Subgroups — #36447598

Order 22784.

454 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#13083574	C₂	1	1	yes	maximal, characteristic
2	#26020273	C₂	1	2	yes	maximal
3	#26019011	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018155	C₂	1	2	yes	maximal, complemented
5	C₄×#107×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13083574	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₇₈	C₄	1	1	yes	characteristic, complemented
8	C₈×Dic₁₇₈	C₄	1	1	yes	characteristic, complemented
9	C₈×D₃₅₆	C₄	1	1	yes	characteristic, complemented
10	C₈×D₃₅₆	C₄	1	1	yes	characteristic, complemented
11	C₈×C₂²⋊D₈₉	—	2	1	no	—
12	C₈×C₂²⋊D₈₉	—	2	1	no	—
13	#19151651	C₂²	1	1	yes	characteristic
14	#19151649	—	2	1	no	—
15	#19151622	C₂²	1	2	yes	—
16	C₂×#15667940	—	2	2	no	—
17	C₄×C₈×D₈₉	—	2	1	no	—
18	#19151358	—	2	2	no	—
19	#19151351	—	2	1	no	—
20	#19151351	—	2	1	no	—
21	C₂×#107×C₈₉	—	2	1	no	—
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
24	#340×C₈₉	—	2	1	no	—
25	C₂×#13083574	D₄	1	1	yes	characteristic
26	C₂×#13083558	—	2	2	no	—
27	C₂×C₈×D₈₉	—	2	1	no	—
28	C₂×C₈×D₈₉	—	2	1	no	—
29	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₇₈	Q₈	1	1	yes	characteristic
31	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
32	C₄×D₃₅₆	Q₈	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₉	—	2	1	no	—
34	C₄×C₂²⋊D₈₉	—	2	1	no	—
35	#15667971	D₄	1	1	yes	characteristic
36	#15667969	—	2	1	no	—
37	C₄²×D₈₉	—	2	1	no	—
38	C₄⋊#10623494	—	2	2	no	—
39	C₄⋊#10623494	—	2	2	no	—
40	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
41	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
42	#15667941	—	2	1	no	—
43	#15667941	—	2	1	no	—
44	#15667940	—	4	4	no	—
45	C₂²⋊#10623494	—	2	1	no	—
46	#15667924	—	2	1	no	—
47	#15667924	—	2	1	no	—
48	C₄×#10623494	—	2	1	no	—
49	C₈×Dic₈₉	—	2	1	no	—
50	C₈×Dic₈₉	—	2	1	no	—
51	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
54	C₂×C₄²×C₈₉	—	2	2	no	—
55	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
56	#107×C₈₉	—	2	4	no	—
57	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
60	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
61	#95×C₈₉	—	2	2	no	—
62	#13083574	—	2	1	no	—
63	#13083574	—	2	1	no	—
64	#13083574	D₈	1	2	yes	—
65	#13083574	Q₁₆	1	2	yes	—
66	C₂×Dic₁₇₈	C₄⋊C₄	1	1	yes	characteristic
67	C₂×D₃₅₆	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	#13083558	—	4	2	no	—
73	#13083558	—	4	2	no	—
74	C₂×#10623494	—	2	1	no	—
75	C₂×#10623494	—	2	1	no	—
76	C₂×#10623494	—	4	2	no	—
77	C₈×D₈₉	—	4	2	no	—
78	C₈×D₈₉	—	4	2	no	—
79	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
80	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	C₂²⋊Dic₈₉	—	2	1	no	—
84	#13083551	—	2	1	no	—
85	#13083551	—	2	1	no	—
86	C₄×Dic₈₉	—	2	1	no	—
87	C₄×Dic₈₉	—	2	1	no	—
88	C₄×Dic₈₉	—	2	1	no	—
89	C₂²×C₄×C₈₉	—	4	2	no	—
90	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
93	C₂×C₈×C₈₉	—	2	2	no	—
94	C₂×C₈×C₈₉	—	4	1	no	—
95	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₈₉	—	2	4	no	—
100	C₄⋊C₄×C₈₉	—	4	2	no	—
101	C₄²×C₈₉	—	4	2	no	—
102	C₄²×C₈₉	—	4	2	no	—
103	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
106	C₄²×C₈₉	#41	1	1	yes	characteristic
107	C₂³×C₈₉	#121	1	1	yes	characteristic
108	C₂²×D₈₉	—	2	1	no	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	4	1	no	—
112	C₂×C₄×C₈₉	—	4	4	no	—
113	C₂×C₄×C₈₉	—	8	2	no	—
114	C₂×C₄×C₈₉	—	8	2	no	—
115	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
124	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
125	C₂²⋊D₈₉	—	2	2	no	—
