Subgroups — C₂²×#455×C₈₉ #36438227

Order 22784.

5110 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#137×C₈₉	C₂	1	2	yes	maximal, complemented
3	C₂²×#124×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#455×C₈₉	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#104×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂²×#100×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×H(5,𝔽₂)×C₈₉	—	2	12	no	—
8	C₂×H(5,𝔽₂)×C₈₉	—	2	6	no	—
9	C₂×H(5,𝔽₂)×C₈₉	C₂²	1	12	yes	—
10	C₂²×#41×C₈₉	—	2	2	no	—
11	C₂²×#41×C₈₉	C₂²	1	2	yes	—
12	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
13	C₂³×D₄×C₈₉	—	2	2	no	—
14	C₂³×D₄×C₈₉	—	2	2	no	—
15	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#137×C₈₉	—	2	12	no	—
17	C₂×#137×C₈₉	C₂²	1	24	yes	complemented
18	C₂²×Q₁₆×C₈₉	—	2	2	no	—
19	C₂²×SD₁₆×C₈₉	—	2	2	no	—
20	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	complemented
21	C₂×#124×C₈₉	—	2	6	no	—
22	C₂×#124×C₈₉	C₂²	1	12	yes	complemented
23	C₂²×C₂²⋊C₄×C₈₉	—	2	2	no	—
24	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic, complemented
25	#455×C₈₉	C₂²	1	64	yes	direct product factor
26	C₂×#104×C₈₉	C₂²	1	24	yes	complemented
27	C₂×#100×C₈₉	C₂²	1	12	yes	complemented
28	C₂⁵×C₈₉	—	2	2	no	—
29	C₂⁵×C₈₉	D₄	1	2	yes	—
30	H(5,𝔽₂)×C₈₉	—	2	48	no	—
31	H(5,𝔽₂)×C₈₉	—	2	48	no	—
32	H(5,𝔽₂)×C₈₉	—	2	24	no	—
33	H(5,𝔽₂)×C₈₉	C₂³	1	16	yes	—
34	C₂×#41×C₈₉	—	2	24	no	—
35	C₂×#41×C₈₉	—	2	24	no	—
36	C₂×#41×C₈₉	—	2	6	no	—
37	C₂×#41×C₈₉	—	2	6	no	—
38	C₂×#41×C₈₉	D₄	1	12	yes	—
39	C₂×#41×C₈₉	C₂³	1	12	yes	—
40	C₂²×Q₈×C₈₉	—	2	3	no	—
41	C₂²×Q₈×C₈₉	—	2	2	no	—
42	C₂²×Q₈×C₈₉	—	2	6	no	—
43	C₂²×Q₈×C₈₉	D₄	1	2	yes	—
44	C₂²×Q₈×C₈₉	C₂³	1	6	yes	complemented
45	C₂²×D₄×C₈₉	—	2	4	no	—
46	C₂²×D₄×C₈₉	—	2	24	no	—
47	C₂²×D₄×C₈₉	—	2	24	no	—
48	C₂²×D₄×C₈₉	—	2	2	no	—
49	C₂²×D₄×C₈₉	—	2	24	no	—
50	C₂²×D₄×C₈₉	—	2	24	no	—
51	C₂²×D₄×C₈₉	—	2	2	no	—
52	C₂²×D₄×C₈₉	—	2	6	no	—
53	C₂²×D₄×C₈₉	D₄	1	6	yes	—
54	C₂²×D₄×C₈₉	D₄	1	2	yes	—
55	C₂²×D₄×C₈₉	C₂³	1	6	yes	—
56	C₂³×C₄×C₈₉	—	2	2	no	—
57	C₂³×C₄×C₈₉	—	2	2	no	—
58	C₂³×C₄×C₈₉	—	2	1	no	—
59	C₂³×C₄×C₈₉	—	2	1	no	—
60	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
61	#137×C₈₉	—	2	48	no	—
62	#137×C₈₉	C₂³	1	32	yes	complemented
63	C₂×Q₁₆×C₈₉	—	2	24	no	—
64	C₂×SD₁₆×C₈₉	—	2	24	no	—
65	C₂×M₁₆×C₈₉	—	2	6	no	—
66	C₂×M₁₆×C₈₉	C₂³	1	12	yes	complemented
