Subgroups — M₁₆×#13083562 #36525864

Order 22784.

716 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#13083562	C₂	1	1	yes	maximal, characteristic
3	#26019140	C₂	1	2	yes	maximal, complemented
4	#26019093	C₂	1	2	yes	maximal, complemented
5	#26019088	C₂	1	2	yes	maximal, complemented
6	C₈×#13083562	C₂	1	2	yes	maximal, complemented
7	#26019007	C₂	1	1	yes	maximal, characteristic, complemented
8	#26018680	C₂	1	1	yes	maximal, characteristic, complemented
9	#26018677	C₂	1	1	yes	maximal, characteristic
10	M₁₆×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×#13083562	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
14	#19151728	C₂²	1	2	yes	complemented
15	#19151726	—	2	2	no	—
16	C₈×Dic₁₇₈	C₂²	1	2	yes	complemented
17	C₄×#13083562	C₄	1	1	yes	characteristic
18	C₄×#13083562	C₄	1	1	yes	characteristic
19	C₄×#13083562	C₄	1	1	yes	characteristic
20	C₄×#13083562	C₄	1	1	yes	characteristic
21	C₄×#13083562	C₂²	1	1	yes	characteristic
22	C₄×#13083562	C₂²	1	1	yes	characteristic
23	C₄×#13083562	C₂²	1	1	yes	characteristic
24	C₄×#13083562	C₂²	1	1	yes	characteristic
25	#19151651	C₂²	1	1	yes	characteristic
26	C₂×#15667943	C₄	1	1	yes	characteristic
27	#19151622	C₂²	1	1	yes	characteristic
28	#19151621	—	2	1	no	—
29	C₂×#15667935	C₄	1	1	yes	characteristic
30	C₂×#15667933	C₄	1	1	yes	characteristic
31	M₁₆×Dic₈₉	—	2	1	no	—
32	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
33	#19151348	C₂²	1	2	yes	complemented
34	#19151348	C₂²	1	2	yes	complemented
35	#19151328	C₂²	1	2	yes	complemented
36	#19151328	C₂²	1	2	yes	complemented
37	D₄⋊#10623494	C₂²	1	2	yes	—
38	D₄⋊#10623494	C₂²	1	2	yes	—
39	#19151319	C₂²	1	2	yes	complemented
40	C₈×#10623494	C₂²	1	2	yes	complemented
41	C₂²×M₁₆×C₈₉	—	2	1	no	—
42	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
43	#436×C₈₉	C₂²	1	2	yes	—
44	#435×C₈₉	—	2	2	no	—
45	C₈×D₄×C₈₉	C₂²	1	2	yes	—
46	#423×C₈₉	C₂²	1	1	yes	characteristic
47	#407×C₈₉	—	2	1	no	—
48	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
49	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
50	C₂×#13083562	—	2	4	no	—
51	C₂×#13083562	C₂×C₄	1	1	yes	characteristic
52	C₂×#13083562	C₂×C₄	1	1	yes	characteristic
53	C₂×#13083562	C₂×C₄	1	1	yes	characteristic
54	C₂×#13083562	C₂×C₄	1	1	yes	characteristic
55	C₂×#13083558	—	2	1	no	—
56	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
57	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₁₇₈	—	2	1	no	—
59	C₄×Dic₁₇₈	—	2	1	no	—
60	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
65	C₂×#13083553	—	2	1	no	—
66	C₂×C₄×Dic₈₉	—	2	1	no	—
67	#15667943	C₂×C₄	1	1	yes	characteristic
68	#15667943	C₂×C₄	1	1	yes	characteristic
69	#15667943	C₂×C₄	1	1	yes	characteristic
70	#15667943	C₂×C₄	1	1	yes	characteristic
71	C₄⋊#10623494	—	2	1	no	—
72	C₄⋊#10623494	—	2	1	no	—
73	C₄⋊#10623494	C₂³	1	2	yes	—
74	#15667941	—	2	2	no	—
75	#15667941	—	2	2	no	—
76	#15667935	C₂×C₄	1	1	yes	characteristic
