Finite Group Databasebeta

← #32999532

Subgroups — #32999532

Order 18688.

306 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 71 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#22622937C211yesmaximal, characteristic
2#22612183C211yesmaximal, characteristic
3#22611978C211yesmaximal, characteristic
4#22607979C211yesmaximal, characteristic
5Q8⋊#11924003C211yesmaximal, characteristic
6#22607894C211yesmaximal, characteristic
7C73×#1670C211yesmaximal, characteristic
8#17989316—21no—
9C2×#14556537—21no—
10C2×#14556537—22no—
11#17988967—21no—
12#17988967C2211yescharacteristic
13#17988798—22no—
14#17988798—21no—
15C4×Dic292—21no—
16C4×Dic292C2211yescharacteristic
17C4⋊#11924003C2211yescharacteristic
18#17988391—21no—
19C4×#11924003C2211yescharacteristic
20#494×C73C2211yescharacteristic
21#475×C73—21no—
22C8×Q8×C73C2211yescharacteristic
23#335×C73C2211yescharacteristic
24#332×C73—21no—
25C2×Dic292—22no—
26C2×Dic292—21no—
27C2×Dic292D411yescharacteristic
28C2×Dic292D412yes—
29#14556591—22no—
30#14556591—21no—
31#14556582—42no—
32#14556582—41no—
33Q8×Dic73—41no—
34C4×Dic146—22no—
35C4×Dic146—21no—
36#14556537—44no—
37#14556537—42no—
38C2×#11924003—21no—
39C2×#11924003—21no—
40#14556525D411yescharacteristic
41#14556516—22no—
42#14556516—21no—
43C2×Q16×C73—21no—
44C2×Q16×C73—21no—
45C4⋊Q8×C73—21no—
46C4×Q8×C73D411yescharacteristic
47#106×C73—41no—
48C4⋊C8×C73—21no—
49C4⋊C8×C73—21no—
50C4⋊C8×C73D411yescharacteristic
51#103×C73—41no—
52C4×C8×C73—21no—
53C4×C8×C73C2311yescharacteristic
54C2×Dic146—21no—
55C2×Dic146—22no—
56Dic292—21no—
57Dic292—21no—
58Dic292—42no—
59Dic292—41no—
60Dic292—41no—
61Dic292—82no—
62C4⋊Dic73—21no—
63C4⋊Dic73—42no—
64C4⋊Dic73—42no—
65C4⋊Dic73—41no—
66C4⋊Dic73—41no—
67#11924007—42no—
68#11924007—41no—
69C4×Dic73—42no—
70C4×Dic73—41no—
71#11924003—22no—
72#11924003—41no—
73C2×Q8×C73—41no—
74C2×Q8×C73D811yescharacteristic
75Q16×C73—42no—
76Q16×C73—41no—
77Q16×C73—41no—
78C2×C8×C73—21no—
79C2×C8×C73—41no—
80C2×C8×C73C2×D411yescharacteristic
81C2×C8×C73C2×D411yescharacteristic
82C4⋊C4×C73—21no—
83C4⋊C4×C73—41no—
84C4⋊C4×C73—41no—
85C4⋊C4×C73D811yescharacteristic
86C42×C73—21no—
87C42×C73C2×D411yescharacteristic
88Q8×C73—21no—
89Q8×C73—81no—
90Q8×C73—81no—
91Q8×C73Q3212yescomplemented
92C2×C4×C73—21no—
93C2×C4×C73—21no—
94C2×C4×C73—81no—
95C2×C4×C73#12011yescharacteristic
96C2×C4×C73C2×D811yescharacteristic
97C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
98C2×Dic73—41no—
99C2×Dic73—42no—
100C2×Dic73—41no—
101Dic146—41no—
102Dic146—42no—
103Dic146—42no—
104Dic146—41no—
105C8×C73—21no—
106C8×C73—41no—
107C8×C73—81no—
108C8×C73#13511yescharacteristic
109C8×C73AGL(1,ℤ/8)11yescharacteristic
110C22×C73#44711yescharacteristic
111C4×C73—22no—
112C4×C73—21no—
113C4×C73—41no—
114C4×C73—161no—
115C4×C73#44911yescharacteristic
116C4×C73#45311yescharacteristic
117C4×C73C2×Q3211yescharacteristic
118C4×C73#51011yescharacteristic
119Dic73—81no—
120Dic73—82no—
121Dic73—81no—
122#12334—731nomaximal, Sylow
123C2×C73#162111yescharacteristic
124C2×C73#218811yescharacteristic
