Finite Group Databasebeta

← #22588393

Subgroups — #22588393

Order 9088.

256 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C2×#14556136C211yesmaximal, characteristic
2C2×#14556063C211yesmaximal, characteristic, complemented
3#17986035C211yesmaximal, characteristic, complemented
4#17986017C211yesmaximal, characteristic
5#17986015C211yesmaximal, characteristic
6#17985771C211yesmaximal, characteristic, complemented
7#344×C71C211yesmaximal, characteristic, complemented
8C2×#11765865C2211yescharacteristic
9#14556136—21no—
10#14556136—21no—
11#14556136C412yescomplemented
12#14556136C412yescomplemented
13C2×#11765849C2211yescharacteristic
14C2×#11765848—21no—
15C22×C71⋊C8C2211yescharacteristic
16C2×C8×D71—21no—
17#14556087C2211yescharacteristic
18#14556063—21no—
19#14556063—21no—
20#14556061—21no—
21#14556059—21no—
22#14556056—22no—
23#14556054—22no—
24C2×M16×C71C2211yescharacteristic
25C2×M16×C71C2211yescharacteristic
26#117×C71C2211yescharacteristic
27#11765865—21no—
28#11765865—21no—
29#11765865C2×C412yes—
30#11765865D412yes—
31C2×Dic142Q811yescharacteristic
32C2×D284Q811yescharacteristic
33C2×C22⋊D71—21no—
34C2×C4×D71—21no—
35#11765849—21no—
36#11765849—21no—
37#11765848—22no—
38#11765848—22no—
39C2×C71⋊C8—21no—
40C2×C71⋊C8C2×C411yescharacteristic
41C2×C71⋊C8C2×C411yescharacteristic
42C2×C71⋊C8D411yescharacteristic
43C2×C71⋊C8D411yescharacteristic
44C2×C71⋊C8D411yescharacteristic
45C2×C71⋊C8D411yescharacteristic
46C8×D71—22no—
47C8×D71—22no—
48C4⋊Dic71—21no—
49C22⋊Dic71—21no—
50C4×Dic71—21no—
51C22×C4×C71C2311yescharacteristic
52M16×C71—21no—
53M16×C71—21no—
54M16×C71—21no—
55M16×C71C2×C412yes—
56C2×C8×C71—21no—
57C2×C8×C71—21no—
58C4⋊C4×C71—21no—
59C22⋊C4×C71—21no—
60C42×C71—21no—
61C23×C71#4111yescharacteristic
62C22×D71—21no—
63C2×C4×C71—41no—
64C2×C4×C71C22×C411yescharacteristic
65C2×C4×C71C2×D411yescharacteristic
66C2×C4×C71C2×D411yescharacteristic
67C2×C4×C71C2×D411yescharacteristic
68C2×C4×C71C2×Q811yescharacteristic
69C2×C4×C71#4111yescharacteristic
70C22⋊D71—22no—
71C22⋊D71—22no—
72C2×Dic71—21no—
73C2×Dic71—41no—
74D284—21no—
75D284C4⋊C412yes—
76C4×D71—22no—
77C4×D71—22no—
78Dic142—21no—
79Dic142C4⋊C412yes—
80C8×C71—22no—
81C8×C71—41no—
82C71⋊C8—41no—
83C71⋊C8C22⋊C411yescharacteristic
84C71⋊C8C22⋊C411yescharacteristic
85C71⋊C8C22⋊C411yescharacteristic
86C71⋊C8C22⋊C411yescharacteristic
87C22×C71—41no—
88C22×C71#11711yescharacteristic
89C22×C71#12111yescharacteristic
90C22×C71C22⋊Q811yescharacteristic
91C2×D71—22no—
92C2×D71—42no—
93C4×C71—81no—
94C4×C71C2×C22⋊C411yescharacteristic
95C4×C71C2×C4⋊C411yescharacteristic
96C4×C71#12111yescharacteristic
97C4×C71C22⋊Q811yescharacteristic
98Dic71—22no—
99Dic71—81no—
100C2×C71—21no—
101C2×C71—21no—
102C2×C71—21no—
103C2×C71#38011yescharacteristic
104D71—42no—
105#1814—711nomaximal, Sylow
106C71#181411yescharacteristic, complemented, Sylow
107C2×#131—711no—
108C2×#108—711no—
109#427—711no—
110#421—711no—
111#421—711no—
112#344—711no—
113#344D7111yescharacteristic, complemented
114C2×#41—711no—
115#131—712no—
116#131—712no—
117#131—1421no—
118#131—1421no—
119C2×M16—711no—
120C2×M16—1421no—
121C2×M16C2×D7111yescharacteristic
122C2×M16C2×D7111yescharacteristic
123C22×C8—711no—
124C22×C8—1421no—
125#117—711no—
126#117C2×D7111yescharacteristic
127#108—1421no—
128#108—1421no—
129#107—1421no—
130#106—1421no—
131#104—1422no—
132#104—1422no—
133#41—712no—
134#41—712no—
135#41—1421no—
136#41—1421no—
137C2×Q8—711no—
138C2×D4—711no—
139C2×D4—1421no—
140C22×C4—1421no—
141C22×C4C22×D7111yescharacteristic
142M16—21no—
143M16—21no—
144M16—21no—
145M16—1422no—
146M16—1422no—
147M16—1421no—
148M16—1421no—
149M16C4×D7112yes—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—711no—
153C2×C8—711no—
154C2×C8—711no—
155C2×C8—711no—
156C2×C8—711no—
157C2×C8—711no—
158C2×C8—1422no—
159C2×C8—1422no—
160C2×C8—1421no—
161C4⋊C4—21no—
162C4⋊C4—1421no—
163C22⋊C4—21no—
164C22⋊C4—1421no—
165C42—21no—
166C42—1421no—
167C23—1421no—
168C23#1176586511yescharacteristic
169Q8—712no—
170Q8—1421no—
171D4—712no—
172D4—1422no—
173D4—1422no—
174D4—1421no—
175C2×C4—41no—
176C2×C4—2841no—
177C2×C4—1421no—
178C2×C4—1422no—
179C2×C4—1422no—
180C2×C4C2×C4×D7111yescharacteristic
181C2×C4D4×D7111yescharacteristic
182C2×C4D4×D7111yescharacteristic
183C2×C4D4×D7111yescharacteristic
184C2×C4Q8×D7111yescharacteristic
185C2×C4#1176586511yescharacteristic
186C8—22no—
187C8—41no—
188C8—2841no—
189C8—711no—
190C8—711no—
191C8—711no—
192C8—711no—
193C22—41no—
194C22—2842no—
195C22—1422no—
196C22#1455608511yescharacteristic
197C22#1455610111yescharacteristic
198C22#1455610611yescharacteristic
199C4—81no—
200C4—5681no—
201C4—1422no—
202C4C22⋊C4×D7111yescharacteristic
203C4C4⋊C4×D7111yescharacteristic
204C4#1455610111yescharacteristic
205C4#1455610611yescharacteristic
206C2—21no—
207C2—21no—
208C2—21no—
209C2—2842no—
210C2#1798585811yescharacteristic
subgroupG/H
centerC4#14556101
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#344×C71C2
socleC2×C71#380
hypercenter#344D71
hypocenterC71#1814
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. C2×#14556063C2semidirect
3. #17986035C2semidirect
6. #17985771C2semidirect
7. #344×C71C2semidirect
11. #14556136C4semidirect
12. #14556136C4semidirect
106. C71#1814semidirect
113. #344D71semidirect