Subgroups — #36711726

Order 22784.

546 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022481	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022460	C₂	1	2	yes	maximal, complemented
4	#26022401	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022363	C₂	1	2	yes	maximal, complemented
6	#26022297	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022168	C₂	1	2	yes	maximal, complemented
8	C₄×#15667956	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022084	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021971	C₂	1	2	yes	maximal, complemented
11	C₈₉×#3341	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
13	D₄×Dic₁₇₈	—	2	1	no	—
14	C₂×#15667956	C₂²	1	1	yes	characteristic
15	C₂×#15667956	C₂²	1	2	yes	—
16	C₂×#15667952	C₂²	1	2	yes	—
17	#19151978	C₂²	1	1	yes	characteristic
18	#19151839	C₂²	1	2	yes	—
19	#19151809	—	2	1	no	—
20	#19151756	—	2	2	no	—
21	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
22	#19151425	C₂²	1	1	yes	characteristic
23	#19151423	C₂²	1	2	yes	—
24	C₄⋊#13083549	C₂²	1	1	yes	characteristic
25	#19151401	C₂²	1	2	yes	—
26	#19151401	C₂²	1	2	yes	—
27	#19151401	C₂²	1	2	yes	—
28	#19151401	C₂²	1	2	yes	—
29	#19151401	C₂²	1	1	yes	characteristic
30	#19151399	C₂²	1	1	yes	characteristic
31	C₂²⋊#13083549	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	#555×C₈₉	—	2	1	no	—
34	#500×C₈₉	C₂²	1	1	yes	characteristic
35	#482×C₈₉	—	2	2	no	—
36	#476×C₈₉	—	2	1	no	—
37	#466×C₈₉	C₂²	1	2	yes	—
38	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
40	#427×C₈₉	C₂²	1	2	yes	—
41	C₂²×Dic₁₇₈	—	2	2	no	—
42	C₂×Dic₃₅₆	—	2	4	no	—
43	C₂×Dic₃₅₆	—	2	2	no	—
44	C₂×Dic₃₅₆	—	2	1	no	—
45	C₂×Dic₃₅₆	—	4	2	no	—
46	C₂×Dic₃₅₆	C₂³	1	2	yes	—
47	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
48	#15668012	—	2	1	no	—
49	#15668011	—	4	1	no	—
50	#15667991	—	2	2	no	—
51	#15667989	—	4	2	no	—
52	C₄×Dic₁₇₈	—	2	1	no	—
53	D₄×Dic₈₉	—	4	1	no	—
54	#15667956	—	2	2	no	—
55	#15667956	—	4	2	no	—
56	#15667952	—	4	2	no	—
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₂×#13083549	C₂³	1	2	yes	—
60	#15667946	C₂³	1	1	yes	characteristic
61	#15667937	—	2	1	no	—
62	#15667937	—	2	2	no	—
63	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₈₉	—	2	1	no	—
66	#124×C₈₉	—	4	2	no	—
67	#121×C₈₉	—	2	2	no	—
68	#121×C₈₉	—	4	1	no	—
69	C₄×D₄×C₈₉	—	2	1	no	—
70	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
71	#117×C₈₉	—	2	2	no	—
72	#117×C₈₉	—	4	1	no	—
73	#107×C₈₉	—	2	1	no	—
74	#107×C₈₉	C₂³	1	1	yes	characteristic
75	#107×C₈₉	C₂³	1	2	yes	—
76	#106×C₈₉	—	2	2	no	—
77	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
78	#102×C₈₉	—	2	1	no	—
79	#102×C₈₉	—	2	2	no	—
