Subgroups — #36536765

Order 22784.

714 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030279	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#26019619	C₂	1	2	yes	maximal, complemented
4	#26019609	C₂	1	2	yes	maximal, complemented
5	#26019597	C₂	1	4	yes	maximal, complemented
6	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018901	C₂	1	2	yes	maximal, complemented
8	C₈₉×#3147	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
10	#19151914	—	2	2	no	—
11	#19151880	C₂²	1	4	yes	—
12	#19151795	C₂²	1	4	yes	—
13	#19151736	—	2	2	no	—
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	C₄×#13083562	—	2	4	no	—
19	#19151651	C₂²	1	2	yes	—
20	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
21	C₂×#15667941	—	2	1	no	—
22	C₂×C₈×Dic₈₉	—	2	1	no	—
23	#19151351	C₂²	1	2	yes	—
24	#19151349	C₂²	1	2	yes	—
25	#19151344	C₂²	1	4	yes	—
26	#19151342	C₂²	1	4	yes	—
27	C₂×#135×C₈₉	—	2	1	no	—
28	C₂×#135×C₈₉	—	2	1	no	—
29	#532×C₈₉	C₂²	1	2	yes	—
30	#495×C₈₉	—	2	1	no	—
31	#495×C₈₉	—	2	1	no	—
32	#494×C₈₉	C₂²	1	1	yes	characteristic
33	#493×C₈₉	C₂²	1	1	yes	characteristic
34	#492×C₈₉	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₂²×Dic₁₇₈	D₄	1	1	yes	characteristic
37	C₂×#13083562	—	2	4	no	—
38	C₂²×#10623494	—	2	1	no	—
39	C₄×Dic₁₇₈	—	2	1	no	—
40	C₄×Dic₁₇₈	—	2	1	no	—
41	C₄×Dic₁₇₈	D₄	1	2	yes	—
42	C₄×Dic₁₇₈	C₂³	1	2	yes	—
43	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
44	C₂×#13083553	—	2	1	no	—
45	C₂×C₄×Dic₈₉	—	2	1	no	—
46	#15667943	—	2	4	no	—
47	C₄⋊#10623494	—	2	1	no	—
48	C₄⋊#10623494	C₂³	1	2	yes	—
49	C₄⋊#10623494	C₂³	1	4	yes	—
50	#15667941	—	2	2	no	—
51	#15667941	—	2	2	no	—
52	#15667935	—	2	4	no	—
53	#15667933	—	2	4	no	—
54	C₂²⋊#10623494	—	2	2	no	—
55	C₄×#10623494	—	2	2	no	—
56	C₈×Dic₈₉	—	2	2	no	—
57	C₈×Dic₈₉	—	2	2	no	—
58	C₂×#41×C₈₉	—	2	2	no	—
59	#135×C₈₉	—	4	4	no	—
60	#135×C₈₉	—	4	4	no	—
61	C₂×Q₁₆×C₈₉	—	2	1	no	—
62	C₂×Q₁₆×C₈₉	—	2	1	no	—
63	C₂×SD₁₆×C₈₉	—	2	2	no	—
64	C₂×SD₁₆×C₈₉	—	2	2	no	—
65	C₂×D₈×C₈₉	—	2	1	no	—
66	C₂×D₈×C₈₉	—	2	1	no	—
67	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
68	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
69	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
70	#127×C₈₉	C₂³	1	2	yes	—
71	#124×C₈₉	—	2	2	no	—
72	#121×C₈₉	—	2	2	no	—
73	#121×C₈₉	—	2	2	no	—
74	C₄×D₄×C₈₉	—	2	2	no	—
75	C₄×D₄×C₈₉	—	2	2	no	—
76	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
78	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
79	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
81	C₂×Dic₁₇₈	—	2	1	no	—
82	C₂×Dic₁₇₈	—	2	4	no	—
83	C₂×Dic₁₇₈	C₂×D₄	1	2	yes	—
84	C₂²×Dic₈₉	—	2	1	no	—
85	C₂²×Dic₈₉	—	2	1	no	—
86	#13083562	—	4	8	no	—
87	C₂×#10623494	—	2	2	no	—
88	C₂×#10623494	—	2	4	no	—
89	C₂×#10623494	—	4	2	no	—
90	C₄⋊Dic₈₉	—	2	1	no	—
91	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
92	#13083553	—	2	2	no	—
93	#13083553	—	2	2	no	—
94	C₄×Dic₈₉	—	2	2	no	—
95	C₄×Dic₈₉	—	2	2	no	—
96	#41×C₈₉	—	4	4	no	—
97	#41×C₈₉	—	4	4	no	—
98	C₂×Q₈×C₈₉	—	2	2	no	—
99	C₂×D₄×C₈₉	—	2	2	no	—
100	C₂×D₄×C₈₉	—	2	2	no	—
101	C₂×D₄×C₈₉	—	2	2	no	—
102	C₂×D₄×C₈₉	—	4	2	no	—
103	C₂²×C₄×C₈₉	—	4	2	no	—
104	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
107	Q₁₆×C₈₉	—	4	2	no	—
108	Q₁₆×C₈₉	—	4	2	no	—
109	SD₁₆×C₈₉	—	4	4	no	—
110	SD₁₆×C₈₉	—	4	4	no	—
111	D₈×C₈₉	—	4	2	no	—
112	D₈×C₈₉	—	4	2	no	—
113	C₂×C₈×C₈₉	—	2	2	no	—
114	C₂×C₈×C₈₉	SD₁₆	1	4	yes	complemented
