Group #36710868 — subgroups

Subgroups — #36710868

Order 22784.

488 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030892	C₂	1	2	yes	maximal, complemented
2	#26022402	C₂	1	2	yes	maximal, complemented
3	#26022388	C₂	1	2	yes	maximal, complemented
4	#26022367	C₂	1	2	yes	maximal, complemented
5	#26022338	C₂	1	1	yes	maximal, characteristic
6	#26022316	C₂	1	2	yes	maximal, complemented
7	#26022310	C₂	1	2	yes	maximal, complemented
8	#26022010	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2216	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152221	—	2	2	no	—
11	C₂×#15667955	C₂²	1	2	yes	—
12	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	#19152002	C₂²	1	2	yes	—
15	#19151983	C₂²	1	2	yes	—
16	#19151959	—	2	2	no	—
17	#19151846	—	2	2	no	—
18	#19151834	C₂²	1	2	yes	—
19	#19151701	—	2	2	no	—
20	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
21	#19151680	C₂²	1	2	yes	—
22	#19151553	C₂²	1	1	yes	characteristic
23	#19151424	C₂²	1	1	yes	characteristic
24	#19151423	C₂²	1	2	yes	—
25	#19151416	C₂²	1	2	yes	—
26	#19151416	C₂²	1	2	yes	—
27	#19151402	C₂²	1	2	yes	—
28	#19151400	C₂²	1	2	yes	—
29	#19151398	C₂²	1	2	yes	—
30	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic
31	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
32	#467×C₈₉	C₂²	1	2	yes	—
33	#366×C₈₉	C₂²	1	1	yes	characteristic
34	#358×C₈₉	C₂²	1	2	yes	—
35	C₂×#13083561	—	2	2	no	—
36	C₂×D₇₁₂	C₂³	1	2	yes	—
37	C₂×#13083558	D₄	1	1	yes	characteristic
38	#15668011	—	2	2	no	—
39	#15668007	—	4	2	no	—
40	#15668001	—	2	2	no	—
41	#15667998	—	4	2	no	—
42	#15667994	—	4	2	no	—
43	#15667992	—	4	2	no	—
44	#15667986	—	2	2	no	—
45	Q₈×Dic₈₉	—	2	2	no	—
46	#15667974	—	2	2	no	—
47	#15667971	D₄	1	1	yes	characteristic
48	#15667955	—	4	2	no	—
49	D₁₄₂₄	—	4	1	no	—
50	#15667952	—	4	1	no	—
51	C₂×#13083549	C₂³	1	2	yes	—
52	#15667946	C₂³	1	2	yes	—
53	#15667945	—	2	2	no	—
54	#15667936	—	2	2	no	—
55	#15667934	—	2	2	no	—
56	#15667932	—	2	2	no	—
57	#15667922	D₄	1	2	yes	—
58	C₈×Dic₈₉	C₂³	1	2	yes	—
59	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
60	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₂²⋊Q₈×C₈₉	—	2	2	no	—
62	Q₃₂×C₈₉	—	4	1	no	—
63	C₂×C₁₆×C₈₉	—	4	1	no	—
64	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
66	#107×C₈₉	C₂³	1	2	yes	—
67	#103×C₈₉	—	2	2	no	—
68	C₂×Dic₁₇₈	—	4	2	no	—
69	C₂×D₃₅₆	—	2	2	no	—
70	C₂×C₂²⋊D₈₉	—	4	2	no	—
71	C₂×C₄×D₈₉	—	4	2	no	—
72	#13083561	—	4	4	no	—
73	D₇₁₂	—	2	2	no	—
74	D₇₁₂	—	2	2	no	—
75	#13083558	—	4	1	no	—
76	C₂×#10623494	C₂×D₄	1	2	yes	—
77	C₄⋊Dic₈₉	—	2	2	no	—
78	C₄⋊Dic₈₉	—	4	2	no	—
79	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
80	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
81	#13083553	—	4	2	no	—
82	C₂²⋊Dic₈₉	D₈	1	2	yes	—
83	#13083551	—	4	2	no	—
84	#13083551	—	4	2	no	—
