Finite Group Databasebeta

← #33002261

Subgroups — #33002261

Order 18688.

342 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 79 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#22618571C211yesmaximal, characteristic
2#22610032C212yesmaximal
3#22608059C212yesmaximal
4#22608054C211yesmaximal, characteristic
5C73×#1391C211yesmaximal, characteristic
6#41×Dic73C2211yescharacteristic
7#17988645—22no—
8#17988637—22no—
9#17988615C411yescharacteristic, complemented
10#17988613C411yescharacteristic, complemented
11#17988571C2212yes—
12#17988554C2211yescharacteristic
13M16×Dic73C411yescharacteristic, complemented
14C2×C8×Dic73C411yescharacteristic, complemented
15#17988346—22no—
16#17988343—22no—
17#410×C73C2212yes—
18#408×C73C2211yescharacteristic
19C2×#11924012—22no—
20C2×#11924012C2×C411yescharacteristic
21C22×#10596831C2×C411yescharacteristic
22Q8×Dic73—21no—
23D4×Dic73—21no—
24D4×Dic73—22no—
25#14556550C2×C411yescharacteristic
26#14556550D412yes—
27C2×C4×Dic73C2×C411yescharacteristic
28C2×C4×Dic73D412yes—
29#14556520—22no—
30#14556520—22no—
31#14556510—22no—
32#14556508—22no—
33C22⋊#10596831—22no—
34#14556501—21no—
35C8×Dic73—21no—
36C8×Dic73—21no—
37C8×Dic73—21no—
38C2×#41×C73C2311yescharacteristic
39C2×M16×C73C2×C411yescharacteristic
40C22×C8×C73C2×C411yescharacteristic
41#117×C73—22no—
42#99×C73—22no—
43#99×C73—22no—
44C22⋊C8×C73—22no—
45C22×Dic73—21no—
46C22×Dic73—22no—
47#11924012—41no—
48#11924012—42no—
49#11924012—42no—
50C2×#10596831—21no—
51C2×#10596831—21no—
52C2×#10596831—41no—
53C2×#10596831—42no—
54C4⋊Dic73—21no—
55C4⋊Dic73—22no—
56C22⋊Dic73—21no—
57C22⋊Dic73—22no—
58C4×Dic73—21no—
59C4×Dic73—22no—
60C4×Dic73C22⋊C411yescharacteristic
61C4×Dic73C22⋊C411yescharacteristic
62C4×Dic73C22⋊C412yes—
63#41×C73—41no—
64#41×C73—41no—
65C2×Q8×C73#4111yescharacteristic
66C2×D4×C73#4111yescharacteristic
67C2×D4×C73#4112yes—
68C22×C4×C73C22×C411yescharacteristic
69C22×C4×C73C2×D412yes—
70M16×C73—41no—
71C2×C8×C73—21no—
72C2×C8×C73—21no—
73C2×C8×C73—41no—
74C4⋊C4×C73—42no—
75C22⋊C4×C73—42no—
76C42×C73—42no—
77C23×C73#11711yescharacteristic
78C23×C73#12312yes—
79Q8×C73—41no—
80D4×C73—41no—
81D4×C73—42no—
82C2×C4×C73—22no—
83C2×C4×C73—22no—
84C2×C4×C73—21no—
85C2×C4×C73—21no—
86C2×C4×C73—82no—
87C2×C4×C73C2×C22⋊C411yescharacteristic
88C2×C4×C73#11711yescharacteristic
89C2×C4×C73#12312yes—
90C2×Dic73—21no—
91C2×Dic73—21no—
92C2×Dic73—22no—
93C2×Dic73—42no—
94C2×Dic73—44no—
95C8×C73—41no—
96C8×C73—41no—
97#10596831—41no—
98#10596831—41no—
99#10596831—82no—
100C22×C73—22no—
101C22×C73—21no—
102C22×C73—41no—
103C22×C73—42no—
104C22×C73#38111yescharacteristic
105C4×C73—22no—
106C4×C73—21no—
107C4×C73—162no—
108C4×C73#41011yescharacteristic
109C4×C73#41011yescharacteristic
110Dic73—22no—
111Dic73—41no—
112Dic73—42no—
113#12864—731nomaximal, Sylow
114C2×C73—21no—
115C2×C73—42no—
