Subgroups — #36525365

Order 22784.

602 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030292	C₂	1	1	yes	maximal, characteristic
2	M₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
3	#26029199	C₂	1	1	yes	maximal, characteristic
4	#26019150	C₂	1	2	yes	maximal, complemented
5	#26019104	C₂	1	2	yes	maximal, complemented
6	#26019104	C₂	1	2	yes	maximal, complemented
7	C₈×Dic₃₅₆	C₂	1	2	yes	maximal, complemented
8	#26019027	C₂	1	1	yes	maximal, characteristic
9	#26018708	C₂	1	1	yes	maximal, characteristic
10	C₂×#19151330	C₂	1	1	yes	maximal, characteristic
11	C₈×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
12	#19152155	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
14	#19151934	C₄	1	2	yes	—
15	#19151894	C₄	1	1	yes	characteristic
16	#19151894	C₄	1	1	yes	characteristic
17	#19151809	C₄	1	1	yes	characteristic
18	#19151809	C₄	1	1	yes	characteristic
19	#19151756	C₄	1	2	yes	—
20	#19151728	C₂²	1	2	yes	—
21	#19151728	C₂²	1	2	yes	—
22	#19151726	—	2	2	no	—
23	#19151726	—	2	2	no	—
24	C₈×Dic₁₇₈	C₂²	1	2	yes	—
25	C₈×Dic₁₇₈	C₂²	1	2	yes	—
26	C₄×Dic₃₅₆	C₂²	1	2	yes	—
27	C₄×Dic₃₅₆	C₂²	1	2	yes	—
28	#19151626	—	2	1	no	—
29	#19151622	C₂²	1	1	yes	characteristic
30	#19151621	—	2	1	no	—
31	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
32	#19151556	—	2	1	no	—
33	M₁₆×Dic₈₉	—	2	1	no	—
34	#19151351	C₂²	1	2	yes	—
35	#19151351	C₂²	1	2	yes	—
36	#19151330	C₂²	1	1	yes	characteristic
37	#19151330	C₂²	1	1	yes	characteristic
38	#19151330	C₂²	1	2	yes	—
39	#19151330	C₂²	1	1	yes	characteristic
40	#19151330	C₂²	1	1	yes	characteristic
41	#19151330	C₂²	1	2	yes	—
42	#431×C₈₉	C₂²	1	1	yes	characteristic
43	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
45	#322×C₈₉	C₂²	1	2	yes	—
46	C₈²×C₈₉	C₂²	1	2	yes	—
47	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
48	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
49	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
50	C₂×#13083558	—	2	1	no	—
51	C₂²×#10623494	—	2	1	no	—
52	#15668012	C₂×C₄	1	1	yes	characteristic
53	#15668012	C₂×C₄	1	1	yes	characteristic
54	#15668004	—	2	1	no	—
55	#15667991	—	2	1	no	—
56	#15667991	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₇₈	—	2	1	no	—
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	2	yes	—
63	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
64	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	#15667971	—	2	1	no	—
66	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
67	C₂×#13083553	—	2	1	no	—
68	C₂×C₄×Dic₈₉	—	2	1	no	—
69	#15667946	C₂×C₄	1	2	yes	—
70	#15667946	C₂×C₄	1	2	yes	—
71	C₄⋊#10623494	—	2	1	no	—
72	C₄⋊#10623494	—	2	1	no	—
73	C₄⋊#10623494	C₂³	1	1	yes	characteristic