67	C₂²×C₈×C₈₉	—	2	2	no	—
68	#124×C₈₉	—	2	24	no	—
69	#124×C₈₉	C₂³	1	16	yes	complemented
70	C₂×C₂²⋊C₄×C₈₉	—	2	24	no	—
71	C₂×C₄²×C₈₉	—	2	3	no	—
72	C₂×C₄²×C₈₉	C₂³	1	6	yes	complemented
73	#104×C₈₉	C₂³	1	32	yes	complemented
74	#100×C₈₉	C₂³	1	16	yes	complemented
75	C₂⁴×C₈₉	—	2	24	no	—
76	C₂⁴×C₈₉	—	2	2	no	—
77	C₂⁴×C₈₉	—	2	2	no	—
78	C₂⁴×C₈₉	—	2	6	no	—
79	C₂⁴×C₈₉	—	4	8	no	—
80	C₂⁴×C₈₉	—	8	8	no	—
81	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂⁴×C₈₉	C₂×D₄	1	12	yes	—
83	#41×C₈₉	—	2	96	no	—
84	#41×C₈₉	—	2	32	no	—
85	#41×C₈₉	—	2	24	no	—
86	#41×C₈₉	—	2	12	no	—
87	#41×C₈₉	—	2	12	no	—
88	#41×C₈₉	C₂×D₄	1	24	yes	—
89	#41×C₈₉	C₂⁴	1	8	yes	—
90	C₂×Q₈×C₈₉	—	2	6	no	—
91	C₂×Q₈×C₈₉	—	2	6	no	—
92	C₂×Q₈×C₈₉	—	2	12	no	—
93	C₂×Q₈×C₈₉	—	2	12	no	—
94	C₂×Q₈×C₈₉	—	4	6	no	—
95	C₂×Q₈×C₈₉	—	4	6	no	—
96	C₂×Q₈×C₈₉	C₂×D₄	1	12	yes	—
97	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	complemented
98	C₂×D₄×C₈₉	—	2	48	no	—
99	C₂×D₄×C₈₉	—	2	96	no	—
100	C₂×D₄×C₈₉	—	2	96	no	—
101	C₂×D₄×C₈₉	—	2	12	no	—
102	C₂×D₄×C₈₉	—	2	12	no	—
103	C₂×D₄×C₈₉	—	2	32	no	—
104	C₂×D₄×C₈₉	—	2	32	no	—
105	C₂×D₄×C₈₉	—	2	12	no	—
106	C₂×D₄×C₈₉	—	2	6	no	—
107	C₂×D₄×C₈₉	—	2	12	no	—
108	C₂×D₄×C₈₉	—	2	12	no	—
109	C₂×D₄×C₈₉	—	2	12	no	—
110	C₂×D₄×C₈₉	C₂×D₄	1	12	yes	—
111	C₂×D₄×C₈₉	C₂×D₄	1	12	yes	—
112	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
113	C₂²×C₄×C₈₉	—	2	24	no	—
114	C₂²×C₄×C₈₉	—	2	2	no	—
115	C₂²×C₄×C₈₉	—	2	12	no	—
116	C₂²×C₄×C₈₉	—	2	12	no	—
117	C₂²×C₄×C₈₉	—	2	6	no	—
118	C₂²×C₄×C₈₉	—	2	6	no	—
119	C₂²×C₄×C₈₉	—	4	4	no	—
120	C₂²×C₄×C₈₉	—	4	3	no	—
121	C₂²×C₄×C₈₉	—	4	1	no	—
122	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
123	C₂²×C₄×C₈₉	C₂×D₄	1	6	yes	—
124	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
126	Q₁₆×C₈₉	—	2	32	no	—
127	SD₁₆×C₈₉	—	2	32	no	—
128	M₁₆×C₈₉	—	2	12	no	—
129	M₁₆×C₈₉	C₂⁴	1	8	yes	complemented
130	C₂×C₈×C₈₉	—	2	12	no	—
131	C₂²⋊C₄×C₈₉	—	2	32	no	—
132	C₄²×C₈₉	—	2	6	no	—
133	C₄²×C₈₉	C₂⁴	1	4	yes	complemented
134	C₂³×C₈₉	—	2	4	no	—
135	C₂³×C₈₉	—	2	32	no	—
136	C₂³×C₈₉	—	2	12	no	—
137	C₂³×C₈₉	—	2	12	no	—
138	C₂³×C₈₉	—	2	12	no	—
139	C₂³×C₈₉	—	4	24	no	—
140	C₂³×C₈₉	—	4	8	no	—
141	C₂³×C₈₉	—	4	8	no	—
142	C₂³×C₈₉	—	8	12	no	—
143	C₂³×C₈₉	—	8	48	no	—
144	C₂³×C₈₉	#120	1	3	yes	—
145	C₂³×C₈₉	#120	1	1	yes	characteristic
146	C₂³×C₈₉	C₂²×D₄	1	3	yes	—
147	C₂³×C₈₉	C₂²×D₄	1	8	yes	—
