Subgroups — C₂×C₈₉×#3509 #36440652

Order 22784.

7022 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3509	C₂	1	32	yes	maximal, direct product factor
2	C₂×#557×C₈₉	C₂	1	4	yes	maximal, complemented
3	C₂×#556×C₈₉	C₂	1	4	yes	maximal, complemented
4	C₂×#552×C₈₉	C₂	1	8	yes	maximal, complemented
5	C₂×#541×C₈₉	C₂	1	4	yes	maximal, complemented
6	C₂×#536×C₈₉	C₂	1	2	yes	maximal
7	C₂×#533×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#533×C₈₉	C₂	1	4	yes	maximal, complemented
9	C₂²×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#519×C₈₉	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₈₉	C₂²	1	2	yes	—
14	#557×C₈₉	C₂²	1	64	yes	complemented
15	#556×C₈₉	C₂²	1	64	yes	complemented
16	#552×C₈₉	C₂²	1	128	yes	complemented
17	#541×C₈₉	C₂²	1	64	yes	complemented
18	#536×C₈₉	C₂²	1	32	yes	—
19	#533×C₈₉	C₂²	1	16	yes	complemented
20	#533×C₈₉	C₂²	1	64	yes	complemented
21	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
22	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
23	C₂×C₄⋊Q₈×C₈₉	C₂²	1	4	yes	—
24	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
25	C₂×C₄⋊Q₈×C₈₉	C₂²	1	4	yes	—
26	C₂×C₄⋊Q₈×C₈₉	C₂²	1	16	yes	complemented
27	C₂×#126×C₈₉	C₂²	1	16	yes	—
28	C₂×#125×C₈₉	C₂²	1	2	yes	—
29	C₂×#125×C₈₉	C₂²	1	8	yes	—
30	C₂×#125×C₈₉	C₂²	1	16	yes	—
31	C₂×#124×C₈₉	C₂²	1	4	yes	—
32	C₂×#123×C₈₉	C₂²	1	8	yes	—
33	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	8	yes	—
34	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	—
35	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	8	yes	—
36	#519×C₈₉	C₂²	1	32	yes	complemented
37	C₄×#41×C₈₉	C₂²	1	16	yes	—
38	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
39	C₂×C₄×Q₈×C₈₉	C₂²	1	8	yes	—
40	C₂×C₄×Q₈×C₈₉	C₂²	1	8	yes	—
41	C₂×C₄×Q₈×C₈₉	C₂²	1	8	yes	—
42	C₂×C₄×Q₈×C₈₉	C₂²	1	4	yes	—
43	C₂×C₄×D₄×C₈₉	C₂²	1	4	yes	—
44	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
45	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
46	C₂×#117×C₈₉	C₂²	1	2	yes	—
47	C₂×#117×C₈₉	C₂²	1	2	yes	—
48	C₂×#117×C₈₉	C₂²	1	8	yes	—
49	C₂×#117×C₈₉	C₂²	1	8	yes	—
50	C₂²×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
51	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
52	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
53	C₂×#41×C₈₉	—	2	8	no	—
54	C₂×#41×C₈₉	C₂³	1	8	yes	—
55	C₂²×Q₈×C₈₉	—	2	16	no	—
56	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
57	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
58	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
59	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₂²×Q₈×C₈₉	C₂³	1	16	yes	—