74	C₄⋊#10623494	C₂³	1	1	yes	characteristic
75	C₄⋊#10623494	C₂³	1	2	yes	—
76	#15667941	—	2	2	no	—
77	#15667941	—	2	2	no	—
78	#15667941	—	2	2	no	—
79	#15667941	—	2	2	no	—
80	#15667937	C₂×C₄	1	2	yes	—
81	#15667937	C₂×C₄	1	2	yes	—
82	#15667937	C₂×C₄	1	2	yes	—
83	#15667937	C₂×C₄	1	2	yes	—
84	#15667922	—	2	2	no	—
85	#15667922	—	2	2	no	—
86	C₈×Dic₈₉	—	2	2	no	—
87	C₈×Dic₈₉	—	2	2	no	—
88	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
89	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
90	C₂²×C₈×C₈₉	—	2	1	no	—
91	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
92	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
93	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
94	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
95	#97×C₈₉	D₄	1	2	yes	—
96	C₄×C₈×C₈₉	—	2	1	no	—
97	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₈₉	C₂³	1	2	yes	—
100	C₄×C₈×C₈₉	C₂³	1	2	yes	—
101	C₂×Dic₁₇₈	—	2	2	no	—
102	C₂×Dic₁₇₈	—	2	1	no	—
103	C₂×Dic₁₇₈	—	2	1	no	—
104	C₂×Dic₁₇₈	—	4	1	no	—
105	C₂×Dic₁₇₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
107	C₂×Dic₁₇₈	C₂²×C₄	1	1	yes	characteristic
108	C₂²×Dic₈₉	—	2	1	no	—
109	C₂²×Dic₈₉	—	2	1	no	—
110	Dic₃₅₆	—	2	4	no	—
111	Dic₃₅₆	—	4	2	no	—
112	#13083558	—	4	2	no	—
113	C₂×#10623494	—	2	1	no	—
114	C₂×#10623494	—	2	1	no	—
115	C₂×#10623494	—	2	2	no	—
116	C₂×#10623494	—	4	2	no	—
117	C₄⋊Dic₈₉	—	2	1	no	—
118	C₄⋊Dic₈₉	—	2	1	no	—
119	C₄⋊Dic₈₉	—	2	1	no	—
120	C₄⋊Dic₈₉	—	2	1	no	—
121	C₄⋊Dic₈₉	C₂²×C₄	1	1	yes	characteristic
122	C₄⋊Dic₈₉	C₂²×C₄	1	1	yes	characteristic
123	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
124	#13083553	—	2	1	no	—
125	#13083553	—	2	1	no	—
126	#13083553	—	2	2	no	—
127	#13083553	—	2	1	no	—
128	#13083553	—	2	1	no	—
129	#13083553	—	2	2	no	—
130	C₂²⋊Dic₈₉	—	2	2	no	—
131	C₄×Dic₈₉	—	2	1	no	—
132	C₄×Dic₈₉	—	2	1	no	—
133	C₄×Dic₈₉	—	2	2	no	—
134	C₄×Dic₈₉	—	2	1	no	—
135	C₄×Dic₈₉	—	2	1	no	—
136	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
139	M₁₆×C₈₉	—	2	2	no	—
140	M₁₆×C₈₉	Q₁₆	1	4	yes	complemented
141	C₂×C₈×C₈₉	—	2	1	no	—
142	C₂×C₈×C₈₉	—	2	1	no	—
143	C₂×C₈×C₈₉	—	4	2	no	—
144	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
145	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
146	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
147	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
148	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
149	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₈₉	#41	1	1	yes	characteristic
151	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
152	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₈₉	—	2	1	no	—
154	C₂×C₄×C₈₉	—	2	1	no	—
155	C₂×C₄×C₈₉	—	4	1	no	—
