Subgroups — #36538267

Order 22784.

710 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	#26029720	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#26019693	C₂	1	4	yes	maximal, complemented
4	#26019643	C₂	1	2	yes	maximal, complemented
5	#26019609	C₂	1	2	yes	maximal, complemented
6	#26019011	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018929	C₂	1	2	yes	maximal, complemented
8	C₈₉×#3169	C₂	1	1	yes	maximal, characteristic
9	C₂×#15668042	—	2	2	no	—
10	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
11	#19151817	—	2	4	no	—
12	#19151805	C₂²	1	4	yes	—
13	#19151795	C₂²	1	4	yes	—
14	C₈×Dic₁₇₈	—	2	1	no	—
15	C₈×Dic₁₇₈	—	2	1	no	—
16	C₈×Dic₁₇₈	C₂²	1	2	yes	—
17	C₈×Dic₁₇₈	C₂²	1	2	yes	—
18	#19151622	C₂²	1	2	yes	—
19	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
20	C₂×#15667941	—	2	1	no	—
21	#19151599	—	2	2	no	—
22	C₂×C₈×Dic₈₉	—	2	1	no	—
23	#19151351	C₂²	1	2	yes	—
24	#19151351	C₂²	1	2	yes	—
25	#19151344	C₂²	1	4	yes	—
26	#19151342	C₂²	1	4	yes	—
27	#532×C₈₉	C₂²	1	2	yes	—
28	#500×C₈₉	C₂²	1	1	yes	characteristic
29	#493×C₈₉	C₂²	1	1	yes	characteristic
30	#486×C₈₉	C₂²	1	2	yes	—
31	#466×C₈₉	—	2	1	no	—
32	#466×C₈₉	—	2	1	no	—
33	C₄×D₈×C₈₉	—	2	1	no	—
34	C₄×D₈×C₈₉	—	2	1	no	—
35	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₂²×Dic₁₇₈	D₄	1	1	yes	characteristic
37	#15668042	—	4	8	no	—
38	C₂×#13083564	—	2	4	no	—
39	C₂×#13083562	—	2	4	no	—
40	C₂×#13083558	—	2	2	no	—
41	C₂²×#10623494	—	2	1	no	—
42	C₄×Dic₁₇₈	—	2	1	no	—
43	C₄×Dic₁₇₈	—	2	1	no	—
44	C₄×Dic₁₇₈	D₄	1	2	yes	—
45	C₄×Dic₁₇₈	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
47	C₂×#13083553	—	2	1	no	—
48	C₂×C₄×Dic₈₉	—	2	1	no	—
49	C₄⋊#10623494	—	2	1	no	—
50	C₄⋊#10623494	—	2	2	no	—
51	C₄⋊#10623494	C₂³	1	2	yes	—
52	C₄⋊#10623494	C₂³	1	4	yes	—
53	#15667941	—	2	2	no	—
54	#15667941	—	2	2	no	—
55	#15667936	—	2	4	no	—
56	#15667933	—	2	4	no	—
57	C₈×Dic₈₉	—	2	2	no	—
58	C₈×Dic₈₉	—	2	2	no	—
59	C₂×#41×C₈₉	—	2	2	no	—
60	C₂×D₈×C₈₉	—	2	1	no	—
61	C₂×D₈×C₈₉	—	2	1	no	—
62	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
64	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
65	#127×C₈₉	C₂³	1	2	yes	—
66	#124×C₈₉	—	2	2	no	—
67	#121×C₈₉	—	2	2	no	—
68	#121×C₈₉	—	2	2	no	—
69	C₄×D₄×C₈₉	—	2	2	no	—
70	C₄×D₄×C₈₉	—	2	2	no	—
71	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
72	#107×C₈₉	—	2	1	no	—
73	#107×C₈₉	—	2	1	no	—
74	#102×C₈₉	—	2	2	no	—
75	#102×C₈₉	—	2	2	no	—
76	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	C₂×Dic₁₇₈	—	2	1	no	—
81	C₂×Dic₁₇₈	—	2	2	no	—
82	C₂×Dic₁₇₈	D₈	1	4	yes	complemented
83	C₂×Dic₁₇₈	C₂×D₄	1	2	yes	—
84	C₂²×Dic₈₉	—	2	1	no	—
85	C₂²×Dic₈₉	—	2	1	no	—
86	#13083564	—	4	8	no	—
87	#13083562	—	4	8	no	—
88	#13083558	—	4	4	no	—
89	C₂×#10623494	—	2	2	no	—
