Finite Group Databasebeta

← #32999684

Subgroups — #32999684

Order 18688.

312 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22622796C211yesmaximal, characteristic, complemented
2#22612155C211yesmaximal, characteristic, complemented
3#22611966C211yesmaximal, characteristic
4#22609292C211yesmaximal, characteristic, complemented
5#22607990C211yesmaximal, characteristic, complemented
6#22607911C211yesmaximal, characteristic, complemented
7C73×#1331C211yesmaximal, characteristic, complemented
8#17989294—21no—
9C2×#14556532—21no—
10#17988992C2211yescharacteristic
11#17988988—21no—
12#17988964C2211yescharacteristic
13#17988885—22no—
14#17988877—21no—
15#17988788—21no—
16C8×Dic146C2211yescharacteristic
17D8×Dic73—21no—
18C4⋊#11924003C2211yescharacteristic
19#17988389—22no—
20#17988387—21no—
21#17988330—21no—
22C4×D8×C73C2211yescharacteristic
23C4×C16×C73C2211yescharacteristic
24#335×C73C2211yescharacteristic
25C2×#11924017—41no—
26C2×D584—21no—
27C2×D584—21no—
28#14556591—21no—
29#14556588—21no—
30#14556582—22no—
31#14556570—41no—
32#14556568—42no—
33C4×Dic146D411yescharacteristic
34D4×Dic73—22no—
35C2×C4⋊Dic73—41no—
36#14556532—42no—
37C2×#11924003—21no—
38#14556525—21no—
39#14556525D412yes—
40#14556522—22no—
41C4⋊#10596831D411yescharacteristic
42#14556520—21no—
43#14556513—41no—
44#14556512—22no—
45C8×Dic73—21no—
46C2×D8×C73D411yescharacteristic
47C4×D4×C73—21no—
48C2×C16×C73—21no—
49#107×C73D411yescharacteristic
50C4⋊C8×C73—21no—
51#102×C73—21no—
52C4×C8×C73C2311yescharacteristic
53C2×Dic146—41no—
54C2×Dic146D811yescharacteristic
55C2×D292—41no—
56C2×D292—41no—
57C2×D292—41no—
58C22×Dic73—42no—
59#11924017—82no—
60D584—42no—
61D584—41no—
62D584—41no—
63C2×#10596831—21no—
64C4⋊Dic73—22no—
65C4⋊Dic73—42no—
66C4⋊Dic73—41no—
67C4⋊Dic73D811yescharacteristic
68#11924007—21no—
69#11924007—42no—
70C22⋊Dic73—42no—
71C4×Dic73—21no—
72#11924003—41no—
73C2×D4×C73—21no—
74C22×C4×C73—41no—
75D8×C73—21no—
76D8×C73SD1612yescomplemented
77C2×C8×C73—41no—
78C2×C8×C73C2×D411yescharacteristic
79C2×C8×C73C2×D411yescharacteristic
80C4⋊C4×C73—21no—
81C22⋊C4×C73—41no—
82C42×C73C2×D411yescharacteristic
83C16×C73—41no—
84C23×C73—41no—
85C22×D73—81no—
86D4×C73—22no—
87D4×C73—41no—
88C2×C4×C73—41no—
89C2×C4×C73—81no—
90C2×C4×C73—81no—
91C2×C4×C73#12011yescharacteristic
92C2×C4×C73C2×D811yescharacteristic
93C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
94C2×Dic73—21no—
95C2×Dic73—21no—
96C2×Dic73—42no—
97C2×Dic73—162no—
98D292—81no—
99D292—81no—
100D292—81no—
101D292—81no—
102D292—161no—
103Dic146—21no—
104Dic146—21no—
105Dic146—161no—
106C8×C73—21no—
107C8×C73—81no—
108C8×C73C2×SD1611yescharacteristic
109C8×C73#13711yescharacteristic
110#10596831—41no—
111C22×C73—42no—
112C22×C73—81no—
113C22×C73—81no—
114C22×C73#44711yescharacteristic
115C2×D73—161no—
116C2×D73—161no—
117C2×D73—161no—
118C4×C73—21no—
119C4×C73—81no—
120C4×C73#44811yescharacteristic
121C4×C73#45411yescharacteristic
122C4×C73#50811yescharacteristic
123C4×C73#50911yescharacteristic
124Dic73—41no—
125Dic73—41no—
126Dic73—82no—
127#12355—731nomaximal, Sylow
128C2×C73—82no—
129C2×C73#162711yescharacteristic
130C2×C73#218611yescharacteristic
131C2×C73#219211yescharacteristic
132D73—321no—
133#3380—731no—
134#2240—731no—
135#2206—731no—
