Finite Group Databasebeta

← #32999712

Subgroups — #32999712

Order 18688.

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

normal subgroups only

This group has 296 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
2#22612159C211yesmaximal, characteristic, complemented
3#22611976C211yesmaximal, characteristic
4#22609302C211yesmaximal, characteristic, complemented
5#22607990C211yesmaximal, characteristic, complemented
6#22607911C211yesmaximal, characteristic, complemented
7C73×#1331C211yesmaximal, characteristic
8#17989294—21no—
9C2×#14556535—21no—
10#17988992C2211yescharacteristic
11#17988988—21no—
12#17988967C2211yescharacteristic
13#17988885—22no—
14#17988877—21no—
15#17988873—21no—
16C8×Dic146C2211yescharacteristic
17D8×Dic73—21no—
18C4⋊#11924003C2211yescharacteristic
19#17988392—22no—
20#17988390—21no—
21#17988332—21no—
22C4×D8×C73C2211yescharacteristic
23C4×C16×C73C2211yescharacteristic
24#335×C73C2211yescharacteristic
25C2×Dic292—21no—
26C2×Dic292—21no—
27#14556591—21no—
28#14556591—21no—
29#14556582—22no—
30#14556570—41no—
31#14556568—42no—
32C4×Dic146D411yescharacteristic
33D4×Dic73—22no—
34C2×C4⋊Dic73—41no—
35#14556535—42no—
36C2×#11924003—21no—
37#14556525—21no—
38#14556525D412yes—
39#14556523—41no—
40#14556522—22no—
41C4⋊#10596831D411yescharacteristic
42#14556520—21no—
43#14556516—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×Dic146—41no—
55C2×Dic146D811yescharacteristic
56C22×Dic73—42no—
57Dic292—42no—
58Dic292—41no—
59Dic292—41no—
60C2×#10596831—21no—
61C4⋊Dic73—22no—
62C4⋊Dic73—42no—
63C4⋊Dic73—41no—
64C4⋊Dic73—41no—
65C4⋊Dic73—41no—
66C4⋊Dic73D811yescharacteristic
67#11924007—21no—
68#11924007—42no—
69C22⋊Dic73—42no—
70C4×Dic73—21no—
71#11924003—41no—
72C2×D4×C73—21no—
73C22×C4×C73—41no—
74D8×C73—21no—
75D8×C73SD1612yes—
76C2×C8×C73—41no—
77C2×C8×C73C2×D411yescharacteristic
78C2×C8×C73C2×D411yescharacteristic
79C4⋊C4×C73—21no—
80C22⋊C4×C73—41no—
81C42×C73C2×D411yescharacteristic
82C16×C73—41no—
83C23×C73—41no—
84D4×C73—22no—
85D4×C73—41no—
86C2×C4×C73—41no—
87C2×C4×C73—81no—
88C2×C4×C73—81no—
89C2×C4×C73#12011yescharacteristic
90C2×C4×C73C2×D811yescharacteristic
91C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
92C2×Dic73—21no—
93C2×Dic73—21no—
94C2×Dic73—42no—
95C2×Dic73—81no—
96C2×Dic73—162no—
97Dic146—21no—
98Dic146—21no—
99Dic146—81no—
100Dic146—81no—
101Dic146—161no—
102C8×C73—21no—
103C8×C73—81no—
104C8×C73C2×SD1611yescharacteristic
105C8×C73#13711yescharacteristic
106#10596831—41no—
107C22×C73—42no—
108C22×C73—81no—
109C22×C73—81no—
110C22×C73#44711yescharacteristic
111C4×C73—21no—
112C4×C73—81no—
113C4×C73#44811yescharacteristic
114C4×C73#45411yescharacteristic
115C4×C73#50811yescharacteristic
116C4×C73#51011yescharacteristic
117Dic73—41no—
118Dic73—41no—
119Dic73—82no—
120Dic73—161no—
121#12359—731nomaximal, Sylow
122C2×C73—82no—
123C2×C73#162711yescharacteristic
124C2×C73#218911yescharacteristic
125C2×C73#219411yescharacteristic
126#3380—731no—