115	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₄⋊C₄×C₈₉	—	2	2	no	—
120	C₄⋊C₄×C₈₉	—	2	2	no	—
121	C₂²⋊C₄×C₈₉	—	4	2	no	—
122	C₂²⋊C₄×C₈₉	—	4	2	no	—
123	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₈₉	#41	1	1	yes	characteristic
126	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
127	C₂³×C₈₉	—	4	2	no	—
128	C₂³×C₈₉	#121	1	1	yes	characteristic
129	Q₈×C₈₉	—	4	2	no	—
130	Q₈×C₈₉	—	4	2	no	—
131	D₄×C₈₉	—	4	2	no	—
132	D₄×C₈₉	—	4	4	no	—
133	D₄×C₈₉	—	4	2	no	—
134	D₄×C₈₉	—	8	2	no	—
135	D₄×C₈₉	—	8	4	no	—
136	C₂×C₄×C₈₉	—	2	2	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×SD₁₆	1	2	yes	—
144	C₂×C₄×C₈₉	AGL(1,ℤ/8)	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	4	no	—
149	C₂×Dic₈₉	—	2	2	no	—
150	C₂×Dic₈₉	—	2	2	no	—
151	C₂×Dic₈₉	—	4	2	no	—
152	Dic₁₇₈	—	2	4	no	—
153	Dic₁₇₈	—	2	2	no	—
154	Dic₁₇₈	#135	1	4	yes	complemented
155	C₈×C₈₉	—	2	1	no	—
156	C₈×C₈₉	—	2	1	no	—
157	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
158	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
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	#460	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	#460	1	2	yes	—
173	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
174	C₄×C₈₉	C₂×#135	1	2	yes	—
175	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
176	Dic₈₉	—	2	4	no	—
177	Dic₈₉	—	4	2	no	—
178	#19114	—	89	1	no	maximal, Sylow
179	C₂×C₈₉	—	2	2	no	—
180	C₂×C₈₉	—	16	2	no	—
181	C₂×C₈₉	C₂×#460	1	1	yes	characteristic
182	C₂×C₈₉	#3050	1	1	yes	characteristic
183	C₂×C₈₉	#3067	1	1	yes	characteristic
184	#3147	D₈₉	1	1	yes	characteristic
185	#3103	—	89	2	no	—
186	C₂×C₈×Q₈	—	89	1	no	—
187	#1670	—	89	2	no	—
188	#1668	—	89	2	no	—
189	#1666	—	89	4	no	—
190	#1564	—	89	1	no	—
191	#1540	—	89	2	no	—
192	C₈₉	#19114	1	1	yes	characteristic, complemented, Sylow
193	C₂×#135	—	2	1	no	—
194	C₂×#135	—	2	1	no	—
195	#532	C₂×D₈₉	1	2	yes	—
196	C₂×C₄×Q₈	—	89	1	no	—
197	#495	—	2	1	no	—
198	#495	—	2	1	no	—
199	#494	C₂×D₈₉	1	1	yes	characteristic
200	#493	C₂×D₈₉	1	1	yes	characteristic
201	#492	C₂×D₈₉	1	2	yes	—
202	#481	—	178	2	no	—
203	#475	—	89	4	no	—
204	#460	—	89	4	no	—
205	#448	—	178	2	no	—
206	C₈×Q₈	—	89	2	no	—
207	C₈×Q₈	—	89	2	no	—
208	C₈×Q₈	—	178	1	no	—
209	C₈×Q₈	—	178	1	no	—
210	C₄×SD₁₆	—	178	4	no	—
211	#431	—	89	2	no	—
212	C₂×C₄⋊C₈	—	89	1	no	—
213	C₂×C₄⋊C₈	—	178	1	no	—
214	C₂×C₄×C₈	—	178	1	no	—
215	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
216	#335	—	89	2	no	—
217	#334	—	89	2	no	—
218	#332	—	89	4	no	—
219	#331	—	89	4	no	—
220	C₂×#41	—	2	2	no	—
221	C₂²×Q₈	—	89	1	no	—
222	#135	—	4	4	no	—
223	#135	—	4	4	no	—
224	C₂×Q₁₆	—	2	1	no	—
225	C₂×Q₁₆	—	2	1	no	—
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×SD₁₆	—	2	2	no	—
228	C₂×SD₁₆	—	178	4	no	—
229	C₂×D₈	—	2	1	no	—
230	C₂×D₈	—	2	1	no	—
231	C₂²×C₈	—	178	1	no	—
232	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
233	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
234	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
235	#127	C₂²×D₈₉	1	2	yes	—
236	#124	—	2	2	no	—
237	#121	—	2	2	no	—
238	#121	—	2	2	no	—
239	C₄×Q₈	—	89	2	no	—
240	C₄×Q₈	—	89	2	no	—
241	C₄×Q₈	—	178	1	no	—
242	C₄×Q₈	—	178	1	no	—
243	C₄×D₄	—	2	2	no	—
244	C₄×D₄	—	2	2	no	—
245	C₂×C₄⋊C₄	—	89	1	no	—
246	C₂×C₄⋊C₄	—	178	1	no	—
247	C₂×C₄²	—	178	1	no	—
248	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
249	#106	—	178	4	no	—
250	C₄⋊C₈	—	89	2	no	—