156	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
160	C₂×C₄×C₈₉	#135	1	2	yes	—
161	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
164	C₂×Dic₈₉	—	2	1	no	—
165	C₂×Dic₈₉	—	2	1	no	—
166	C₂×Dic₈₉	—	2	1	no	—
167	C₂×Dic₈₉	—	2	1	no	—
168	C₂×Dic₈₉	—	2	2	no	—
169	C₂×Dic₈₉	—	2	2	no	—
170	C₂×Dic₈₉	—	4	2	no	—
171	C₂×Dic₈₉	—	8	1	no	—
172	Dic₁₇₈	—	2	1	no	—
173	Dic₁₇₈	—	2	2	no	—
174	Dic₁₇₈	—	2	1	no	—
175	Dic₁₇₈	—	2	1	no	—
176	Dic₁₇₈	—	2	2	no	—
177	Dic₁₇₈	—	4	2	no	—
178	Dic₁₇₈	—	8	1	no	—
179	Dic₁₇₈	#131	1	2	yes	—
180	C₈×C₈₉	—	2	2	no	—
181	C₈×C₈₉	—	2	2	no	—
182	C₈×C₈₉	—	2	2	no	—
183	C₈×C₈₉	—	2	2	no	—
184	C₈×C₈₉	—	4	1	no	—
185	C₈×C₈₉	C₂×Q₁₆	1	4	yes	—
186	#10623494	—	4	1	no	—
187	#10623494	—	4	1	no	—
188	#10623494	—	4	2	no	—
189	C₂²×C₈₉	—	2	1	no	—
190	C₂²×C₈₉	—	2	1	no	—
191	C₂²×C₈₉	C₄×Q₁₆	1	2	yes	—
192	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
193	C₄×C₈₉	—	2	1	no	—
194	C₄×C₈₉	—	2	1	no	—
195	C₄×C₈₉	—	2	1	no	—
196	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
197	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
198	C₄×C₈₉	#568	1	1	yes	characteristic
199	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
200	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
201	Dic₈₉	—	2	2	no	—
202	Dic₈₉	—	4	1	no	—
203	Dic₈₉	—	4	2	no	—
204	Dic₈₉	—	4	2	no	—
205	Dic₈₉	—	4	1	no	—
206	Dic₈₉	—	4	1	no	—
207	#17998	—	89	1	no	maximal, Sylow
208	C₂×C₈₉	—	2	2	no	—
209	C₂×C₈₉	#2933	1	1	yes	characteristic
210	C₂×C₈₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
211	C₂×C₈₉	#2956	1	1	yes	characteristic
212	#3104	—	89	1	no	—
213	Q₈×M₁₆	—	89	1	no	—
214	#2962	—	89	1	no	—
215	#1591	—	89	2	no	—
216	#1584	—	89	2	no	—
217	#1584	—	89	2	no	—
218	C₈×Q₁₆	—	89	2	no	—
219	#1571	—	89	1	no	—
220	#1493	—	89	1	no	—
221	C₂×Q₈⋊C₈	—	89	1	no	—
222	C₈×M₁₆	D₈₉	1	1	yes	characteristic
223	C₈₉	#17998	1	1	yes	characteristic, complemented, Sylow
224	#533	—	89	1	no	—
225	C₂×C₄×Q₈	—	89	1	no	—
226	#484	—	89	2	no	—
227	#477	—	89	1	no	—
228	#477	—	89	1	no	—
229	#462	—	89	1	no	—
230	#462	—	89	1	no	—
231	#451	—	89	2	no	—
232	#446	—	89	2	no	—
233	#446	—	89	2	no	—
234	#446	—	178	2	no	—
235	#446	—	178	2	no	—
236	C₈×Q₈	—	89	2	no	—
237	C₈×Q₈	—	89	2	no	—
238	C₄×Q₁₆	—	89	2	no	—
239	C₄×Q₁₆	—	89	2	no	—
240	#431	C₂×D₈₉	1	1	yes	characteristic
241	#424	—	178	1	no	—
242	#423	—	89	1	no	—
243	#423	—	178	1	no	—
244	C₂×C₄⋊C₈	—	89	1	no	—
245	#405	—	178	1	no	—
246	C₄×M₁₆	—	178	1	no	—
247	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
248	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
249	#335	—	89	2	no	—
250	#335	—	89	2	no	—
251	Q₈⋊C₈	—	89	1	no	—
252	Q₈⋊C₈	—	89	2	no	—
253	Q₈⋊C₈	—	89	1	no	—
254	Q₈⋊C₈	—	89	1	no	—
255	Q₈⋊C₈	—	89	2	no	—