148	Q₈×C₈₉	—	2	12	no	—
149	Q₈×C₈₉	—	2	8	no	—
150	Q₈×C₈₉	—	4	12	no	—
151	Q₈×C₈₉	C₂²×D₄	1	8	yes	—
152	D₄×C₈₉	—	2	64	no	—
153	D₄×C₈₉	—	2	24	no	—
154	D₄×C₈₉	—	2	8	no	—
155	D₄×C₈₉	—	2	24	no	—
156	D₄×C₈₉	—	2	12	no	—
157	D₄×C₈₉	—	2	8	no	—
158	D₄×C₈₉	C₂²×D₄	1	8	yes	—
159	C₂×C₄×C₈₉	—	2	32	no	—
160	C₂×C₄×C₈₉	—	2	12	no	—
161	C₂×C₄×C₈₉	—	2	24	no	—
162	C₂×C₄×C₈₉	—	2	8	no	—
163	C₂×C₄×C₈₉	—	2	4	no	—
164	C₂×C₄×C₈₉	—	2	4	no	—
165	C₂×C₄×C₈₉	—	4	12	no	—
166	C₂×C₄×C₈₉	—	4	6	no	—
167	C₂×C₄×C₈₉	—	4	6	no	—
168	C₂×C₄×C₈₉	#120	1	6	yes	—
169	C₂×C₄×C₈₉	#120	1	6	yes	—
170	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
171	C₂×C₄×C₈₉	C₂²×D₄	1	6	yes	—
172	C₂×C₄×C₈₉	C₂²×D₄	1	6	yes	—
173	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
174	C₈×C₈₉	—	2	8	no	—
175	C₂²×C₈₉	—	2	6	no	—
176	C₂²×C₈₉	—	2	6	no	—
177	C₂²×C₈₉	—	2	8	no	—
178	C₂²×C₈₉	—	2	8	no	—
179	C₂²×C₈₉	—	4	8	no	—
180	C₂²×C₈₉	—	4	24	no	—
181	C₂²×C₈₉	—	4	24	no	—
182	C₂²×C₈₉	—	8	12	no	—
183	C₂²×C₈₉	—	8	32	no	—
184	C₂²×C₈₉	#455	1	4	yes	direct product factor
185	C₂²×C₈₉	C₂×#120	1	3	yes	—
186	C₂²×C₈₉	C₂×#120	1	3	yes	—
187	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
188	C₄×C₈₉	—	2	8	no	—
189	C₄×C₈₉	—	4	4	no	—
190	C₄×C₈₉	—	4	4	no	—
191	C₄×C₈₉	C₂×#120	1	6	yes	—
192	C₄×C₈₉	C₂³×D₄	1	2	yes	—
193	C₂²×#455	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂×C₈₉	—	2	1	no	—
195	C₂×C₈₉	—	2	3	no	—
196	C₂×C₈₉	—	4	8	no	—
197	C₂×C₈₉	—	4	8	no	—
198	C₂×C₈₉	C₂×#455	1	6	yes	direct product factor
199	C₂×C₈₉	C₂²×#120	1	1	yes	characteristic
200	C₂²×H(5,𝔽₂)	C₂×C₈₉	1	1	yes	characteristic
201	C₂²×#137	C₂×C₈₉	1	2	yes	complemented
202	C₂²×#124	C₂×C₈₉	1	1	yes	characteristic, complemented
203	C₂×#455	C₂×C₈₉	1	24	yes	direct product factor
204	C₂²×#104	C₂×C₈₉	1	2	yes	complemented
205	C₂²×#100	C₂×C₈₉	1	1	yes	characteristic, complemented
206	C₈₉	C₂²×#455	1	1	yes	characteristic, direct product factor, Sylow
207	C₂×H(5,𝔽₂)	—	2	12	no	—
208	C₂×H(5,𝔽₂)	—	2	6	no	—
209	C₂×H(5,𝔽₂)	C₂²×C₈₉	1	12	yes	—
210	C₂²×#41	—	2	2	no	—
211	C₂²×#41	C₂²×C₈₉	1	2	yes	—
212	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
213	C₂³×D₄	—	2	2	no	—
214	C₂³×D₄	—	2	2	no	—
215	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
216	C₂×#137	—	2	12	no	—
217	C₂×#137	C₂²×C₈₉	1	24	yes	complemented
218	C₂²×Q₁₆	—	2	2	no	—
219	C₂²×SD₁₆	—	2	2	no	—
