Subgroups — #36574131

Order 22784.

710 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15668034	C₂	1	1	yes	maximal, characteristic
2	#26028917	C₂	1	2	yes	maximal, complemented
3	#26021145	C₂	1	2	yes	maximal
4	#26020901	C₂	1	4	yes	maximal, complemented
5	#26020424	C₂	1	1	yes	maximal, characteristic, complemented
6	#26020273	C₂	1	2	yes	maximal, complemented
7	#26020136	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668034	C₂²	1	1	yes	characteristic
10	C₄×#13083574	C₂²	1	2	yes	—
11	C₄×Dic₃₅₆	C₄	1	1	yes	characteristic, complemented
12	C₄×Dic₃₅₆	C₄	1	1	yes	characteristic, complemented
13	C₄×#13083563	C₄	1	2	yes	complemented
14	C₄×#13083563	C₄	1	2	yes	complemented
15	C₄×D₇₁₂	C₄	1	1	yes	characteristic, complemented
16	C₄×D₇₁₂	C₄	1	1	yes	characteristic, complemented
17	#19151658	—	2	4	no	—
18	#19151654	—	2	2	no	—
19	#19151650	—	2	2	no	—
20	C₂×#15667948	C₂²	1	2	yes	—
21	C₂×#15667948	C₂²	1	2	yes	—
22	#19151636	C₂²	1	1	yes	characteristic
23	#19151622	C₂²	1	2	yes	—
24	C₂×#15667940	C₂²	1	4	yes	—
25	#19151595	C₂²	1	2	yes	—
26	#19151564	C₂²	1	4	yes	—
27	#19151553	C₂²	1	4	yes	—
28	#19151358	C₂²	1	4	yes	—
29	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×#97×C₈₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
33	#336×C₈₉	C₂²	1	2	yes	—
34	C₂×#13083574	C₂³	1	2	yes	—
35	#15668034	—	2	2	no	—
36	#15668034	—	2	2	no	—
37	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
38	C₂×#13083563	C₂×C₄	1	2	yes	—
39	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
40	C₂×#13083558	C₂³	1	4	yes	—
41	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
43	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
44	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
45	C₄×C₂²⋊D₈₉	—	2	2	no	—
46	C₄×C₂²⋊D₈₉	—	2	2	no	—
47	#15667971	C₂³	1	2	yes	—
48	#15667969	—	2	2	no	—
49	C₄²×D₈₉	—	2	2	no	—
50	#15667948	—	2	4	no	—
51	#15667948	—	4	2	no	—
52	#15667946	C₂×C₄	1	1	yes	characteristic
53	#15667946	C₂×C₄	1	1	yes	characteristic
54	#15667944	C₂×C₄	1	2	yes	—
55	C₄⋊#10623494	—	2	2	no	—
56	C₄⋊#10623494	—	2	2	no	—
57	#15667940	—	4	4	no	—
58	#15667937	C₂×C₄	1	2	yes	—
59	#15667937	C₂×C₄	1	2	yes	—
60	#15667934	C₂×C₄	1	2	yes	—
61	#15667934	C₂×C₄	1	2	yes	—
62	#15667924	—	2	4	no	—
63	#15667924	—	2	4	no	—
64	#15667922	—	2	4	no	—
65	C₈×Dic₈₉	—	2	4	no	—
66	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
68	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
70	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
71	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
73	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
74	#97×C₈₉	—	2	4	no	—
75	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
79	#95×C₈₉	—	2	1	no	—
80	#13083574	—	2	2	no	—
81	#13083574	—	2	2	no	—
82	#13083574	—	2	2	no	—
83	#13083574	—	2	2	no	—
84	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
85	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
