Subgroups — C₂×C₈₉×#2357 #36411968

Order 22784.

3158 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2357	C₂	1	16	yes	maximal, direct product factor
3	C₂×#390×C₈₉	C₂	1	2	yes	maximal
4	C₂×#388×C₈₉	C₂	1	4	yes	maximal, complemented
5	C₂×#382×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#381×C₈₉	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₄⋊C₄×C₈₉	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×C₂²⋊C₄×C₈₉	C₂	1	2	yes	maximal, complemented
10	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₈₉	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₈₉	C₄	1	16	yes	complemented
13	C₂×C₄×D₄×C₈₉	C₂²	1	8	yes	—
14	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
18	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
19	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
20	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
21	C₂²×C₄²×C₈₉	C₂²	1	4	yes	—
22	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	#390×C₈₉	C₂²	1	16	yes	—
24	#388×C₈₉	C₂²	1	32	yes	complemented
25	#382×C₈₉	C₂²	1	8	yes	complemented
26	#381×C₈₉	C₂²	1	16	yes	complemented
27	C₄×C₄⋊C₄×C₈₉	C₂²	1	8	yes	complemented
28	C₄×C₄⋊C₄×C₈₉	C₂²	1	16	yes	complemented
29	C₄×C₂²⋊C₄×C₈₉	C₂²	1	16	yes	complemented
30	C₂×#95×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#95×C₈₉	C₂²	1	2	yes	—
32	C₂×#95×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#95×C₈₉	C₂²	1	4	yes	—
34	C₂×#95×C₈₉	C₂²	1	2	yes	—
35	C₂⁵×C₈₉	C₂³	1	2	yes	—
36	C₂²×D₄×C₈₉	—	2	4	no	—
37	C₂²×D₄×C₈₉	—	2	4	no	—
38	C₂²×D₄×C₈₉	C₂×C₄	1	8	yes	—
39	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
40	C₂³×C₄×C₈₉	—	4	8	no	—
41	C₂³×C₄×C₈₉	C₂×C₄	1	16	yes	—
42	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
43	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
44	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
46	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
48	C₂³×C₄×C₈₉	C₂³	1	8	yes	—
49	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
50	C₄×D₄×C₈₉	C₂×C₄	1	64	yes	complemented
51	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
52	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
53	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	8	yes	—
54	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
55	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
56	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
57	C₂×C₂²⋊C₄×C₈₉	—	2	16	no	—
58	C₂×C₂²⋊C₄×C₈₉	—	4	8	no	—
59	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	16	yes	—
60	C₂×C₂²⋊C₄×C₈₉	C₂³	1	16	yes	—
61	C₂×C₂²⋊C₄×C₈₉	C₂³	1	16	yes	—
62	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
63	C₂×C₄²×C₈₉	—	2	8	no	—
64	C₂×C₄²×C₈₉	—	4	8	no	—
65	C₂×C₄²×C₈₉	C₂×C₄	1	8	yes	—
66	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
67	C₂×C₄²×C₈₉	C₂³	1	16	yes	—
68	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
69	#95×C₈₉	C₂³	1	4	yes	—
70	#95×C₈₉	C₂³	1	8	yes	—
71	#95×C₈₉	C₂³	1	4	yes	—
72	#95×C₈₉	C₂³	1	16	yes	—
73	#95×C₈₉	C₂³	1	8	yes	—
74	C₂⁴×C₈₉	—	2	8	no	—
75	C₂⁴×C₈₉	—	2	4	no	—
76	C₂⁴×C₈₉	—	2	4	no	—
77	C₂⁴×C₈₉	—	2	8	no	—
78	C₂⁴×C₈₉	C₂²×C₄	1	8	yes	—
