Subgroups — #26031916

Order 11392.

482 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668043	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15668041	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152266	C₂	1	1	yes	maximal, characteristic
4	Q₈×C₂²⋊D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151969	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151847	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151842	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151807	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151796	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151762	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151757	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151738	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151690	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151673	C₂	1	1	yes	maximal, characteristic, complemented
15	#448×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	C₂²	1	1	yes	characteristic
17	C₂×#13083573	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
19	#15668043	—	2	4	no	—
20	#15668041	—	2	4	no	—
21	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
22	C₂×#13083564	C₂²	1	1	yes	characteristic
23	C₂×#13083564	C₂²	1	1	yes	characteristic
24	C₂×#13083563	C₂²	1	1	yes	characteristic
25	C₂×#13083562	C₂²	1	1	yes	characteristic
26	C₂×#13083561	C₂²	1	1	yes	characteristic
27	C₂×#13083561	C₂²	1	1	yes	characteristic
28	C₂×#13083558	C₂²	1	1	yes	characteristic
29	C₂×#13083557	C₂²	1	1	yes	characteristic
30	#15668010	—	2	1	no	—
31	#15668010	C₂²	1	1	yes	characteristic
32	#15668008	—	2	1	no	—
33	#15668008	C₂²	1	1	yes	characteristic
34	#15667991	—	2	1	no	—
35	#15667991	C₂²	1	1	yes	characteristic
36	#15667988	—	2	1	no	—
37	#15667988	C₂²	1	1	yes	characteristic
38	#15667987	C₂²	1	1	yes	characteristic
39	#15667985	—	2	1	no	—
40	#15667984	—	2	1	no	—
41	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
42	#15667977	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊D₈₉	—	2	1	no	—
44	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
45	#15667972	—	2	1	no	—
46	#15667945	C₂²	1	1	yes	characteristic
47	#15667941	C₂²	1	1	yes	characteristic
48	#15667937	C₂²	1	1	yes	characteristic
49	#15667935	C₂²	1	1	yes	characteristic
50	#15667934	C₂²	1	1	yes	characteristic
51	#15667933	C₂²	1	1	yes	characteristic
52	#15667924	C₂²	1	1	yes	characteristic
53	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
54	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
56	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
57	#121×C₈₉	C₂²	1	1	yes	characteristic
58	#103×C₈₉	C₂²	1	1	yes	characteristic
59	#103×C₈₉	C₂²	1	1	yes	characteristic
60	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
61	#13083574	—	2	2	no	—
62	#13083574	—	4	1	no	—
63	#13083573	—	2	2	no	—
64	#13083573	—	4	1	no	—
65	C₂×Dic₁₇₈	—	2	1	no	—
66	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
68	Q₈×D₈₉	—	2	2	no	—
69	Q₈×D₈₉	—	4	1	no	—
70	C₂×D₃₅₆	—	2	1	no	—
71	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
72	C₂×C₂²⋊D₈₉	—	2	1	no	—
73	C₂×C₂²⋊D₈₉	—	2	1	no	—
74	C₂×C₂²⋊D₈₉	—	2	1	no	—
75	C₂×C₂²⋊D₈₉	—	2	1	no	—
76	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