256	Q₈⋊C₈	—	89	1	no	—
257	#322	C₂×D₈₉	1	2	yes	—
258	C₈²	C₂×D₈₉	1	2	yes	—
259	C₂²×Q₈	—	89	1	no	—
260	C₂×Q₁₆	—	89	2	no	—
261	C₂×Q₁₆	—	89	2	no	—
262	C₂×M₁₆	—	178	1	no	—
263	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
264	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
265	C₂²×C₈	—	2	1	no	—
266	C₂²×C₈	—	178	1	no	—
267	C₄⋊Q₈	—	89	1	no	—
268	C₄⋊Q₈	—	89	1	no	—
269	#125	—	178	1	no	—
270	C₂²⋊Q₈	—	89	2	no	—
271	C₂²⋊Q₈	—	178	1	no	—
272	C₄×Q₈	—	89	1	no	—
273	C₄×Q₈	—	89	1	no	—
274	C₄×Q₈	—	89	1	no	—
275	C₄×Q₈	—	89	1	no	—
276	C₄×Q₈	—	89	2	no	—
277	C₄×Q₈	—	178	1	no	—
278	C₄×Q₈	—	178	1	no	—
279	C₄×Q₈	—	178	1	no	—
280	#117	—	178	1	no	—
281	C₂×C₄⋊C₄	—	89	1	no	—
282	C₂×C₄⋊C₄	—	178	1	no	—
283	C₂×C₄²	—	178	1	no	—
284	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
285	#107	—	89	2	no	—
286	#107	—	89	2	no	—
287	C₄⋊C₈	—	89	1	no	—
288	C₄⋊C₈	—	89	1	no	—
289	C₄⋊C₈	—	89	2	no	—
290	C₄⋊C₈	—	178	2	no	—
291	C₄⋊C₈	—	178	2	no	—
292	C₄⋊C₈	—	178	2	no	—
293	C₄⋊C₈	—	178	2	no	—
294	C₄⋊C₈	—	178	1	no	—
295	C₄⋊C₈	—	178	1	no	—
296	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
297	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
298	#103	—	89	2	no	—
299	#103	—	89	2	no	—
300	#103	—	89	2	no	—
301	#103	—	89	2	no	—
302	C₂²⋊C₈	C₄×D₈₉	1	2	yes	—
303	#97	—	178	2	no	—
304	#97	—	178	2	no	—
305	#97	D₃₅₆	1	2	yes	—
306	C₄×C₈	—	2	1	no	—
307	C₄×C₈	—	178	2	no	—
308	C₄×C₈	—	178	2	no	—
309	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
310	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
311	C₄×C₈	C₂²×D₈₉	1	2	yes	—
312	C₄×C₈	C₂²×D₈₉	1	2	yes	—
313	C₂×Q₈	—	89	1	no	—
314	C₂×Q₈	—	89	1	no	—
315	C₂×Q₈	—	89	2	no	—
316	C₂×Q₈	—	356	1	no	—
317	C₂×Q₈	—	178	2	no	—
318	C₂×Q₈	—	178	1	no	—
319	C₂×Q₈	—	178	1	no	—
320	C₂²×C₄	—	178	1	no	—
321	C₂²×C₄	—	178	1	no	—
322	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
323	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
324	C₂²×C₄	#13083574	1	1	yes	characteristic
325	Q₁₆	—	356	2	no	—
326	Q₁₆	—	178	4	no	—
327	M₁₆	—	2	2	no	—
328	M₁₆	—	356	2	no	—
329	M₁₆	Dic₃₅₆	1	4	yes	complemented
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	4	2	no	—
333	C₂×C₈	—	356	2	no	—
334	C₂×C₈	—	178	1	no	—
335	C₂×C₈	—	178	2	no	—
336	C₂×C₈	—	178	1	no	—
337	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
338	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
339	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
340	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
341	C₄⋊C₄	—	89	1	no	—
342	C₄⋊C₄	—	89	1	no	—
343	C₄⋊C₄	—	89	2	no	—
344	C₄⋊C₄	—	178	1	no	—
345	C₄⋊C₄	—	178	1	no	—
346	C₄⋊C₄	—	178	1	no	—
347	C₄⋊C₄	—	178	1	no	—
348	C₄⋊C₄	—	178	1	no	—
349	C₄⋊C₄	—	178	2	no	—