126	C₂²⋊D₈₉	—	2	2	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	—	2	1	no	—
130	D₃₅₆	—	2	1	no	—
131	D₃₅₆	#107	1	2	yes	complemented
132	C₄×D₈₉	—	2	2	no	—
133	C₄×D₈₉	—	2	2	no	—
134	C₄×D₈₉	—	4	2	no	—
135	C₄×D₈₉	—	4	1	no	—
136	C₄×D₈₉	—	4	1	no	—
137	Dic₁₇₈	—	2	1	no	—
138	Dic₁₇₈	#107	1	2	yes	complemented
139	C₈×C₈₉	—	2	1	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	#104	1	2	yes	complemented
143	#10623494	—	4	1	no	—
144	#10623494	—	4	1	no	—
145	#10623494	—	8	2	no	—
146	C₂²×C₈₉	—	2	1	no	—
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	#380	1	1	yes	characteristic
149	C₂²×C₈₉	#466	1	1	yes	characteristic
150	C₂²×C₈₉	#467	1	1	yes	characteristic
151	C₂×D₈₉	—	2	2	no	—
152	C₂×D₈₉	—	4	1	no	—
153	C₂×D₈₉	—	4	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	8	4	no	—
158	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
159	C₄×C₈₉	#421	1	1	yes	characteristic
160	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
161	C₄×C₈₉	#427	1	1	yes	characteristic
162	C₄×C₈₉	#466	1	1	yes	characteristic
163	C₄×C₈₉	#467	1	1	yes	characteristic
164	Dic₈₉	—	2	2	no	—
165	Dic₈₉	—	4	1	no	—
166	Dic₈₉	—	4	1	no	—
167	#8667	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	—	2	1	no	—
169	C₂×C₈₉	—	2	1	no	—
170	C₂×C₈₉	#1806	1	1	yes	characteristic
171	C₂×C₈₉	#1808	1	1	yes	characteristic
172	C₂×C₈₉	#1814	1	1	yes	characteristic
173	D₈₉	—	4	2	no	—
174	C₈×#41	—	89	1	no	—
175	#1806	—	89	2	no	—
176	C₄×#107	D₈₉	1	1	yes	characteristic, complemented
177	#1567	—	89	1	no	—
178	#1303	—	89	2	no	—
179	C₈₉	#8667	1	1	yes	characteristic, complemented, Sylow
180	C₄×#41	—	89	1	no	—
181	C₈×Q₈	—	89	1	no	—
182	C₈×Q₈	—	89	1	no	—
183	C₈×D₄	—	89	1	no	—
184	C₈×D₄	—	89	1	no	—
185	C₈×D₄	—	178	1	no	—
186	C₈×D₄	—	178	1	no	—
187	#431	—	89	1	no	—
188	#431	—	178	1	no	—
189	C₂×#107	—	2	1	no	—
190	#423	—	89	2	no	—
191	C₂×#104	—	178	2	no	—
192	C₂×C₄×C₈	—	178	1	no	—
193	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
194	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
195	#340	—	2	1	no	—
196	#339	—	178	2	no	—
197	#335	—	178	1	no	—
198	#335	—	178	1	no	—
199	C₂×#41	—	89	1	no	—
200	C₂×M₁₆	—	178	2	no	—
201	C₂²×C₈	—	178	1	no	—
202	C₂²×C₈	—	178	1	no	—
203	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
204	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
205	C₄×Q₈	—	89	1	no	—
206	C₄×Q₈	—	89	1	no	—
207	C₄×D₄	—	89	1	no	—
208	C₄×D₄	—	89	1	no	—
209	C₄×D₄	—	178	1	no	—
210	C₄×D₄	—	178	1	no	—
211	#117	—	89	1	no	—
212	#117	—	178	1	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	178	1	no	—
216	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
217	#107	—	2	4	no	—
218	C₄⋊C₈	—	89	1	no	—
219	C₄⋊C₈	—	89	1	no	—
220	C₄⋊C₈	—	178	1	no	—
221	C₄⋊C₈	—	178	1	no	—
222	C₄⋊C₈	—	178	2	no	—
223	C₄⋊C₈	—	178	2	no	—
224	#104	—	356	4	no	—
225	C₂²⋊C₈	—	178	1	no	—
226	C₂²⋊C₈	—	178	1	no	—
227	C₂²⋊C₈	—	178	1	no	—
228	C₄×C₈	—	178	1	no	—
229	C₄×C₈	—	178	1	no	—
230	C₄×C₈	—	178	1	no	—
231	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
232	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
233	C₄×C₈	D₃₅₆	1	1	yes	characteristic
234	C₄×C₈	D₃₅₆	1	1	yes	characteristic
235	#95	—	2	2	no	—
236	#41	—	89	2	no	—
237	#41	—	89	2	no	—
238	#41	—	178	1	no	—
239	#41	—	178	1	no	—
240	C₂×Q₈	—	89	1	no	—
241	C₂×D₄	—	89	1	no	—
242	C₂×D₄	—	178	1	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	178	1	no	—
245	C₂²×C₄	—	178	1	no	—
246	C₂²×C₄	—	178	1	no	—
247	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
249	C₂²×C₄	#13083573	1	1	yes	characteristic
250	M₁₆	—	356	2	no	—
