Finite Group Databasebeta

← #33005627

Subgroups — #33005627

Order 18688.

318 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 99 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#22620467C211yesmaximal, characteristic
2#22611717C211yesmaximal, characteristic
3#22610714C211yesmaximal, characteristic
4#22608039C211yesmaximal, characteristic
5#22608027C211yesmaximal, characteristic
6#22608019C211yesmaximal, characteristic
7C2×#367×C73C211yesmaximal, characteristic
8#17989218—21no—
9C2×#14556531C2211yescharacteristic
10#17988995—22no—
11#17988991—21no—
12#17988988—21no—
13C2×#14556527C2211yescharacteristic
14#17988627—21no—
15#17988627—21no—
16#17988625C2211yescharacteristic
17#17988542C2211yescharacteristic
18#17988414—21no—
19#17988412—21no—
20#17988356—21no—
21C22×C16×C73C2211yescharacteristic
22C2×#107×C73C2211yescharacteristic
23C2×#107×C73C2211yescharacteristic
24#367×C73—22no—
25C2×#11924012—21no—
26C2×#11924012D411yescharacteristic
27#14556590—22no—
28#14556590—41no—
29#14556582—22no—
30#14556582—41no—
31#14556570—42no—
32#14556550—21no—
33#14556550D411yescharacteristic
34#14556531—21no—
35#14556531Q812yescomplemented
36C2×#11924003—21no—
37#14556527—21no—
38#14556527Q812yescomplemented
39#14556525D411yescharacteristic
40#14556525D411yescharacteristic
41#14556524—22no—
42#14556524—41no—
43#14556523—21no—
44#14556522—22no—
45#14556522—41no—
46#14556501—21no—
47C8×Dic73—21no—
48C22×C8×C73C2311yescharacteristic
49C2×C4⋊C4×C73—21no—
50C2×C4⋊C4×C73—21no—
51C2×C16×C73—21no—
52C2×C16×C73—21no—
53C2×C16×C73Q812yescomplemented
54#107×C73—21no—
55#107×C73—21no—
56#107×C73—21no—
57#107×C73D412yes—
58C2×Dic146—42no—
59#11924012—21no—
60#11924012—22no—
61#11924012—41no—
62#11924012Q1612yes—
63C2×#10596831—21no—
64C2×#10596831—41no—
65C4⋊Dic73—21no—
66C4⋊Dic73—21no—
67C4⋊Dic73—42no—
68C4⋊Dic73D811yescharacteristic
69C4⋊Dic73D811yescharacteristic
70#11924007—42no—
71#11924007—42no—
72C22⋊Dic73—21no—
73C22⋊Dic73—41no—
74C4×Dic73—21no—
75C4×Dic73—41no—
76#11924003—22no—
77C22×C4×C73—41no—
78C22×C4×C73—41no—
79C22×C4×C73C2×D411yescharacteristic
80C2×C8×C73C2×D411yescharacteristic
81C2×C8×C73C2×D411yescharacteristic
82C2×C8×C73C2×Q811yescharacteristic
83C2×C8×C73C2×Q811yescharacteristic
84C2×C8×C73C2×Q811yescharacteristic
85C2×C8×C73#4111yescharacteristic
86C4⋊C4×C73—22no—
87C4⋊C4×C73—41no—
88C4⋊C4×C73—41no—
89C4⋊C4×C73—41no—
90C16×C73—22no—
91C23×C73AGL(1,ℤ/8)11yescharacteristic
92C2×C4×C73—42no—
93C2×C4×C73—81no—
94C2×C4×C73—161no—
95C2×C4×C73—161no—
96C2×C4×C73#12011yescharacteristic
97C2×C4×C73C22⋊Q811yescharacteristic
98C2×C4×C73C4⋊Q811yescharacteristic
99C2×C4×C73C2×D811yescharacteristic
100C2×C4×C73C2×Q1611yescharacteristic
101C2×C4×C73#13711yescharacteristic
102C2×Dic73—21no—
103C2×Dic73—21no—
104C2×Dic73—41no—
105C2×Dic73—41no—
106Dic146—162no—
107C8×C73C22⋊Q811yescharacteristic
108C8×C73C22⋊Q811yescharacteristic
109C8×C73C4⋊Q811yescharacteristic
110C8×C73C4⋊Q811yescharacteristic
111#10596831—22no—
112#10596831—42no—
113C22×C73—21no—
114C22×C73—21no—
115C22×C73#44711yescharacteristic
116C22×C73#47711yescharacteristic
117C22×C73#50111yescharacteristic
118C4×C73—82no—
119C4×C73—161no—
120C4×C73#39511yescharacteristic
121C4×C73#45111yescharacteristic
122C4×C73#47411yescharacteristic
123C4×C73#50011yescharacteristic
124Dic73—41no—
125Dic73—41no—
126Dic73—81no—
127Dic73—81no—
128#13456—731nomaximal, Sylow
129C2×C73—21no—
130C2×C73—21no—
131C2×C73#205711yescharacteristic
132C2×C73#219311yescharacteristic
