Finite Group Databasebeta

← #34011636

Subgroups — #34011636

Order 18688.

296 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 84 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#22626290C211yesmaximal, characteristic
2#22625487C211yesmaximal, characteristic
3#22624947C214yesmaximal, complemented
4#22620442C211yesmaximal, characteristic
5#17989591—21no—
6#17989508C2212yes—
7#17989482—21no—
8C4×#11924031—21no—
9#17989449—21no—
10#17989449C2212yes—
11#17989142—21no—
12#17988992C411yescharacteristic
13#17988988—21no—
14#17988988C2212yes—
15#17988983—21no—
16#17988624—21no—
17#17988624C412yes—
18C4×C8×D73C2211yescharacteristic
19#500×C73C411yescharacteristic
20C2×#11924038—41no—
21C2×#11924031—22no—
22C4⋊#10596833—22no—
23C4⋊#10596833—41no—
24C2×C8×D73C2×C411yescharacteristic
25C2×C8×D73D411yescharacteristic
26#14556591—21no—
27#14556582—22no—
28#14556571—41no—
29#14556567—41no—
30Q8×Dic73—41no—
31#14556556—41no—
32#14556549—22no—
33C42×D73D411yescharacteristic
34#14556525—21no—
35#14556525C2×C412yes—
36#14556522—22no—
37#14556522—22no—
38C4×#10596831D411yescharacteristic
39C8×Dic73D411yescharacteristic
40C8×Dic73C2311yescharacteristic
41C4⋊Q8×C73—21no—
42#107×C73—21no—
43#107×C73C2×C412yes—
44C4×C8×C73C2×C411yescharacteristic
45#11924038—82no—
46C2×#10596833—42no—
47#11924031—42no—
48C2×Dic146—41no—
49C2×C4×D73C22⋊C411yescharacteristic
50C2×C4×D73D811yescharacteristic
51C2×C4×D73SD1611yescharacteristic
52C2×#10596831C22⋊C411yescharacteristic
53C2×#10596831C2×D411yescharacteristic
54C8×D73—22no—
55C8×D73—22no—
56C4⋊Dic73—22no—
57C4⋊Dic73—41no—
58C4⋊Dic73—42no—
59#11924007—42no—
60#11924005—42no—
61C4×Dic73—42no—
62C4×Dic73D811yescharacteristic
63C4×Dic73SD1611yescharacteristic
64C4×Dic73C2×D411yescharacteristic
65C2×Q8×C73—41no—
66C2×C8×C73C22⋊C411yescharacteristic
67C2×C8×C73C22×C411yescharacteristic
68C4⋊C4×C73—22no—
69C4⋊C4×C73—41no—
70C42×C73C22⋊C411yescharacteristic
71C22×D73#10211yescharacteristic
72Q8×C73—82no—
73C2×C4×C73—42no—
74C2×C4×C73#10211yescharacteristic
75C2×C4×C73#10211yescharacteristic
76C2×C4×C73C2×C22⋊C411yescharacteristic
77C2×Dic73—42no—
78C2×Dic73#10211yescharacteristic
79C2×Dic73C2×D811yescharacteristic
80C2×Dic73C2×SD1611yescharacteristic
81C4×D73—22no—
82C4×D73—22no—
83C4×D73—21no—
84C4×D73—21no—
85Dic146—82no—
86#10596833—82no—
87C8×C73—21no—
88C8×C73—21no—
89#10596831#10112yes—
90#10596831#13712yes—
91C22×C73C2×#10211yescharacteristic
92C2×D73—21no—
93C2×D73—21no—
94C2×D73—21no—
95C4×C73—82no—
96C4×C73#35912yes—
97C4×C73#36012yes—
98C4×C73C2×#10111yescharacteristic
99C4×C73#41911yescharacteristic
100Dic73—21no—
101Dic73—21no—
102Dic73—21no—
103Dic73—82no—
104#11968—731nomaximal, Sylow
105C2×C73#154611yescharacteristic