90	C₂×#10623494	—	2	4	no	—
91	C₂×#10623494	—	4	2	no	—
92	C₄⋊Dic₈₉	—	2	1	no	—
93	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
94	#13083553	—	2	2	no	—
95	#13083553	—	2	2	no	—
96	C₄×Dic₈₉	—	2	2	no	—
97	C₄×Dic₈₉	—	2	2	no	—
98	#41×C₈₉	—	4	4	no	—
99	#41×C₈₉	—	4	4	no	—
100	C₂×Q₈×C₈₉	—	2	2	no	—
101	C₂×D₄×C₈₉	—	2	2	no	—
102	C₂×D₄×C₈₉	—	2	2	no	—
103	C₂×D₄×C₈₉	—	2	2	no	—
104	C₂×D₄×C₈₉	—	4	2	no	—
105	C₂²×C₄×C₈₉	—	4	2	no	—
106	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
109	D₈×C₈₉	—	4	2	no	—
110	D₈×C₈₉	—	4	2	no	—
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	—	4	1	no	—
113	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₄⋊C₄×C₈₉	—	2	2	no	—
118	C₄⋊C₄×C₈₉	—	2	2	no	—
119	C₂²⋊C₄×C₈₉	—	4	2	no	—
120	C₂²⋊C₄×C₈₉	—	4	2	no	—
121	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₈₉	#41	1	1	yes	characteristic
124	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
125	C₂³×C₈₉	—	4	2	no	—
126	C₂³×C₈₉	#121	1	1	yes	characteristic
127	Q₈×C₈₉	—	4	2	no	—
128	Q₈×C₈₉	—	4	2	no	—
129	D₄×C₈₉	—	4	2	no	—
130	D₄×C₈₉	—	4	2	no	—
131	D₄×C₈₉	—	4	2	no	—
132	D₄×C₈₉	—	4	2	no	—
133	D₄×C₈₉	—	8	2	no	—
134	D₄×C₈₉	—	8	4	no	—
135	C₂×C₄×C₈₉	—	2	1	no	—
136	C₂×C₄×C₈₉	—	2	1	no	—
137	C₂×C₄×C₈₉	—	4	1	no	—
138	C₂×C₄×C₈₉	—	4	2	no	—
139	C₂×C₄×C₈₉	—	8	4	no	—
140	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
141	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
142	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₈₉	#135	1	2	yes	—
145	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₈₉	—	2	2	no	—
149	C₂×Dic₈₉	—	2	4	no	—
150	C₂×Dic₈₉	—	2	2	no	—
151	C₂×Dic₈₉	—	4	2	no	—
152	Dic₁₇₈	—	2	2	no	—
153	Dic₁₇₈	—	2	4	no	—
154	Dic₁₇₈	C₂×D₈	1	4	yes	—
155	C₈×C₈₉	—	2	1	no	—
156	C₈×C₈₉	—	2	1	no	—
157	C₈×C₈₉	—	2	1	no	—
158	C₈×C₈₉	#137	1	2	yes	complemented
159	#10623494	—	4	4	no	—
160	#10623494	—	4	2	no	—
161	C₂²×C₈₉	—	2	1	no	—
162	C₂²×C₈₉	—	2	1	no	—
163	C₂²×C₈₉	—	8	2	no	—
164	C₂²×C₈₉	—	8	2	no	—
165	C₂²×C₈₉	—	8	2	no	—
166	C₂²×C₈₉	#466	1	2	yes	—
167	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	2	1	no	—
171	C₄×C₈₉	—	8	2	no	—
172	C₄×C₈₉	#466	1	2	yes	—
173	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
174	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
175	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
176	C₄×C₈₉	#575	1	1	yes	characteristic
177	Dic₈₉	—	2	4	no	—
178	Dic₈₉	—	4	2	no	—
179	#19261	—	89	1	no	maximal, Sylow
180	C₂×C₈₉	—	2	2	no	—
181	C₂×C₈₉	—	16	2	no	—
182	C₂×C₈₉	#3037	1	1	yes	characteristic
183	C₂×C₈₉	C₂×#466	1	1	yes	characteristic
184	C₂×C₈₉	#3067	1	1	yes	characteristic
185	#3169	D₈₉	1	1	yes	characteristic
186	#3037	—	89	2	no	—
187	C₂×C₈×Q₈	—	89	1	no	—
188	#1682	—	89	4	no	—
189	#1674	—	89	2	no	—
190	#1668	—	89	2	no	—