85	#13083551	—	4	2	no	—
86	#13083551	—	4	2	no	—
87	C₄×Dic₈₉	—	4	2	no	—
88	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
89	#13083549	—	2	2	no	—
90	C₂×Q₈×C₈₉	—	2	2	no	—
91	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₈₉	—	2	2	no	—
93	Q₁₆×C₈₉	—	2	2	no	—
94	C₂×C₈×C₈₉	—	2	1	no	—
95	C₂×C₈×C₈₉	—	2	1	no	—
96	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	—	2	2	no	—
99	C₄⋊C₄×C₈₉	—	4	2	no	—
100	C₂²⋊C₄×C₈₉	—	4	2	no	—
101	C₁₆×C₈₉	—	2	1	no	—
102	C₁₆×C₈₉	—	2	1	no	—
103	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
104	C₂²×D₈₉	—	4	2	no	—
105	Q₈×C₈₉	—	4	2	no	—
106	Q₈×C₈₉	—	4	2	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	1	no	—
109	C₂×C₄×C₈₉	—	4	2	no	—
110	C₂×C₄×C₈₉	—	4	2	no	—
111	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
113	C₂²⋊D₈₉	—	16	2	no	—
114	C₂×Dic₈₉	—	4	2	no	—
115	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
116	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
117	D₃₅₆	—	4	2	no	—
118	D₃₅₆	—	4	2	no	—
119	C₄×D₈₉	—	16	2	no	—
120	Dic₁₇₈	—	16	2	no	—
121	C₈×C₈₉	—	2	1	no	—
122	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
123	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
124	#10623494	—	2	2	no	—
125	C₂²×C₈₉	—	2	1	no	—
126	C₂²×C₈₉	—	2	1	no	—
127	C₂²×C₈₉	—	2	1	no	—
128	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
129	C₂×D₈₉	—	8	2	no	—
130	C₂×D₈₉	—	8	2	no	—
131	C₂×D₈₉	—	8	2	no	—
132	C₄×C₈₉	—	2	1	no	—
133	C₄×C₈₉	—	8	2	no	—
134	C₄×C₈₉	—	8	2	no	—
135	C₄×C₈₉	#537	1	1	yes	characteristic
136	C₄×C₈₉	#575	1	1	yes	characteristic
137	Dic₈₉	—	2	2	no	—
138	Dic₈₉	—	2	2	no	—
139	Dic₈₉	—	8	2	no	—
140	#33828	—	89	1	no	maximal, Sylow
141	C₂×C₈₉	—	4	1	no	—
142	C₂×C₈₉	#3245	1	1	yes	characteristic
143	C₂×C₈₉	#3415	1	1	yes	characteristic
144	C₂×C₈₉	#3417	1	1	yes	characteristic
145	D₈₉	—	16	2	no	—
146	#3175	—	89	2	no	—
147	#2227	—	89	2	no	—
148	#2224	—	89	2	no	—
149	#2220	—	89	2	no	—
150	#2216	D₈₉	1	1	yes	characteristic, complemented
151	#2215	—	89	1	no	—
152	#2211	—	89	2	no	—
153	#2210	—	89	2	no	—
154	#2151	—	89	1	no	—
155	C₈₉	#33828	1	1	yes	characteristic, complemented, Sylow
156	#542	—	178	2	no	—
157	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
158	C₂×SD₃₂	—	89	2	no	—
159	C₂×D₁₆	—	89	1	no	—
160	C₂×M₃₂	—	89	1	no	—
161	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
162	#500	—	89	2	no	—
163	#495	—	89	2	no	—
164	#489	—	178	2	no	—
165	#468	—	178	2	no	—
166	#467	C₂×D₈₉	1	2	yes	—
167	#466	—	89	2	no	—
168	#440	—	178	2	no	—
169	C₄×Q₁₆	—	89	2	no	—
170	C₄×D₈	—	89	2	no	—
171	#405	—	89	1	no	—
172	#366	—	89	1	no	—
173	#366	—	89	2	no	—
174	#366	C₂×D₈₉	1	1	yes	characteristic
175	C₄⋊C₁₆	—	89	2	no	—
176	C₄⋊C₁₆	—	89	2	no	—
177	#358	—	89	2	no	—
178	#358	C₂×D₈₉	1	2	yes	—
179	#357	—	89	2	no	—
180	#357	—	89	2	no	—
