Finite Group Databasebeta

← #33005074

Subgroups — #33005074

Order 18688.

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

normal subgroups only

This group has 370 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#22620455C211yesmaximal, characteristic
2#22611652C212yesmaximal
3#22610802C211yesmaximal, characteristic
4#22608037C212yesmaximal
5C73×#1377C211yesmaximal, characteristic
6C22×Dic292C2211yescharacteristic
7#17989144—21no—
8#17988996—24no—
9#17988625—21no—
10#17988625C2212yes—
11C2×#14556523C2211yescharacteristic
12C2×#14556516—21no—
13#17988493—21no—
14#17988392—24no—
15C2×M32×C73C2212yes—
16C2×#97×C73C2211yescharacteristic
17C22×Dic146—21no—
18C2×Dic292—22no—
19C2×Dic292—22no—
20C2×Dic292—21no—
21C2×Dic292Q812yescomplemented
22#14556591—22no—
23#14556583—22no—
24#14556570—42no—
25C4×Dic146—42no—
26#14556550—22no—
27C2×C4⋊Dic73—21no—
28#14556525—22no—
29#14556525D414yes—
30#14556523—22no—
31#14556523—22no—
32#14556523—22no—
33#14556516—42no—
34#14556498—42no—
35C22×C8×C73D411yescharacteristic
36C22×C8×C73C2311yescharacteristic
37C2×C42×C73D411yescharacteristic
38M32×C73—22no—
39M32×C73—22no—
40C2×C16×C73—22no—
41#97×C73—22no—
42C2×Dic146—22no—
43C2×Dic146—41no—
44C2×Dic146—41no—
45C2×Dic146—41no—
46C2×Dic146—41no—
47C2×Dic146—42no—
48C2×Dic146—41no—
49C22×Dic73—41no—
50C22×Dic73—41no—
51Dic292—24no—
52Dic292—42no—
53C4⋊Dic73—24no—
54C4⋊Dic73—42no—
55C4⋊Dic73—41no—
56C4⋊Dic73—41no—
57#11924007—44no—
58C22⋊Dic73—42no—
59C4×Dic73—42no—
60C22×C4×C73D812yes—
61C22×C4×C73C2×D411yescharacteristic
62C2×C8×C73—21no—
63C2×C8×C73—21no—
64C2×C8×C73SD1612yes—
65C2×C8×C73C2×D412yes—
66C2×C8×C73C2×Q811yescharacteristic
67C2×C8×C73#4111yescharacteristic
68C2×C8×C73#4112yes—
69C42×C73—21no—
70C42×C73—21no—
71C16×C73—42no—
72C23×C73C2×D811yescharacteristic
73C2×C4×C73—22no—
74C2×C4×C73—42no—
75C2×C4×C73#12011yescharacteristic
76C2×C4×C73C22⋊Q811yescharacteristic
77C2×C4×C73#12411yescharacteristic
78C2×C4×C73C2×SD1611yescharacteristic
79C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
80C2×C4×C73#13711yescharacteristic
81C2×Dic73—44no—
82C2×Dic73—42no—
83C2×Dic73—81no—
84C2×Dic73—81no—
85C2×Dic73—82no—
86C2×Dic73—81no—
87Dic146—42no—
88Dic146—42no—
89Dic146—84no—
90Dic146—81no—
91Dic146—81no—
92C8×C73—22no—
93C8×C73C22⋊Q812yes—
94C8×C73#12412yes—
95C22×C73—22no—
96C22×C73#44711yescharacteristic
97C22×C73#47411yescharacteristic
98C22×C73#48611yescharacteristic
99C4×C73—42no—
100C4×C73#39311yescharacteristic
101C4×C73#44811yescharacteristic
102C4×C73#47611yescharacteristic
103C4×C73#48911yescharacteristic
104Dic73—84no—
105Dic73—82no—
106Dic73—161no—
107#13368—731nomaximal, Sylow
108C2×C73—21no—
109C2×C73—21no—
110C2×C73#202311yescharacteristic
111C2×C73#219112yes—
112#3162—731no—
113#2141—732no—
114#1971—731no—
115#1386—732no—