77	#15667935	C₂×C₄	1	1	yes	characteristic
78	#15667935	C₂×C₄	1	1	yes	characteristic
79	#15667935	C₂×C₄	1	1	yes	characteristic
80	#15667933	C₂×C₄	1	1	yes	characteristic
81	#15667933	C₂×C₄	1	1	yes	characteristic
82	#15667933	C₂×C₄	1	1	yes	characteristic
83	#15667933	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊#10623494	—	2	1	no	—
85	#15667922	—	2	2	no	—
86	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
87	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
88	C₄×#10623494	C₂³	1	2	yes	—
89	C₄×#10623494	C₂³	1	1	yes	characteristic
90	C₄×#10623494	C₂³	1	1	yes	characteristic
91	C₈×Dic₈₉	—	2	2	no	—
92	C₂²×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
93	C₂³×C₄×C₈₉	—	2	1	no	—
94	C₂×M₁₆×C₈₉	—	2	1	no	—
95	C₂×M₁₆×C₈₉	—	2	2	no	—
96	C₂×M₁₆×C₈₉	—	4	4	no	—
97	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
98	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
99	C₂²×C₈×C₈₉	—	2	2	no	—
100	C₄×D₄×C₈₉	—	2	1	no	—
101	C₄×D₄×C₈₉	—	2	1	no	—
102	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
103	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
104	C₄×D₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₄×D₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
107	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
108	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
109	C₄⋊C₈×C₈₉	—	2	1	no	—
110	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
111	C₂²⋊C₈×C₈₉	—	2	2	no	—
112	C₂²⋊C₈×C₈₉	—	2	2	no	—
113	#97×C₈₉	D₄	1	2	yes	—
114	C₄×C₈×C₈₉	C₂³	1	2	yes	—
115	C₂×Dic₁₇₈	—	2	2	no	—
116	C₂×Dic₁₇₈	—	2	1	no	—
117	C₂×Dic₁₇₈	—	4	1	no	—
118	C₂×Dic₁₇₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₁₇₈	C₂²×C₄	1	1	yes	characteristic
120	C₂²×Dic₈₉	—	2	1	no	—
121	C₂²×Dic₈₉	—	2	1	no	—
122	#13083562	—	2	2	no	—
123	#13083562	—	2	2	no	—
124	#13083562	—	2	2	no	—
125	#13083562	M₁₆	1	4	yes	direct product factor
126	#13083558	—	4	2	no	—
127	C₂×#10623494	—	2	2	no	—
128	C₂×#10623494	—	2	2	no	—
129	C₂×#10623494	—	2	2	no	—
130	C₂×#10623494	—	4	1	no	—
131	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
132	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
133	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
134	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
135	C₄⋊Dic₈₉	—	2	1	no	—
136	C₄⋊Dic₈₉	C₂²×C₄	1	1	yes	characteristic
137	C₄⋊Dic₈₉	C₂²×C₄	1	1	yes	characteristic
138	#13083553	—	2	1	no	—
139	#13083553	—	2	1	no	—
140	#13083553	—	2	1	no	—
141	#13083553	—	2	1	no	—
142	C₄×Dic₈₉	—	2	1	no	—
143	C₄×Dic₈₉	—	2	1	no	—
144	C₄×Dic₈₉	—	2	1	no	—
145	C₄×Dic₈₉	—	2	1	no	—
146	C₂⁴×C₈₉	—	2	1	no	—
147	C₂×D₄×C₈₉	—	2	2	no	—
148	C₂×D₄×C₈₉	—	2	1	no	—
149	C₂×D₄×C₈₉	—	4	1	no	—
150	C₂×D₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
151	C₂×D₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₈₉	—	2	1	no	—
153	C₂²×C₄×C₈₉	—	2	1	no	—
154	C₂²×C₄×C₈₉	—	2	1	no	—