80	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₇₈	—	2	1	no	—
83	C₂×Dic₁₇₈	—	2	2	no	—
84	C₂×Dic₁₇₈	—	4	4	no	—
85	C₂×Dic₁₇₈	—	4	2	no	—
86	C₂×Dic₁₇₈	—	8	2	no	—
87	C₂²×Dic₈₉	—	4	2	no	—
88	Dic₃₅₆	—	2	2	no	—
89	Dic₃₅₆	—	2	1	no	—
90	Dic₃₅₆	—	2	2	no	—
91	Dic₃₅₆	—	4	4	no	—
92	Dic₃₅₆	—	8	1	no	—
93	Dic₃₅₆	#41	1	2	yes	—
94	C₄⋊Dic₈₉	—	2	1	no	—
95	C₄⋊Dic₈₉	—	4	1	no	—
96	C₄⋊Dic₈₉	—	4	1	no	—
97	#13083553	—	4	1	no	—
98	#13083553	—	4	2	no	—
99	C₂²⋊Dic₈₉	—	4	2	no	—
100	C₄×Dic₈₉	—	4	1	no	—
101	#13083549	—	2	1	no	—
102	#13083549	—	2	1	no	—
103	#13083549	—	2	2	no	—
104	C₂×Q₈×C₈₉	—	4	1	no	—
105	C₂×D₄×C₈₉	—	2	1	no	—
106	C₂×D₄×C₈₉	—	4	2	no	—
107	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
108	C₂²×C₄×C₈₉	—	4	1	no	—
109	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
110	D₈×C₈₉	—	2	1	no	—
111	D₈×C₈₉	—	2	1	no	—
112	C₂×C₈×C₈₉	—	2	2	no	—
113	C₂×C₈×C₈₉	—	2	2	no	—
114	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₈₉	—	2	1	no	—
118	C₄⋊C₄×C₈₉	—	2	2	no	—
119	C₄⋊C₄×C₈₉	—	4	1	no	—
120	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
121	C₂²⋊C₄×C₈₉	—	4	2	no	—
122	C₂²⋊C₄×C₈₉	—	4	2	no	—
123	C₂²⋊C₄×C₈₉	—	4	1	no	—
124	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
125	C₄²×C₈₉	—	4	2	no	—
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂³×C₈₉	—	4	1	no	—
128	C₂³×C₈₉	C₂×D₈	1	2	yes	—
129	Q₈×C₈₉	—	16	1	no	—
130	D₄×C₈₉	—	2	2	no	—
131	D₄×C₈₉	—	4	1	no	—
132	D₄×C₈₉	—	4	1	no	—
133	D₄×C₈₉	—	16	2	no	—
134	C₂×C₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	2	2	no	—
136	C₂×C₄×C₈₉	—	4	1	no	—
137	C₂×C₄×C₈₉	—	4	2	no	—
138	C₂×C₄×C₈₉	—	8	2	no	—
139	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
143	C₂×Dic₈₉	—	4	1	no	—
144	C₂×Dic₈₉	—	4	1	no	—
145	C₂×Dic₈₉	—	4	2	no	—
146	C₂×Dic₈₉	—	8	4	no	—
147	Dic₁₇₈	—	2	2	no	—
148	Dic₁₇₈	—	4	1	no	—
149	Dic₁₇₈	—	4	2	no	—
150	Dic₁₇₈	—	4	2	no	—
151	Dic₁₇₈	—	8	4	no	—
152	C₈×C₈₉	—	2	2	no	—
153	C₈×C₈₉	—	2	1	no	—
154	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
155	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²×C₈₉	—	2	2	no	—
157	C₂²×C₈₉	—	2	2	no	—
158	C₂²×C₈₉	—	2	2	no	—
159	C₂²×C₈₉	—	8	1	no	—
160	C₂²×C₈₉	—	8	1	no	—
161	C₂²×C₈₉	—	8	1	no	—
162	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₈₉	—	2	1	no	—
164	C₄×C₈₉	—	2	2	no	—
165	C₄×C₈₉	—	8	1	no	—
166	C₄×C₈₉	—	8	2	no	—
167	C₄×C₈₉	#510	1	2	yes	—
168	C₄×C₈₉	#546	1	1	yes	characteristic
169	C₄×C₈₉	#577	1	1	yes	characteristic
170	Dic₈₉	—	4	2	no	—
171	Dic₈₉	—	8	1	no	—
172	Dic₈₉	—	8	2	no	—
173	#33896	—	89	1	no	maximal, Sylow
174	C₂×C₈₉	—	4	2	no	—
175	C₂×C₈₉	—	16	1	no	—
176	C₂×C₈₉	#3294	1	1	yes	characteristic
177	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
178	C₂×C₈₉	#3416	1	1	yes	characteristic