127#2241—731no—
128#2208—731no—
129#1670—731no—
130#1365—731no—
131Q8⋊C16—731no—
132#1331D7311yescharacteristic
133C73#1235911yescharacteristic, complemented, Sylow
134#554—1461no—
135C2×SD32—1461no—
136#500—731no—
137#499—1461no—
138#494—731no—
139#478—1462no—
140#476—1461no—
141#475—1461no—
142C8×Q8—731no—
143C4×D8—1461no—
144C4×D8C2×D7311yescharacteristic
145C4⋊C16—731no—
146#358—1462no—
147#358—1461no—
148C4×C16C2×D7311yescharacteristic
149#335C2×D7311yescharacteristic
150#332—1461no—
151C2×Q16—1461no—
152C2×Q16—1461no—
153C2×D8D29211yescharacteristic
154C4⋊Q8—1461no—
155C4⋊Q8—1461no—
156#125—1462no—
157C22⋊Q8—2922no—
158C22⋊Q8—2921no—
159C4×Q8—731no—
160C4×D4—21no—
161C4×D4—1462no—
162C2×C4⋊C4—2921no—
163SD32—2922no—
164C2×C16—21no—
165C2×C16—1461no—
166#107—732no—
167#107—1461no—
168#107D29211yescharacteristic
169#106—2921no—
170#106—1462no—
171C4⋊C8—21no—
172C4⋊C8—731no—
173C4⋊C8—1461no—
174#103—2921no—
175#102—21no—
176#102—1462no—
177C4×C8—1461no—
178C4×C8C22×D7311yescharacteristic
179C2×Q8—2921no—
180C2×Q8—2921no—
181C2×Q8—731no—
182C2×D4—21no—
183C22×C4—41no—
184C22×C4—2922no—
185Q16—2922no—
186Q16—2921no—
187Q16—2921no—
188D8—21no—
189D8#1192401712yes—
190C2×C8—41no—
191C2×C8—1461no—
192C2×C8D4×D7311yescharacteristic
193C2×C8C2×D29211yescharacteristic
194C4⋊C4—21no—
195C4⋊C4—2922no—
196C4⋊C4—2922no—
197C4⋊C4—2921no—
198C4⋊C4—2921no—
199C4⋊C4—2921no—
200C4⋊C4—731no—
201C4⋊C4—1462no—
202C4⋊C4—1461no—
203C22⋊C4—41no—
204C22⋊C4—2922no—
205C42—1461no—
206C42D4×D7311yescharacteristic
207C16—41no—
208C16—2921no—
209C23—41no—
210Q8—5841no—
211Q8—5841no—
212Q8—11681no—
213Q8—1461no—
214Q8—1461no—
215D4—22no—
216D4—41no—
217C2×C4—41no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—2922no—
221C2×C4—5841no—
222C2×C4—11682no—
223C2×C4—1461no—
224C2×C4—1461no—
225C2×C4#1455656111yescharacteristic
226C2×C4D8×D7311yescharacteristic
227C2×C4#1455661811yescharacteristic
228C8—21no—
229C8—81no—
230C8—2921no—
231C8C2×#1192401711yescharacteristic
232C8#1455662411yescharacteristic
233C22—42no—
234C22—81no—
235C22—81no—
236C22#1798872211yescharacteristic
237C4—21no—
238C4—81no—
239C4—2921no—
240C4—2921no—
241C4—5842no—
242C4—11681no—
243C4#1798872811yescharacteristic
244C4#1798876111yescharacteristic
245C4#1798901611yescharacteristic
246C4#1798902611yescharacteristic
247C2—82no—
248C2#2260903711yescharacteristic
249C2#2261187211yescharacteristic
250C2#2261190311yescharacteristic
subgroupG/H
centerC22#17988722
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#1331C2
socleC22×C73#447
prime radicalC2×D8×C73D4
hypercenter#1331D73
hypocenterC73#12359
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
2. #22612159C2semidirect
4. #22609302C2semidirect
5. #22607990C2semidirect
6. #22607911C2semidirect
133. C73#12359semidirect

Maximal subgroups

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

subgroupindex
#226227962
#226121592
#226119762
#226093022
#226079902
#226079112
C73×#13312
#1235973