86	C₂×C₂²⋊D₈₉	—	2	2	no	—
87	C₂×C₄×D₈₉	—	2	2	no	—
88	C₂×C₄×D₈₉	—	2	2	no	—
89	C₂×C₄×D₈₉	—	2	2	no	—
90	Dic₃₅₆	—	2	2	no	—
91	#13083563	—	2	4	no	—
92	D₇₁₂	—	2	2	no	—
93	#13083558	—	2	4	no	—
94	#13083558	—	2	4	no	—
95	C₂×#10623494	—	2	4	no	—
96	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
97	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
98	#13083553	—	2	2	no	—
99	#13083553	—	2	2	no	—
100	C₂²⋊Dic₈₉	—	2	2	no	—
101	#13083551	—	2	2	no	—
102	#13083551	—	2	2	no	—
103	C₄×Dic₈₉	—	2	2	no	—
104	C₄×Dic₈₉	—	2	2	no	—
105	C₄×Dic₈₉	—	2	2	no	—
106	C₂²×C₄×C₈₉	—	2	2	no	—
107	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
108	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
109	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
110	C₂×C₈×C₈₉	—	2	2	no	—
111	C₂×C₈×C₈₉	—	2	1	no	—
112	C₂×C₈×C₈₉	—	2	1	no	—
113	C₂×C₈×C₈₉	—	2	2	no	—
114	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
119	C₂×C₈×C₈₉	#41	1	2	yes	—
120	C₂×C₈×C₈₉	#41	1	2	yes	—
121	C₄⋊C₄×C₈₉	—	4	2	no	—
122	C₄²×C₈₉	—	2	2	no	—
123	C₄²×C₈₉	—	2	2	no	—
124	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
129	C₂²×D₈₉	—	2	2	no	—
130	C₂×C₄×C₈₉	—	2	2	no	—
131	C₂×C₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	—	4	2	no	—
134	C₂×C₄×C₈₉	—	4	2	no	—
135	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
145	C₂²⋊D₈₉	—	4	2	no	—
146	C₂²⋊D₈₉	—	4	2	no	—
147	C₂×Dic₈₉	—	2	2	no	—
148	C₂×Dic₈₉	—	2	2	no	—
149	C₂×Dic₈₉	—	2	2	no	—
150	D₃₅₆	—	2	2	no	—
151	D₃₅₆	—	2	2	no	—
152	C₄×D₈₉	—	4	2	no	—
153	C₄×D₈₉	—	4	2	no	—
154	C₄×D₈₉	—	4	4	no	—
155	C₄×D₈₉	—	8	2	no	—
156	Dic₁₇₈	—	2	2	no	—
157	Dic₁₇₈	—	2	2	no	—
158	C₈×C₈₉	—	2	1	no	—
159	C₈×C₈₉	—	2	1	no	—
160	C₈×C₈₉	—	2	4	no	—
161	C₈×C₈₉	#117	1	2	yes	—
162	C₈×C₈₉	#117	1	2	yes	—
163	#10623494	—	4	4	no	—
164	C₂²×C₈₉	—	2	1	no	—
165	C₂²×C₈₉	—	2	1	no	—
166	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₈₉	#555	1	1	yes	characteristic
168	C₂²×C₈₉	#556	1	1	yes	characteristic
169	C₂×D₈₉	—	4	2	no	—
170	C₂×D₈₉	—	4	2	no	—
171	C₂×D₈₉	—	4	2	no	—
172	C₄×C₈₉	—	2	1	no	—
173	C₄×C₈₉	—	2	1	no	—
174	C₄×C₈₉	—	2	1	no	—
175	C₄×C₈₉	—	8	2	no	—
176	C₄×C₈₉	#444	1	2	yes	—
177	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
178	C₄×C₈₉	#520	1	1	yes	characteristic
179	C₄×C₈₉	#547	1	1	yes	characteristic
180	C₄×C₈₉	#550	1	1	yes	characteristic
181	Dic₈₉	—	4	2	no	—
182	Dic₈₉	—	4	2	no	—
183	Dic₈₉	—	4	2	no	—
184	#22803	—	89	1	no	maximal, Sylow
185	C₂×C₈₉	—	2	1	no	—
186	C₂×C₈₉	—	2	1	no	—
187	C₂×C₈₉	#2354	1	1	yes	characteristic
188	C₂×C₈₉	C₂×#444	1	1	yes	characteristic
189	C₂×C₈₉	#2957	1	1	yes	characteristic
190	D₈₉	—	8	2	no	—
191	C₄×#135	—	89	1	no	—
192	#2925	—	89	2	no	—
193	#1975	—	89	2	no	—
194	#1929	—	89	4	no	—
195	#1848	—	89	1	no	—
196	#1806	—	89	2	no	—
197	#1773	—	89	2	no	—
198	#1770	D₈₉	1	1	yes	characteristic, complemented
199	C₈₉	#22803	1	1	yes	characteristic, complemented, Sylow
200	C₂×#135	—	89	1	no	—
201	C₄×#41	—	89	2	no	—
202	C₄×Q₁₆	—	89	1	no	—
