Subgroups — #26031892

Order 11392.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668041	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₂²⋊D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152247	C₂	1	1	yes	maximal, characteristic
4	#19151976	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151856	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151853	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151806	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151801	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151760	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151757	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151754	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151752	C₂	1	1	yes	maximal, characteristic, complemented
13	SD₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151674	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#137×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083573	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
18	#15668041	—	2	4	no	—
19	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
20	C₂×#13083564	C₂²	1	1	yes	characteristic
21	C₂×#13083564	C₂²	1	1	yes	characteristic
22	C₂×#13083564	C₂²	1	1	yes	characteristic
23	C₂×#13083562	C₂²	1	1	yes	characteristic
24	C₂×#13083557	C₂²	1	1	yes	characteristic
25	#15668010	—	2	1	no	—
26	#15668010	C₂²	1	1	yes	characteristic
27	#15668010	C₂²	1	1	yes	characteristic
28	#15667995	—	2	1	no	—
29	#15667993	—	2	1	no	—
30	#15667991	—	2	1	no	—
31	#15667991	C₂²	1	1	yes	characteristic
32	#15667991	C₂²	1	1	yes	characteristic
33	#15667988	—	2	1	no	—
34	#15667988	C₂²	1	1	yes	characteristic
35	#15667988	C₂²	1	1	yes	characteristic
36	#15667984	—	2	1	no	—
37	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
38	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊D₈₉	—	2	1	no	—
40	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083553	—	2	1	no	—
42	#15667944	C₂²	1	1	yes	characteristic
43	#15667941	C₂²	1	1	yes	characteristic
44	#15667937	C₂²	1	1	yes	characteristic
45	#15667937	C₂²	1	1	yes	characteristic
46	#15667936	C₂²	1	1	yes	characteristic
47	#15667936	C₂²	1	1	yes	characteristic
48	#15667936	C₂²	1	1	yes	characteristic
49	#15667933	C₂²	1	1	yes	characteristic
50	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
51	#15667924	C₂²	1	1	yes	characteristic
52	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
53	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
54	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	#137×C₈₉	—	2	4	no	—
56	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
57	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
58	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
60	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
61	#13083573	—	2	2	no	—
62	#13083573	—	4	1	no	—
63	C₂×Dic₁₇₈	—	2	1	no	—
64	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
66	Q₈×D₈₉	—	2	2	no	—
67	Q₈×D₈₉	—	4	1	no	—
68	C₂×C₂²⋊D₈₉	—	2	1	no	—
69	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
70	C₂²×Dic₈₉	—	2	1	no	—
71	C₂²×Dic₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	—	2	1	no	—
73	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
74	Dic₃₅₆	—	2	2	no	—