61	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
62	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
64	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
65	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
66	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
67	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₈₉	C₂³	1	16	yes	—
69	C₄⋊Q₈×C₈₉	C₂³	1	16	yes	—
70	C₄⋊Q₈×C₈₉	C₂³	1	32	yes	—
71	C₄⋊Q₈×C₈₉	C₂³	1	16	yes	—
72	C₄⋊Q₈×C₈₉	C₂³	1	32	yes	—
73	#126×C₈₉	C₂³	1	128	yes	—
74	#125×C₈₉	C₂³	1	16	yes	—
75	#125×C₈₉	C₂³	1	64	yes	—
76	#125×C₈₉	C₂³	1	128	yes	—
77	#124×C₈₉	C₂³	1	32	yes	—
78	#123×C₈₉	C₂³	1	64	yes	—
79	C₂²⋊Q₈×C₈₉	C₂³	1	64	yes	—
80	C₂²⋊Q₈×C₈₉	C₂³	1	32	yes	—
81	C₂²⋊Q₈×C₈₉	C₂³	1	64	yes	—
82	C₄×Q₈×C₈₉	C₂³	1	16	yes	—
83	C₄×Q₈×C₈₉	C₂³	1	64	yes	—
84	C₄×Q₈×C₈₉	C₂³	1	64	yes	—
85	C₄×Q₈×C₈₉	C₂³	1	64	yes	—
86	C₄×Q₈×C₈₉	C₂³	1	32	yes	—
87	C₄×D₄×C₈₉	C₂³	1	32	yes	—
88	C₄×D₄×C₈₉	C₂³	1	16	yes	—
89	#117×C₈₉	C₂³	1	8	yes	—
90	#117×C₈₉	C₂³	1	16	yes	—
91	#117×C₈₉	C₂³	1	16	yes	—
92	#117×C₈₉	C₂³	1	64	yes	—
93	#117×C₈₉	C₂³	1	64	yes	—
94	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
95	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
96	C₂×C₄⋊C₄×C₈₉	C₂³	1	32	yes	—
97	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₈₉	C₂³	1	16	yes	—
102	C₂×C₄⋊C₄×C₈₉	C₂³	1	16	yes	—
103	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
104	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₈₉	C₂³	1	16	yes	—
108	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
110	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
112	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
113	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
114	C₂×C₄²×C₈₉	C₂³	1	16	yes	—
115	C₂×C₄²×C₈₉	C₂³	1	16	yes	—
116	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
117	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
118	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
119	#41×C₈₉	—	2	32	no	—
120	C₂×Q₈×C₈₉	—	2	16	no	—
121	C₂×Q₈×C₈₉	—	2	16	no	—
122	C₂×Q₈×C₈₉	—	2	64	no	—
123	C₂×Q₈×C₈₉	—	2	4	no	—
124	C₂×Q₈×C₈₉	—	2	16	no	—
125	C₂×Q₈×C₈₉	C₂⁴	1	16	yes	—
126	C₂×Q₈×C₈₉	C₂⁴	1	16	yes	—
127	C₂×Q₈×C₈₉	C₂⁴	1	16	yes	—
128	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
129	C₂×D₄×C₈₉	—	2	8	no	—
130	C₂×D₄×C₈₉	—	2	4	no	—
131	C₂×D₄×C₈₉	C₂⁴	1	8	yes	—
132	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₈₉	—	2	16	no	—
134	C₂²×C₄×C₈₉	—	2	8	no	—
135	C₂²×C₄×C₈₉	—	2	8	no	—
136	C₂²×C₄×C₈₉	—	2	16	no	—
137	C₂²×C₄×C₈₉	#41	1	8	yes	—
