Finite Group Databasebeta

← #34034370

Subgroups — #34034370

Order 18688.

334 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 97 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#22627121C211yesmaximal, characteristic
2#22627087C211yesmaximal, characteristic
3#22627011C214yesmaximal
4#22626720C211yesmaximal, characteristic
5C2×#14556688C2212yes—
6C2×#14556686C2212yes—
7#17989690—21no—
8#17989690—21no—
9#17989681—21no—
10#17989681—21no—
11#17989638—21no—
12#17989637—21no—
13#17989636—21no—
14Q8×#10596833—21no—
15#17989594C412yes—
16#17989591C411yescharacteristic
17#17989552C2212yes—
18C4×#11924036C2211yescharacteristic
19#17989142C411yescharacteristic
20#14556688—24no—
21#14556686—22no—
22#14556686Q814yescomplemented
23C2×#11924040—22no—
24C2×#11924040—22no—
25C2×#11924038C2×C412yes—
26C2×#11924036C2×C411yescharacteristic
27C2×#11924036C2311yescharacteristic
28#14556658—22no—
29C4⋊#10596833—22no—
30C4⋊#10596833—21no—
31C4⋊#10596833—21no—
32#14556646—22no—
33C4×#10596833—22no—
34C4×#10596833D411yescharacteristic
35C4×#10596833D411yescharacteristic
36C4×#10596833D411yescharacteristic
37C4×#10596833D411yescharacteristic
38#14556591C811yescharacteristic
39#14556582C812yes—
40#14556571—21no—
41#14556567—21no—
42Q8×Dic73—21no—
43#14556556—21no—
44#14556549C2×C412yes—
45C42×D73C2×C411yescharacteristic
46C4⋊Q8×C73C811yescharacteristic
47#11924040—24no—
48#11924040—42no—
49#11924038—24no—
50#11924036—22no—
51#11924036C2×Q812yes—
52#11924036#4112yes—
53C2×#10596833—22no—
54C2×#10596833C22⋊C411yescharacteristic
55C2×#10596833C22⋊C411yescharacteristic
56C2×#10596833C2×D411yescharacteristic
57C2×#10596833C2×D411yescharacteristic
58C2×Dic146—21no—
59C2×C4×D73M1611yescharacteristic
60C2×C4×D73M1611yescharacteristic
61C2×C4×D73C22×C411yescharacteristic
62C4⋊Dic73—21no—
63C4⋊Dic73—22no—
64C4⋊Dic73C2×C812yes—
65#11924007—22no—
66#11924005—22no—
67C4×Dic73—22no—
68C4×Dic73C22⋊C411yescharacteristic
69C4×Dic73C22⋊C411yescharacteristic
70C4×Dic73C2×C811yescharacteristic
71C2×Q8×C73—21no—
72C4⋊C4×C73—21no—
73C4⋊C4×C73C2×C812yes—
74C42×C73C2×C811yescharacteristic
75C22×D73C2×M1611yescharacteristic
76Q8×C73—42no—
77C2×C4×C73—22no—
78C2×C4×C73C22⋊C811yescharacteristic
79C2×C4×C73C22⋊C811yescharacteristic
80C2×C4×C73C22×C811yescharacteristic
81C2×Dic73—22no—
82C2×Dic73C22⋊C811yescharacteristic
83C2×Dic73C22⋊C811yescharacteristic
84C2×Dic73C2×C22⋊C411yescharacteristic
85C4×D73—41no—
86C4×D73—41no—
87C4×D73C4×Q812yes—
88C4×D73#13112yes—
89Dic146—42no—
90#10596833—21no—
91#10596833—21no—
92#10596833—42no—
93#10596833C22⋊Q812yes—
94#10596833#12411yescharacteristic
95#10596833C4⋊Q811yescharacteristic
96C22×C73C2×C22⋊C811yescharacteristic
97C2×D73—21no—
98C2×D73—21no—
99C2×D73#44612yes—
100C4×C73—21no—
101C4×C73—21no—
102C4×C73—42no—
103C4×C73C8×Q812yes—