179	#3341	D₈₉	1	1	yes	characteristic
180	D₄×Q₁₆	—	89	1	no	—
181	#2241	—	89	1	no	—
182	#2237	—	89	2	no	—
183	#2227	—	89	1	no	—
184	#2220	—	89	2	no	—
185	#2208	—	89	1	no	—
186	#2185	—	89	2	no	—
187	C₄×SD₃₂	—	89	1	no	—
188	#2169	—	89	1	no	—
189	#2141	—	89	2	no	—
190	C₈₉	#33896	1	1	yes	characteristic, complemented, Sylow
191	C₂²×Q₁₆	—	89	2	no	—
192	#555	—	2	1	no	—
193	D₄×Q₈	—	178	1	no	—
194	C₂×SD₃₂	—	89	2	no	—
195	C₂×SD₃₂	—	89	1	no	—
196	C₂×M₃₂	—	89	2	no	—
197	#500	C₂×D₈₉	1	1	yes	characteristic
198	#494	—	89	1	no	—
199	#482	—	2	2	no	—
200	#476	—	2	1	no	—
201	#467	—	89	2	no	—
202	#466	C₂×D₈₉	1	2	yes	—
203	#462	—	178	1	no	—
204	#451	—	178	2	no	—
205	C₄×Q₁₆	—	89	1	no	—
206	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
207	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
208	#427	C₂×D₈₉	1	2	yes	—
209	#367	—	89	1	no	—
210	#366	—	89	2	no	—
211	C₄⋊C₁₆	—	89	1	no	—
212	#358	—	89	1	no	—
213	#358	—	89	2	no	—
214	#358	—	89	2	no	—
215	#358	—	89	2	no	—
216	#358	—	89	2	no	—
217	#357	—	89	1	no	—
218	C₂²⋊C₁₆	—	89	2	no	—
219	C₄×C₁₆	—	89	1	no	—
220	C₂²×Q₈	—	178	2	no	—
221	C₂×Q₁₆	—	89	1	no	—
222	C₂×Q₁₆	—	89	2	no	—
223	C₂×Q₁₆	—	356	2	no	—
224	C₂×Q₁₆	—	178	4	no	—
225	C₂×Q₁₆	—	178	2	no	—
226	C₂×Q₁₆	—	178	1	no	—
227	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	356	1	no	—
231	C₄⋊Q₈	—	178	1	no	—
232	#124	—	4	2	no	—
233	C₂²⋊Q₈	—	356	2	no	—
234	C₂²⋊Q₈	—	178	2	no	—
235	#121	—	2	2	no	—
236	#121	—	4	1	no	—
237	C₄×Q₈	—	178	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	—	356	1	no	—
240	C₄×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
241	#117	—	2	2	no	—
242	#117	—	4	1	no	—
243	SD₃₂	—	356	2	no	—
244	SD₃₂	—	178	2	no	—
245	M₃₂	—	356	2	no	—
246	C₂×C₁₆	—	89	1	no	—
247	C₂×C₁₆	—	89	1	no	—
248	C₂×C₁₆	—	89	2	no	—
249	#107	—	2	1	no	—
250	#107	—	89	1	no	—
251	#107	C₂²×D₈₉	1	2	yes	—
252	#107	C₂²×D₈₉	1	1	yes	characteristic
253	#106	—	2	2	no	—
254	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
255	#103	—	178	1	no	—
256	#103	—	178	2	no	—
257	#102	—	2	1	no	—
258	#102	—	2	2	no	—
259	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
260	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
261	C₂×Q₈	—	4	1	no	—
262	C₂×Q₈	—	356	4	no	—
263	C₂×Q₈	—	356	2	no	—
264	C₂×Q₈	—	178	2	no	—
265	C₂×Q₈	—	178	1	no	—
266	C₂×Q₈	—	712	2	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	4	2	no	—
269	C₂×D₄	#13083559	1	1	yes	characteristic
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	356	2	no	—
272	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
273	Q₁₆	—	89	2	no	—
274	Q₁₆	—	356	4	no	—
275	Q₁₆	—	178	2	no	—
276	Q₁₆	—	178	2	no	—
277	Q₁₆	—	178	1	no	—
278	Q₁₆	—	712	1	no	—