125C2×C73#219511yescharacteristic
126#3395—731no—
127#2247—731no—
128#2209—731no—
129#1670D7311yescharacteristic
130#1362—731no—
131Q8⋊C16—731no—
132#1332—731no—
133C73#1233411yescharacteristic, complemented, Sylow
134#557—1461no—
135C2×Q32—1461no—
136C2×Q32—1462no—
137#494—731no—
138#494—1461no—
139#494C2×D7311yescharacteristic
140#475—21no—
141#462—1462no—
142#462—1461no—
143C8×Q8C2×D7311yescharacteristic
144C4×Q16—731no—
145C4×Q16—1461no—
146C4⋊C16—731no—
147#358—1461no—
148C4×C16—731no—
149#335C2×D7311yescharacteristic
150#332—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—732no—
154C2×Q16—731no—
155C2×Q16—1462no—
156C2×Q16—1461no—
157C4⋊Q8—21no—
158C4⋊Q8—1462no—
159C4⋊Q8—1461no—
160#125—2922no—
161#125—2921no—
162C4×Q8—2921no—
163C4×Q8—1462no—
164C4×Q8—1461no—
165C4×Q8C22⋊D7311yescharacteristic
166Q32—2922no—
167Q32—2924no—
168C2×C16—1461no—
169C2×C16—1461no—
170#107—731no—
171#106—41no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8C22⋊D7311yescharacteristic
175#103—41no—
176#103—1462no—
177#103—1461no—
178C4×C8—21no—
179C4×C8C22×D7311yescharacteristic
180C2×Q8—41no—
181C2×Q8—1462no—
182C2×Q8—1461no—
183C2×Q8#1192401311yescharacteristic
184Q16—42no—
185Q16—41no—
186Q16—41no—
187Q16—2922no—
188Q16—2921no—
189Q16—2921no—
190Q16—5842no—
191Q16—1461no—
192Q16—1461no—
193C2×C8—21no—
194C2×C8—41no—
195C2×C8D4×D7311yescharacteristic
196C2×C8D4×D7311yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—2922no—
201C4⋊C4—2922no—
202C4⋊C4—2922no—
203C4⋊C4—2921no—
204C4⋊C4—2921no—
205C4⋊C4—2921no—
206C4⋊C4—1461no—
207C4⋊C4#1192401311yescharacteristic
208C42—21no—
209C42—2922no—
210C42—2921no—
211C42C2×C22⋊D7311yescharacteristic
212C16—2921no—
213C16—1462no—
214Q8—21no—
215Q8—81no—
216Q8—81no—
217Q8—2922no—
218Q8—2921no—
219Q8—2922no—
220Q8—2921no—
221Q8#1455653712yescomplemented
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—81no—
225C2×C4—2921no—
226C2×C4—2922no—
227C2×C4—2921no—
228C2×C4#1455655911yescharacteristic
229C2×C4C2×#1192401311yescharacteristic
230C2×C4#1455661611yescharacteristic
231C8—21no—
232C8—41no—
233C8—81no—
234C8#1455661211yescharacteristic
235C8#1455661811yescharacteristic
236C22#1798872011yescharacteristic
237C4—22no—
238C4—21no—
239C4—41no—
240C4—161no—
241C4—5842no—
242C4—5841no—
243C4—5841no—
244C4#1798873111yescharacteristic
245C4#1798875311yescharacteristic
246C4C2×#1455653711yescharacteristic
247C4#1798902711yescharacteristic
248C2#2260899411yescharacteristic
249C2#2261186411yescharacteristic
250C2#2261190711yescharacteristic
subgroupG/H
centerC22#17988720
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#1670C2
socle, prime radicalC22×C73#447
hypercenter#1670D73
hypocenterC73#12334
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1670

Complements

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

NH ≅ G/Ntype
91. Q8×C73Q32semidirect
133. C73#12334semidirect
221. Q8#14556537semidirect

Maximal subgroups

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

subgroupindex
#226229372
#226121832
#226119782
#226079792
Q8⋊#119240032
#226078942
C73×#16702
#1233473