136#1668—731no—
137#1365—731no—
138Q8⋊C16—731no—
139#1331D7311yescharacteristic, complemented
140C73#1235511yescharacteristic, complemented, Sylow
141#554—1461no—
142C2×D16—1461no—
143#500—731no—
144#499—1461no—
145#493—731no—
146#478—1462no—
147#476—1461no—
148#460—1461no—
149C8×Q8—731no—
150C4×D8—1461no—
151C4×D8C2×D7311yescharacteristic
152C4⋊C16—731no—
153#357—1462no—
154#357—1461no—
155C4×C16C2×D7311yescharacteristic
156#335C2×D7311yescharacteristic
157#331—1461no—
158C2×SD16—2921no—
159C2×D8—1461no—
160C2×D8—1461no—
161C2×D8D29211yescharacteristic
162C4⋊Q8—1461no—
163#127—1461no—
164#125—1462no—
165C22⋊Q8—2922no—
166C22⋊Q8—2921no—
167C4×Q8—731no—
168C4×D4—21no—
169C4×D4—1462no—
170C2×C4⋊C4—2921no—
171D16—2922no—
172C2×C16—21no—
173C2×C16—1461no—
174#107—732no—
175#107—1461no—
176#107D29211yescharacteristic
177#106—1462no—
178C4⋊C8—21no—
179C4⋊C8—731no—
180C4⋊C8—1461no—
181#102—21no—
182#102—2921no—
183#102—1462no—
184C4×C8—1461no—
185C4×C8C22×D7311yescharacteristic
186C2×Q8—2921no—
187C2×Q8—731no—
188C2×D4—21no—
189C2×D4—2921no—
190C2×D4—2921no—
191C2×D4—2921no—
192C22×C4—41no—
193C22×C4—2922no—
194SD16—5842no—
195D8—21no—
196D8—2922no—
197D8—2921no—
198D8—2921no—
199D8#1192401712yescomplemented
200C2×C8—41no—
201C2×C8—1461no—
202C2×C8D4×D7311yescharacteristic
203C2×C8C2×D29211yescharacteristic
204C4⋊C4—21no—
205C4⋊C4—2922no—
206C4⋊C4—2922no—
207C4⋊C4—2921no—
208C4⋊C4—731no—
209C4⋊C4—1462no—
210C4⋊C4—1461no—
211C22⋊C4—41no—
212C22⋊C4—2922no—
213C42—1461no—
214C42D4×D7311yescharacteristic
215C16—41no—
216C16—2921no—
217C23—41no—
218C23—5841no—
219Q8—11681no—
220Q8—1461no—
221Q8—1461no—
222D4—22no—
223D4—41no—
224D4—5841no—
225D4—5841no—
226D4—5841no—
227D4—5841no—
228D4—11681no—
229C2×C4—41no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—2922no—
233C2×C4—11682no—
234C2×C4—1461no—
235C2×C4—1461no—
236C2×C4#1455656111yescharacteristic
237C2×C4D8×D7311yescharacteristic
238C2×C4#1455661811yescharacteristic
239C8—21no—
240C8—81no—
241C8—2921no—
242C8C2×#1192401711yescharacteristic
243C8#1455662411yescharacteristic
244C22—42no—
245C22—81no—
246C22—81no—
247C22—11681no—
248C22—11681no—
249C22—11681no—
250C22#1798872211yescharacteristic
251C4—21no—
252C4—81no—
253C4—2921no—
254C4—2921no—
255C4—5842no—
256C4#1798872811yescharacteristic
257C4#1798876111yescharacteristic
258C4#1798901811yescharacteristic
259C4#1798902111yescharacteristic
260C2—82no—
261C2—23361no—
262C2#2260903711yescharacteristic
263C2#2261185511yescharacteristic
264C2#2261188911yescharacteristic
subgroupG/H
centerC22#17988722
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#1331C2
socleC22×C73#447
prime radical#22611966C2
hypercenter#1331D73
hypocenterC73#12355
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1331

Complements

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

NH ≅ G/Ntype
1. #22622796C2semidirect
2. #22612155C2semidirect
4. #22609292C2semidirect
5. #22607990C2semidirect
6. #22607911C2semidirect
7. C73×#1331C2semidirect
76. D8×C73SD16semidirect
139. #1331D73semidirect
140. C73#12355semidirect
199. D8#11924017semidirect

Maximal subgroups

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

subgroupindex
#226227962
#226121552
#226119662
#226092922
#226079902
#226079112
C73×#13312
#1235573