Subgroups — #36573291

Order 22784.

774 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151728	C₂	1	1	yes	maximal, characteristic
2	#26023207	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
4	#26021019	C₂	1	2	yes	maximal
5	#26021015	C₂	1	1	yes	maximal, characteristic
6	#26021014	C₂	1	2	yes	maximal
7	#26021010	C₂	1	1	yes	maximal, characteristic
8	#26020419	C₂	1	2	yes	maximal
9	#26020409	C₂	1	1	yes	maximal, characteristic
10	#26020151	C₂	1	2	yes	maximal
11	C₄×#107×C₈₉	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₁₇₈	—	2	1	no	—
13	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
14	#19151728	C₄	1	8	yes	complemented
15	C₂×#15667945	C₂²	1	2	yes	—
16	C₂×#15667944	C₂²	1	2	yes	—
17	C₂×#15667944	C₂²	1	2	yes	—
18	C₂×#15667943	C₂²	1	2	yes	—
19	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
20	C₂×#15667941	C₂²	1	2	yes	—
21	C₂×#15667922	C₂²	1	2	yes	—
22	#19151504	—	2	1	no	—
23	#19151496	—	2	1	no	—
24	#19151495	—	2	2	no	—
25	#19151485	—	2	2	no	—
26	#19151468	C₂²	1	1	yes	characteristic
27	#19151468	C₂²	1	1	yes	characteristic
28	#19151467	C₂²	1	2	yes	—
29	#19151463	C₂²	1	2	yes	—
30	C₄⋊C₄×Dic₈₉	C₂²	1	2	yes	—
31	C₄⋊C₄×Dic₈₉	C₂²	1	2	yes	—
32	C₄×C₄⋊Dic₈₉	—	2	1	no	—
33	#19151360	C₂²	1	2	yes	—
34	#19151360	C₂²	1	2	yes	—
35	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
40	#340×C₈₉	C₂²	1	1	yes	characteristic
41	#340×C₈₉	C₂²	1	1	yes	characteristic
42	C₂²×Dic₁₇₈	D₄	1	1	yes	characteristic
43	C₂²×#10623494	C₂³	1	2	yes	—
44	C₄×Dic₁₇₈	—	4	4	no	—
45	C₄×Dic₁₇₈	C₂×C₄	1	4	yes	—
46	C₄×Dic₁₇₈	Q₈	1	4	yes	—
47	C₂×C₄⋊Dic₈₉	—	2	1	no	—
48	C₂×C₄⋊Dic₈₉	—	2	1	no	—
49	C₂×C₄⋊Dic₈₉	—	2	1	no	—
50	C₂×C₄⋊Dic₈₉	—	2	1	no	—
51	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
52	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
53	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
54	C₂×#13083553	—	2	2	no	—
55	C₂×#13083553	—	2	2	no	—
56	C₂×#13083553	D₄	1	2	yes	—
57	C₂×C₄×Dic₈₉	—	2	2	no	—
58	C₂×C₄×Dic₈₉	—	2	2	no	—
59	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
60	#15667945	—	4	2	no	—
61	#15667944	—	2	4	no	—
62	#15667944	—	4	2	no	—
63	#15667943	—	4	2	no	—
64	C₄⋊#10623494	C₂×C₄	1	4	yes	—
65	#15667941	C₂×C₄	1	8	yes	—
66	#15667922	C₂×C₄	1	8	yes	—
67	#15667919	—	2	2	no	—
68	#15667919	—	2	2	no	—
69	#15667919	—	2	2	no	—
70	#15667919	—	2	2	no	—
71	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₈₉	—	2	1	no	—
78	C₂×C₄²×C₈₉	—	2	1	no	—
79	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
80	#107×C₈₉	—	2	2	no	—
81	#107×C₈₉	Q₈	1	4	yes	complemented
82	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
83	#95×C₈₉	—	2	1	no	—
84	#95×C₈₉	—	2	1	no	—
85	C₂×Dic₁₇₈	—	2	2	no	—
86	C₂×Dic₁₇₈	—	2	2	no	—
87	C₂×Dic₁₇₈	C₄⋊C₄	1	4	yes	—
88	C₂²×Dic₈₉	—	2	2	no	—
89	C₂²×Dic₈₉	—	2	2	no	—
90	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
91	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
92	C₂×#10623494	—	2	2	no	—
93	C₂×#10623494	—	2	2	no	—
94	C₂×#10623494	C₂²×C₄	1	4	yes	—
95	C₄⋊Dic₈₉	—	2	2	no	—
96	C₄⋊Dic₈₉	—	2	2	no	—
97	C₄⋊Dic₈₉	—	2	2	no	—
98	C₄⋊Dic₈₉	—	2	2	no	—
99	C₄⋊Dic₈₉	—	4	2	no	—
100	C₄⋊Dic₈₉	—	4	2	no	—
101	C₄⋊Dic₈₉	—	8	1	no	—
102	C₄⋊Dic₈₉	—	8	1	no	—