133C2×C73#219411yescharacteristic
134#3164—731no—
135#2159—731no—
136C2×#367D7311yescharacteristic
137#1952—731no—
138#1387—731no—
139#1382—731no—
140#1379—731no—
141C73#1345611yescharacteristic, complemented, Sylow
142#544—1461no—
143C2×M32—731no—
144C22×C16C2×D7311yescharacteristic
145#501—1462no—
146#500—1461no—
147#499—1461no—
148C2×#108—731no—
149#428—1461no—
150#428—1461no—
151#427—731no—
152C2×#107C2×D7311yescharacteristic
153C2×#107C2×D7311yescharacteristic
154#405—731no—
155#367—22no—
156#367—1461no—
157#366—1461no—
158#343—1461no—
159C2×M16—731no—
160C2×M16—1461no—
161C22×C8C22×D7311yescharacteristic
162C4⋊Q8—2921no—
163C4⋊Q8—1462no—
164#125—2921no—
165#125—1462no—
166C22⋊Q8—2922no—
167#117—731no—
168#117—1461no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171M32—732no—
172M32—1461no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—1461no—
176C2×C16Dic14612yescomplemented
177#108—732no—
178#108—1461no—
179#107—21no—
180#107—21no—
181#107—21no—
182#107—2921no—
183#107—731no—
184#107—731no—
185#107—1462no—
186#107C22⋊D7312yes—
187#106—2921no—
188#106—1462no—
189#106—1461no—
190#97—1461no—
191C4×C8—1461no—
192C2×Q8—2922no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4D4×D7311yescharacteristic
196M16—2921no—
197M16—732no—
198M16—1462no—
199M16—1461no—
200C2×C8—2921no—
201C2×C8—1461no—
202C2×C8D4×D7311yescharacteristic
203C2×C8C2×C22⋊D7311yescharacteristic
204C2×C8Q8×D7311yescharacteristic
205C2×C8Q8×D7311yescharacteristic
206C2×C8C2×Dic14611yescharacteristic
207C2×C8#1192402811yescharacteristic
208C4⋊C4—22no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—2922no—
213C4⋊C4—2922no—
214C4⋊C4—2922no—
215C4⋊C4—731no—
216C4⋊C4—731no—
217C4⋊C4—1461no—
218C4⋊C4—1461no—
219C22⋊C4—2921no—
220C22⋊C4—1461no—
221C42—2921no—
222C42—1461no—
223C16—22no—
224C16—1462no—
225C23#1455661511yescharacteristic
226Q8—11682no—
227C2×C4—42no—
228C2×C4—81no—
229C2×C4—161no—
230C2×C4—161no—
231C2×C4—2921no—
232C2×C4—2921no—
233C2×C4—1461no—
234C2×C4—1461no—
235C2×C4#1455655911yescharacteristic
236C2×C4#1455656911yescharacteristic
237C2×C4#1455659011yescharacteristic
238C2×C4D8×D7311yescharacteristic
239C2×C4Q16×D7311yescharacteristic
240C2×C4#1455662311yescharacteristic
241C8—2922no—
242C8—1462no—
243C8#1455656911yescharacteristic
244C8#1455657011yescharacteristic
245C8#1455658911yescharacteristic
246C8#1455659011yescharacteristic
247C22—21no—
248C22—21no—
249C22#1798871911yescharacteristic
250C22#1798888111yescharacteristic
251C22#1798899511yescharacteristic
252C4—82no—
253C4—161no—
254C4—2921no—
255C4—2921no—
256C4—5841no—
257C4—5841no—
258C4#1798850711yescharacteristic
259C4#1798874111yescharacteristic
260C4#1798886611yescharacteristic
261C4#1798899111yescharacteristic
262C2—21no—
263C2—21no—
264C2#2261125711yescharacteristic
265C2#2261189411yescharacteristic
266C2#2261190111yescharacteristic
subgroupG/H
centerC22#17988719
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC2×#367×C73C2
socleC22×C73#447
prime radicalC23×C73AGL(1,ℤ/8)
hypercenterC2×#367D73
hypocenterC73#13456
LCSG ▷ C22×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#367

Complements

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

NH ≅ G/Ntype
35. #14556531Q8semidirect
38. #14556527Q8semidirect
53. C2×C16×C73Q8semidirect
141. C73#13456semidirect
176. C2×C16Dic146semidirect

Maximal subgroups

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

subgroupindex
#226204672
#226117172
#226107142
#226080392
#226080272
#226080192
C2×#367×C732
#1345673