155	C₂²×C₄×C₈₉	—	2	1	no	—
156	C₂²×C₄×C₈₉	—	2	1	no	—
157	C₂²×C₄×C₈₉	—	4	2	no	—
158	C₂²×C₄×C₈₉	—	4	2	no	—
159	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
162	M₁₆×C₈₉	—	2	2	no	—
163	M₁₆×C₈₉	—	4	2	no	—
164	M₁₆×C₈₉	—	4	4	no	—
165	M₁₆×C₈₉	SD₁₆	1	4	yes	complemented
166	C₂×C₈×C₈₉	—	2	2	no	—
167	C₂×C₈×C₈₉	—	4	4	no	—
168	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
169	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
170	C₄⋊C₄×C₈₉	—	2	1	no	—
171	C₄⋊C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
172	C₄⋊C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
173	C₂²⋊C₄×C₈₉	—	2	1	no	—
174	C₂²⋊C₄×C₈₉	—	2	1	no	—
175	C₂²⋊C₄×C₈₉	—	2	1	no	—
176	C₂²⋊C₄×C₈₉	—	2	1	no	—
177	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₈₉	#41	1	1	yes	characteristic
180	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
181	C₂³×C₈₉	—	2	1	no	—
182	C₂³×C₈₉	—	2	1	no	—
183	C₂³×C₈₉	—	4	2	no	—
184	C₂³×C₈₉	—	4	2	no	—
185	C₂³×C₈₉	—	8	1	no	—
186	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
187	D₄×C₈₉	—	2	1	no	—
188	D₄×C₈₉	—	2	1	no	—
189	D₄×C₈₉	—	2	1	no	—
190	D₄×C₈₉	—	4	2	no	—
191	D₄×C₈₉	C₂×M₁₆	1	2	yes	—
192	C₂×C₄×C₈₉	—	2	1	no	—
193	C₂×C₄×C₈₉	—	2	1	no	—
194	C₂×C₄×C₈₉	—	2	1	no	—
195	C₂×C₄×C₈₉	—	2	1	no	—
196	C₂×C₄×C₈₉	—	4	1	no	—
197	C₂×C₄×C₈₉	—	4	2	no	—
198	C₂×C₄×C₈₉	—	4	1	no	—
199	C₂×C₄×C₈₉	—	4	1	no	—
200	C₂×C₄×C₈₉	—	4	1	no	—
201	C₂×C₄×C₈₉	—	4	1	no	—
202	C₂×C₄×C₈₉	—	4	2	no	—
203	C₂×C₄×C₈₉	—	4	2	no	—
204	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
205	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
206	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
207	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
208	C₂×C₄×C₈₉	#135	1	2	yes	—
209	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
210	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
212	C₂×Dic₈₉	—	2	1	no	—
213	C₂×Dic₈₉	—	2	1	no	—
214	C₂×Dic₈₉	—	2	1	no	—
215	C₂×Dic₈₉	—	2	1	no	—
216	C₂×Dic₈₉	—	4	2	no	—
217	C₂×Dic₈₉	—	8	1	no	—
218	Dic₁₇₈	—	2	1	no	—
219	Dic₁₇₈	—	2	1	no	—
220	Dic₁₇₈	—	2	1	no	—
221	Dic₁₇₈	—	4	2	no	—
222	Dic₁₇₈	C₂×M₁₆	1	2	yes	complemented
223	C₈×C₈₉	—	2	2	no	—
224	C₈×C₈₉	—	4	2	no	—
225	C₈×C₈₉	C₂×SD₁₆	1	4	yes	complemented
226	#10623494	—	2	1	no	—
227	#10623494	—	2	1	no	—
228	#10623494	—	2	1	no	—
229	#10623494	—	4	2	no	—
230	#10623494	—	4	2	no	—
231	#10623494	C₂×M₁₆	1	2	yes	complemented
232	C₂²×C₈₉	—	2	1	no	—
233	C₂²×C₈₉	—	2	1	no	—
234	C₂²×C₈₉	—	2	2	no	—
235	C₂²×C₈₉	—	4	1	no	—
236	C₂²×C₈₉	—	4	1	no	—
237	C₂²×C₈₉	—	4	1	no	—
238	C₂²×C₈₉	—	4	1	no	—
239	C₂²×C₈₉	—	4	1	no	—
240	C₂²×C₈₉	—	8	4	no	—
241	C₂²×C₈₉	C₄×SD₁₆	1	2	yes	—
242	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
243	C₄×C₈₉	—	2	1	no	—
244	C₄×C₈₉	—	2	1	no	—
245	C₄×C₈₉	—	2	1	no	—
246	C₄×C₈₉	—	4	1	no	—
247	C₄×C₈₉	—	4	1	no	—
248	C₄×C₈₉	C₄×SD₁₆	1	2	yes	—
249	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
250	C₄×C₈₉	#568	1	1	yes	characteristic