103	C₄⋊Dic₈₉	C₄⋊C₄	1	4	yes	—
104	#13083553	—	4	4	no	—
105	#13083553	—	8	2	no	—
106	#13083553	C₄⋊C₄	1	8	yes	—
107	C₄×Dic₈₉	—	4	4	no	—
108	C₄×Dic₈₉	—	8	2	no	—
109	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
110	C₄×Dic₈₉	C₂×Q₈	1	4	yes	—
111	C₂²×C₄×C₈₉	—	2	1	no	—
112	C₂²×C₄×C₈₉	—	2	1	no	—
113	C₂²×C₄×C₈₉	—	2	1	no	—
114	C₂²×C₄×C₈₉	—	2	1	no	—
115	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
118	C₂×C₈×C₈₉	—	2	1	no	—
119	C₂×C₈×C₈₉	—	2	1	no	—
120	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
121	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
123	C₂×C₈×C₈₉	#41	1	2	yes	—
124	C₄⋊C₄×C₈₉	—	2	1	no	—
125	C₄⋊C₄×C₈₉	—	2	1	no	—
126	C₄⋊C₄×C₈₉	—	2	1	no	—
127	C₄⋊C₄×C₈₉	—	2	1	no	—
128	C₄⋊C₄×C₈₉	—	2	1	no	—
129	C₄⋊C₄×C₈₉	—	2	1	no	—
130	C₄⋊C₄×C₈₉	C₂×Q₈	1	2	yes	—
131	C₄⋊C₄×C₈₉	C₂×Q₈	1	2	yes	—
132	C₄²×C₈₉	—	4	2	no	—
133	C₄²×C₈₉	—	4	2	no	—
134	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
135	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
136	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	—	2	1	no	—
138	C₂×C₄×C₈₉	—	2	1	no	—
139	C₂×C₄×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	—	2	2	no	—
141	C₂×C₄×C₈₉	—	4	1	no	—
142	C₂×C₄×C₈₉	—	4	1	no	—
143	C₂×C₄×C₈₉	—	4	1	no	—
144	C₂×C₄×C₈₉	—	4	1	no	—
145	C₂×C₄×C₈₉	—	4	1	no	—
146	C₂×C₄×C₈₉	—	4	1	no	—
147	C₂×C₄×C₈₉	—	4	1	no	—
148	C₂×C₄×C₈₉	—	4	1	no	—
149	C₂×C₄×C₈₉	—	4	1	no	—
150	C₂×C₄×C₈₉	—	4	1	no	—
151	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
152	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
153	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
154	C₂×C₄×C₈₉	#137	1	2	yes	—
155	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
161	C₂×Dic₈₉	—	2	2	no	—
162	C₂×Dic₈₉	—	2	2	no	—
163	C₂×Dic₈₉	—	2	2	no	—
164	C₂×Dic₈₉	—	2	2	no	—
165	C₂×Dic₈₉	—	4	2	no	—
166	C₂×Dic₈₉	—	4	2	no	—
167	C₂×Dic₈₉	—	4	2	no	—
168	C₂×Dic₈₉	—	4	2	no	—
169	C₂×Dic₈₉	—	4	2	no	—
170	C₂×Dic₈₉	—	4	2	no	—
171	C₂×Dic₈₉	C₂×C₄⋊C₄	1	4	yes	—
172	C₂×Dic₈₉	C₂×C₄⋊C₄	1	4	yes	—
173	Dic₁₇₈	—	2	8	no	—
174	C₈×C₈₉	—	2	2	no	—
175	C₈×C₈₉	C₄×Q₈	1	4	yes	complemented
176	#10623494	—	2	4	no	—
177	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
179	C₂²×C₈₉	#550	1	1	yes	characteristic
180	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
182	C₂²×C₈₉	#577	1	1	yes	characteristic
183	C₂²×C₈₉	#577	1	1	yes	characteristic
184	C₄×C₈₉	—	2	2	no	—
185	C₄×C₈₉	—	4	2	no	—
186	C₄×C₈₉	—	4	2	no	—
187	C₄×C₈₉	—	8	1	no	—
188	C₄×C₈₉	—	8	1	no	—
189	C₄×C₈₉	#428	1	4	yes	—
190	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
191	C₄×C₈₉	#520	1	1	yes	characteristic
192	C₄×C₈₉	#557	1	1	yes	characteristic
193	C₄×C₈₉	Q₈²	1	1	yes	characteristic
194	Dic₈₉	—	2	4	no	—
195	Dic₈₉	—	2	4	no	—
196	Dic₈₉	—	8	2	no	—
197	Dic₈₉	—	8	2	no	—
198	#22729	—	89	1	no	maximal, Sylow
199	C₂×C₈₉	Q₈×C₄⋊C₄	1	1	yes	characteristic
200	C₂×C₈₉	C₂×#428	1	1	yes	characteristic
201	C₂×C₈₉	#2912	1	1	yes	characteristic
202	C₂×C₈₉	#3384	1	1	yes	characteristic
203	C₂×C₈₉	#3387	1	1	yes	characteristic
204	C₂×C₈₉	#3397	1	1	yes	characteristic
205	C₂×C₈₉	#3400	1	1	yes	characteristic
206	C₂×#446	—	89	1	no	—
207	#2355	—	89	1	no	—
208	Q₈×C₄⋊C₄	—	89	1	no	—