79	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
80	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
81	C₂×D₄×C₈₉	—	2	32	no	—
82	C₂×D₄×C₈₉	—	2	8	no	—
83	C₂×D₄×C₈₉	—	2	8	no	—
84	C₂×D₄×C₈₉	C₂²×C₄	1	16	yes	—
85	C₂²×C₄×C₈₉	—	2	4	no	—
86	C₂²×C₄×C₈₉	—	2	4	no	—
87	C₂²×C₄×C₈₉	—	2	4	no	—
88	C₂²×C₄×C₈₉	—	2	8	no	—
89	C₂²×C₄×C₈₉	—	2	4	no	—
90	C₂²×C₄×C₈₉	—	2	4	no	—
91	C₂²×C₄×C₈₉	—	2	8	no	—
92	C₂²×C₄×C₈₉	—	2	4	no	—
93	C₂²×C₄×C₈₉	—	2	8	no	—
94	C₂²×C₄×C₈₉	—	2	8	no	—
95	C₂²×C₄×C₈₉	—	2	4	no	—
96	C₂²×C₄×C₈₉	—	4	32	no	—
97	C₂²×C₄×C₈₉	—	4	16	no	—
98	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
99	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
100	C₂²×C₄×C₈₉	C₂²×C₄	1	8	yes	—
101	C₂²×C₄×C₈₉	C₂²×C₄	1	32	yes	—
102	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
103	C₂²×C₄×C₈₉	#41	1	8	yes	—
104	C₂²×C₄×C₈₉	#41	1	4	yes	—
105	C₂²×C₄×C₈₉	#41	1	4	yes	—
106	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
107	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
108	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
109	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
110	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
111	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
112	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
113	C₄⋊C₄×C₈₉	—	2	16	no	—
114	C₄⋊C₄×C₈₉	—	2	16	no	—
115	C₄⋊C₄×C₈₉	C₂²×C₄	1	16	yes	—
116	C₂²⋊C₄×C₈₉	—	2	32	no	—
117	C₂²⋊C₄×C₈₉	—	4	16	no	—
118	C₂²⋊C₄×C₈₉	C₂²×C₄	1	32	yes	—
119	C₄²×C₈₉	—	2	16	no	—
120	C₄²×C₈₉	—	4	16	no	—
121	C₄²×C₈₉	C₂²×C₄	1	16	yes	—
122	C₂³×C₈₉	—	2	16	no	—
123	C₂³×C₈₉	—	2	16	no	—
124	C₂³×C₈₉	—	2	16	no	—
125	C₂³×C₈₉	—	2	8	no	—
126	C₂³×C₈₉	—	2	4	no	—
127	C₂³×C₈₉	—	2	4	no	—
128	C₂³×C₈₉	—	2	8	no	—
129	C₂³×C₈₉	—	4	16	no	—
130	C₂³×C₈₉	—	4	16	no	—
131	C₂³×C₈₉	C₂³×C₄	1	8	yes	—
132	C₂³×C₈₉	C₂³×C₄	1	2	yes	—
133	C₂³×C₈₉	C₂×#41	1	2	yes	—
134	C₂³×C₈₉	C₂×#41	1	2	yes	—
135	C₂³×C₈₉	C₂×#41	1	4	yes	—
136	C₂³×C₈₉	H(5,𝔽₂)	1	4	yes	—
137	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
138	D₄×C₈₉	—	2	32	no	—
139	C₂×C₄×C₈₉	—	2	8	no	—
140	C₂×C₄×C₈₉	—	2	8	no	—
141	C₂×C₄×C₈₉	—	2	16	no	—
142	C₂×C₄×C₈₉	—	2	16	no	—
143	C₂×C₄×C₈₉	—	2	4	no	—
144	C₂×C₄×C₈₉	—	2	4	no	—
145	C₂×C₄×C₈₉	—	2	4	no	—
146	C₂×C₄×C₈₉	—	2	8	no	—
147	C₂×C₄×C₈₉	—	2	4	no	—
148	C₂×C₄×C₈₉	—	2	4	no	—
149	C₂×C₄×C₈₉	—	2	8	no	—
150	C₂×C₄×C₈₉	—	2	4	no	—
151	C₂×C₄×C₈₉	—	2	8	no	—
152	C₂×C₄×C₈₉	—	2	8	no	—
153	C₂×C₄×C₈₉	—	2	4	no	—
154	C₂×C₄×C₈₉	—	4	32	no	—
155	C₂×C₄×C₈₉	—	4	8	no	—
156	C₂×C₄×C₈₉	—	4	16	no	—
157	C₂×C₄×C₈₉	#117	1	16	yes	—
158	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
159	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
160	C₂×C₄×C₈₉	C₂³×C₄	1	8	yes	—
161	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
162	C₂×C₄×C₈₉	C₂×#41	1	8	yes	—
163	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
164	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
165	C₂²×C₈₉	—	2	8	no	—
166	C₂²×C₈₉	—	2	8	no	—
167	C₂²×C₈₉	—	2	8	no	—
168	C₂²×C₈₉	—	4	32	no	—
169	C₂²×C₈₉	—	4	8	no	—
170	C₂²×C₈₉	—	4	8	no	—
171	C₂²×C₈₉	C₂×#117	1	4	yes	—
172	C₂²×C₈₉	C₄×#41	1	4	yes	—