251	M₁₆	—	356	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	356	2	no	—
255	C₂×C₈	—	356	2	no	—
256	C₂×C₈	—	356	2	no	—
257	C₂×C₈	—	178	1	no	—
258	C₂×C₈	—	178	1	no	—
259	C₂×C₈	#13083551	1	1	yes	characteristic
260	C₂×C₈	#13083551	1	1	yes	characteristic
261	C₂×C₈	#13083551	1	1	yes	characteristic
262	C₂×C₈	#13083551	1	1	yes	characteristic
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	89	1	no	—
266	C₄⋊C₄	—	89	1	no	—
267	C₄⋊C₄	—	178	1	no	—
268	C₄⋊C₄	—	178	1	no	—
269	C₂²⋊C₄	—	178	1	no	—
270	C₂²⋊C₄	—	178	1	no	—
271	C₂²⋊C₄	—	178	1	no	—
272	C₄²	—	4	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	—	178	1	no	—
275	C₄²	—	178	1	no	—
276	C₄²	—	178	1	no	—
277	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
278	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
279	C₄²	Q₈×D₈₉	1	1	yes	characteristic
280	C₄²	#13083575	1	1	yes	characteristic
281	C₂³	—	178	1	no	—
282	C₂³	#15667983	1	1	yes	characteristic
283	Q₈	—	89	2	no	—
284	Q₈	—	178	1	no	—
285	D₄	—	89	2	no	—
286	D₄	—	178	2	no	—
287	D₄	—	178	2	no	—
288	D₄	—	178	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	—	8	2	no	—
295	C₂×C₄	—	356	2	no	—
296	C₂×C₄	—	356	1	no	—
297	C₂×C₄	—	356	1	no	—
298	C₂×C₄	—	178	1	no	—
299	C₂×C₄	—	178	1	no	—
300	C₂×C₄	—	178	1	no	—
301	C₂×C₄	—	178	2	no	—
302	C₂×C₄	—	178	2	no	—
303	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
304	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
305	C₂×C₄	#15667970	1	1	yes	characteristic
306	C₂×C₄	#15667987	1	1	yes	characteristic
307	C₂×C₄	#15667988	1	1	yes	characteristic
308	C₂×C₄	#15667992	1	1	yes	characteristic
309	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
310	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
311	C₂×C₄	#15668031	1	1	yes	characteristic
312	C₂×C₄	#15668035	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	356	1	no	—
317	C₈	—	356	1	no	—
318	C₈	—	712	2	no	—
319	C₈	#15667940	1	2	yes	complemented
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	356	1	no	—
323	C₂²	—	356	1	no	—
324	C₂²	—	178	2	no	—
325	C₂²	#19151455	1	1	yes	characteristic
326	C₂²	#19151831	1	1	yes	characteristic
327	C₂²	#19151836	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	8	4	no	—
332	C₄	—	356	1	no	—
333	C₄	—	356	1	no	—
334	C₄	—	178	2	no	—
335	C₄	C₂×#15667940	1	1	yes	characteristic
336	C₄	#19151616	1	1	yes	characteristic
337	C₄	#107×D₈₉	1	1	yes	characteristic
338	C₄	#19151635	1	1	yes	characteristic
339	C₄	#19151835	1	1	yes	characteristic
340	C₄	#19151840	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	356	2	no	—
344	C₂	#26020273	1	1	yes	characteristic
345	C₂	#26020278	1	1	yes	characteristic
346	C₂	#26020303	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667987
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#107×C₈₉	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂×#13083574	D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#107	D₈₉
hypocenter	C₈₉	#8667
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
3. #26019011	C₂	semidirect
4. #26018155	C₂	semidirect
5. C₄×#107×C₈₉	C₂	semidirect
7. C₈×Dic₁₇₈	C₄	semidirect
8. C₈×Dic₁₇₈	C₄	semidirect
9. C₈×D₃₅₆	C₄	semidirect
10. C₈×D₃₅₆	C₄	semidirect
131. D₃₅₆	#107	semidirect
138. Dic₁₇₈	#107	semidirect
142. C₈×C₈₉	#104	semidirect
176. C₄×#107	D₈₉	semidirect
179. C₈₉	#8667	semidirect
319. C₈	#15667940	semidirect

Maximal subgroups

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

subgroup	index
C₈×#13083574	2
#26020273	2
#26019011	2
#26018155	2
C₄×#107×C₈₉	2
#8667	89