75	#13083564	—	2	2	no	—
76	#13083564	—	4	1	no	—
77	#13083564	—	4	1	no	—
78	#13083562	—	2	2	no	—
79	#13083557	—	2	2	no	—
80	C₂×#10623494	C₂³	1	1	yes	characteristic
81	C₂×#10623494	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₉	—	2	1	no	—
83	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
85	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083553	—	2	1	no	—
88	#13083553	—	2	1	no	—
89	#13083553	—	2	1	no	—
90	#13083553	—	2	1	no	—
91	#13083553	—	2	1	no	—
92	#13083553	—	2	1	no	—
93	#13083553	D₄	1	1	yes	characteristic
94	C₂²⋊Dic₈₉	—	2	1	no	—
95	C₂²⋊Dic₈₉	—	2	1	no	—
96	C₂²⋊Dic₈₉	—	2	1	no	—
97	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
98	#13083551	—	2	1	no	—
99	#13083551	—	2	1	no	—
100	#13083551	D₄	1	1	yes	characteristic
101	C₄×Dic₈₉	—	2	1	no	—
102	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
103	#41×C₈₉	—	2	2	no	—
104	#41×C₈₉	—	4	1	no	—
105	C₂×Q₈×C₈₉	—	2	1	no	—
106	C₂×Q₈×C₈₉	—	2	2	no	—
107	C₂×Q₈×C₈₉	—	4	1	no	—
108	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
109	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
110	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
111	C₂×D₄×C₈₉	—	2	1	no	—
112	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
113	C₂²×C₄×C₈₉	—	2	1	no	—
114	C₂²×C₄×C₈₉	—	2	1	no	—
115	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
116	Q₁₆×C₈₉	—	2	2	no	—
117	Q₁₆×C₈₉	—	2	2	no	—
118	SD₁₆×C₈₉	—	2	2	no	—
119	SD₁₆×C₈₉	D₄	1	4	yes	complemented
120	M₁₆×C₈₉	—	2	2	no	—
121	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
122	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
123	C₂³×C₈₉	—	2	1	no	—
124	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
126	Q₈×C₈₉	—	2	1	no	—
127	Q₈×C₈₉	—	2	1	no	—
128	Q₈×C₈₉	—	2	1	no	—
129	Q₈×C₈₉	—	2	1	no	—
130	Q₈×C₈₉	—	2	1	no	—
131	Q₈×C₈₉	—	4	2	no	—
132	Q₈×C₈₉	C₂×D₄	1	2	yes	—
133	D₄×C₈₉	—	2	1	no	—
134	D₄×C₈₉	—	4	2	no	—
135	D₄×C₈₉	C₂×D₄	1	2	yes	—
136	C₂×C₄×C₈₉	—	2	1	no	—
137	C₂×C₄×C₈₉	—	2	1	no	—
138	C₂×C₄×C₈₉	—	2	1	no	—
139	C₂×C₄×C₈₉	—	2	1	no	—
140	C₂×C₄×C₈₉	—	2	1	no	—
141	C₂×C₄×C₈₉	—	4	2	no	—
142	C₂×C₄×C₈₉	—	4	2	no	—
143	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
145	C₂²⋊D₈₉	—	2	2	no	—
146	C₂²⋊D₈₉	—	4	2	no	—
147	C₂×Dic₈₉	—	2	1	no	—
148	C₂×Dic₈₉	—	2	1	no	—
149	C₂×Dic₈₉	—	2	1	no	—
150	C₂×Dic₈₉	—	2	1	no	—
151	C₂×Dic₈₉	—	4	2	no	—
152	C₂×Dic₈₉	—	4	2	no	—
153	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
154	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
155	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
156	C₄×D₈₉	—	2	1	no	—
157	C₄×D₈₉	—	2	1	no	—
158	C₄×D₈₉	—	4	2	no	—
159	Dic₁₇₈	—	2	1	no	—
160	Dic₁₇₈	—	2	1	no	—
161	Dic₁₇₈	—	2	1	no	—
162	Dic₁₇₈	—	2	1	no	—
163	Dic₁₇₈	—	4	2	no	—
164	C₈×C₈₉	—	2	1	no	—
165	C₈×C₈₉	C₂×D₄	1	2	yes	—
166	#10623494	—	2	1	no	—
167	#10623494	—	2	1	no	—
168	C₂²×C₈₉	—	2	2	no	—
169	C₂²×C₈₉	—	2	1	no	—
170	C₂²×C₈₉	—	2	1	no	—
171	C₂²×C₈₉	—	2	1	no	—
172	C₂²×C₈₉	—	4	1	no	—
173	C₂²×C₈₉	—	4	1	no	—
174	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
175	C₂×D₈₉	—	2	1	no	—
176	C₂×D₈₉	—	2	1	no	—
177	C₂×D₈₉	—	2	1	no	—
178	C₄×C₈₉	—	2	2	no	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	4	1	no	—
181	C₄×C₈₉	—	4	1	no	—
182	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