173	C₂²×C₈₉	#518	1	4	yes	—
174	C₂²×C₈₉	#553	1	8	yes	—
175	C₂²×C₈₉	#559	1	8	yes	—
176	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
177	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
178	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
179	C₂²×C₈₉	C₂²×#41	1	2	yes	—
180	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
181	C₄×C₈₉	—	2	4	no	—
182	C₄×C₈₉	—	2	4	no	—
183	C₄×C₈₉	—	2	8	no	—
184	C₄×C₈₉	—	2	8	no	—
185	C₄×C₈₉	—	4	4	no	—
186	C₄×C₈₉	—	4	8	no	—
187	C₄×C₈₉	C₂×#117	1	8	yes	—
188	C₂×#2357	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂×C₈₉	—	4	8	no	—
190	C₂×C₈₉	#2357	1	8	yes	direct product factor
191	C₂×C₈₉	C₂²×#117	1	1	yes	characteristic
192	C₂×C₈₉	C₂×C₄×#41	1	1	yes	characteristic
193	C₂×C₈₉	C₂×#518	1	1	yes	characteristic
194	C₂×C₈₉	C₂×#553	1	2	yes	—
195	C₂×C₈₉	C₂×#559	1	2	yes	—
196	C₂²×C₄×D₄	C₂×C₈₉	1	1	yes	characteristic
197	#2357	C₂×C₈₉	1	16	yes	direct product factor
198	C₂×#390	C₂×C₈₉	1	2	yes	—
199	C₂×#388	C₂×C₈₉	1	4	yes	complemented
200	C₂×#382	C₂×C₈₉	1	1	yes	characteristic, complemented
201	C₂×#381	C₂×C₈₉	1	2	yes	complemented
202	C₂×C₄×C₄⋊C₄	C₂×C₈₉	1	2	yes	complemented
203	C₂×C₄×C₄⋊C₄	C₂×C₈₉	1	1	yes	characteristic, complemented
204	C₂×C₄×C₂²⋊C₄	C₂×C₈₉	1	2	yes	complemented
205	C₈₉	C₂×#2357	1	1	yes	characteristic, direct product factor, Sylow
206	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
207	C₂⁴×C₄	C₂²×C₈₉	1	2	yes	—
208	C₂×C₄×D₄	C₄×C₈₉	1	16	yes	complemented
209	C₂×C₄×D₄	C₂²×C₈₉	1	8	yes	—
210	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
211	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
212	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
213	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
214	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
215	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
216	C₂²×C₄²	C₂²×C₈₉	1	2	yes	—
217	C₂²×C₄²	C₂²×C₈₉	1	4	yes	—
218	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
219	#390	C₂²×C₈₉	1	16	yes	—
220	#388	C₂²×C₈₉	1	32	yes	complemented
221	#382	C₂²×C₈₉	1	8	yes	complemented
222	#381	C₂²×C₈₉	1	16	yes	complemented
223	C₄×C₄⋊C₄	C₂²×C₈₉	1	8	yes	complemented
224	C₄×C₄⋊C₄	C₂²×C₈₉	1	16	yes	complemented
225	C₄×C₂²⋊C₄	C₂²×C₈₉	1	16	yes	complemented
226	C₂×#95	C₂²×C₈₉	1	1	yes	characteristic
227	C₂×#95	C₂²×C₈₉	1	2	yes	—
228	C₂×#95	C₂²×C₈₉	1	1	yes	characteristic
229	C₂×#95	C₂²×C₈₉	1	4	yes	—
230	C₂×#95	C₂²×C₈₉	1	2	yes	—
231	C₂⁵	C₂³×C₈₉	1	2	yes	—
232	C₂²×D₄	—	2	4	no	—
233	C₂²×D₄	—	2	4	no	—
234	C₂²×D₄	C₂×C₄×C₈₉	1	8	yes	—
235	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
236	C₂³×C₄	—	4	8	no	—
237	C₂³×C₄	C₂×C₄×C₈₉	1	16	yes	—
238	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
239	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
240	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
241	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
242	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
243	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
244	C₂³×C₄	C₂³×C₈₉	1	8	yes	—
245	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
246	C₄×D₄	C₂×C₄×C₈₉	1	64	yes	complemented