220	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	complemented
221	C₂×#124	—	2	6	no	—
222	C₂×#124	C₂²×C₈₉	1	12	yes	complemented
223	C₂²×C₂²⋊C₄	—	2	2	no	—
224	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic, complemented
225	#455	C₂²×C₈₉	1	64	yes	direct product factor
226	C₂×#104	C₂²×C₈₉	1	24	yes	complemented
227	C₂×#100	C₂²×C₈₉	1	12	yes	complemented
228	C₂⁵	—	2	2	no	—
229	C₂⁵	D₄×C₈₉	1	2	yes	—
230	H(5,𝔽₂)	—	2	48	no	—
231	H(5,𝔽₂)	—	2	48	no	—
232	H(5,𝔽₂)	—	2	24	no	—
233	H(5,𝔽₂)	C₂³×C₈₉	1	16	yes	—
234	C₂×#41	—	2	24	no	—
235	C₂×#41	—	2	24	no	—
236	C₂×#41	—	2	6	no	—
237	C₂×#41	—	2	6	no	—
238	C₂×#41	D₄×C₈₉	1	12	yes	—
239	C₂×#41	C₂³×C₈₉	1	12	yes	—
240	C₂²×Q₈	—	2	3	no	—
241	C₂²×Q₈	—	2	2	no	—
242	C₂²×Q₈	—	2	6	no	—
243	C₂²×Q₈	D₄×C₈₉	1	2	yes	—
244	C₂²×Q₈	C₂³×C₈₉	1	6	yes	complemented
245	C₂²×D₄	—	2	4	no	—
246	C₂²×D₄	—	2	24	no	—
247	C₂²×D₄	—	2	24	no	—
248	C₂²×D₄	—	2	2	no	—
249	C₂²×D₄	—	2	24	no	—
250	C₂²×D₄	—	2	24	no	—
251	C₂²×D₄	—	2	2	no	—
252	C₂²×D₄	—	2	6	no	—
253	C₂²×D₄	D₄×C₈₉	1	6	yes	—
254	C₂²×D₄	D₄×C₈₉	1	2	yes	—
255	C₂²×D₄	C₂³×C₈₉	1	6	yes	—
256	C₂³×C₄	—	2	2	no	—
257	C₂³×C₄	—	2	2	no	—
258	C₂³×C₄	—	2	1	no	—
259	C₂³×C₄	—	2	1	no	—
260	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
261	#137	—	2	48	no	—
262	#137	C₂³×C₈₉	1	32	yes	complemented
263	C₂×Q₁₆	—	2	24	no	—
264	C₂×SD₁₆	—	2	24	no	—
265	C₂×M₁₆	—	2	6	no	—
266	C₂×M₁₆	C₂³×C₈₉	1	12	yes	complemented
267	C₂²×C₈	—	2	2	no	—
268	#124	—	2	24	no	—
269	#124	C₂³×C₈₉	1	16	yes	complemented
270	C₂×C₂²⋊C₄	—	2	24	no	—
271	C₂×C₄²	—	2	3	no	—
272	C₂×C₄²	C₂³×C₈₉	1	6	yes	complemented
273	#104	C₂³×C₈₉	1	32	yes	complemented
274	#100	C₂³×C₈₉	1	16	yes	complemented
275	C₂⁴	—	2	24	no	—
276	C₂⁴	—	2	2	no	—
277	C₂⁴	—	2	2	no	—
278	C₂⁴	—	2	6	no	—
279	C₂⁴	—	4	8	no	—
280	C₂⁴	—	8	8	no	—
281	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
282	C₂⁴	C₂×D₄×C₈₉	1	12	yes	—
283	#41	—	2	96	no	—
284	#41	—	2	32	no	—
285	#41	—	2	24	no	—
286	#41	—	2	12	no	—
287	#41	—	2	12	no	—
288	#41	C₂×D₄×C₈₉	1	24	yes	—
289	#41	C₂⁴×C₈₉	1	8	yes	—
290	C₂×Q₈	—	2	6	no	—
291	C₂×Q₈	—	2	6	no	—
292	C₂×Q₈	—	2	12	no	—
293	C₂×Q₈	—	2	12	no	—
294	C₂×Q₈	—	4	6	no	—
295	C₂×Q₈	—	4	6	no	—
296	C₂×Q₈	C₂×D₄×C₈₉	1	12	yes	—
297	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	complemented
298	C₂×D₄	—	2	48	no	—
299	C₂×D₄	—	2	96	no	—
300	C₂×D₄	—	2	96	no	—