138	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
139	C₂²×C₄×C₈₉	C₂⁴	1	16	yes	—
140	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
141	C₂²×C₄×C₈₉	C₂⁴	1	16	yes	—
142	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
144	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
145	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
146	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
150	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
151	C₄⋊C₄×C₈₉	C₂⁴	1	32	yes	—
152	C₄⋊C₄×C₈₉	C₂⁴	1	32	yes	—
153	C₄⋊C₄×C₈₉	C₂⁴	1	16	yes	—
154	C₄⋊C₄×C₈₉	C₂⁴	1	8	yes	—
155	C₄⋊C₄×C₈₉	C₂⁴	1	16	yes	—
156	C₄⋊C₄×C₈₉	C₂⁴	1	16	yes	—
157	C₄⋊C₄×C₈₉	C₂⁴	1	64	yes	—
158	C₄⋊C₄×C₈₉	C₂⁴	1	64	yes	—
159	C₄⋊C₄×C₈₉	C₂⁴	1	32	yes	—
160	C₂²⋊C₄×C₈₉	C₂⁴	1	8	yes	—
161	C₂²⋊C₄×C₈₉	C₂⁴	1	16	yes	—
162	C₂²⋊C₄×C₈₉	C₂⁴	1	16	yes	—
163	C₂²⋊C₄×C₈₉	C₂⁴	1	64	yes	—
164	C₄²×C₈₉	C₂⁴	1	4	yes	—
165	C₄²×C₈₉	C₂⁴	1	8	yes	—
166	C₄²×C₈₉	C₂⁴	1	4	yes	—
167	C₄²×C₈₉	C₂⁴	1	16	yes	—
168	C₄²×C₈₉	C₂⁴	1	8	yes	—
169	C₄²×C₈₉	C₂⁴	1	64	yes	—
170	C₄²×C₈₉	C₂⁴	1	16	yes	—
171	C₂³×C₈₉	—	2	4	no	—
172	C₂³×C₈₉	—	2	4	no	—
173	C₂³×C₈₉	—	2	8	no	—
174	C₂³×C₈₉	C₂×#41	1	4	yes	—
175	C₂³×C₈₉	C₂⁵	1	4	yes	—
176	C₂³×C₈₉	C₂⁵	1	2	yes	—
177	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
178	Q₈×C₈₉	—	2	32	no	—
179	Q₈×C₈₉	—	2	32	no	—
180	Q₈×C₈₉	—	2	8	no	—
181	Q₈×C₈₉	—	2	32	no	—
182	D₄×C₈₉	—	2	16	no	—
183	D₄×C₈₉	—	2	8	no	—
184	C₂×C₄×C₈₉	—	2	4	no	—
185	C₂×C₄×C₈₉	—	2	32	no	—
186	C₂×C₄×C₈₉	—	2	16	no	—
187	C₂×C₄×C₈₉	—	2	8	no	—
188	C₂×C₄×C₈₉	—	2	4	no	—
189	C₂×C₄×C₈₉	—	2	16	no	—
190	C₂×C₄×C₈₉	—	2	16	no	—
191	C₂×C₄×C₈₉	—	2	16	no	—
192	C₂×C₄×C₈₉	—	2	32	no	—
193	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
194	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
195	C₂×C₄×C₈₉	C₂×#41	1	16	yes	—
196	C₂×C₄×C₈₉	2₋¹⁺⁴	1	8	yes	—
197	C₂×C₄×C₈₉	C₂⁵	1	8	yes	—
198	C₂×C₄×C₈₉	C₂⁵	1	16	yes	—
199	C₂×C₄×C₈₉	C₂⁵	1	4	yes	—
200	C₂×C₄×C₈₉	C₂⁵	1	16	yes	—
201	C₂×C₄×C₈₉	C₂⁵	1	4	yes	—
202	C₂×C₄×C₈₉	C₂⁵	1	2	yes	—
203	C₂×C₄×C₈₉	C₂⁵	1	2	yes	—
204	C₂×C₄×C₈₉	C₂⁵	1	4	yes	—
205	C₂²×C₈₉	—	2	4	no	—
206	C₂²×C₈₉	—	2	8	no	—
207	C₂²×C₈₉	—	2	4	no	—
208	C₂²×C₈₉	—	2	8	no	—
209	C₂²×C₈₉	—	4	8	no	—
210	C₂²×C₈₉	C₂²×#41	1	4	yes	—
211	C₂²×C₈₉	C₂²×#41	1	2	yes	—
212	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	4	yes	—
213	C₂²×C₈₉	C₂⁶	1	1	yes	characteristic
214	C₄×C₈₉	—	2	4	no	—
215	C₄×C₈₉	—	2	16	no	—
216	C₄×C₈₉	—	2	8	no	—
217	C₄×C₈₉	—	2	4	no	—
218	C₄×C₈₉	—	2	16	no	—
219	C₄×C₈₉	C₂²×#41	1	4	yes	—
220	C₄×C₈₉	C₂²×#41	1	4	yes	—
221	C₄×C₈₉	C₂×2₋¹⁺⁴	1	8	yes	—
222	C₂×#3509	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
