29	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
30	C₄³×C₈₉	C₂²	1	1	yes	characteristic
31	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×#10623494	C₂³	1	2	yes	—
33	C₂²×#10623494	C₂³	1	2	yes	—
34	C₄×Dic₁₇₈	—	2	4	no	—
35	C₄×Dic₁₇₈	—	2	2	no	—
36	C₄×Dic₁₇₈	—	2	2	no	—
37	C₄×Dic₁₇₈	C₈	1	8	yes	complemented
38	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂×#13083553	C₂×C₄	1	2	yes	—
42	C₂×#13083553	C₂×C₄	1	2	yes	—
43	C₂×#13083553	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
45	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
49	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
50	C₄⋊#10623494	—	2	2	no	—
51	C₄⋊#10623494	Q₈	1	4	yes	complemented
52	C₄×#10623494	—	2	4	no	—
53	C₈×Dic₈₉	—	4	4	no	—
54	#15667919	C₂×C₄	1	2	yes	—
55	#15667919	C₂×C₄	1	2	yes	—
56	#15667919	C₂×C₄	1	2	yes	—
57	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
58	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₈₉	—	2	4	no	—
66	C₄⋊C₈×C₈₉	—	2	4	no	—
67	C₄⋊C₈×C₈₉	—	4	2	no	—
68	C₂×Dic₁₇₈	—	2	2	no	—
69	C₂×Dic₁₇₈	—	2	2	no	—
70	C₂×Dic₁₇₈	C₂×C₈	1	4	yes	—
71	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
72	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
73	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
74	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
75	C₂×#10623494	—	2	2	no	—
76	C₂×#10623494	—	2	2	no	—
77	C₂×#10623494	—	2	2	no	—
78	C₂×#10623494	—	2	2	no	—
79	C₂×#10623494	—	2	2	no	—
80	C₂×#10623494	C₂×Q₈	1	4	yes	—
81	C₄⋊Dic₈₉	—	2	2	no	—
82	C₄⋊Dic₈₉	C₂×C₈	1	4	yes	—
83	C₄⋊Dic₈₉	M₁₆	1	4	yes	—
84	#13083553	—	2	4	no	—
85	#13083553	—	2	4	no	—
86	#13083553	C₂×C₈	1	8	yes	—
87	C₄×Dic₈₉	—	2	4	no	—
88	C₄×Dic₈₉	—	2	2	no	—
89	C₄×Dic₈₉	—	2	2	no	—
90	C₄×Dic₈₉	—	2	2	no	—
91	C₄×Dic₈₉	—	2	2	no	—
92	C₄×Dic₈₉	—	2	4	no	—
93	C₄×Dic₈₉	—	4	2	no	—
94	C₄×Dic₈₉	C₂×C₈	1	8	yes	—
95	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₈₉	—	2	4	no	—
103	C₂×C₈×C₈₉	—	2	4	no	—
104	C₂×C₈×C₈₉	—	2	4	no	—
105	C₄²×C₈₉	—	2	2	no	—
106	C₄²×C₈₉	—	2	2	no	—
107	C₄²×C₈₉	—	2	1	no	—
108	C₄²×C₈₉	—	2	1	no	—
109	C₄²×C₈₉	—	4	1	no	—
110	C₄²×C₈₉	C₂×C₈	1	4	yes	—
111	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
112	C₄²×C₈₉	#41	1	2	yes	—
113	C₄²×C₈₉	#41	1	2	yes	—
114	C₄²×C₈₉	#41	1	2	yes	—
115	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	#117	1	2	yes	—
126	C₂×C₄×C₈₉	#117	1	2	yes	—
127	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₈₉	#131	1	2	yes	—
134	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂×Dic₈₉	—	2	2	no	—
141	C₂×Dic₈₉	—	2	2	no	—
142	C₂×Dic₈₉	—	2	2	no	—
143	C₂×Dic₈₉	—	2	2	no	—
144	C₂×Dic₈₉	—	2	2	no	—
145	C₂×Dic₈₉	—	2	2	no	—
146	C₂×Dic₈₉	—	2	2	no	—
147	C₂×Dic₈₉	—	2	2	no	—
148	C₂×Dic₈₉	—	2	2	no	—
149	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
150	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
151	C₂×Dic₈₉	C₂×M₁₆	1	4	yes	—
152	Dic₁₇₈	—	2	8	no	—
153	C₈×C₈₉	—	4	4	no	—
154	#10623494	—	2	4	no	—
155	#10623494	—	4	2	no	—
156	C₂²×C₈₉	C₂×#117	1	1	yes	characteristic
157	C₂²×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₂²×C₈₉	#520	1	1	yes	characteristic
159	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