203	C₄×Q₁₆	—	89	1	no	—
204	C₄×SD₁₆	—	89	2	no	—
205	C₄×SD₁₆	—	89	2	no	—
206	C₄×D₈	—	89	1	no	—
207	C₄×D₈	—	89	1	no	—
208	#433	—	178	4	no	—
209	#432	—	178	2	no	—
210	#431	—	178	2	no	—
211	C₂×#108	—	89	2	no	—
212	C₂×#108	—	89	2	no	—
213	#427	—	89	1	no	—
214	#423	—	89	2	no	—
215	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
216	C₂×#104	—	89	4	no	—
217	#416	—	89	2	no	—
218	#408	—	89	4	no	—
219	#405	—	89	4	no	—
220	C₂×#97	C₂×D₈₉	1	2	yes	—
221	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
222	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
223	#339	—	89	4	no	—
224	#336	C₂×D₈₉	1	2	yes	—
225	C₂×#41	—	89	2	no	—
226	#135	—	178	2	no	—
227	#135	—	178	2	no	—
228	C₂×Q₁₆	—	89	1	no	—
229	C₂×SD₁₆	—	89	2	no	—
230	C₂×D₈	—	89	1	no	—
231	C₂×M₁₆	—	89	4	no	—
232	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
234	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
235	C₄×Q₈	—	89	2	no	—
236	C₄×Q₈	—	89	2	no	—
237	C₄×D₄	—	89	2	no	—
238	C₄×D₄	—	89	2	no	—
239	C₄×D₄	—	178	2	no	—
240	C₄×D₄	—	178	2	no	—
241	#117	—	89	2	no	—
242	#117	—	178	2	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄²	—	178	2	no	—
245	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
246	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
247	#108	—	356	2	no	—
248	#108	—	178	4	no	—
249	#107	—	89	1	no	—
250	#107	—	89	1	no	—
251	#106	—	89	2	no	—
252	C₄⋊C₈	—	178	2	no	—
253	C₄⋊C₈	—	178	2	no	—
254	C₄⋊C₈	D₃₅₆	1	2	yes	—
255	C₄⋊C₈	D₃₅₆	1	2	yes	—
256	#104	—	356	4	no	—
257	#103	—	89	2	no	—
258	#103	—	89	2	no	—
259	#102	—	89	2	no	—
260	#102	—	89	2	no	—
261	C₂²⋊C₈	—	178	4	no	—
262	C₂²⋊C₈	—	178	4	no	—
263	#97	—	2	4	no	—
264	#97	—	178	4	no	—
265	C₄×C₈	—	178	4	no	—
266	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
267	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
268	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
269	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
270	#95	—	2	1	no	—
271	#41	—	178	2	no	—
272	#41	—	178	2	no	—
273	#41	—	178	2	no	—
274	#41	—	178	2	no	—
275	C₂×Q₈	—	89	2	no	—
276	C₂×D₄	—	89	2	no	—
277	C₂×D₄	—	178	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	178	2	no	—
280	C₂²×C₄	—	178	2	no	—
281	C₂²×C₄	—	178	2	no	—
282	C₂²×C₄	#13083574	1	1	yes	characteristic
283	C₂²×C₄	#13083574	1	1	yes	characteristic
284	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
285	Q₁₆	—	178	2	no	—
286	SD₁₆	—	178	4	no	—
287	D₈	—	178	2	no	—
288	M₁₆	—	178	4	no	—
289	M₁₆	—	178	4	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	178	4	no	—
295	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
299	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
300	C₂×C₈	#13083573	1	2	yes	—
301	C₂×C₈	#13083575	1	2	yes	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	89	2	no	—
304	C₄⋊C₄	—	89	2	no	—
305	C₄⋊C₄	—	178	2	no	—
306	C₄⋊C₄	—	178	2	no	—
307	C₂²⋊C₄	—	178	2	no	—
308	C₂²⋊C₄	—	178	2	no	—