209	#1949	—	89	2	no	—
210	#1948	—	89	1	no	—
211	#1948	—	89	2	no	—
212	#1947	—	89	1	no	—
213	#1847	—	89	2	no	—
214	#1844	—	89	1	no	—
215	#1776	—	89	2	no	—
216	C₄×#107	D₈₉	1	1	yes	characteristic
217	C₈₉	#22729	1	1	yes	characteristic, complemented, Sylow
218	C₂×C₄×Q₈	—	89	1	no	—
219	C₂×C₄×Q₈	—	178	1	no	—
220	#446	—	89	8	no	—
221	C₂×#107	—	89	2	no	—
222	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
223	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
224	C₂×#106	—	89	2	no	—
225	C₂×#106	—	89	2	no	—
226	C₂×#106	—	89	2	no	—
227	C₂×C₄⋊C₈	—	89	2	no	—
228	C₂×C₄⋊C₈	—	89	1	no	—
229	C₂×#97	—	89	2	no	—
230	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
231	#391	—	178	1	no	—
232	#389	—	178	2	no	—
233	#389	—	178	1	no	—
234	#387	—	178	2	no	—
235	#384	—	89	2	no	—
236	#384	—	89	1	no	—
237	#384	—	89	1	no	—
238	#382	—	89	2	no	—
239	C₄×C₄⋊C₄	—	89	2	no	—
240	C₄×C₄⋊C₄	—	89	2	no	—
241	C₄×C₄⋊C₄	—	178	1	no	—
242	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
243	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
244	#340	—	89	2	no	—
245	#340	—	89	2	no	—
246	#340	C₂×D₈₉	1	1	yes	characteristic
247	#340	C₂×D₈₉	1	1	yes	characteristic
248	C₂²×Q₈	—	89	1	no	—
249	C₂²×C₈	—	89	2	no	—
250	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
251	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
252	C₄×Q₈	—	89	4	no	—
253	C₄×Q₈	—	89	4	no	—
254	C₄×Q₈	—	356	4	no	—
255	C₂×C₄⋊C₄	—	89	1	no	—
256	C₂×C₄⋊C₄	—	89	2	no	—
257	C₂×C₄⋊C₄	—	89	2	no	—
258	C₂×C₄⋊C₄	—	89	2	no	—
259	C₂×C₄⋊C₄	—	178	2	no	—
260	C₂×C₄⋊C₄	—	178	1	no	—
261	C₂×C₄⋊C₄	—	178	1	no	—
262	C₂×C₄⋊C₄	—	178	2	no	—
263	C₂×C₄⋊C₄	—	178	1	no	—
264	C₂×C₄⋊C₄	—	178	1	no	—
265	C₂×C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
266	C₂×C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
267	C₂×C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
268	C₂×C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
269	C₂×C₄²	—	2	1	no	—
270	C₂×C₄²	—	2	1	no	—
271	C₂×C₄²	—	89	2	no	—
272	C₂×C₄²	—	178	2	no	—
273	C₂×C₄²	—	178	2	no	—
274	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
275	#107	—	2	2	no	—
276	#107	—	356	2	no	—
277	#107	Dic₁₇₈	1	4	yes	complemented
278	#106	—	356	2	no	—
279	#106	—	356	2	no	—
280	#106	—	178	4	no	—
281	C₄⋊C₈	—	89	8	no	—
282	C₄⋊C₈	—	89	4	no	—
283	#97	—	89	8	no	—
284	C₄×C₈	C₄×D₈₉	1	4	yes	—
285	#95	—	2	1	no	—
286	#95	—	2	1	no	—
287	#95	—	178	2	no	—
288	#95	—	178	2	no	—
289	#95	—	178	2	no	—
290	#95	—	178	2	no	—
291	C₂×Q₈	—	89	4	no	—
292	C₂×Q₈	—	178	2	no	—
293	C₂×Q₈	—	178	2	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	89	2	no	—
299	C₂²×C₄	—	89	2	no	—
300	C₂²×C₄	—	178	2	no	—
301	C₂²×C₄	—	178	2	no	—
302	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
303	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
304	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	89	4	no	—
308	C₂×C₈	—	178	2	no	—
309	C₂×C₈	—	178	2	no	—
310	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
311	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
312	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
313	C₂×C₈	#13083574	1	2	yes	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	1	no	—
