Finite Group Databasebeta

← #32999185

Subgroups — #32999185

Order 18688.

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

normal subgroups only

This group has 396 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#22612109C212yesmaximal, complemented
2#22611354C211yesmaximal, characteristic, complemented
3#22607940C212yesmaximal, complemented
4#22607855C211yesmaximal, characteristic, complemented
5C73×#3186C211yesmaximal, characteristic
6C2×#14556525C2211yescharacteristic, complemented
7#17988447—21no—
8#17988412—22no—
9#17988388—22no—
10C22⋊#11924003C2212yescomplemented
11#17988350—21no—
12#17988350C2211yescharacteristic, complemented
13C2×#121×C73—21no—
14#480×C73—22no—
15#466×C73—21no—
16#466×C73C2212yes—
17#447×C73—22no—
18C2×C22⋊C8×C73C2211yescharacteristic
19C2×C4⋊Dic73—21no—
20C2×C4⋊Dic73—21no—
21C2×C22⋊Dic73—41no—
22C2×#11924003—22no—
23#14556525—22no—
24#14556498—41no—
25C22×D4×C73—21no—
26C22×D4×C73—41no—
27C23×C4×C73D411yescharacteristic
28C2×D8×C73—21no—
29C2×D8×C73D412yes—
30C22×C8×C73D411yescharacteristic
31C22×C8×C73C2311yescharacteristic
32#121×C73—22no—
33#121×C73—22no—
34#121×C73—41no—
35#121×C73—41no—
36C2×C4⋊C4×C73—21no—
37C2×C22⋊C4×C73—41no—
38#107×C73—21no—
39#107×C73D412yes—
40#102×C73—22no—
41#102×C73—22no—
42C22⋊C8×C73—22no—
43C22×Dic73—41no—
44C22×Dic73—41no—
45C4⋊Dic73—41no—
46C4⋊Dic73—41no—
47C4⋊Dic73—41no—
48C4⋊Dic73—41no—
49C22⋊Dic73—82no—
50#11924003—42no—
51C24×C73—41no—
52C24×C73D811yescharacteristic
53C2×D4×C73—22no—
54C2×D4×C73—41no—
55C2×D4×C73—42no—
56C2×D4×C73—42no—
57C2×D4×C73—81no—
58C2×D4×C73—81no—
59C2×D4×C73—82no—
60C2×D4×C73—82no—
61C22×C4×C73—21no—
62C22×C4×C73—21no—
63C22×C4×C73—41no—
64C22×C4×C73—41no—
65C22×C4×C73—41no—
66C22×C4×C73D811yescharacteristic
67C22×C4×C73C2×D411yescharacteristic
68D8×C73—42no—
69D8×C73—81no—
70C2×C8×C73—21no—
71C2×C8×C73—41no—
72C2×C8×C73—41no—
73C2×C8×C73C2×D412yes—
74C4⋊C4×C73—22no—
75C4⋊C4×C73—41no—
76C22⋊C4×C73—42no—
77C22⋊C4×C73—42no—
78C23×C73—21no—
79C23×C73—21no—
80C23×C73—41no—
81C23×C73—41no—
82C23×C73—42no—
83C23×C73—82no—
84C23×C73—82no—
85C23×C73—82no—
86C23×C73D1612yes—
87C23×C73C2×D811yescharacteristic
88D4×C73—42no—
89D4×C73—42no—
90D4×C73—82no—
91D4×C73—84no—
92D4×C73—82no—
93D4×C73—82no—
94C2×C4×C73—21no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—41no—
98C2×C4×C73—41no—
99C2×C4×C73—41no—
100C2×C4×C73—42no—
101C2×C4×C73—81no—
102C2×C4×C73—82no—
103C2×C4×C73#12011yescharacteristic
104C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
105C2×Dic73—81no—
106C2×Dic73—81no—
107C2×Dic73—81no—
108C2×Dic73—81no—
109C2×Dic73—81no—
110C2×Dic73—81no—
111C8×C73—22no—
112C8×C73—41no—
113C22×C73—21no—
114C22×C73—21no—
115C22×C73—22no—
116C22×C73—42no—
117C22×C73—41no—
118C22×C73—41no—
119C22×C73—81no—
120C22×C73—82no—
121C22×C73—82no—
122C22×C73—82no—
123C22×C73—164no—
124C22×C73#44711yescharacteristic
125C22×C73C2×D1611yescharacteristic
126C22×C73#50911yescharacteristic
127C4×C73—21no—
128C4×C73—41no—
129C4×C73—82no—
130C4×C73#45511yescharacteristic
131C4×C73#45811yescharacteristic
132Dic73—161no—
133Dic73—161no—
134#12278—731nomaximal, Sylow
135C2×C73—21no—
136C2×C73—21no—
137C2×C73—42no—
138C2×C73—162no—