309	C₂²⋊C₄	—	178	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	178	2	no	—
313	C₄²	—	178	2	no	—
314	C₄²	—	178	2	no	—
315	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
316	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
317	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
318	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
319	C₂³	—	178	2	no	—
320	C₂³	C₂×#13083574	1	1	yes	characteristic
321	Q₈	—	178	2	no	—
322	Q₈	—	178	2	no	—
323	D₄	—	356	2	no	—
324	D₄	—	356	2	no	—
325	D₄	—	178	2	no	—
326	D₄	—	178	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	356	4	no	—
333	C₂×C₄	—	356	2	no	—
334	C₂×C₄	—	356	2	no	—
335	C₂×C₄	—	178	2	no	—
336	C₂×C₄	—	178	2	no	—
337	C₂×C₄	—	178	2	no	—
338	C₂×C₄	—	712	2	no	—
339	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
340	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
341	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
342	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
343	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
344	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
345	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
346	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
347	C₂×C₄	#15668058	1	1	yes	characteristic
348	C₂×C₄	#15668059	1	1	yes	characteristic
349	C₈	—	2	4	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	356	4	no	—
353	C₈	#15667970	1	2	yes	—
354	C₈	#15667970	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	356	2	no	—
358	C₂²	—	356	2	no	—
359	C₂²	—	356	2	no	—
360	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
361	C₂²	#19152313	1	1	yes	characteristic
362	C₂²	#19152321	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	8	2	no	—
367	C₄	—	356	2	no	—
368	C₄	—	356	2	no	—
369	C₄	—	356	2	no	—
370	C₄	#19151721	1	2	yes	—
371	C₄	C₂×#15667970	1	1	yes	characteristic
372	C₄	#19152078	1	1	yes	characteristic
373	C₄	#19152252	1	1	yes	characteristic
374	C₄	#19152270	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	712	2	no	—
378	C₂	#26023199	1	1	yes	characteristic
379	C₂	C₂×#19151721	1	1	yes	characteristic
380	C₂	#26029170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1770	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂×#15668034	C₂²
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1770	D₈₉
hypocenter	C₈₉	#22803
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
2. #26028917	C₂	semidirect
4. #26020901	C₂	semidirect
5. #26020424	C₂	semidirect
6. #26020273	C₂	semidirect
7. #26020136	C₂	semidirect
8. C₈₉×#1770	C₂	semidirect
11. C₄×Dic₃₅₆	C₄	semidirect
12. C₄×Dic₃₅₆	C₄	semidirect
13. C₄×#13083563	C₄	semidirect
14. C₄×#13083563	C₄	semidirect
15. C₄×D₇₁₂	C₄	semidirect
16. C₄×D₇₁₂	C₄	semidirect
198. #1770	D₈₉	semidirect
199. C₈₉	#22803	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15668034	2
#26028917	2
#26021145	2
#26020901	2
#26020424	2
#26020273	2
#26020136	2
C₈₉×#1770	2
#22803	89