160	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
161	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
162	C₂²×C₈₉	#568	1	1	yes	characteristic
163	C₄×C₈₉	—	2	2	no	—
164	C₄×C₈₉	—	2	2	no	—
165	C₄×C₈₉	—	2	2	no	—
166	C₄×C₈₉	—	4	1	no	—
167	C₄×C₈₉	#431	1	4	yes	—
168	C₄×C₈₉	C₈×Q₈	1	4	yes	—
169	C₄×C₈₉	#554	1	1	yes	characteristic
170	C₄×C₈₉	#556	1	1	yes	characteristic
171	C₄×C₈₉	#557	1	1	yes	characteristic
172	C₄×C₈₉	#559	1	1	yes	characteristic
173	Dic₈₉	—	2	4	no	—
174	Dic₈₉	—	2	4	no	—
175	Dic₈₉	—	2	4	no	—
176	Dic₈₉	—	4	2	no	—
177	#22821	—	89	1	no	maximal, Sylow
178	C₂×C₈₉	#2355	1	1	yes	characteristic
179	C₂×C₈₉	C₂×#431	1	1	yes	characteristic
180	C₂×C₈₉	#2925	1	1	yes	characteristic
181	C₂×C₈₉	C₂×C₈×Q₈	1	1	yes	characteristic
182	C₂×C₈₉	Q₈×M₁₆	1	1	yes	characteristic
183	C₂×C₈₉	#2988	1	1	yes	characteristic
184	C₂×C₈₉	#2990	1	1	yes	characteristic
185	C₄²×Q₈	—	89	1	no	—
186	#1984	—	89	2	no	—
187	#1916	—	89	4	no	—
188	#1842	D₈₉	1	1	yes	characteristic
189	#1831	—	89	2	no	—
190	C₈×C₄⋊C₄	—	89	2	no	—
191	C₄×C₄⋊C₈	—	89	2	no	—
192	C₄×C₄⋊C₈	—	89	1	no	—
193	C₈₉	#22821	1	1	yes	characteristic, complemented, Sylow
194	C₂×C₄×Q₈	—	89	1	no	—
195	C₂×C₄×Q₈	—	89	1	no	—
196	C₂×C₄×Q₈	—	89	1	no	—
197	C₂×C₄⋊C₈	—	89	1	no	—
198	C₂×C₄⋊C₈	—	89	1	no	—
199	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
200	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
201	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
202	C₂×C₄×C₈	—	89	4	no	—
203	C₂×C₄×C₈	—	89	2	no	—
204	C₄×C₄⋊C₄	—	89	1	no	—
205	C₄×C₄⋊C₄	—	89	1	no	—
206	C₄×C₄⋊C₄	—	89	2	no	—
207	C₄×C₄⋊C₄	—	89	2	no	—
208	C₄×C₄⋊C₄	—	89	1	no	—
209	C₄×C₄⋊C₄	—	89	2	no	—
210	C₄³	—	89	2	no	—
211	C₄³	C₂×D₈₉	1	1	yes	characteristic
212	#336	—	89	4	no	—
213	#336	—	89	4	no	—
214	#336	—	89	4	no	—
215	#336	—	89	2	no	—
216	C₂²×Q₈	—	89	1	no	—
217	C₂²×C₈	—	89	2	no	—
218	C₂²×C₈	—	89	2	no	—
219	C₂²×C₈	C₂²×D₈₉	1	4	yes	—
220	C₄×Q₈	—	89	8	no	—
221	C₄×Q₈	—	178	4	no	—
222	C₄×Q₈	—	178	2	no	—
223	C₄×Q₈	—	178	2	no	—
224	C₂×C₄⋊C₄	—	89	2	no	—
225	C₂×C₄⋊C₄	—	89	1	no	—
226	C₂×C₄⋊C₄	—	89	1	no	—
227	C₂×C₄⋊C₄	—	89	2	no	—
228	C₂×C₄⋊C₄	—	89	2	no	—
229	C₂×C₄⋊C₄	—	89	1	no	—
230	C₂×C₄²	—	89	2	no	—
231	C₂×C₄²	—	89	2	no	—
232	C₂×C₄²	—	89	2	no	—
233	C₂×C₄²	—	89	2	no	—
234	C₂×C₄²	—	89	2	no	—
235	C₂×C₄²	—	89	2	no	—
236	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
237	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
238	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
239	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
240	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
243	C₄⋊C₈	—	2	4	no	—
244	C₄⋊C₈	—	2	4	no	—
245	C₄⋊C₈	—	4	2	no	—
246	C₄⋊C₈	—	89	4	no	—
247	C₄⋊C₈	—	178	2	no	—
248	C₄×C₈	—	356	4	no	—
249	C₄×C₈	—	178	4	no	—
250	#95	—	89	2	no	—
251	#95	—	89	2	no	—
252	#95	—	89	2	no	—
253	C₂×Q₈	—	89	4	no	—
254	C₂×Q₈	—	178	2	no	—
255	C₂×Q₈	—	178	2	no	—
256	C₂²×C₄	—	89	2	no	—
257	C₂²×C₄	—	89	2	no	—
258	C₂²×C₄	—	89	2	no	—
259	C₂²×C₄	—	89	2	no	—
260	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
266	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	89	4	no	—
271	C₂×C₈	—	178	2	no	—
272	C₂×C₈	—	178	2	no	—
273	C₂×C₈	—	178	2	no	—
274	C₂×C₈	—	178	2	no	—
275	C₂×C₈	—	178	2	no	—