77	C₂²×Dic₈₉	—	2	1	no	—
78	C₂×C₄×D₈₉	—	2	1	no	—
79	C₂×C₄×D₈₉	—	2	1	no	—
80	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
81	Dic₃₅₆	—	2	2	no	—
82	#13083564	—	2	2	no	—
83	#13083564	D₄	1	4	yes	complemented
84	#13083563	—	4	1	no	—
85	#13083562	—	2	2	no	—
86	#13083561	—	2	2	no	—
87	#13083561	—	2	2	no	—
88	#13083558	—	2	2	no	—
89	#13083557	—	2	2	no	—
90	C₂×#10623494	C₂³	1	1	yes	characteristic
91	C₂×#10623494	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₈₉	—	2	1	no	—
93	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
94	#13083553	—	2	1	no	—
95	#13083553	—	2	1	no	—
96	#13083553	—	2	1	no	—
97	#13083553	—	2	1	no	—
98	#13083553	—	2	1	no	—
99	#13083553	D₄	1	1	yes	characteristic
100	C₂²⋊Dic₈₉	—	2	1	no	—
101	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
102	#13083551	—	2	1	no	—
103	#13083551	—	2	1	no	—
104	#13083551	D₄	1	1	yes	characteristic
105	C₄×Dic₈₉	—	2	1	no	—
106	C₄×Dic₈₉	—	2	1	no	—
107	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
108	C₂×Q₈×C₈₉	—	2	1	no	—
109	C₂×Q₈×C₈₉	—	2	2	no	—
110	C₂×Q₈×C₈₉	—	4	1	no	—
111	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
112	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
113	C₂×D₄×C₈₉	—	2	1	no	—
114	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
115	C₂²×C₄×C₈₉	—	2	1	no	—
116	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
117	SD₁₆×C₈₉	—	2	2	no	—
118	SD₁₆×C₈₉	—	4	1	no	—
119	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
121	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
122	C₂²⋊C₄×C₈₉	—	2	1	no	—
123	C₂³×C₈₉	—	2	1	no	—
124	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂²×D₈₉	—	2	1	no	—
126	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
127	Q₈×C₈₉	—	2	1	no	—
128	Q₈×C₈₉	—	2	1	no	—
129	Q₈×C₈₉	—	2	1	no	—
130	Q₈×C₈₉	—	4	2	no	—
131	Q₈×C₈₉	C₂×D₄	1	2	yes	—
132	D₄×C₈₉	—	2	1	no	—
133	D₄×C₈₉	—	2	1	no	—
134	D₄×C₈₉	—	8	1	no	—
135	C₂×C₄×C₈₉	—	2	1	no	—
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	—	4	2	no	—
141	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
143	C₂²⋊D₈₉	—	2	2	no	—
144	C₂²⋊D₈₉	—	4	2	no	—
145	C₂²⋊D₈₉	—	4	2	no	—
146	C₂²⋊D₈₉	—	4	2	no	—
147	C₂²⋊D₈₉	—	8	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	C₂×D₄	1	1	yes	characteristic
153	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
154	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
155	D₃₅₆	—	2	1	no	—
156	D₃₅₆	—	2	1	no	—
157	D₃₅₆	—	8	1	no	—
158	C₄×D₈₉	—	2	1	no	—
159	C₄×D₈₉	—	2	1	no	—
160	C₄×D₈₉	—	4	2	no	—
161	C₄×D₈₉	—	4	2	no	—
162	Dic₁₇₈	—	2	1	no	—
163	Dic₁₇₈	—	2	1	no	—
164	Dic₁₇₈	—	2	1	no	—
165	Dic₁₇₈	—	4	2	no	—
166	Dic₁₇₈	C₂×D₄	1	2	yes	—
167	C₈×C₈₉	—	2	1	no	—
168	C₈×C₈₉	—	2	1	no	—
169	#10623494	—	2	1	no	—
170	#10623494	C₂×D₄	1	2	yes	—
171	C₂²×C₈₉	—	2	1	no	—
172	C₂²×C₈₉	—	2	1	no	—
173	C₂²×C₈₉	—	2	1	no	—
174	C₂²×C₈₉	—	4	1	no	—
175	C₂²×C₈₉	—	4	1	no	—
176	C₂²×C₈₉	—	4	1	no	—
177	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
178	C₂×D₈₉	—	2	1	no	—
179	C₂×D₈₉	—	2	1	no	—
180	C₂×D₈₉	—	2	1	no	—
181	C₂×D₈₉	—	4	1	no	—
182	C₂×D₈₉	—	4	1	no	—
183	C₂×D₈₉	—	4	1	no	—
184	C₄×C₈₉	—	2	2	no	—
185	C₄×C₈₉	—	2	1	no	—
186	C₄×C₈₉	—	4	1	no	—
187	C₄×C₈₉	—	4	1	no	—
188	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
189	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