139C2×C73#160711yescharacteristic
140C2×C73#218411yescharacteristic
141C2×C73#219111yescharacteristic
142#3186D7311yescharacteristic
143#2232—732no—
144#2075—731no—
145#1347—732no—
146#1314—731no—
147C73#1227811yescharacteristic, complemented, Sylow
148C2×#121—21no—
149#480—22no—
150#466—21no—
151#466C2×D7312yes—
152#447—22no—
153C2×#107—731no—
154C2×C22⋊C8C2×D7311yescharacteristic
155#380—1461no—
156#366—1462no—
157#357—1462no—
158C22⋊C16—732no—
159#340—731no—
160#340—1461no—
161C22×D4—21no—
162C22×D4—41no—
163C23×C4C22⋊D7311yescharacteristic
164C2×D8—21no—
165C2×D8C22⋊D7312yes—
166C22×C8C22⋊D7311yescharacteristic
167C22×C8C22×D7311yescharacteristic
168#121—22no—
169#121—22no—
170#121—41no—
171#121—41no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—1461no—
174C2×C4⋊C4—1461no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—2921no—
177C2×C16—1462no—
178#107—21no—
179#107—1462no—
180#107C22⋊D7312yes—
181#102—22no—
182#102—22no—
183C22⋊C8—22no—
184#95—2921no—
185C24—41no—
186C24#1192401311yescharacteristic
187C2×D4—22no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4—82no—
194C2×D4—82no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—2921no—
201C22×C4—2921no—
202C22×C4#1192401311yescharacteristic
203C22×C4C2×C22⋊D7311yescharacteristic
204D8—42no—
205D8—81no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8C2×C22⋊D7312yes—
210C4⋊C4—22no—
211C4⋊C4—41no—
212C4⋊C4—2921no—
213C4⋊C4—2921no—
214C4⋊C4—2921no—
215C4⋊C4—2921no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—5842no—
219C16—2922no—
220C23—21no—
221C23—21no—
222C23—41no—
223C23—41no—
224C23—42no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23#1455653212yes—
229C23C2×#1192401311yescharacteristic
230D4—42no—
231D4—42no—
232D4—82no—
233D4—84no—
234D4—82no—
235D4—82no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—81no—
244C2×C4—82no—
245C2×C4—5841no—
246C2×C4—5841no—
247C2×C4—5841no—
248C2×C4—5841no—
249C2×C4—5841no—
250C2×C4—5841no—
251C2×C4#1455656011yescharacteristic
252C2×C4#1455661711yescharacteristic
253C8—22no—
254C8—41no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—164no—
266C22#1798872111yescharacteristic
267C22C2×#1455653211yescharacteristic
268C22#1798902311yescharacteristic
269C4—21no—
270C4—41no—
271C4—82no—
272C4—11681no—
273C4—11681no—
274C4#1798876511yescharacteristic
275C4#1798877711yescharacteristic
276C2—21no—
277C2—21no—
278C2—42no—
279C2—162no—
280C2#2260891711yescharacteristic
281C2#2261184311yescharacteristic
282C2#2261188211yescharacteristic
subgroupG/H
centerC22#17988721
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
Fitting, prime radicalC73×#3186C2
socleC22×C73#447
hypercenter#3186D73
hypocenterC73#12278
LCSG ▷ C22×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #3186

Complements

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

NH ≅ G/Ntype
1. #22612109C2semidirect
2. #22611354C2semidirect
3. #22607940C2semidirect
4. #22607855C2semidirect
6. C2×#14556525C22semidirect
10. C22⋊#11924003C22semidirect
12. #17988350C22semidirect
147. C73#12278semidirect

Maximal subgroups

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

subgroupindex
#226121092
#226113542
#226079402
#226078552
C73×#31862
#1227873