276	C₄⋊C₄	—	89	8	no	—
277	C₄⋊C₄	—	89	4	no	—
278	C₄⋊C₄	—	89	4	no	—
279	C₄⋊C₄	—	178	4	no	—
280	C₄⋊C₄	—	178	4	no	—
281	C₄⋊C₄	—	178	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	4	1	no	—
287	C₄²	—	89	8	no	—
288	C₄²	—	356	2	no	—
289	C₄²	—	178	4	no	—
290	C₄²	—	178	4	no	—
291	C₄²	—	178	2	no	—
292	C₄²	—	178	2	no	—
293	C₄²	—	178	2	no	—
294	C₄²	—	178	2	no	—
295	C₄²	C₈×D₈₉	1	4	yes	—
296	C₄²	Q₈×D₈₉	1	2	yes	—
297	C₄²	#13083573	1	2	yes	—
298	C₄²	#13083573	1	2	yes	—
299	C₄²	#13083575	1	2	yes	—
300	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
301	Q₈	—	178	8	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	89	4	no	—
312	C₂×C₄	—	89	4	no	—
313	C₂×C₄	—	89	4	no	—
314	C₂×C₄	—	178	2	no	—
315	C₂×C₄	—	178	2	no	—
316	C₂×C₄	—	178	2	no	—
317	C₂×C₄	—	178	2	no	—
318	C₂×C₄	—	178	2	no	—
319	C₂×C₄	—	178	2	no	—
320	C₂×C₄	—	178	2	no	—
321	C₂×C₄	—	178	2	no	—
322	C₂×C₄	—	178	2	no	—
323	C₂×C₄	#15667970	1	2	yes	—
324	C₂×C₄	#15667970	1	2	yes	—
325	C₂×C₄	#15667977	1	2	yes	—
326	C₂×C₄	#15667977	1	2	yes	—
327	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
328	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
329	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
330	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
331	C₂×C₄	#15668020	1	2	yes	—
332	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
333	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
334	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
335	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
336	C₂×C₄	#15668057	1	1	yes	characteristic
337	C₂×C₄	#15668060	1	1	yes	characteristic
338	C₈	—	4	4	no	—
339	C₈	—	356	2	no	—
340	C₈	—	178	4	no	—
341	C₂²	C₂×#15667970	1	1	yes	characteristic
342	C₂²	C₂×#15667977	1	1	yes	characteristic
343	C₂²	#19152079	1	1	yes	characteristic
344	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
345	C₂²	C₂×M₁₆×D₈₉	1	1	yes	characteristic
346	C₂²	C₂×#15668020	1	1	yes	characteristic
347	C₂²	#19152364	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	4	1	no	—
352	C₄	—	356	2	no	—
353	C₄	—	178	4	no	—
354	C₄	—	178	4	no	—
355	C₄	—	178	4	no	—
356	C₄	#19151650	1	4	yes	—
357	C₄	#19151722	1	4	yes	—
358	C₄	#19152309	1	1	yes	characteristic
359	C₄	#19152322	1	1	yes	characteristic
360	C₄	#19152323	1	1	yes	characteristic
361	C₄	#19152334	1	1	yes	characteristic
362	C₂	#26023211	1	1	yes	characteristic
363	C₂	C₂×#19151650	1	1	yes	characteristic
364	C₂	#26028920	1	1	yes	characteristic
365	C₂	C₂×#19151722	1	1	yes	characteristic
366	C₂	#26029203	1	1	yes	characteristic
367	C₂	#26029361	1	1	yes	characteristic
368	C₂	#26029378	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#1842	C₂
socle, prime radical	C₂³×C₈₉	C₂³×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1842	D₈₉
hypocenter	C₈₉	#22821
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #1842

Complements

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

N	H ≅ G/N	type
37. C₄×Dic₁₇₈	C₈	semidirect
51. C₄⋊#10623494	Q₈	semidirect
193. C₈₉	#22821	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Dic₁₇₈	2
#26021187	2
#26020806	2
#26020368	2
#26020117	2
C₄⋊C₈×Dic₈₉	2
C₄×C₄⋊#10623494	2
C₈₉×#1842	2
#22821	89