190	Dic₈₉	—	2	2	no	—
191	Dic₈₉	—	2	1	no	—
192	Dic₈₉	—	2	1	no	—
193	Dic₈₉	—	4	1	no	—
194	Dic₈₉	—	4	1	no	—
195	Dic₈₉	#137	1	2	yes	complemented
196	C₂×C₈₉	—	4	1	no	—
197	C₂×C₈₉	—	8	1	no	—
198	C₂×C₈₉	D₄²	1	1	yes	characteristic
199	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
200	C₂×C₈₉	#577	1	1	yes	characteristic
201	D₈₉	—	4	1	no	—
202	D₈₉	—	8	1	no	—
203	#3278	—	89	1	no	maximal, Sylow
204	C₈₉	#3278	1	1	yes	characteristic, complemented, Sylow
205	C₂×#137	—	89	1	no	—
206	C₂×#137	—	89	1	no	—
207	#550	—	89	1	no	—
208	D₄×Q₈	—	89	1	no	—
209	#492	—	89	1	no	—
210	#468	—	89	1	no	—
211	#468	—	89	1	no	—
212	#462	—	89	1	no	—
213	#460	—	89	1	no	—
214	#452	—	89	1	no	—
215	#452	—	89	1	no	—
216	#448	—	89	1	no	—
217	#448	D₈₉	1	1	yes	characteristic, complemented
218	C₄×Q₁₆	—	89	1	no	—
219	#436	—	89	1	no	—
220	C₂×#41	—	89	1	no	—
221	C₂×#41	—	89	1	no	—
222	C₂²×Q₈	—	89	1	no	—
223	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
224	#137	—	178	4	no	—
225	#137	—	178	4	no	—
226	C₂×Q₁₆	—	89	1	no	—
227	C₂×Q₁₆	—	89	1	no	—
228	C₂×Q₁₆	—	89	1	no	—
229	C₂×SD₁₆	—	89	1	no	—
230	C₂×SD₁₆	—	89	1	no	—
231	C₂×SD₁₆	—	89	1	no	—
232	C₂×SD₁₆	—	89	1	no	—
233	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
234	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
235	C₂×M₁₆	—	89	1	no	—
236	C₂×M₁₆	—	89	1	no	—
237	C₄⋊Q₈	—	89	1	no	—
238	C₄⋊Q₈	—	178	1	no	—
239	#127	—	89	1	no	—
240	#127	—	178	1	no	—
241	C₂²⋊Q₈	—	89	1	no	—
242	C₂²⋊Q₈	—	89	1	no	—
243	C₂²⋊Q₈	—	178	1	no	—
244	C₂²⋊Q₈	—	178	1	no	—
245	#121	—	89	1	no	—
246	#121	—	178	1	no	—
247	#121	—	178	1	no	—
248	#121	C₂×D₈₉	1	1	yes	characteristic
249	C₄×Q₈	—	89	1	no	—
250	C₄×Q₈	—	89	1	no	—
251	C₄×D₄	—	89	1	no	—
252	C₄×D₄	—	178	1	no	—
253	C₄×D₄	—	178	1	no	—
254	#107	—	89	1	no	—
255	C₄⋊C₈	—	89	1	no	—
256	#103	—	89	1	no	—
257	#103	—	89	1	no	—
258	#103	C₂×D₈₉	1	1	yes	characteristic
259	#103	C₂×D₈₉	1	1	yes	characteristic
260	#102	—	89	1	no	—
261	#102	—	89	1	no	—
262	C₂²⋊C₈	—	89	1	no	—
263	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
264	C₄×C₈	—	89	1	no	—
265	#41	—	356	1	no	—
266	#41	—	356	1	no	—
267	#41	—	178	2	no	—
268	#41	—	178	2	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	4	1	no	—
272	C₂×Q₈	—	89	1	no	—
273	C₂×Q₈	—	89	1	no	—
274	C₂×Q₈	—	356	1	no	—
275	C₂×Q₈	—	178	1	no	—
276	C₂×Q₈	—	178	2	no	—
277	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
278	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
279	C₂×D₄	—	2	1	no	—
280	C₂×D₄	—	89	1	no	—
281	C₂×D₄	—	89	1	no	—
282	C₂×D₄	—	178	1	no	—
283	C₂×D₄	—	178	1	no	—
284	C₂×D₄	—	178	1	no	—
285	C₂×D₄	—	178	1	no	—
286	C₂×D₄	—	178	1	no	—
287	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	89	1	no	—
290	C₂²×C₄	—	178	1	no	—
291	C₂²×C₄	—	178	1	no	—
292	C₂²×C₄	—	178	1	no	—
293	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
294	Q₁₆	—	89	4	no	—
295	Q₁₆	—	178	2	no	—
296	Q₁₆	—	178	2	no	—
297	SD₁₆	—	2	2	no	—
298	SD₁₆	—	4	1	no	—
299	SD₁₆	—	356	1	no	—
300	SD₁₆	—	178	2	no	—
301	SD₁₆	—	178	2	no	—
302	SD₁₆	—	178	2	no	—
303	M₁₆	—	178	2	no	—
304	M₁₆	—	178	2	no	—
305	C₂×C₈	—	89	1	no	—
306	C₂×C₈	—	89	1	no	—
307	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
308	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