251	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
252	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
253	Dic₈₉	—	2	2	no	—
254	Dic₈₉	—	4	1	no	—
255	Dic₈₉	—	4	1	no	—
256	Dic₈₉	—	4	1	no	—
257	M₁₆×SD₁₆	—	89	1	no	maximal, Sylow
258	C₂×C₈₉	—	2	2	no	—
259	C₂×C₈₉	—	4	2	no	—
260	C₂×C₈₉	—	8	1	no	—
261	C₂×C₈₉	D₄×M₁₆	1	1	yes	characteristic
262	C₂×C₈₉	C₂×C₄×SD₁₆	1	1	yes	characteristic
263	C₂×C₈₉	#2957	1	1	yes	characteristic
264	Q₈×M₁₆	—	89	1	no	—
265	C₂×C₄×SD₁₆	—	89	1	no	—
266	D₄×M₁₆	D₈₉	1	1	yes	characteristic
267	#1590	—	89	2	no	—
268	#1583	—	89	2	no	—
269	#1582	—	89	2	no	—
270	C₈×SD₁₆	—	89	2	no	—
271	#1566	—	89	1	no	—
272	#1487	—	89	1	no	—
273	#1486	—	89	1	no	—
274	C₈×M₁₆	—	89	1	no	—
275	C₈₉	M₁₆×SD₁₆	1	1	yes	characteristic, complemented, Sylow
276	C₂²×SD₁₆	—	89	1	no	—
277	C₂²×M₁₆	—	2	1	no	—
278	C₂×C₄×Q₈	—	89	1	no	—
279	C₂×C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
280	#446	—	89	2	no	—
281	#446	—	178	2	no	—
282	C₈×Q₈	—	89	2	no	—
283	C₄×SD₁₆	—	89	1	no	—
284	C₄×SD₁₆	—	89	1	no	—
285	C₄×SD₁₆	—	89	1	no	—
286	C₄×SD₁₆	—	89	1	no	—
287	C₄×SD₁₆	—	89	1	no	—
288	C₄×SD₁₆	—	89	1	no	—
289	C₄×SD₁₆	—	89	1	no	—
290	C₄×SD₁₆	—	89	1	no	—
291	#436	C₂×D₈₉	1	2	yes	—
292	#435	—	2	2	no	—
293	C₈×D₄	C₂×D₈₉	1	2	yes	—
294	#431	—	89	1	no	—
295	C₂×#106	—	89	1	no	—
296	#423	—	89	1	no	—
297	#423	—	178	1	no	—
298	#423	C₂×D₈₉	1	1	yes	characteristic
299	C₂×#103	—	89	1	no	—
300	C₂×#102	—	89	1	no	—
301	#407	—	2	1	no	—
302	C₄×M₁₆	—	178	1	no	—
303	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
304	C₂×C₄×C₈	—	89	1	no	—
305	#334	—	89	2	no	—
306	#334	—	89	2	no	—
307	Q₈⋊C₈	—	89	2	no	—
308	Q₈⋊C₈	—	89	2	no	—
309	D₄⋊C₈	—	89	2	no	—
310	D₄⋊C₈	—	89	2	no	—
311	#322	—	89	2	no	—
312	C₈²	—	89	2	no	—
313	C₂²×Q₈	—	89	1	no	—
314	C₂²×D₄	C₄×D₈₉	1	1	yes	characteristic
315	C₂³×C₄	—	2	1	no	—
316	C₂×SD₁₆	—	89	1	no	—
317	C₂×SD₁₆	—	89	1	no	—
318	C₂×SD₁₆	—	89	1	no	—
319	C₂×SD₁₆	—	89	1	no	—
320	C₂×SD₁₆	—	178	4	no	—
321	C₂×M₁₆	—	2	1	no	—
322	C₂×M₁₆	—	2	2	no	—
323	C₂×M₁₆	—	4	4	no	—
324	C₂×M₁₆	—	178	1	no	—
325	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
326	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
327	C₂²×C₈	—	2	2	no	—
328	C₂²×C₈	—	89	1	no	—
329	C₂²×C₈	—	89	1	no	—
330	C₄×Q₈	—	89	1	no	—
331	C₄×Q₈	—	89	1	no	—
332	C₄×Q₈	—	89	1	no	—
333	C₄×Q₈	—	89	1	no	—
334	C₄×Q₈	—	178	1	no	—
335	C₄×Q₈	—	178	1	no	—
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	—	2	1	no	—
338	C₄×D₄	C₄×D₈₉	1	1	yes	characteristic
339	C₄×D₄	C₄×D₈₉	1	1	yes	characteristic
340	C₄×D₄	C₂²×D₈₉	1	1	yes	characteristic
341	C₄×D₄	C₂²×D₈₉	1	1	yes	characteristic
342	C₂×C₄⋊C₄	—	89	1	no	—
343	C₂×C₄⋊C₄	—	178	1	no	—
344	C₂×C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
345	C₂×C₂²⋊C₄	—	2	1	no	—
346	C₂×C₄²	—	178	1	no	—
347	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
348	#106	—	89	1	no	—
349	#106	—	89	1	no	—
350	#106	—	89	1	no	—
351	#106	—	89	1	no	—