181	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
182	C₂×SD₁₆	—	178	2	no	—
183	C₂×D₈	—	89	2	no	—
184	C₂×M₁₆	—	89	1	no	—
185	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
186	C₄⋊Q₈	—	178	2	no	—
187	#126	—	356	2	no	—
188	#124	—	356	2	no	—
189	#124	—	178	2	no	—
190	#123	—	356	2	no	—
191	C₂²⋊Q₈	—	2	2	no	—
192	C₂²⋊Q₈	—	356	2	no	—
193	#121	—	178	2	no	—
194	C₄×Q₈	—	178	2	no	—
195	C₄×D₄	—	178	2	no	—
196	#117	—	89	1	no	—
197	Q₃₂	—	4	1	no	—
198	SD₃₂	—	356	2	no	—
199	D₁₆	—	356	1	no	—
200	M₃₂	—	356	1	no	—
201	C₂×C₁₆	—	4	1	no	—
202	C₂×C₁₆	—	89	2	no	—
203	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
204	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
205	#107	—	89	2	no	—
206	#107	—	178	2	no	—
207	#107	C₂²×D₈₉	1	2	yes	—
208	#103	—	2	2	no	—
209	#103	—	178	2	no	—
210	#102	—	178	2	no	—
211	#102	—	178	2	no	—
212	#97	—	89	2	no	—
213	C₄×C₈	—	89	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	356	2	no	—
216	C₂×D₄	—	356	2	no	—
217	C₂×D₄	—	178	2	no	—
218	C₂²×C₄	—	356	2	no	—
219	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
220	Q₁₆	—	2	2	no	—
221	Q₁₆	—	2	2	no	—
222	SD₁₆	—	356	4	no	—
223	D₈	—	178	2	no	—
224	D₈	—	178	2	no	—
225	M₁₆	—	356	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	89	2	no	—
229	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
230	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	89	1	no	—
234	C₄⋊C₄	—	89	1	no	—
235	C₄⋊C₄	—	356	2	no	—
236	C₄⋊C₄	—	356	2	no	—
237	C₄⋊C₄	—	178	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	89	2	no	—
240	C₂²⋊C₄	—	356	2	no	—
241	C₂²⋊C₄	—	356	2	no	—
242	C₂²⋊C₄	—	356	2	no	—
243	C₂²⋊C₄	—	356	2	no	—
244	C₄²	—	89	2	no	—
245	C₄²	—	356	2	no	—
246	C₁₆	—	2	1	no	—
247	C₁₆	—	2	1	no	—
248	C₁₆	—	178	2	no	—
249	C₂³	—	356	2	no	—
250	C₂³	D₈×D₈₉	1	1	yes	characteristic
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	Q₈	—	1424	2	no	—
254	D₄	—	356	2	no	—
255	D₄	—	356	2	no	—
256	D₄	—	1424	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	89	2	no	—
262	C₂×C₄	—	89	2	no	—
263	C₂×C₄	—	356	2	no	—
264	C₂×C₄	—	1424	2	no	—
265	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
266	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	178	2	no	—
269	C₈	#15668057	1	1	yes	characteristic
270	C₈	#15668059	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	712	2	no	—
275	C₂²	—	712	2	no	—
276	C₂²	—	712	2	no	—
277	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	8	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	178	2	no	—
282	C₄	—	178	2	no	—
283	C₄	—	712	2	no	—
284	C₄	#19152175	1	1	yes	characteristic
285	C₄	#19152395	1	1	yes	characteristic
286	C₂	—	4	1	no	—
287	C₂	—	1424	2	no	—
288	C₂	#26031558	1	1	yes	characteristic
289	C₂	#26033390	1	1	yes	characteristic
290	C₂	#26033401	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26030892