191	#1567	—	89	1	no	—
192	#1547	—	89	2	no	—
193	C₈₉	#19261	1	1	yes	characteristic, complemented, Sylow
194	C₂×#137	—	178	2	no	—
195	#532	C₂×D₈₉	1	2	yes	—
196	C₂×C₄×Q₈	—	89	1	no	—
197	#500	C₂×D₈₉	1	1	yes	characteristic
198	#493	C₂×D₈₉	1	1	yes	characteristic
199	#486	C₂×D₈₉	1	2	yes	—
200	#466	—	2	1	no	—
201	#466	—	2	1	no	—
202	#464	—	178	4	no	—
203	#462	—	89	4	no	—
204	#460	—	89	4	no	—
205	C₈×Q₈	—	89	2	no	—
206	C₈×Q₈	—	89	2	no	—
207	C₈×Q₈	—	178	1	no	—
208	C₈×Q₈	—	178	1	no	—
209	C₄×D₈	—	2	1	no	—
210	C₄×D₈	—	2	1	no	—
211	#423	—	89	2	no	—
212	C₂×C₄⋊C₈	—	89	1	no	—
213	C₂×C₄⋊C₈	—	178	1	no	—
214	#417	—	178	2	no	—
215	C₂×C₄×C₈	—	178	1	no	—
216	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
217	#335	—	89	2	no	—
218	#335	—	89	2	no	—
219	#332	—	89	4	no	—
220	#331	—	89	4	no	—
221	C₂×#41	—	2	2	no	—
222	C₂²×Q₈	—	89	1	no	—
223	#137	—	356	8	no	—
224	C₂×Q₁₆	—	178	4	no	—
225	C₂×SD₁₆	—	178	4	no	—
226	C₂×D₈	—	2	1	no	—
227	C₂×D₈	—	2	1	no	—
228	C₂×M₁₆	—	178	2	no	—
229	C₂²×C₈	—	178	1	no	—
230	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
231	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
232	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
233	#127	C₂²×D₈₉	1	2	yes	—
234	#124	—	2	2	no	—
235	#121	—	2	2	no	—
236	#121	—	2	2	no	—
237	C₄×Q₈	—	89	2	no	—
238	C₄×Q₈	—	89	2	no	—
239	C₄×Q₈	—	178	1	no	—
240	C₄×Q₈	—	178	1	no	—
241	C₄×D₄	—	2	2	no	—
242	C₄×D₄	—	2	2	no	—
243	C₂×C₄⋊C₄	—	89	1	no	—
244	C₂×C₄⋊C₄	—	178	1	no	—
245	C₂×C₄²	—	178	1	no	—
246	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
247	#107	—	2	1	no	—
248	#107	—	2	1	no	—
249	C₄⋊C₈	—	89	2	no	—
250	C₄⋊C₈	—	89	4	no	—
251	C₄⋊C₈	—	178	2	no	—
252	C₄⋊C₈	—	178	2	no	—
253	C₄⋊C₈	—	178	2	no	—
254	C₄⋊C₈	—	178	1	no	—
255	#103	—	178	4	no	—
256	#102	—	2	2	no	—
257	#102	—	2	2	no	—
258	#102	—	178	4	no	—
259	C₄×C₈	—	178	2	no	—
260	C₄×C₈	—	178	2	no	—
261	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
262	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
265	#41	—	4	4	no	—
266	#41	—	4	4	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	89	4	no	—
269	C₂×Q₈	—	89	2	no	—
270	C₂×Q₈	—	178	1	no	—
271	C₂×Q₈	—	178	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	4	2	no	—
276	C₂²×C₄	—	4	2	no	—
277	C₂²×C₄	—	178	1	no	—
278	C₂²×C₄	—	178	1	no	—
279	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
280	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
281	C₂²×C₄	#13083573	1	1	yes	characteristic
282	Q₁₆	—	356	8	no	—
283	SD₁₆	—	356	8	no	—
284	D₈	—	4	2	no	—
285	D₈	—	4	2	no	—
286	M₁₆	—	356	4	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	356	2	no	—
290	C₂×C₈	—	178	4	no	—
291	C₂×C₈	—	178	2	no	—
292	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