352	C₄⋊C₈	—	2	1	no	—
353	C₄⋊C₈	—	89	2	no	—
354	C₄⋊C₈	—	178	2	no	—
355	C₄⋊C₈	—	178	2	no	—
356	C₄⋊C₈	—	178	1	no	—
357	C₄⋊C₈	—	178	1	no	—
358	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
359	#103	—	89	1	no	—
360	#103	—	89	1	no	—
361	#103	—	89	1	no	—
362	#103	—	89	1	no	—
363	#102	—	89	1	no	—
364	#102	—	89	1	no	—
365	#102	—	89	1	no	—
366	#102	—	89	1	no	—
367	C₂²⋊C₈	—	2	2	no	—
368	C₂²⋊C₈	—	2	2	no	—
369	C₂²⋊C₈	—	178	1	no	—
370	#97	—	178	2	no	—
371	#97	C₂²⋊D₈₉	1	2	yes	—
372	C₄×C₈	—	89	1	no	—
373	C₄×C₈	—	89	1	no	—
374	C₄×C₈	—	89	1	no	—
375	C₄×C₈	—	89	1	no	—
376	C₄×C₈	—	89	2	no	—
377	C₄×C₈	—	178	2	no	—
378	C₄×C₈	C₂²×D₈₉	1	2	yes	—
379	C₂⁴	—	2	1	no	—
380	C₂×Q₈	—	89	1	no	—
381	C₂×Q₈	—	89	1	no	—
382	C₂×Q₈	—	356	1	no	—
383	C₂×Q₈	—	178	2	no	—
384	C₂×Q₈	—	178	1	no	—
385	C₂×D₄	—	2	2	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	4	1	no	—
388	C₂×D₄	C₂×C₄×D₈₉	1	1	yes	characteristic
389	C₂×D₄	C₂×C₄×D₈₉	1	1	yes	characteristic
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	—	4	2	no	—
397	C₂²×C₄	—	178	1	no	—
398	C₂²×C₄	—	178	1	no	—
399	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
400	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
401	C₂²×C₄	#13083574	1	1	yes	characteristic
402	SD₁₆	—	89	4	no	—
403	SD₁₆	—	178	2	no	—
404	SD₁₆	—	178	2	no	—
405	SD₁₆	—	178	2	no	—
406	M₁₆	—	2	2	no	—
407	M₁₆	—	4	2	no	—
408	M₁₆	—	4	4	no	—
409	M₁₆	—	356	2	no	—
410	M₁₆	#13083562	1	4	yes	direct product factor
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	4	4	no	—
413	C₂×C₈	—	89	1	no	—
414	C₂×C₈	—	89	1	no	—
415	C₂×C₈	—	89	1	no	—
416	C₂×C₈	—	89	1	no	—
417	C₂×C₈	—	356	1	no	—
418	C₂×C₈	—	178	2	no	—
419	C₂×C₈	—	178	2	no	—
420	C₂×C₈	—	178	2	no	—
421	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
422	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	89	1	no	—
425	C₄⋊C₄	—	89	1	no	—
426	C₄⋊C₄	—	178	1	no	—
427	C₄⋊C₄	—	178	1	no	—
428	C₄⋊C₄	—	178	1	no	—
429	C₄⋊C₄	—	178	1	no	—
430	C₄⋊C₄	—	178	1	no	—
431	C₄⋊C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
432	C₄⋊C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
433	C₂²⋊C₄	—	2	1	no	—
434	C₂²⋊C₄	—	2	1	no	—
435	C₂²⋊C₄	—	2	1	no	—
436	C₂²⋊C₄	—	2	1	no	—
437	C₄²	—	178	1	no	—
438	C₄²	—	178	1	no	—
439	C₄²	—	178	1	no	—
440	C₄²	—	178	1	no	—
441	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
442	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
443	C₄²	#13083574	1	1	yes	characteristic
444	C₄²	C₂³×D₈₉	1	1	yes	characteristic
445	C₂³	—	2	1	no	—
446	C₂³	—	2	1	no	—
447	C₂³	—	4	2	no	—
448	C₂³	—	4	2	no	—
449	C₂³	—	8	1	no	—
450	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
451	Q₈	—	89	2	no	—
452	Q₈	—	356	2	no	—
453	Q₈	—	178	1	no	—
454	Q₈	—	178	1	no	—
455	Q₈	—	178	1	no	—
456	D₄	—	2	1	no	—
457	D₄	—	2	1	no	—
458	D₄	—	2	1	no	—
459	D₄	—	4	2	no	—
460	D₄	M₁₆×D₈₉	1	2	yes	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	356	2	no	—