C₂

yes

maximal, complemented

#26022402

C₂

yes

maximal, complemented

#26022388

C₂

yes

maximal, complemented

#26022367

C₂

yes

maximal, complemented

#26022338

C₂

yes

maximal, characteristic

#26022316

C₂

yes

maximal, complemented

#26022310

C₂

yes

maximal, complemented

#26022010

C₂

yes

maximal, characteristic, complemented

C₈₉×#2216

C₂

yes

maximal, characteristic, complemented

#19152221

—

C₂×#15667955

C₂²

yes

—

C₂×D₁₄₂₄

C₂²

yes

characteristic

C₂×#15667952

C₂²

yes

characteristic

#19152002

C₂²

yes

—

#19151983

C₂²

yes

—

#19151959

—

#19151846

—

#19151834

C₂²

yes

—

#19151701

—

Q₁₆×Dic₈₉

C₂²

yes

—

#19151680

C₂²

yes

—

#19151553

C₂²

yes

characteristic

#19151424

C₂²

yes

characteristic

#19151423

C₂²

yes

—

#19151416

C₂²

yes

—

#19151416

C₂²

yes

—

#19151402

C₂²

yes

—

#19151400

C₂²

yes

—

#19151398

C₂²

yes

—

C₂×Q₃₂×C₈₉

C₂²

yes

characteristic

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

#467×C₈₉

C₂²

yes

—

#366×C₈₉

C₂²

yes

characteristic

#358×C₈₉

C₂²

yes

—

C₂×#13083561

—

C₂×D₇₁₂

C₂³

yes

—

C₂×#13083558

D₄

yes

—

—

—

—

—

—

—

Q₈×Dic₈₉

—

#15667974

—

#15667971

D₄

yes

characteristic

#15667955

—

D₁₄₂₄

—

#15667952

—

C₂×#13083549

C₂³

yes

—

#15667946

C₂³

yes

—

—

—

—

—

D₄

yes

—

C₈×Dic₈₉

C₂³

yes

—

C₂×Q₁₆×C₈₉

C₂³

yes

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂²⋊Q₈×C₈₉

—

Q₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

#107×C₈₉

C₂³

yes

—

#103×C₈₉

—

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

#13083561

—

D₇₁₂

—

D₇₁₂

—

#13083558

—

C₂×#10623494

C₂×D₄

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₈

yes

characteristic

C₄⋊Dic₈₉

D₈

yes

characteristic

#13083553

—

C₂²⋊Dic₈₉

D₈

yes

—

—

—

—

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₂×D₄

yes

—

#13083549

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

100

C₂²⋊C₄×C₈₉

—

101

C₁₆×C₈₉

—

102

C₁₆×C₈₉

—

103

C₂³×C₈₉

C₂×D₈

yes

characteristic

104

C₂²×D₈₉

—

105

Q₈×C₈₉

—

106

Q₈×C₈₉

—

107

C₂×C₄×C₈₉

—

108

C₂×C₄×C₈₉

—

109

C₂×C₄×C₈₉

—

110

C₂×C₄×C₈₉

—

111

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

112

C₂×C₄×C₈₉

C₂²×D₄

yes

characteristic

113

C₂²⋊D₈₉

—

114

C₂×Dic₈₉

—

115

C₂×Dic₈₉

C₂×D₈

yes

—

116

C₂×Dic₈₉

C₂×D₈

yes

—

117

D₃₅₆

—

118

D₃₅₆

—

119

C₄×D₈₉

—

120

Dic₁₇₈

—

121

C₈×C₈₉

—

122

C₈×C₈₉

H(5,𝔽₂)