116C2×C73—41no—
117C2×C73#182011yescharacteristic
118#2922—731no—
119#1827—732no—
120#1395—732no—
121#1392—731no—
122#1391D7311yescharacteristic
123C73#1286411yescharacteristic, complemented, Sylow
124C4×#41—731no—
125#433—1462no—
126#431—1462no—
127#424—731no—
128#424—731no—
129#413—732no—
130#410C2×D7312yes—
131#408—731no—
132#408C2×D7311yescharacteristic
133C4×M16—731no—
134C2×C4×C8—731no—
135#339—1462no—
136H(3,ℤ/4)—1462no—
137C2×#41C22×D7311yescharacteristic
138C2×M16—731no—
139C2×M16—1462no—
140C2×M16C4×D7311yescharacteristic
141C22×C8—731no—
142C22×C8C4×D7311yescharacteristic
143C4×Q8—1461no—
144C4×D4—1461no—
145C4×D4—1462no—
146#117—22no—
147#117—731no—
148#117—732no—
149C2×C42—731no—
150C2×C42—732no—
151C4⋊C8—1462no—
152C4⋊C8—1462no—
153#101—1462no—
154#100—1462no—
155#99—22no—
156#99—22no—
157C22⋊C8—22no—
158C22⋊C8—1462no—
159#97—1461no—
160C4×C8—1461no—
161C4×C8—1461no—
162C4×C8—1461no—
163#41—41no—
164#41—41no—
165C2×Q8#1192402911yescharacteristic
166C2×D4#1192402711yescharacteristic
167C2×D4#1192402812yes—
168C22×C4—1461no—
169C22×C4—1462no—
170C22×C4C2×C4×D7311yescharacteristic
171C22×C4D4×D7312yes—
172M16—41no—
173M16—2921no—
174M16—2922no—
175M16—2922no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—41no—
179C2×C8—2921no—
180C2×C8—2922no—
181C2×C8—1461no—
182C2×C8—1461no—
183C4⋊C4—42no—
184C4⋊C4—1461no—
185C4⋊C4—1462no—
186C22⋊C4—42no—
187C22⋊C4—1461no—
188C22⋊C4—1462no—
189C42—42no—
190C42—731no—
191C42—731no—
192C42—732no—
193C42—1461no—
194C42—1462no—
195C23#1455654811yescharacteristic
196C23#1455657312yes—
197Q8—41no—
198D4—41no—
199D4—42no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—82no—
205C2×C4—2922no—
206C2×C4—2924no—
207C2×C4—1461no—
208C2×C4—1462no—
209C2×C4—1461no—
210C2×C4C22⋊C4×D7311yescharacteristic
211C2×C4#1455654911yescharacteristic
212C2×C4#1455657612yes—
213C8—41no—
214C8—41no—
215C8—2921no—
216C8—2921no—
217C8—5842no—
218C22—22no—
219C22—21no—
220C22—41no—
221C22—42no—
222C22#1798844911yescharacteristic
223C4—22no—
224C4—21no—
225C4—162no—
226C4—2921no—
227C4—2922no—
228C4—1462no—
229C4#1798855911yescharacteristic
230C4#1798855911yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2#2261000711yescharacteristic
subgroupG/H
centerC4#17988559
commutatorC22×C4×C73C22×C4
FrattiniC2×#41C22×D73
FittingC73×#1391C2
socleC2×C73#1820
prime radical, cosocleC2×#41×C73C23
hypercenter#1391D73
hypocenterC73#12864
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1391

Complements

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

NH ≅ G/Ntype
9. #17988615C4semidirect
10. #17988613C4semidirect
13. M16×Dic73C4semidirect
14. C2×C8×Dic73C4semidirect
123. C73#12864semidirect

Maximal subgroups

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

subgroupindex
#226185712
#226100322
#226080592
#226080542
C73×#13912
#1286473