223	C₂×C₈₉	—	2	4	no	—
224	C₂×C₈₉	—	4	4	no	—
225	C₂×C₈₉	#3509	1	4	yes	direct product factor
226	C₂×C₈₉	C₂³×#41	1	1	yes	characteristic
227	C₂×C₈₉	C₂²×2₋¹⁺⁴	1	2	yes	—
228	#3509	C₂×C₈₉	1	32	yes	direct product factor
229	C₂×#557	C₂×C₈₉	1	4	yes	complemented
230	C₂×#556	C₂×C₈₉	1	4	yes	complemented
231	C₂×#552	C₂×C₈₉	1	8	yes	complemented
232	C₂×#541	C₂×C₈₉	1	4	yes	complemented
233	C₂×#536	C₂×C₈₉	1	2	yes	—
234	C₂×#533	C₂×C₈₉	1	1	yes	characteristic, complemented
235	C₂×#533	C₂×C₈₉	1	4	yes	complemented
236	C₂²×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
237	C₂×#519	C₂×C₈₉	1	2	yes	complemented
238	C₂×C₄×#41	C₂×C₈₉	1	1	yes	characteristic
239	C₈₉	C₂×#3509	1	1	yes	characteristic, direct product factor, Sylow
240	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
241	C₂³×Q₈	C₂²×C₈₉	1	2	yes	—
242	#557	C₂²×C₈₉	1	64	yes	complemented
243	#556	C₂²×C₈₉	1	64	yes	complemented
244	#552	C₂²×C₈₉	1	128	yes	complemented
245	#541	C₂²×C₈₉	1	64	yes	complemented
246	#536	C₂²×C₈₉	1	32	yes	—
247	#533	C₂²×C₈₉	1	16	yes	complemented
248	#533	C₂²×C₈₉	1	64	yes	complemented
249	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
250	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
251	C₂×C₄⋊Q₈	C₂²×C₈₉	1	4	yes	—
252	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
253	C₂×C₄⋊Q₈	C₂²×C₈₉	1	4	yes	—
254	C₂×C₄⋊Q₈	C₂²×C₈₉	1	16	yes	complemented
255	C₂×#126	C₂²×C₈₉	1	16	yes	—
256	C₂×#125	C₂²×C₈₉	1	2	yes	—
257	C₂×#125	C₂²×C₈₉	1	8	yes	—
258	C₂×#125	C₂²×C₈₉	1	16	yes	—
259	C₂×#124	C₂²×C₈₉	1	4	yes	—
260	C₂×#123	C₂²×C₈₉	1	8	yes	—
261	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	8	yes	—
262	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	—
263	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	8	yes	—
264	#519	C₂²×C₈₉	1	32	yes	complemented
265	C₄×#41	C₂²×C₈₉	1	16	yes	—
266	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
267	C₂×C₄×Q₈	C₂²×C₈₉	1	8	yes	—
268	C₂×C₄×Q₈	C₂²×C₈₉	1	8	yes	—
269	C₂×C₄×Q₈	C₂²×C₈₉	1	8	yes	—
270	C₂×C₄×Q₈	C₂²×C₈₉	1	4	yes	—
271	C₂×C₄×D₄	C₂²×C₈₉	1	4	yes	—
272	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
273	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
274	C₂×#117	C₂²×C₈₉	1	2	yes	—
275	C₂×#117	C₂²×C₈₉	1	2	yes	—
276	C₂×#117	C₂²×C₈₉	1	8	yes	—
277	C₂×#117	C₂²×C₈₉	1	8	yes	—
278	C₂²×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
279	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
280	C₂²×C₄²	C₂²×C₈₉	1	2	yes	—
281	C₂×#41	—	2	8	no	—
282	C₂×#41	C₂³×C₈₉	1	8	yes	—
283	C₂²×Q₈	—	2	16	no	—