yes

characteristic

123

C₈×C₈₉

2₋¹⁺⁴

yes

characteristic

124

#10623494

—

125

C₂²×C₈₉

—

126

C₂²×C₈₉

—

127

C₂²×C₈₉

—

128

C₂²×C₈₉

C₂²×D₈

yes

characteristic

129

C₂×D₈₉

—

130

C₂×D₈₉

—

131

C₂×D₈₉

—

132

C₄×C₈₉

—

133

C₄×C₈₉

—

134

C₄×C₈₉

—

135

C₄×C₈₉

#537

yes

characteristic

136

C₄×C₈₉

#575

yes

characteristic

137

Dic₈₉

—

138

Dic₈₉

—

139

Dic₈₉

—

140

#33828

—

maximal, Sylow

141

C₂×C₈₉

—

142

C₂×C₈₉

#3245

yes

characteristic

143

C₂×C₈₉

#3415

yes

characteristic

144

C₂×C₈₉

#3417

yes

characteristic

145

D₈₉

—

146

#3175

—

147

#2227

—

148

#2224

—

149

#2220

—

150

#2216

D₈₉

yes

characteristic, complemented

151

#2215

—

152

#2211

—

153

#2210

—

154

#2151

—

155

C₈₉

#33828

yes

characteristic, complemented, Sylow

156

#542

—

178

—

157

C₂×Q₃₂

C₂×D₈₉

yes

characteristic

158

C₂×SD₃₂

—

159

C₂×D₁₆

—

160

C₂×M₃₂

—

161

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

162

#500

—

163

#495

—

164

#489

—

178

—

165

#468

—

178

—

166

#467

C₂×D₈₉

yes

—

167

#466

—

168

#440

—

178

—

169

C₄×Q₁₆

—

170

C₄×D₈

—

171

#405

—

172

#366

—

173

#366

—

174

#366

C₂×D₈₉

yes

characteristic

175

C₄⋊C₁₆

—

176

C₄⋊C₁₆

—

177

#358

—

178

#358

C₂×D₈₉

yes

—

179

#357

—

180

#357

—

181

C₂×Q₁₆

C₂²×D₈₉

yes

—

182

C₂×SD₁₆

—

178

—

183

C₂×D₈

—

184

C₂×M₁₆

—

185

C₂²×C₈

C₂²×D₈₉

yes

characteristic

186

C₄⋊Q₈

—

178

—

187

#126

—

356

—

188

#124

—

356

—

189

#124

—

178

—

190

#123

—

356

—

191

C₂²⋊Q₈

—

192

C₂²⋊Q₈

—

356

—

193

#121

—

178

—

194

C₄×Q₈

—

178

—

195

C₄×D₄

—

178

—

196

#117

—

197

Q₃₂

—

198

SD₃₂

—

356

—

199

D₁₆

—

356

—

200

M₃₂

—

356

—

201

C₂×C₁₆

—

202

C₂×C₁₆

—

203

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

204

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

205

#107

—

206

#107

—

178

—

207

#107

C₂²×D₈₉

yes

—

208

#103

—

209

#103

—

178

—

210

#102

—

178

—

211

#102

—

178

—

212

#97

—

213

C₄×C₈

—

214

C₂×Q₈

—

215

C₂×Q₈

—

356

—

216

C₂×D₄

—

356

—

217

C₂×D₄

—

178

—

218

C₂²×C₄

—

356

—

219

C₂²×C₄

D₄×D₈₉

yes

characteristic

220

Q₁₆

—

221

Q₁₆

—

222

SD₁₆

—

356

—

223

D₈

—

178

—

224

D₈

—

178

—

225

M₁₆

—

356

—

226

C₂×C₈

—

227

C₂×C₈

—

228

C₂×C₈

—

229

C₂×C₈

D₄×D₈₉

yes

characteristic

230

C₂×C₈

C₂³×D₈₉

yes

characteristic

231

C₄⋊C₄

—

232

C₄⋊C₄

—

233

C₄⋊C₄

—

234

C₄⋊C₄

—

235

C₄⋊C₄

—

356

—