251	C₄⋊C₈	—	89	4	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	—	178	4	no	—
257	C₂²⋊C₈	—	178	2	no	—
258	C₄×C₈	—	178	2	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	2	no	—
269	C₂×Q₈	—	178	4	no	—
270	C₂×Q₈	—	178	1	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	4	2	no	—
275	C₂²×C₄	—	4	2	no	—
276	C₂²×C₄	—	178	1	no	—
277	C₂²×C₄	—	178	1	no	—
278	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
280	C₂²×C₄	#13083573	1	1	yes	characteristic
281	Q₁₆	—	4	2	no	—
282	Q₁₆	—	4	2	no	—
283	SD₁₆	—	4	4	no	—
284	SD₁₆	—	4	4	no	—
285	SD₁₆	—	356	8	no	—
286	D₈	—	4	2	no	—
287	D₈	—	4	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	356	2	no	—
290	C₂×C₈	—	178	4	no	—
291	C₂×C₈	—	178	2	no	—
292	C₂×C₈	#13083562	1	4	yes	complemented
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	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	89	2	no	—
300	C₄⋊C₄	—	178	1	no	—
301	C₄⋊C₄	—	178	2	no	—
302	C₄⋊C₄	—	178	2	no	—
303	C₂²⋊C₄	—	4	2	no	—
304	C₂²⋊C₄	—	4	2	no	—
305	C₄²	—	178	2	no	—
306	C₄²	—	178	2	no	—
307	C₄²	D₄×D₈₉	1	1	yes	characteristic
308	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
309	C₄²	#13083573	1	1	yes	characteristic
310	C₄²	C₂³×D₈₉	1	1	yes	characteristic
311	C₂³	—	4	2	no	—
312	C₂³	#15667983	1	1	yes	characteristic
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	Q₈	—	89	4	no	—
316	Q₈	—	178	4	no	—
317	Q₈	—	178	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	4	4	no	—
321	D₄	—	8	2	no	—
322	D₄	—	8	4	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	8	4	no	—
327	C₂×C₄	—	356	2	no	—
328	C₂×C₄	—	178	2	no	—
329	C₂×C₄	—	178	4	no	—
330	C₂×C₄	—	178	2	no	—
331	C₂×C₄	#15667983	1	1	yes	characteristic
332	C₂×C₄	#15667983	1	1	yes	characteristic
333	C₂×C₄	#15667983	1	1	yes	characteristic
334	C₂×C₄	C₂×#13083562	1	2	yes	—
335	C₂×C₄	#15668038	1	2	yes	—
336	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
337	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
338	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	356	2	no	—
342	C₈	—	356	4	no	—
343	C₈	C₂×#13083562	1	2	yes	—
344	C₈	C₂×#13083562	1	2	yes	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	#19151795	1	2	yes	—
351	C₂²	C₂×#15667983	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	8	2	no	—
356	C₄	—	356	2	no	—
357	C₄	—	178	4	no	—
358	C₄	#19151795	1	2	yes	—
359	C₄	C₂²×#13083562	1	1	yes	characteristic
360	C₄	#135×D₈₉	1	2	yes	—
361	C₄	C₂×#15668038	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	16	2	no	—
364	C₂	C₂×#19151795	1	1	yes	characteristic
365	C₂	#26029829	1	1	yes	characteristic
366	C₂	#26029968	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667983
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting, prime radical	C₈₉×#3147	C₂
socle	C₂²×C₈₉	C₂×#121
hypercenter	#3147	D₈₉
hypocenter	C₈₉	#19114
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #26030279	C₂	semidirect
2. C₂×C₈×Dic₁₇₈	C₂	semidirect
3. #26019619	C₂	semidirect
4. #26019609	C₂	semidirect
5. #26019597	C₂	semidirect
6. #26019000	C₂	semidirect
7. #26018901	C₂	semidirect
114. C₂×C₈×C₈₉	SD₁₆	semidirect
154. Dic₁₇₈	#135	semidirect
192. C₈₉	#19114	semidirect
292. C₂×C₈	#13083562	semidirect

Maximal subgroups

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

subgroup	index
#26030279	2
C₂×C₈×Dic₁₇₈	2
#26019619	2
#26019609	2
#26019597	2
#26019000	2
#26018901	2
C₈₉×#3147	2
#19114	89