284	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
285	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
286	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
287	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
288	C₂²×Q₈	C₂³×C₈₉	1	16	yes	—
289	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
290	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
291	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
292	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
293	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
294	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
295	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
296	C₄⋊Q₈	C₂³×C₈₉	1	16	yes	—
297	C₄⋊Q₈	C₂³×C₈₉	1	16	yes	—
298	C₄⋊Q₈	C₂³×C₈₉	1	32	yes	—
299	C₄⋊Q₈	C₂³×C₈₉	1	16	yes	—
300	C₄⋊Q₈	C₂³×C₈₉	1	32	yes	—
301	#126	C₂³×C₈₉	1	128	yes	—
302	#125	C₂³×C₈₉	1	16	yes	—
303	#125	C₂³×C₈₉	1	64	yes	—
304	#125	C₂³×C₈₉	1	128	yes	—
305	#124	C₂³×C₈₉	1	32	yes	—
306	#123	C₂³×C₈₉	1	64	yes	—
307	C₂²⋊Q₈	C₂³×C₈₉	1	64	yes	—
308	C₂²⋊Q₈	C₂³×C₈₉	1	32	yes	—
309	C₂²⋊Q₈	C₂³×C₈₉	1	64	yes	—
310	C₄×Q₈	C₂³×C₈₉	1	16	yes	—
311	C₄×Q₈	C₂³×C₈₉	1	64	yes	—
312	C₄×Q₈	C₂³×C₈₉	1	64	yes	—
313	C₄×Q₈	C₂³×C₈₉	1	64	yes	—
314	C₄×Q₈	C₂³×C₈₉	1	32	yes	—
315	C₄×D₄	C₂³×C₈₉	1	32	yes	—
316	C₄×D₄	C₂³×C₈₉	1	16	yes	—
317	#117	C₂³×C₈₉	1	8	yes	—
318	#117	C₂³×C₈₉	1	16	yes	—
319	#117	C₂³×C₈₉	1	16	yes	—
320	#117	C₂³×C₈₉	1	64	yes	—
321	#117	C₂³×C₈₉	1	64	yes	—
322	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
323	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
324	C₂×C₄⋊C₄	C₂³×C₈₉	1	32	yes	—
325	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
326	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
328	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
329	C₂×C₄⋊C₄	C₂³×C₈₉	1	16	yes	—
330	C₂×C₄⋊C₄	C₂³×C₈₉	1	16	yes	—
331	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
332	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
333	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
334	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
335	C₂×C₂²⋊C₄	C₂³×C₈₉	1	16	yes	—
336	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
337	C₂×C₄²	C₂³×C₈₉	1	16	yes	—
338	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
339	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
340	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
341	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
342	C₂×C₄²	C₂³×C₈₉	1	8	yes	—
343	C₂×C₄²	C₂³×C₈₉	1	16	yes	—
344	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