247	C₂×C₄⋊C₄	—	2	8	no	—
248	C₂×C₄⋊C₄	—	2	8	no	—
249	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	8	yes	—
250	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
251	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
252	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
253	C₂×C₂²⋊C₄	—	2	16	no	—
254	C₂×C₂²⋊C₄	—	4	8	no	—
255	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	16	yes	—
256	C₂×C₂²⋊C₄	C₂³×C₈₉	1	16	yes	—
257	C₂×C₂²⋊C₄	C₂³×C₈₉	1	16	yes	—
258	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
259	C₂×C₄²	—	2	8	no	—
260	C₂×C₄²	—	4	8	no	—
261	C₂×C₄²	C₂×C₄×C₈₉	1	8	yes	—
262	C₂×C₄²	C₂³×C₈₉	1	8	yes	—
263	C₂×C₄²	C₂³×C₈₉	1	16	yes	—
264	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
265	#95	C₂³×C₈₉	1	4	yes	—
266	#95	C₂³×C₈₉	1	8	yes	—
267	#95	C₂³×C₈₉	1	4	yes	—
268	#95	C₂³×C₈₉	1	16	yes	—
269	#95	C₂³×C₈₉	1	8	yes	—
270	C₂⁴	—	2	8	no	—
271	C₂⁴	—	2	4	no	—
272	C₂⁴	—	2	4	no	—
273	C₂⁴	—	2	8	no	—
274	C₂⁴	C₂²×C₄×C₈₉	1	8	yes	—
275	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
276	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
277	C₂×D₄	—	2	32	no	—
278	C₂×D₄	—	2	8	no	—
279	C₂×D₄	—	2	8	no	—
280	C₂×D₄	C₂²×C₄×C₈₉	1	16	yes	—
281	C₂²×C₄	—	2	4	no	—
282	C₂²×C₄	—	2	4	no	—
283	C₂²×C₄	—	2	4	no	—
284	C₂²×C₄	—	2	8	no	—
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	2	4	no	—
287	C₂²×C₄	—	2	8	no	—
288	C₂²×C₄	—	2	4	no	—
289	C₂²×C₄	—	2	8	no	—
290	C₂²×C₄	—	2	8	no	—
291	C₂²×C₄	—	2	4	no	—
292	C₂²×C₄	—	4	32	no	—
293	C₂²×C₄	—	4	16	no	—
294	C₂²×C₄	C₂²×C₄×C₈₉	1	4	yes	—
295	C₂²×C₄	C₂²×C₄×C₈₉	1	4	yes	—
296	C₂²×C₄	C₂²×C₄×C₈₉	1	8	yes	—
297	C₂²×C₄	C₂²×C₄×C₈₉	1	32	yes	—
298	C₂²×C₄	C₂²×C₄×C₈₉	1	4	yes	—
299	C₂²×C₄	#41×C₈₉	1	8	yes	—
300	C₂²×C₄	#41×C₈₉	1	4	yes	—
301	C₂²×C₄	#41×C₈₉	1	4	yes	—
302	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
303	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
304	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
305	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
306	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
307	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
308	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
309	C₄⋊C₄	—	2	16	no	—
310	C₄⋊C₄	—	2	16	no	—
311	C₄⋊C₄	C₂²×C₄×C₈₉	1	16	yes	—
312	C₂²⋊C₄	—	2	32	no	—
313	C₂²⋊C₄	—	4	16	no	—
314	C₂²⋊C₄	C₂²×C₄×C₈₉	1	32	yes	—
315	C₄²	—	2	16	no	—
316	C₄²	—	4	16	no	—
317	C₄²	C₂²×C₄×C₈₉	1	16	yes	—
318	C₂³	—	2	16	no	—
319	C₂³	—	2	16	no	—
320	C₂³	—	2	16	no	—
321	C₂³	—	2	8	no	—
322	C₂³	—	2	4	no	—
323	C₂³	—	2	4	no	—
324	C₂³	—	2	8	no	—
325	C₂³	—	4	16	no	—
326	C₂³	—	4	16	no	—
327	C₂³	C₂³×C₄×C₈₉	1	8	yes	—
328	C₂³	C₂³×C₄×C₈₉	1	2	yes	—
329	C₂³	C₂×#41×C₈₉	1	2	yes	—
330	C₂³	C₂×#41×C₈₉	1	2	yes	—
331	C₂³	C₂×#41×C₈₉	1	4	yes	—
332	C₂³	H(5,𝔽₂)×C₈₉	1	4	yes	—
333	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
334	D₄	—	2	32	no	—
335	C₂×C₄	—	2	8	no	—
336	C₂×C₄	—	2	8	no	—
337	C₂×C₄	—	2	16	no	—
338	C₂×C₄	—	2	16	no	—
339	C₂×C₄	—	2	4	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	2	8	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	4	no	—