301	C₂×D₄	—	2	12	no	—
302	C₂×D₄	—	2	12	no	—
303	C₂×D₄	—	2	32	no	—
304	C₂×D₄	—	2	32	no	—
305	C₂×D₄	—	2	12	no	—
306	C₂×D₄	—	2	6	no	—
307	C₂×D₄	—	2	12	no	—
308	C₂×D₄	—	2	12	no	—
309	C₂×D₄	—	2	12	no	—
310	C₂×D₄	C₂×D₄×C₈₉	1	12	yes	—
311	C₂×D₄	C₂×D₄×C₈₉	1	12	yes	—
312	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
313	C₂²×C₄	—	2	24	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	12	no	—
316	C₂²×C₄	—	2	12	no	—
317	C₂²×C₄	—	2	6	no	—
318	C₂²×C₄	—	2	6	no	—
319	C₂²×C₄	—	4	4	no	—
320	C₂²×C₄	—	4	3	no	—
321	C₂²×C₄	—	4	1	no	—
322	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
323	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
324	C₂²×C₄	C₂×D₄×C₈₉	1	6	yes	—
325	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
326	Q₁₆	—	2	32	no	—
327	SD₁₆	—	2	32	no	—
328	M₁₆	—	2	12	no	—
329	M₁₆	C₂⁴×C₈₉	1	8	yes	complemented
330	C₂×C₈	—	2	12	no	—
331	C₂²⋊C₄	—	2	32	no	—
332	C₄²	—	2	6	no	—
333	C₄²	C₂⁴×C₈₉	1	4	yes	complemented
334	C₂³	—	2	4	no	—
335	C₂³	—	2	32	no	—
336	C₂³	—	2	12	no	—
337	C₂³	—	2	12	no	—
338	C₂³	—	2	12	no	—
339	C₂³	—	4	24	no	—
340	C₂³	—	4	8	no	—
341	C₂³	—	4	8	no	—
342	C₂³	—	8	12	no	—
343	C₂³	—	8	48	no	—
344	C₂³	#120×C₈₉	1	3	yes	—
345	C₂³	#120×C₈₉	1	1	yes	characteristic
346	C₂³	C₂²×D₄×C₈₉	1	3	yes	—
347	C₂³	C₂²×D₄×C₈₉	1	8	yes	—
348	Q₈	—	2	12	no	—
349	Q₈	—	2	8	no	—
350	Q₈	—	4	12	no	—
351	Q₈	C₂²×D₄×C₈₉	1	8	yes	—
352	D₄	—	2	64	no	—
353	D₄	—	2	24	no	—
354	D₄	—	2	8	no	—
355	D₄	—	2	24	no	—
356	D₄	—	2	12	no	—
357	D₄	—	2	8	no	—
358	D₄	C₂²×D₄×C₈₉	1	8	yes	—
359	C₂×C₄	—	2	32	no	—
360	C₂×C₄	—	2	12	no	—
361	C₂×C₄	—	2	24	no	—
362	C₂×C₄	—	2	8	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	4	12	no	—
366	C₂×C₄	—	4	6	no	—
367	C₂×C₄	—	4	6	no	—
368	C₂×C₄	#120×C₈₉	1	6	yes	—
369	C₂×C₄	#120×C₈₉	1	6	yes	—
370	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
371	C₂×C₄	C₂²×D₄×C₈₉	1	6	yes	—
372	C₂×C₄	C₂²×D₄×C₈₉	1	6	yes	—
373	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
374	C₈	—	2	8	no	—
375	C₂²	—	2	6	no	—
376	C₂²	—	2	6	no	—
377	C₂²	—	2	8	no	—
378	C₂²	—	2	8	no	—
379	C₂²	—	4	8	no	—
380	C₂²	—	4	24	no	—
381	C₂²	—	4	24	no	—
382	C₂²	—	8	12	no	—
383	C₂²	—	8	32	no	—
384	C₂²	#455×C₈₉	1	4	yes	direct product factor
385	C₂²	C₂×#120×C₈₉	1	3	yes	—
386	C₂²	C₂×#120×C₈₉	1	3	yes	—
387	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