350	C₄⋊C₄	—	178	1	no	—
351	C₄⋊C₄	—	178	1	no	—
352	C₄⋊C₄	—	178	1	no	—
353	C₄⋊C₄	—	178	2	no	—
354	C₂²⋊C₄	—	178	2	no	—
355	C₄²	—	178	1	no	—
356	C₄²	—	178	1	no	—
357	C₄²	—	178	1	no	—
358	C₄²	—	178	2	no	—
359	C₄²	—	178	1	no	—
360	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
361	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
362	C₄²	#13083574	1	1	yes	characteristic
363	C₄²	C₂³×D₈₉	1	1	yes	characteristic
364	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
365	Q₈	—	89	2	no	—
366	Q₈	—	356	2	no	—
367	Q₈	—	178	2	no	—
368	Q₈	—	178	1	no	—
369	Q₈	—	178	1	no	—
370	Q₈	—	178	2	no	—
371	Q₈	—	178	1	no	—
372	Q₈	—	712	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	356	2	no	—
377	C₂×C₄	—	178	1	no	—
378	C₂×C₄	—	178	1	no	—
379	C₂×C₄	—	178	1	no	—
380	C₂×C₄	—	178	2	no	—
381	C₂×C₄	—	178	2	no	—
382	C₂×C₄	—	178	1	no	—
383	C₂×C₄	—	712	1	no	—
384	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
385	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
386	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
387	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
388	C₂×C₄	#15668034	1	2	yes	—
389	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
390	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
391	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
392	C₈	—	2	2	no	—
393	C₈	—	2	2	no	—
394	C₈	—	2	2	no	—
395	C₈	—	2	2	no	—
396	C₈	—	4	1	no	—
397	C₈	—	356	1	no	—
398	C₈	—	356	1	no	—
399	C₈	—	356	2	no	—
400	C₈	C₂×Dic₃₅₆	1	4	yes	—
401	C₂²	—	2	1	no	—
402	C₂²	—	2	1	no	—
403	C₂²	C₄×Dic₃₅₆	1	2	yes	—
404	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
405	C₄	—	2	1	no	—
406	C₄	—	2	1	no	—
407	C₄	—	2	1	no	—
408	C₄	—	356	1	no	—
409	C₄	—	356	1	no	—
410	C₄	—	356	2	no	—
411	C₄	—	356	1	no	—
412	C₄	—	356	2	no	—
413	C₄	—	178	2	no	—
414	C₄	C₄×Dic₃₅₆	1	2	yes	—
415	C₄	#131×D₈₉	1	1	yes	characteristic
416	C₄	#19152363	1	1	yes	characteristic
417	C₄	C₂²×Dic₃₅₆	1	1	yes	characteristic
418	C₄	C₂×#15668034	1	1	yes	characteristic
419	C₂	—	2	2	no	—
420	C₂	#26028980	1	1	yes	characteristic
421	C₂	C₂×C₄×Dic₃₅₆	1	1	yes	characteristic
422	C₂	#26029163	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₈×M₁₆	D₈₉
hypocenter	C₈₉	#17998
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
4. #26019150	C₂	semidirect
5. #26019104	C₂	semidirect
6. #26019104	C₂	semidirect
7. C₈×Dic₃₅₆	C₂	semidirect
140. M₁₆×C₈₉	Q₁₆	semidirect
223. C₈₉	#17998	semidirect
329. M₁₆	Dic₃₅₆	semidirect

Maximal subgroups

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

subgroup	index
#26030292	2
M₁₆×Dic₁₇₈	2
#26029199	2
#26019150	2
#26019104	2
C₈×Dic₃₅₆	2
#26019027	2
#26018708	2
C₂×#19151330	2
C₈×M₁₆×C₈₉	2
#17998	89