309	C₄⋊C₄	—	89	1	no	—
310	C₄⋊C₄	—	89	1	no	—
311	C₄⋊C₄	—	178	1	no	—
312	C₄⋊C₄	—	178	1	no	—
313	C₄⋊C₄	—	178	1	no	—
314	C₄⋊C₄	—	178	1	no	—
315	C₄⋊C₄	—	178	1	no	—
316	C₄⋊C₄	—	178	1	no	—
317	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
318	C₂²⋊C₄	—	2	1	no	—
319	C₂²⋊C₄	—	89	1	no	—
320	C₂²⋊C₄	—	89	1	no	—
321	C₂²⋊C₄	—	178	1	no	—
322	C₂²⋊C₄	—	178	1	no	—
323	C₂²⋊C₄	—	178	1	no	—
324	C₄²	—	89	1	no	—
325	C₄²	—	178	1	no	—
326	C₄²	—	178	1	no	—
327	C₂³	—	2	1	no	—
328	C₂³	—	89	1	no	—
329	C₂³	—	178	1	no	—
330	C₂³	D₄×D₈₉	1	1	yes	characteristic
331	Q₈	—	2	1	no	—
332	Q₈	—	2	1	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	4	2	no	—
335	Q₈	—	89	2	no	—
336	Q₈	—	356	2	no	—
337	Q₈	—	178	1	no	—
338	Q₈	—	178	1	no	—
339	Q₈	—	178	1	no	—
340	Q₈	D₄×D₈₉	1	2	yes	—
341	D₄	—	2	1	no	—
342	D₄	—	2	1	no	—
343	D₄	—	8	1	no	—
344	D₄	—	356	2	no	—
345	D₄	—	356	2	no	—
346	D₄	—	356	2	no	—
347	D₄	—	178	2	no	—
348	D₄	—	178	1	no	—
349	D₄	—	178	1	no	—
350	D₄	—	712	1	no	—
351	D₄	—	712	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	89	1	no	—
359	C₂×C₄	—	89	1	no	—
360	C₂×C₄	—	89	1	no	—
361	C₂×C₄	—	356	2	no	—
362	C₂×C₄	—	356	2	no	—
363	C₂×C₄	—	356	2	no	—
364	C₂×C₄	—	178	1	no	—
365	C₂×C₄	—	178	1	no	—
366	C₂×C₄	—	178	1	no	—
367	C₂×C₄	—	178	1	no	—
368	C₂×C₄	—	178	1	no	—
369	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
370	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	89	2	no	—
374	C₈	—	178	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	4	1	no	—
379	C₂²	—	4	1	no	—
380	C₂²	—	4	1	no	—
381	C₂²	—	356	1	no	—
382	C₂²	—	356	1	no	—
383	C₂²	—	356	1	no	—
384	C₂²	—	178	1	no	—
385	C₂²	—	178	1	no	—
386	C₂²	—	178	1	no	—
387	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
388	C₄	—	2	1	no	—
389	C₄	—	2	2	no	—
390	C₄	—	4	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	89	2	no	—
393	C₄	—	356	1	no	—
394	C₄	—	356	1	no	—
395	C₄	—	178	1	no	—
396	C₄	—	178	1	no	—
397	C₄	—	178	2	no	—
398	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
399	C₄	#15668057	1	1	yes	characteristic
400	C₂	—	4	1	no	—
401	C₂	—	8	1	no	—
402	C₂	—	356	1	no	—
403	C₂	—	712	1	no	—
404	C₂	#19152230	1	1	yes	characteristic
405	C₂	#137×D₈₉	1	1	yes	characteristic
406	C₂	#19152404	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#448×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152266	C₂
hypercenter	#448	D₈₉
hypocenter	C₈₉	#3278
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #448

Complements

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

N	H ≅ G/N	type
1. C₂×#15668043	C₂	semidirect
2. C₂×#15668041	C₂	semidirect
4. Q₈×C₂²⋊D₈₉	C₂	semidirect
5. #19151969	C₂	semidirect
6. #19151847	C₂	semidirect
7. #19151842	C₂	semidirect
8. #19151807	C₂	semidirect
9. #19151796	C₂	semidirect
10. #19151762	C₂	semidirect
11. #19151757	C₂	semidirect
12. #19151738	C₂	semidirect
13. #19151690	C₂	semidirect
14. #19151673	C₂	semidirect
15. #448×C₈₉	C₂	semidirect
83. #13083564	D₄	semidirect
195. Dic₈₉	#137	semidirect
204. C₈₉	#3278	semidirect
217. #448	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668043	2
C₂×#15668041	2
#19152266	2
Q₈×C₂²⋊D₈₉	2
#19151969	2
#19151847	2
#19151842	2
#19151807	2
#19151796	2
#19151762	2
#19151757	2
#19151738	2
#19151690	2
#19151673	2
#448×C₈₉	2
#3278	89