388	C₄	—	2	8	no	—
389	C₄	—	4	4	no	—
390	C₄	—	4	4	no	—
391	C₄	C₂×#120×C₈₉	1	6	yes	—
392	C₄	C₂³×D₄×C₈₉	1	2	yes	—
393	C₂	—	2	1	no	—
394	C₂	—	2	3	no	—
395	C₂	—	4	8	no	—
396	C₂	—	4	8	no	—
397	C₂	C₂×#455×C₈₉	1	6	yes	direct product factor
398	C₂	C₂²×#120×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₂²×H(5,𝔽₂)×C₈₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂³×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#137×C₈₉	C₂	semidirect
3. C₂²×#124×C₈₉	C₂	semidirect
4. C₂×#455×C₈₉	C₂	direct (N × H)
5. C₂²×#104×C₈₉	C₂	semidirect
6. C₂²×#100×C₈₉	C₂	semidirect
12. C₂³×Q₈×C₈₉	C₂²	semidirect
17. C₂×#137×C₈₉	C₂²	semidirect
20. C₂²×M₁₆×C₈₉	C₂²	semidirect
22. C₂×#124×C₈₉	C₂²	semidirect
24. C₂²×C₄²×C₈₉	C₂²	semidirect
25. #455×C₈₉	C₂²	direct (N × H)
26. C₂×#104×C₈₉	C₂²	semidirect
27. C₂×#100×C₈₉	C₂²	semidirect
44. C₂²×Q₈×C₈₉	C₂³	semidirect
62. #137×C₈₉	C₂³	semidirect
66. C₂×M₁₆×C₈₉	C₂³	semidirect
69. #124×C₈₉	C₂³	semidirect
72. C₂×C₄²×C₈₉	C₂³	semidirect
73. #104×C₈₉	C₂³	semidirect
74. #100×C₈₉	C₂³	semidirect
97. C₂×Q₈×C₈₉	C₂⁴	semidirect
129. M₁₆×C₈₉	C₂⁴	semidirect
133. C₄²×C₈₉	C₂⁴	semidirect
184. C₂²×C₈₉	#455	direct (N × H)
193. C₂²×#455	C₈₉	direct (N × H)
198. C₂×C₈₉	C₂×#455	direct (N × H)
201. C₂²×#137	C₂×C₈₉	semidirect
202. C₂²×#124	C₂×C₈₉	semidirect
203. C₂×#455	C₂×C₈₉	direct (N × H)
204. C₂²×#104	C₂×C₈₉	semidirect
205. C₂²×#100	C₂×C₈₉	semidirect
206. C₈₉	C₂²×#455	direct (N × H)
212. C₂³×Q₈	C₂²×C₈₉	semidirect
217. C₂×#137	C₂²×C₈₉	semidirect
220. C₂²×M₁₆	C₂²×C₈₉	semidirect
222. C₂×#124	C₂²×C₈₉	semidirect
224. C₂²×C₄²	C₂²×C₈₉	semidirect
225. #455	C₂²×C₈₉	direct (N × H)
226. C₂×#104	C₂²×C₈₉	semidirect
227. C₂×#100	C₂²×C₈₉	semidirect
244. C₂²×Q₈	C₂³×C₈₉	semidirect
262. #137	C₂³×C₈₉	semidirect
266. C₂×M₁₆	C₂³×C₈₉	semidirect
269. #124	C₂³×C₈₉	semidirect
272. C₂×C₄²	C₂³×C₈₉	semidirect
273. #104	C₂³×C₈₉	semidirect
274. #100	C₂³×C₈₉	semidirect
297. C₂×Q₈	C₂⁴×C₈₉	semidirect
329. M₁₆	C₂⁴×C₈₉	semidirect
333. C₄²	C₂⁴×C₈₉	semidirect
384. C₂²	#455×C₈₉	direct (N × H)
397. C₂	C₂×#455×C₈₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×H(5,𝔽₂)×C₈₉	2
C₂²×#137×C₈₉	2
C₂²×#124×C₈₉	2
C₂×#455×C₈₉	2
C₂²×#104×C₈₉	2
C₂²×#100×C₈₉	2
C₂²×#455	89

Subgroups — C22×#455×C89 #36438227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#455×C₈₉ #36438227