474	C₂×C₄	—	178	1	no	—
475	C₂×C₄	—	178	1	no	—
476	C₂×C₄	—	178	1	no	—
477	C₂×C₄	—	178	1	no	—
478	C₂×C₄	—	712	1	no	—
479	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
480	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
481	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
482	C₂×C₄	C₂×#13083562	1	2	yes	—
483	C₂×C₄	#15668032	1	2	yes	—
484	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
485	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
486	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
487	C₈	—	2	2	no	—
488	C₈	—	4	2	no	—
489	C₈	—	89	2	no	—
490	C₈	—	356	2	no	—
491	C₈	—	356	2	no	—
492	C₈	—	178	1	no	—
493	C₈	—	178	1	no	—
494	C₈	—	178	1	no	—
495	C₈	C₂×#13083562	1	4	yes	complemented
496	C₂²	—	2	1	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	1	no	—
501	C₂²	—	4	1	no	—
502	C₂²	—	4	1	no	—
503	C₂²	—	4	1	no	—
504	C₂²	—	8	4	no	—
505	C₂²	C₄×#13083562	1	2	yes	—
506	C₂²	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
507	C₄	—	2	1	no	—
508	C₄	—	2	1	no	—
509	C₄	—	2	1	no	—
510	C₄	—	4	1	no	—
511	C₄	—	4	1	no	—
512	C₄	—	356	1	no	—
513	C₄	—	356	1	no	—
514	C₄	—	356	1	no	—
515	C₄	—	178	2	no	—
516	C₄	C₄×#13083562	1	2	yes	—
517	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
518	C₄	#19152364	1	1	yes	characteristic
519	C₄	C₂²×#13083562	1	1	yes	characteristic
520	C₄	C₂×#15668032	1	1	yes	characteristic
521	C₂	—	2	2	no	—
522	C₂	—	4	2	no	—
523	C₂	—	8	1	no	—
524	C₂	M₁₆×C₂²⋊D₈₉	1	1	yes	characteristic
525	C₂	C₂×C₄×#13083562	1	1	yes	characteristic
526	C₂	#26029167	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	D₄×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₄×C₈₉	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	D₄×M₁₆	D₈₉
hypocenter	C₈₉	M₁₆×SD₁₆
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ D₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. M₁₆×Dic₁₇₈	C₂	semidirect
3. #26019140	C₂	semidirect
4. #26019093	C₂	semidirect
5. #26019088	C₂	semidirect
6. C₈×#13083562	C₂	semidirect
7. #26019007	C₂	semidirect
8. #26018680	C₂	semidirect
10. M₁₆×#10623494	C₂	semidirect
14. #19151728	C₂²	semidirect
16. C₈×Dic₁₇₈	C₂²	semidirect
33. #19151348	C₂²	semidirect
34. #19151348	C₂²	semidirect
35. #19151328	C₂²	semidirect
36. #19151328	C₂²	semidirect
39. #19151319	C₂²	semidirect
40. C₈×#10623494	C₂²	semidirect
125. #13083562	M₁₆	direct (N × H)
165. M₁₆×C₈₉	SD₁₆	semidirect
222. Dic₁₇₈	C₂×M₁₆	semidirect
225. C₈×C₈₉	C₂×SD₁₆	semidirect
231. #10623494	C₂×M₁₆	semidirect
275. C₈₉	M₁₆×SD₁₆	semidirect
410. M₁₆	#13083562	direct (N × H)
495. C₈	C₂×#13083562	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₇₈	2
C₂×C₄×#13083562	2
#26019140	2
#26019093	2
#26019088	2
C₈×#13083562	2
#26019007	2
#26018680	2
#26018677	2
M₁₆×#10623494	2
D₄×M₁₆×C₈₉	2
M₁₆×SD₁₆	89

Subgroups — M16×#13083562 #36525864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×#13083562 #36525864