183	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
184	Dic₈₉	—	2	1	no	—
185	Dic₈₉	—	2	1	no	—
186	Dic₈₉	—	4	1	no	—
187	Dic₈₉	—	4	1	no	—
188	Dic₈₉	—	4	1	no	—
189	Dic₈₉	—	4	1	no	—
190	Dic₈₉	#137	1	2	yes	complemented
191	C₂×C₈₉	—	4	1	no	—
192	C₂×C₈₉	—	4	2	no	—
193	C₂×C₈₉	D₄²	1	1	yes	characteristic
194	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
195	C₂×C₈₉	#578	1	1	yes	characteristic
196	D₈₉	—	4	1	no	—
197	#3276	—	89	1	no	maximal, Sylow
198	C₈₉	#3276	1	1	yes	characteristic, complemented, Sylow
199	C₂×#137	—	89	1	no	—
200	C₂×#137	D₈₉	1	1	yes	characteristic, complemented
201	D₄×Q₈	—	89	1	no	—
202	#547	—	89	1	no	—
203	#494	—	89	1	no	—
204	#470	—	89	1	no	—
205	#470	—	89	1	no	—
206	#462	—	89	1	no	—
207	#461	—	89	1	no	—
208	#452	—	89	1	no	—
209	#452	—	89	1	no	—
210	#451	—	89	1	no	—
211	#451	—	89	1	no	—
212	C₄×SD₁₆	—	89	1	no	—
213	#436	—	89	1	no	—
214	C₂×#41	—	89	1	no	—
215	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
216	C₂²×Q₈	—	89	1	no	—
217	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
218	#137	—	2	4	no	—
219	#137	—	178	4	no	—
220	C₂×Q₁₆	—	89	1	no	—
221	C₂×Q₁₆	—	89	1	no	—
222	C₂×Q₁₆	—	89	1	no	—
223	C₂×Q₁₆	—	89	1	no	—
224	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
225	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
226	C₂×SD₁₆	—	89	1	no	—
227	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
228	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
229	C₂×M₁₆	—	89	1	no	—
230	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
231	C₄⋊Q₈	—	89	1	no	—
232	C₄⋊Q₈	—	89	1	no	—
233	C₄⋊Q₈	—	178	1	no	—
234	#123	—	178	1	no	—
235	#123	—	178	1	no	—
236	C₂²⋊Q₈	—	89	1	no	—
237	C₂²⋊Q₈	—	89	1	no	—
238	C₂²⋊Q₈	—	89	1	no	—
239	C₂²⋊Q₈	—	89	1	no	—
240	C₂²⋊Q₈	—	178	1	no	—
241	C₂²⋊Q₈	—	178	1	no	—
242	#121	—	178	1	no	—
243	C₄×Q₈	—	89	1	no	—
244	C₄×D₄	—	89	1	no	—
245	C₄×D₄	—	89	1	no	—
246	C₄×D₄	—	178	1	no	—
247	C₂×C₄⋊C₄	—	178	1	no	—
248	#106	—	89	1	no	—
249	C₄⋊C₈	—	89	1	no	—
250	#103	—	89	1	no	—
251	#103	—	89	1	no	—
252	#103	—	89	1	no	—
253	#103	—	89	1	no	—
254	#103	—	89	1	no	—
255	#102	—	89	1	no	—
256	C₂²⋊C₈	—	89	1	no	—
257	C₂²⋊C₈	—	89	1	no	—
258	C₄×C₈	—	89	1	no	—
259	#41	—	2	2	no	—
260	#41	—	4	1	no	—
261	#41	—	356	1	no	—
262	#41	—	178	2	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	89	1	no	—
267	C₂×Q₈	—	89	1	no	—
268	C₂×Q₈	—	356	1	no	—
269	C₂×Q₈	—	178	1	no	—
270	C₂×Q₈	—	178	2	no	—
271	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
272	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
273	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	89	1	no	—
276	C₂×D₄	—	178	1	no	—
277	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
278	C₂²×C₄	—	2	1	no	—
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	89	1	no	—
281	C₂²×C₄	—	178	1	no	—
282	C₂²×C₄	—	178	1	no	—
283	C₂²×C₄	—	178	1	no	—
284	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
285	Q₁₆	—	2	2	no	—
286	Q₁₆	—	2	2	no	—
287	Q₁₆	—	356	1	no	—
288	Q₁₆	—	356	1	no	—
289	Q₁₆	—	178	2	no	—
290	Q₁₆	—	178	2	no	—
291	SD₁₆	—	2	2	no	—
292	SD₁₆	—	178	2	no	—
293	SD₁₆	C₂²⋊D₈₉	1	4	yes	complemented
294	M₁₆	—	2	2	no	—
295	M₁₆	—	178	2	no	—
296	C₂×C₈	—	89	1	no	—
297	C₂×C₈	—	89	1	no	—
298	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