116#1377D7311yescharacteristic
117C73#1336811yescharacteristic, complemented, Sylow
118C22×Q16—731no—
119#533—1461no—
120C2×M32C2×D7312yes—
121#501—1464no—
122#427—732no—
123#427—1461no—
124C2×#106—731no—
125C2×#103—1461no—
126C2×#97C2×D7311yescharacteristic
127#391—1461no—
128#358—1464no—
129C22×Q8—1461no—
130C2×Q16—732no—
131C2×Q16—1462no—
132C2×Q16—1461no—
133C2×Q16—1462no—
134C22×C8D29211yescharacteristic
135C22×C8C22×D7311yescharacteristic
136C4⋊Q8—1462no—
137#125—1462no—
138C22⋊Q8—2922no—
139C4×Q8—2922no—
140#117—1462no—
141C2×C4⋊C4—1461no—
142C2×C42D29211yescharacteristic
143M32—22no—
144M32—22no—
145C2×C16—22no—
146#107—734no—
147#107—1462no—
148#106—1462no—
149#106—1462no—
150#106—1462no—
151#103—2922no—
152#97—22no—
153#95—2922no—
154C2×Q8—2922no—
155C2×Q8—2921no—
156C2×Q8—2921no—
157C2×Q8—2921no—
158C2×Q8—2921no—
159C2×Q8—2921no—
160C2×Q8—1462no—
161C22×C4—2921no—
162C22×C4—2921no—
163C22×C4D58412yes—
164C22×C4C2×D29211yescharacteristic
165Q16—2922no—
166Q16—1464no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#1192401712yes—
170C2×C8D4×D7312yes—
171C2×C8Q8×D7311yescharacteristic
172C2×C8#1192402711yescharacteristic
173C2×C8#1192402812yes—
174C4⋊C4—2924no—
175C4⋊C4—2922no—
176C4⋊C4—2921no—
177C4⋊C4—2921no—
178C4⋊C4—1464no—
179C22⋊C4—2922no—
180C42—21no—
181C42—21no—
182C42—2922no—
183C16—42no—
184C23C2×D58411yescharacteristic
185Q8—2922no—
186Q8—2922no—
187Q8—5844no—
188Q8—5841no—
189Q8—5841no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—2922no—
193C2×C4—2924no—
194C2×C4—5841no—
195C2×C4—5841no—
196C2×C4—5842no—
197C2×C4—5841no—
198C2×C4#1455656111yescharacteristic
199C2×C4#1455657111yescharacteristic
200C2×C4#1455657911yescharacteristic
201C2×C4C2×#1192401711yescharacteristic
202C2×C4#1455661911yescharacteristic
203C2×C4#1455662411yescharacteristic
204C8—22no—
205C8#1455656912yes—
206C8#1455657712yes—
207C22—22no—
208C22#1798872311yescharacteristic
209C22#1798886811yescharacteristic
210C22#1798893411yescharacteristic
211C4—42no—
212C4—5842no—
213C4—5844no—
214C4—11681no—
215C4#1798850011yescharacteristic
216C4#1798872811yescharacteristic
217C4#1798887811yescharacteristic
218C4#1798894711yescharacteristic
219C2—21no—
220C2—21no—
221C2#2261106811yescharacteristic
222C2#2261188412yes—
subgroupG/H
centerC22#17988723
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC73×#1377C2
socleC22×C73#447
prime radicalC23×C73C2×D8
hypercenter#1377D73
hypocenterC73#13368
LCSG ▷ C22×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #1377

Complements

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

NH ≅ G/Ntype
21. C2×Dic292Q8semidirect
117. C73#13368semidirect

Maximal subgroups

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

subgroupindex
#226204552
#226116522
#226108022
#226080372
C73×#13772
#1336873