236

C₄⋊C₄

—

356

—

237

C₄⋊C₄

—

178

—

238

C₂²⋊C₄

—

239

C₂²⋊C₄

—

240

C₂²⋊C₄

—

356

—

241

C₂²⋊C₄

—

356

—

242

C₂²⋊C₄

—

356

—

243

C₂²⋊C₄

—

356

—

244

C₄²

—

245

C₄²

—

356

—

246

C₁₆

—

247

C₁₆

—

248

C₁₆

—

178

—

249

C₂³

—

356

—

250

C₂³

D₈×D₈₉

yes

characteristic

251

Q₈

—

252

Q₈

—

253

Q₈

—

1424

—

254

D₄

—

356

—

255

D₄

—

356

—

256

D₄

—

1424

—

257

C₂×C₄

—

258

C₂×C₄

—

259

C₂×C₄

—

260

C₂×C₄

—

261

C₂×C₄

—

262

C₂×C₄

—

263

C₂×C₄

—

356

—

264

C₂×C₄

—

1424

—

265

C₂×C₄

D₈×D₈₉

yes

characteristic

266

C₂×C₄

C₂×D₄×D₈₉

yes

characteristic

267

C₈

—

268

C₈

—

178

—

269

C₈

#15668057

yes

characteristic

270

C₈

#15668059

yes

characteristic

271

C₂²

—

272

C₂²

—

273

C₂²

—

274

C₂²

—

712

—

275

C₂²

—

712

—

276

C₂²

—

712

—

277

C₂²

C₂×D₈×D₈₉

yes

characteristic

278

C₄

—

279

C₄

—

280

C₄

—

281

C₄

—

178

—

282

C₄

—

178

—

283

C₄

—

712

—

284

C₄

#19152175

yes

characteristic

285

C₄

#19152395

yes

characteristic

286

C₂

—

287

C₂

—

1424

—

288

C₂

#26031558

yes

characteristic

289

C₂

#26033390

yes

characteristic

290

C₂

#26033401

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2216	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26022338	C₂
hypercenter	#2216	D₈₉
hypocenter	C₈₉	#33828
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

subgroup

G/H

center

C₂²

C₂×D₈×D₈₉

commutator, cosocle

C₂×C₈×C₈₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₈₉

Fitting

C₈₉×#2216

C₂

socle

C₂²×C₈₉

C₂²×D₈

C₂

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

N	H ≅ G/N	type
1. #26030892	C₂	semidirect
2. #26022402	C₂	semidirect
3. #26022388	C₂	semidirect
4. #26022367	C₂	semidirect
6. #26022316	C₂	semidirect
7. #26022310	C₂	semidirect
8. #26022010	C₂	semidirect
9. C₈₉×#2216	C₂	semidirect
150. #2216	D₈₉	semidirect
155. C₈₉	#33828	semidirect

H ≅ G/N

type

1. #26030892

C₂

semidirect

2. #26022402

C₂

semidirect

3. #26022388

C₂

semidirect

4. #26022367

C₂

semidirect

6. #26022316

C₂

semidirect

7. #26022310

C₂

semidirect

8. #26022010

C₂

semidirect

9. C₈₉×#2216

C₂

semidirect

150. #2216

D₈₉

semidirect

155. C₈₉

#33828

semidirect

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

subgroup	index
#26030892	2
#26022402	2
#26022388	2
#26022367	2
#26022338	2
#26022316	2
#26022310	2
#26022010	2
C₈₉×#2216	2
#33828	89

subgroup

index

C₈₉×#2216

Subgroups — #36710868

Conjugacy classes of subgroups

Complements

Maximal subgroups