316	C₄⋊C₄	—	2	1	no	—
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	2	1	no	—
320	C₄⋊C₄	—	89	8	no	—
321	C₄⋊C₄	—	89	4	no	—
322	C₄⋊C₄	—	356	4	no	—
323	C₄⋊C₄	—	356	2	no	—
324	C₄⋊C₄	—	356	2	no	—
325	C₄⋊C₄	—	178	2	no	—
326	C₄⋊C₄	—	178	2	no	—
327	C₄⋊C₄	—	178	2	no	—
328	C₄⋊C₄	—	178	2	no	—
329	C₄⋊C₄	—	712	1	no	—
330	C₄⋊C₄	—	712	1	no	—
331	C₄⋊C₄	—	712	2	no	—
332	C₄⋊C₄	C₂×Dic₁₇₈	1	2	yes	—
333	C₄⋊C₄	C₂×Dic₁₇₈	1	2	yes	—
334	C₄²	—	4	2	no	—
335	C₄²	—	4	2	no	—
336	C₄²	—	89	4	no	—
337	C₄²	—	89	4	no	—
338	C₄²	—	356	4	no	—
339	C₄²	—	712	2	no	—
340	C₄²	C₂×C₄×D₈₉	1	2	yes	—
341	C₄²	Q₈×D₈₉	1	2	yes	—
342	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
343	Q₈	—	178	8	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	—	4	1	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	89	4	no	—
359	C₂×C₄	—	89	4	no	—
360	C₂×C₄	—	356	2	no	—
361	C₂×C₄	—	356	2	no	—
362	C₂×C₄	—	356	2	no	—
363	C₂×C₄	—	356	2	no	—
364	C₂×C₄	—	356	2	no	—
365	C₂×C₄	—	356	2	no	—
366	C₂×C₄	—	178	2	no	—
367	C₂×C₄	—	178	2	no	—
368	C₂×C₄	—	178	2	no	—
369	C₂×C₄	—	178	2	no	—
370	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
371	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
372	C₂×C₄	#15668036	1	2	yes	—
373	C₂×C₄	#15668044	1	2	yes	—
374	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
375	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
376	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
377	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
378	C₂×C₄	#15668058	1	1	yes	characteristic
379	C₂×C₄	#15668059	1	1	yes	characteristic
380	C₈	—	2	2	no	—
381	C₈	—	178	4	no	—
382	C₈	C₄×Dic₁₇₈	1	4	yes	complemented
383	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
384	C₂²	D₄×Dic₁₇₈	1	1	yes	characteristic
385	C₂²	#19152269	1	1	yes	characteristic
386	C₂²	C₂×#15668036	1	1	yes	characteristic
387	C₂²	C₂×#15668044	1	1	yes	characteristic
388	C₂²	#19152405	1	1	yes	characteristic
389	C₂²	#19152405	1	1	yes	characteristic
390	C₄	—	2	2	no	—
391	C₄	—	4	2	no	—
392	C₄	—	4	2	no	—
393	C₄	—	8	1	no	—
394	C₄	—	8	1	no	—
395	C₄	—	178	4	no	—
396	C₄	—	178	4	no	—
397	C₄	—	712	2	no	—
398	C₄	—	712	2	no	—
399	C₄	#19151639	1	4	yes	—
400	C₄	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
401	C₄	#19152078	1	1	yes	characteristic
402	C₄	#19152326	1	1	yes	characteristic
403	C₄	Q₈×Dic₁₇₈	1	1	yes	characteristic
404	C₂	C₄⋊C₄×Dic₁₇₈	1	1	yes	characteristic
405	C₂	C₂×#19151639	1	1	yes	characteristic
406	C₂	#26028844	1	1	yes	characteristic
407	C₂	#26033149	1	1	yes	characteristic
408	C₂	#26033182	1	1	yes	characteristic
409	C₂	#26033270	1	1	yes	characteristic
410	C₂	#26033303	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄×#107×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄×#107	D₈₉
hypocenter	C₈₉	#22729
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
14. #19151728	C₄	semidirect
81. #107×C₈₉	Q₈	semidirect
175. C₈×C₈₉	C₄×Q₈	semidirect
217. C₈₉	#22729	semidirect
277. #107	Dic₁₇₈	semidirect
382. C₈	C₄×Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151728	2
#26023207	2
C₄⋊C₄×Dic₁₇₈	2
#26021019	2
#26021015	2
#26021014	2
#26021010	2
#26020419	2
#26020409	2
#26020151	2
C₄×#107×C₈₉	2
#22729	89