293	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
294	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
295	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	89	2	no	—
299	C₄⋊C₄	—	178	1	no	—
300	C₄⋊C₄	—	178	2	no	—
301	C₄⋊C₄	—	178	2	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	4	2	no	—
304	C₄²	—	178	2	no	—
305	C₄²	—	178	2	no	—
306	C₄²	D₄×D₈₉	1	1	yes	characteristic
307	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
308	C₄²	#13083573	1	1	yes	characteristic
309	C₄²	C₂³×D₈₉	1	1	yes	characteristic
310	C₂³	—	4	2	no	—
311	C₂³	#15667983	1	1	yes	characteristic
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	89	4	no	—
315	Q₈	—	178	4	no	—
316	Q₈	—	178	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	4	2	no	—
321	D₄	—	8	2	no	—
322	D₄	—	8	4	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	8	4	no	—
328	C₂×C₄	—	356	2	no	—
329	C₂×C₄	—	178	2	no	—
330	C₂×C₄	—	178	4	no	—
331	C₂×C₄	—	178	2	no	—
332	C₂×C₄	#15667983	1	1	yes	characteristic
333	C₂×C₄	#15667983	1	1	yes	characteristic
334	C₂×C₄	#15667983	1	1	yes	characteristic
335	C₂×C₄	D₈×D₈₉	1	2	yes	—
336	C₂×C₄	#15668031	1	2	yes	—
337	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
338	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
339	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	356	2	no	—
344	C₈	—	356	4	no	—
345	C₈	#15668042	1	2	yes	complemented
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	8	2	no	—
351	C₂²	#19151831	1	2	yes	—
352	C₂²	C₂×#15667983	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	8	2	no	—
357	C₄	—	356	2	no	—
358	C₄	—	178	4	no	—
359	C₄	#19151831	1	2	yes	—
360	C₄	C₂×D₈×D₈₉	1	1	yes	characteristic
361	C₄	C₂×#15668031	1	1	yes	characteristic
362	C₄	C₂×#15668042	1	1	yes	characteristic
363	C₄	#19152396	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	16	2	no	—
366	C₂	#26029720	1	1	yes	characteristic
367	C₂	C₂×#19151831	1	1	yes	characteristic
368	C₂	#26029968	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667983
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting, prime radical	C₈₉×#3169	C₂
socle	C₂²×C₈₉	C₂×#121
hypercenter	#3169	D₈₉
hypocenter	C₈₉	#19261
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #26029720	C₂	semidirect
2. C₂×C₈×Dic₁₇₈	C₂	semidirect
3. #26019693	C₂	semidirect
4. #26019643	C₂	semidirect
5. #26019609	C₂	semidirect
6. #26019011	C₂	semidirect
7. #26018929	C₂	semidirect
82. C₂×Dic₁₇₈	D₈	semidirect
158. C₈×C₈₉	#137	semidirect
193. C₈₉	#19261	semidirect
345. C₈	#15668042	semidirect

Maximal subgroups

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

subgroup	index
#26029720	2
C₂×C₈×Dic₁₇₈	2
#26019693	2
#26019643	2
#26019609	2
#26019011	2
#26018929	2
C₈₉×#3169	2
#19261	89