279	D₈	—	2	1	no	—
280	D₈	—	2	1	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
284	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
285	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	356	1	no	—
290	C₄⋊C₄	—	356	1	no	—
291	C₄⋊C₄	—	356	1	no	—
292	C₄⋊C₄	—	356	2	no	—
293	C₄⋊C₄	—	178	1	no	—
294	C₄⋊C₄	#13083559	1	1	yes	characteristic
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	—	356	2	no	—
299	C₂²⋊C₄	#13083559	1	2	yes	—
300	C₄²	—	4	2	no	—
301	C₄²	—	356	1	no	—
302	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
303	C₁₆	—	178	1	no	—
304	C₁₆	—	178	1	no	—
305	C₁₆	—	178	2	no	—
306	C₂³	—	4	1	no	—
307	C₂³	C₂×#13083559	1	2	yes	—
308	Q₈	—	16	1	no	—
309	Q₈	—	356	1	no	—
310	Q₈	—	356	2	no	—
311	Q₈	—	356	2	no	—
312	Q₈	—	178	2	no	—
313	Q₈	—	712	4	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	1	no	—
316	D₄	—	4	1	no	—
317	D₄	—	16	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	356	1	no	—
324	C₂×C₄	—	356	1	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	712	4	no	—
327	C₂×C₄	C₂×#13083559	1	2	yes	—
328	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
329	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	2	no	—
333	C₈	#41×D₈₉	1	1	yes	characteristic
334	C₈	#15668058	1	1	yes	characteristic
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	8	1	no	—
339	C₂²	—	8	1	no	—
340	C₂²	—	8	1	no	—
341	C₂²	C₂²×#13083559	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	1	no	—
344	C₄	—	8	1	no	—
345	C₄	—	8	2	no	—
346	C₄	—	356	2	no	—
347	C₄	—	712	2	no	—
348	C₄	—	712	1	no	—
349	C₄	#19152038	1	2	yes	—
350	C₄	#19152236	1	1	yes	characteristic
351	C₄	#19152406	1	1	yes	characteristic
352	C₂	—	4	2	no	—
353	C₂	—	16	1	no	—
354	C₂	#26032059	1	1	yes	characteristic
355	C₂	C₂×#19152038	1	1	yes	characteristic
356	C₂	#26033399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting, prime radical	C₈₉×#3341	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#3341	D₈₉
hypocenter	C₈₉	#33896
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3341

Complements

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

N	H ≅ G/N	type
1. D₄×Dic₃₅₆	C₂	semidirect
2. #26022481	C₂	semidirect
3. #26022460	C₂	semidirect
4. #26022401	C₂	semidirect
5. #26022363	C₂	semidirect
6. #26022297	C₂	semidirect
7. #26022168	C₂	semidirect
8. C₄×#15667956	C₂	semidirect
9. #26022084	C₂	semidirect
10. #26021971	C₂	semidirect
190. C₈₉	#33896	semidirect

Maximal subgroups

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

subgroup	index
D₄×Dic₃₅₆	2
#26022481	2
#26022460	2
#26022401	2
#26022363	2
#26022297	2
#26022168	2
C₄×#15667956	2
#26022084	2
#26021971	2
C₈₉×#3341	2
#33896	89