345	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
346	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
347	#41	—	2	32	no	—
348	C₂×Q₈	—	2	16	no	—
349	C₂×Q₈	—	2	16	no	—
350	C₂×Q₈	—	2	64	no	—
351	C₂×Q₈	—	2	4	no	—
352	C₂×Q₈	—	2	16	no	—
353	C₂×Q₈	C₂⁴×C₈₉	1	16	yes	—
354	C₂×Q₈	C₂⁴×C₈₉	1	16	yes	—
355	C₂×Q₈	C₂⁴×C₈₉	1	16	yes	—
356	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
357	C₂×D₄	—	2	8	no	—
358	C₂×D₄	—	2	4	no	—
359	C₂×D₄	C₂⁴×C₈₉	1	8	yes	—
360	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
361	C₂²×C₄	—	2	16	no	—
362	C₂²×C₄	—	2	8	no	—
363	C₂²×C₄	—	2	8	no	—
364	C₂²×C₄	—	2	16	no	—
365	C₂²×C₄	#41×C₈₉	1	8	yes	—
366	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
367	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
368	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
369	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
370	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
371	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
372	C₂²×C₄	C₂⁴×C₈₉	1	16	yes	—
373	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
374	C₂²×C₄	C₂⁴×C₈₉	1	16	yes	—
375	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
376	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
377	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
378	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
379	C₄⋊C₄	C₂⁴×C₈₉	1	32	yes	—
380	C₄⋊C₄	C₂⁴×C₈₉	1	32	yes	—
381	C₄⋊C₄	C₂⁴×C₈₉	1	16	yes	—
382	C₄⋊C₄	C₂⁴×C₈₉	1	8	yes	—
383	C₄⋊C₄	C₂⁴×C₈₉	1	16	yes	—
384	C₄⋊C₄	C₂⁴×C₈₉	1	16	yes	—
385	C₄⋊C₄	C₂⁴×C₈₉	1	64	yes	—
386	C₄⋊C₄	C₂⁴×C₈₉	1	64	yes	—
387	C₄⋊C₄	C₂⁴×C₈₉	1	32	yes	—
388	C₂²⋊C₄	C₂⁴×C₈₉	1	8	yes	—
389	C₂²⋊C₄	C₂⁴×C₈₉	1	16	yes	—
390	C₂²⋊C₄	C₂⁴×C₈₉	1	16	yes	—
391	C₂²⋊C₄	C₂⁴×C₈₉	1	64	yes	—
392	C₄²	C₂⁴×C₈₉	1	4	yes	—
393	C₄²	C₂⁴×C₈₉	1	8	yes	—
394	C₄²	C₂⁴×C₈₉	1	4	yes	—
395	C₄²	C₂⁴×C₈₉	1	16	yes	—
396	C₄²	C₂⁴×C₈₉	1	8	yes	—
397	C₄²	C₂⁴×C₈₉	1	64	yes	—
398	C₄²	C₂⁴×C₈₉	1	16	yes	—
399	C₂³	—	2	4	no	—
400	C₂³	—	2	4	no	—
401	C₂³	—	2	8	no	—
402	C₂³	C₂×#41×C₈₉	1	4	yes	—
403	C₂³	C₂⁵×C₈₉	1	4	yes	—
404	C₂³	C₂⁵×C₈₉	1	2	yes	—
405	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
406	Q₈	—	2	32	no	—
407	Q₈	—	2	32	no	—
408	Q₈	—	2	8	no	—
409	Q₈	—	2	32	no	—
410	D₄	—	2	16	no	—
411	D₄	—	2	8	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	32	no	—
414	C₂×C₄	—	2	16	no	—
415	C₂×C₄	—	2	8	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	16	no	—
418	C₂×C₄	—	2	16	no	—
419	C₂×C₄	—	2	16	no	—
420	C₂×C₄	—	2	32	no	—
421	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
422	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