345	C₂×C₄	—	2	8	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	8	no	—
348	C₂×C₄	—	2	8	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	4	32	no	—
351	C₂×C₄	—	4	8	no	—
352	C₂×C₄	—	4	16	no	—
353	C₂×C₄	#117×C₈₉	1	16	yes	—
354	C₂×C₄	C₂³×C₄×C₈₉	1	4	yes	—
355	C₂×C₄	C₂³×C₄×C₈₉	1	4	yes	—
356	C₂×C₄	C₂³×C₄×C₈₉	1	8	yes	—
357	C₂×C₄	C₂³×C₄×C₈₉	1	4	yes	—
358	C₂×C₄	C₂×#41×C₈₉	1	8	yes	—
359	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
360	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
361	C₂²	—	2	8	no	—
362	C₂²	—	2	8	no	—
363	C₂²	—	2	8	no	—
364	C₂²	—	4	32	no	—
365	C₂²	—	4	8	no	—
366	C₂²	—	4	8	no	—
367	C₂²	C₂×#117×C₈₉	1	4	yes	—
368	C₂²	C₄×#41×C₈₉	1	4	yes	—
369	C₂²	#518×C₈₉	1	4	yes	—
370	C₂²	#553×C₈₉	1	8	yes	—
371	C₂²	#559×C₈₉	1	8	yes	—
372	C₂²	C₂⁴×C₄×C₈₉	1	1	yes	characteristic
373	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
374	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
375	C₂²	C₂²×#41×C₈₉	1	2	yes	—
376	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
377	C₄	—	2	4	no	—
378	C₄	—	2	4	no	—
379	C₄	—	2	8	no	—
380	C₄	—	2	8	no	—
381	C₄	—	4	4	no	—
382	C₄	—	4	8	no	—
383	C₄	C₂×#117×C₈₉	1	8	yes	—
384	C₂	—	4	8	no	—
385	C₂	C₈₉×#2357	1	8	yes	direct product factor
386	C₂	C₂²×#117×C₈₉	1	1	yes	characteristic
387	C₂	C₂×C₄×#41×C₈₉	1	1	yes	characteristic
388	C₂	C₂×#518×C₈₉	1	1	yes	characteristic
389	C₂	C₂×#553×C₈₉	1	2	yes	—
390	C₂	C₂×#559×C₈₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₈₉	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₈₉
Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂³×D₄×C₈₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2357	C₂	direct (N × H)
4. C₂×#388×C₈₉	C₂	semidirect
5. C₂×#382×C₈₉	C₂	semidirect
6. C₂×#381×C₈₉	C₂	semidirect
7. C₂×C₄×C₄⋊C₄×C₈₉	C₂	semidirect
8. C₂×C₄×C₄⋊C₄×C₈₉	C₂	semidirect
9. C₂×C₄×C₂²⋊C₄×C₈₉	C₂	semidirect
12. C₂×C₄×D₄×C₈₉	C₄	semidirect
24. #388×C₈₉	C₂²	semidirect
25. #382×C₈₉	C₂²	semidirect
26. #381×C₈₉	C₂²	semidirect
27. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
28. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
29. C₄×C₂²⋊C₄×C₈₉	C₂²	semidirect
50. C₄×D₄×C₈₉	C₂×C₄	semidirect
188. C₂×#2357	C₈₉	direct (N × H)
190. C₂×C₈₉	#2357	direct (N × H)
197. #2357	C₂×C₈₉	direct (N × H)
199. C₂×#388	C₂×C₈₉	semidirect
200. C₂×#382	C₂×C₈₉	semidirect
201. C₂×#381	C₂×C₈₉	semidirect
202. C₂×C₄×C₄⋊C₄	C₂×C₈₉	semidirect
203. C₂×C₄×C₄⋊C₄	C₂×C₈₉	semidirect
204. C₂×C₄×C₂²⋊C₄	C₂×C₈₉	semidirect
205. C₈₉	C₂×#2357	direct (N × H)
208. C₂×C₄×D₄	C₄×C₈₉	semidirect
220. #388	C₂²×C₈₉	semidirect
221. #382	C₂²×C₈₉	semidirect
222. #381	C₂²×C₈₉	semidirect
223. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
224. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
225. C₄×C₂²⋊C₄	C₂²×C₈₉	semidirect
246. C₄×D₄	C₂×C₄×C₈₉	semidirect
385. C₂	C₈₉×#2357	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×C₄×D₄×C₈₉	2
C₈₉×#2357	2
C₂×#390×C₈₉	2
C₂×#388×C₈₉	2
C₂×#382×C₈₉	2
C₂×#381×C₈₉	2
C₂×C₄×C₄⋊C₄×C₈₉	2
C₂×C₄×C₂²⋊C₄×C₈₉	2
C₂×#2357	89

Subgroups — C2×C89×#2357 #36411968

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#2357 #36411968