299	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
300	C₄⋊C₄	—	89	1	no	—
301	C₄⋊C₄	—	89	1	no	—
302	C₄⋊C₄	—	89	1	no	—
303	C₄⋊C₄	—	178	1	no	—
304	C₄⋊C₄	—	178	1	no	—
305	C₄⋊C₄	—	178	1	no	—
306	C₄⋊C₄	—	178	1	no	—
307	C₄⋊C₄	—	178	1	no	—
308	C₄⋊C₄	—	178	1	no	—
309	C₄⋊C₄	—	178	1	no	—
310	C₄⋊C₄	—	178	1	no	—
311	C₄⋊C₄	—	178	1	no	—
312	C₂²⋊C₄	—	89	1	no	—
313	C₂²⋊C₄	—	89	1	no	—
314	C₂²⋊C₄	—	178	1	no	—
315	C₂²⋊C₄	—	178	1	no	—
316	C₂²⋊C₄	—	178	1	no	—
317	C₂²⋊C₄	—	178	1	no	—
318	C₂²⋊C₄	—	178	1	no	—
319	C₄²	—	89	1	no	—
320	C₄²	—	178	1	no	—
321	C₂³	—	2	1	no	—
322	C₂³	—	89	1	no	—
323	C₂³	D₄×D₈₉	1	1	yes	characteristic
324	Q₈	—	2	1	no	—
325	Q₈	—	2	1	no	—
326	Q₈	—	2	1	no	—
327	Q₈	—	2	1	no	—
328	Q₈	—	2	1	no	—
329	Q₈	—	4	2	no	—
330	Q₈	—	356	2	no	—
331	Q₈	—	178	1	no	—
332	Q₈	—	178	1	no	—
333	Q₈	—	178	1	no	—
334	Q₈	—	178	1	no	—
335	Q₈	C₂×C₂²⋊D₈₉	1	2	yes	—
336	D₄	—	2	1	no	—
337	D₄	—	4	2	no	—
338	D₄	—	356	2	no	—
339	D₄	—	178	2	no	—
340	D₄	C₂×C₂²⋊D₈₉	1	2	yes	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	89	1	no	—
349	C₂×C₄	—	89	1	no	—
350	C₂×C₄	—	89	1	no	—
351	C₂×C₄	—	356	2	no	—
352	C₂×C₄	—	356	2	no	—
353	C₂×C₄	—	356	2	no	—
354	C₂×C₄	—	178	1	no	—
355	C₂×C₄	—	178	1	no	—
356	C₂×C₄	—	178	1	no	—
357	C₂×C₄	—	178	1	no	—
358	C₂×C₄	—	178	1	no	—
359	C₂×C₄	—	178	1	no	—
360	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
361	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
362	C₈	—	2	1	no	—
363	C₈	—	178	1	no	—
364	C₈	—	178	1	no	—
365	C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	2	2	no	—
370	C₂²	—	4	1	no	—
371	C₂²	—	4	1	no	—
372	C₂²	—	178	1	no	—
373	C₂²	—	178	1	no	—
374	C₂²	—	178	1	no	—
375	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	2	2	no	—
378	C₄	—	4	1	no	—
379	C₄	—	4	1	no	—
380	C₄	—	89	2	no	—
381	C₄	—	356	1	no	—
382	C₄	—	356	1	no	—
383	C₄	—	356	1	no	—
384	C₄	—	356	1	no	—
385	C₄	—	178	1	no	—
386	C₄	—	178	1	no	—
387	C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
388	C₄	#15668057	1	1	yes	characteristic
389	C₂	—	4	1	no	—
390	C₂	—	4	2	no	—
391	C₂	—	356	1	no	—
392	C₂	D₄×C₂²⋊D₈₉	1	1	yes	characteristic
393	C₂	#137×D₈₉	1	1	yes	characteristic
394	C₂	#19152414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×#137×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152247	C₂
hypercenter	C₂×#137	D₈₉
hypocenter	C₈₉	#3276
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#137

Complements

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

N	H ≅ G/N	type
1. C₂×#15668041	C₂	semidirect
2. Q₈×C₂²⋊D₈₉	C₂	semidirect
4. #19151976	C₂	semidirect
5. #19151856	C₂	semidirect
6. #19151853	C₂	semidirect
7. #19151806	C₂	semidirect
8. #19151801	C₂	semidirect
9. #19151760	C₂	semidirect
10. #19151757	C₂	semidirect
11. #19151754	C₂	semidirect
12. #19151752	C₂	semidirect
13. SD₁₆×Dic₈₉	C₂	semidirect
14. #19151674	C₂	semidirect
15. C₂×#137×C₈₉	C₂	semidirect
119. SD₁₆×C₈₉	D₄	semidirect
190. Dic₈₉	#137	semidirect
198. C₈₉	#3276	semidirect
200. C₂×#137	D₈₉	semidirect
293. SD₁₆	C₂²⋊D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668041	2
Q₈×C₂²⋊D₈₉	2
#19152247	2
#19151976	2
#19151856	2
#19151853	2
#19151806	2
#19151801	2
#19151760	2
#19151757	2
#19151754	2
#19151752	2
SD₁₆×Dic₈₉	2
#19151674	2
C₂×#137×C₈₉	2
#3276	89