423	C₂×C₄	C₂×#41×C₈₉	1	16	yes	—
424	C₂×C₄	2₋¹⁺⁴×C₈₉	1	8	yes	—
425	C₂×C₄	C₂⁵×C₈₉	1	8	yes	—
426	C₂×C₄	C₂⁵×C₈₉	1	16	yes	—
427	C₂×C₄	C₂⁵×C₈₉	1	4	yes	—
428	C₂×C₄	C₂⁵×C₈₉	1	16	yes	—
429	C₂×C₄	C₂⁵×C₈₉	1	4	yes	—
430	C₂×C₄	C₂⁵×C₈₉	1	2	yes	—
431	C₂×C₄	C₂⁵×C₈₉	1	2	yes	—
432	C₂×C₄	C₂⁵×C₈₉	1	4	yes	—
433	C₂²	—	2	4	no	—
434	C₂²	—	2	8	no	—
435	C₂²	—	2	4	no	—
436	C₂²	—	2	8	no	—
437	C₂²	—	4	8	no	—
438	C₂²	C₂²×#41×C₈₉	1	4	yes	—
439	C₂²	C₂²×#41×C₈₉	1	2	yes	—
440	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	4	yes	—
441	C₂²	C₂⁶×C₈₉	1	1	yes	characteristic
442	C₄	—	2	4	no	—
443	C₄	—	2	16	no	—
444	C₄	—	2	8	no	—
445	C₄	—	2	4	no	—
446	C₄	—	2	16	no	—
447	C₄	C₂²×#41×C₈₉	1	4	yes	—
448	C₄	C₂²×#41×C₈₉	1	4	yes	—
449	C₄	C₂×2₋¹⁺⁴×C₈₉	1	8	yes	—
450	C₂	—	2	4	no	—
451	C₂	—	4	4	no	—
452	C₂	C₈₉×#3509	1	4	yes	direct product factor
453	C₂	C₂³×#41×C₈₉	1	1	yes	characteristic
454	C₂	C₂²×2₋¹⁺⁴×C₈₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₈₉
prime radical	C₂²×#41×C₈₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3509	C₂	direct (N × H)
2. C₂×#557×C₈₉	C₂	semidirect
3. C₂×#556×C₈₉	C₂	semidirect
4. C₂×#552×C₈₉	C₂	semidirect
5. C₂×#541×C₈₉	C₂	semidirect
7. C₂×#533×C₈₉	C₂	semidirect
8. C₂×#533×C₈₉	C₂	semidirect
9. C₂²×C₄⋊Q₈×C₈₉	C₂	semidirect
10. C₂×#519×C₈₉	C₂	semidirect
14. #557×C₈₉	C₂²	semidirect
15. #556×C₈₉	C₂²	semidirect
16. #552×C₈₉	C₂²	semidirect
17. #541×C₈₉	C₂²	semidirect
19. #533×C₈₉	C₂²	semidirect
20. #533×C₈₉	C₂²	semidirect
26. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
36. #519×C₈₉	C₂²	semidirect
222. C₂×#3509	C₈₉	direct (N × H)
225. C₂×C₈₉	#3509	direct (N × H)
228. #3509	C₂×C₈₉	direct (N × H)
229. C₂×#557	C₂×C₈₉	semidirect
230. C₂×#556	C₂×C₈₉	semidirect
231. C₂×#552	C₂×C₈₉	semidirect
232. C₂×#541	C₂×C₈₉	semidirect
234. C₂×#533	C₂×C₈₉	semidirect
235. C₂×#533	C₂×C₈₉	semidirect
236. C₂²×C₄⋊Q₈	C₂×C₈₉	semidirect
237. C₂×#519	C₂×C₈₉	semidirect
239. C₈₉	C₂×#3509	direct (N × H)
242. #557	C₂²×C₈₉	semidirect
243. #556	C₂²×C₈₉	semidirect
244. #552	C₂²×C₈₉	semidirect
245. #541	C₂²×C₈₉	semidirect
247. #533	C₂²×C₈₉	semidirect
248. #533	C₂²×C₈₉	semidirect
254. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
264. #519	C₂²×C₈₉	semidirect
452. C₂	C₈₉×#3509	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×#3509	2
C₂×#557×C₈₉	2
C₂×#556×C₈₉	2
C₂×#552×C₈₉	2
C₂×#541×C₈₉	2
C₂×#536×C₈₉	2
C₂×#533×C₈₉	2
C₂²×C₄⋊Q₈×C₈₉	2
C₂×#519×C₈₉	2
C₂×C₄×#41×C₈₉	2
C₂×#3509	89

Subgroups — C2×C89×#3509 #36440652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3509 #36440652