106C2×C73C2×#35911yescharacteristic
107C2×C73C2×#36011yescharacteristic
108D73—41no—
109#3160—731no—
110#2162—731no—
111#1566—731no—
112#1363—734no—
113C73#1196811yescharacteristic, complemented, Sylow
114#533—1461no—
115#500—731no—
116#500#6440811yescharacteristic
117#499—732no—
118#499—1461no—
119#498—1461no—
120#427—732no—
121#427—1461no—
122#423—1461no—
123C2×C4×C8—731no—
124C4⋊C16—732no—
125C22⋊C16—1461no—
126C4×C16—1461no—
127#334—732no—
128#334—1461no—
129C2×M16—2921no—
130C22×C8—731no—
131C22×C8—731no—
132C4⋊Q8—21no—
133C4⋊Q8—1461no—
134#125—1462no—
135C22⋊Q8—2921no—
136C22⋊Q8—2921no—
137C4×Q8—2921no—
138C4×Q8—2921no—
139#117—1462no—
140C2×C42—731no—
141C2×C16—1462no—
142#107—21no—
143#107—732no—
144#107—1461no—
145#107C2×#6440812yes—
146#106—1462no—
147#106—1462no—
148C4⋊C8—2921no—
149C4⋊C8—1462no—
150C4×C8—731no—
151C4×C8—731no—
152C4×C8—731no—
153C4×C8C2×#6440811yescharacteristic
154C2×Q8—41no—
155C2×Q8—2921no—
156C22×C4—731no—
157C22×C4—731no—
158C22×C4—731no—
159M16—5842no—
160C2×C8—2922no—
161C2×C8—731no—
162C2×C8—731no—
163C2×C8—1462no—
164C2×C8—1462no—
165C2×C8C22⋊#6440811yescharacteristic
166C2×C8C22×#6440811yescharacteristic
167C4⋊C4—22no—
168C4⋊C4—41no—
169C4⋊C4—2922no—
170C4⋊C4—2922no—
171C4⋊C4—2921no—
172C4⋊C4—1462no—
173C22⋊C4—2922no—
174C42—2922no—
175C42—731no—
176C42—731no—
177C42—731no—
178C42C22⋊#6440811yescharacteristic
179C16—2922no—
180C23—731no—
181Q8—82no—
182Q8—5842no—
183C2×C4—42no—
184C2×C4—2922no—
185C2×C4—731no—
186C2×C4—731no—
187C2×C4—731no—
188C2×C4—1462no—
189C2×C4—1462no—
190C2×C4—1461no—
191C2×C4—1461no—
192C2×C4#1455665311yescharacteristic
193C2×C4#1455665311yescharacteristic
194C2×C4C2×C22⋊#6440811yescharacteristic
195C8—21no—
196C8—21no—
197C8—5842no—
198C8—732no—
199C8—732no—
200C22—1461no—
201C22—1461no—
202C22—1461no—
203C22C2×#1455665311yescharacteristic
204C4—82no—
205C4—5842no—
206C4—1461no—
207C4—1461no—
208C4—1461no—
209C4#1798950112yes—
210C4#1798950412yes—
211C4D73⋊#10111yescharacteristic
212C4#1798958011yescharacteristic
213C2—2921no—
214C2#2262544111yescharacteristic
215C2C2×#1798950111yescharacteristic
216C2C2×#1798950411yescharacteristic
subgroupG/H
centerC22C2×#14556653
commutatorC2×C8×C73C22×C4
FrattiniC2×C8C22×#64408
Fitting#500×C73C4
socleC22×C73C2×#102
prime radicalC22×D73#102
cosocleC8×Dic73C23
hypercenter#500#64408
hypocenterC73#11968
LCSG ▷ C2×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #500

Complements

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

NH ≅ G/Ntype
3. #22624947C2semidirect
113. C73#11968semidirect

Maximal subgroups

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

subgroupindex
#226262902
#226254872
#226249472
#226204422
#1196873