104Dic73—21no—
105Dic73—21no—
106Dic73—42no—
107Dic73#39111yescharacteristic
108Dic73#40811yescharacteristic
109#13267—731nomaximal, Sylow
110C2×C73#198411yescharacteristic
111C2×C73#211711yescharacteristic
112C2×C73#211711yescharacteristic
113D73—22no—
114#2960—731no—
115#2150—731no—
116C2×#346—731no—
117#1379—734no—
118C73#1326711yescharacteristic, complemented, Sylow
119#533—731no—
120C2×M32—732no—
121C22×C16—732no—
122#446—1461no—
123#446—1461no—
124C8×Q8—1461no—
125C8×Q8—1461no—
126#424—732no—
127#423—731no—
128#405—732no—
129C2×C4×C8—731no—
130C4⋊C16—1461no—
131C4⋊C16—1461no—
132#346—1461no—
133#346—1461no—
134C2×M16—732no—
135C22×C8—731no—
136C22×C8—731no—
137C4⋊Q8—731no—
138C4⋊Q8#1059684211yescharacteristic
139#125—732no—
140C22⋊Q8—1461no—
141C22⋊Q8—1461no—
142C4×Q8—1461no—
143C4×Q8—1461no—
144#117—732no—
145C2×C42—731no—
146M32—1464no—
147C2×C16—734no—
148C2×C16—1462no—
149C2×C16—1462no—
150C2×C16—1462no—
151C4⋊C8—1461no—
152C4⋊C8—1461no—
153C4⋊C8—1462no—
154C4⋊C8—1462no—
155#97—1462no—
156C4×C8—731no—
157C4×C8—731no—
158C4×C8—731no—
159C4×C8—731no—
160C4×C8—1462no—
161C2×Q8—21no—
162C2×Q8—1461no—
163C22×C4—731no—
164C22×C4—731no—
165C22×C4—731no—
166M16—1464no—
167C2×C8—731no—
168C2×C8—731no—
169C2×C8—732no—
170C2×C8—732no—
171C2×C8—731no—
172C2×C8—731no—
173C2×C8—1462no—
174C2×C8—1462no—
175C4⋊C4—21no—
176C4⋊C4—732no—
177C4⋊C4—1462no—
178C4⋊C4—1462no—
179C4⋊C4—1461no—
180C4⋊C4C2×#1059684212yes—
181C22⋊C4—1462no—
182C42—731no—
183C42—731no—
184C42—731no—
185C42—1462no—
186C42C2×#1059684211yescharacteristic
187C16—2922no—
188C16—1464no—
189C23—731no—
190Q8—42no—
191Q8—2922no—
192C2×C4—22no—
193C2×C4—2921no—
194C2×C4—2921no—
195C2×C4—731no—
196C2×C4—732no—
197C2×C4—732no—
198C2×C4—731no—
199C2×C4—731no—
200C2×C4—1462no—
201C2×C4C22⋊#1059684211yescharacteristic
202C2×C4C22⋊#1059684211yescharacteristic
203C2×C4C22×#1059684211yescharacteristic
204C8—2922no—
205C8—731no—
206C8—732no—
207C8—731no—
208C8—1461no—
209C8—1461no—
210C22—732no—
211C22—1461no—
212C22—1461no—
213C22C2×C22⋊#1059684211yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—2922no—
218C4—731no—
219C4—731no—
220C4—1461no—
221C4—1461no—
222C4Q8×#1059684212yes—
223C2—1462no—
224C2(C2×#10596842)⋊Q811yescharacteristic
225C2#2262711311yescharacteristic
226C2#2262711511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10596842
commutatorC2×C4×C73C22×C8
FrattiniC2×C4C22×#10596842
FittingC4⋊Q8×C73C8
socleC22×C73C2×C22⋊C8
prime radicalC22×D73C2×M16
cosocleC2×#11924036C23
hypercenterC4⋊Q8#10596842
hypocenterC73#13267
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
22. #14556686Q8semidirect
118. C73#13267semidirect

Maximal subgroups

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

subgroupindex
#226271212
#226270872
#226270112
#226267202
#1326773