Finite Group Databasebeta

← #22601272

Subgroups — #22601272

Order 9088.

326 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 133 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
1Q8×Dic142C211yesmaximal, characteristic
2#17986409C212yesmaximal
3#17986395C211yesmaximal, characteristic
4#17986303C212yesmaximal
5#17986294C211yesmaximal, characteristic
6#17986284C212yesmaximal
7#17986281C211yesmaximal, characteristic
8#17986130C211yesmaximal, characteristic
9#17986104C212yesmaximal
10C4×#11765855C211yesmaximal, characteristic
11#557×C71C211yesmaximal, characteristic
12C2×#11765855C2211yescharacteristic
13#14556128C2211yescharacteristic
14#14556128C2212yes—
15#14556127—21no—
16#14556127—22no—
17Q8×Dic71—21no—
18C4×Dic142C2211yescharacteristic
19C4×Dic142C2212yes—
20#14556061C2211yescharacteristic
21#14556061C2212yes—
22#14556059C2212yes—
23#14556059C2212yes—
24#14556059C2212yes—
25C4⋊(C71⋊C8)C2211yescharacteristic
26C4⋊(C71⋊C8)C2212yes—
27#14556052C2211yescharacteristic
28#14556052C2212yes—
29#14556051C2211yescharacteristic
30#14556051C2212yes—
31#14556039C2212yes—
32C4×C71⋊C8C2211yescharacteristic
33C4⋊Q8×C71—21no—
34C4⋊Q8×C71C2211yescharacteristic
35#125×C71—22no—
36#125×C71C2212yes—
37C4×Q8×C71—21no—
38C4×Q8×C71C2211yescharacteristic
39C4×Q8×C71C2212yes—
40C4×Q8×C71C2211yescharacteristic
41C2×Dic142—22no—
42C2×Dic142—21no—
43C2×Dic142C2311yescharacteristic
44#11765855Q814yescomplemented
45C2×C71⋊C8C2311yescharacteristic
46C2×C71⋊C8C2312yes—
47C2×C71⋊C8C2311yescharacteristic
48C4⋊Dic71—22no—
49C4⋊Dic71—21no—
50C4⋊Dic71C2311yescharacteristic
51C4⋊Dic71C2312yes—
52Dic71⋊C4—22no—
53Dic71⋊C4—21no—
54C4×Dic71—22no—
55C4×Dic71—21no—
56C2×Q8×C71—21no—
57C2×Q8×C71D411yescharacteristic
58C2×Q8×C71C2311yescharacteristic
59C4⋊C4×C71—22no—
60C4⋊C4×C71—21no—
61C4⋊C4×C71—22no—
62C4⋊C4×C71—21no—
63C4⋊C4×C71—22no—
64C4⋊C4×C71D411yescharacteristic
65C4⋊C4×C71D412yes—
66C4⋊C4×C71C2311yescharacteristic
67C4⋊C4×C71C2312yes—
68C42×C71—22no—
69C42×C71—21no—
70C42×C71C2312yes—
71C42×C71C2311yescharacteristic
72Q8×C71—22no—
73Q8×C71—21no—
74Q8×C71—42no—
75Q8×C71C2×Q812yes—
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—22no—
79C2×C4×C71C2×D412yes—
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2×D412yes—
82C2×C4×C71C2×D411yescharacteristic
83C2×C4×C71C2411yescharacteristic
84C2×Dic71—21no—
85C2×Dic71—22no—
86C2×Dic71—21no—
87Dic142—21no—
88Dic142—82no—
89Dic142—81no—
90Dic142C2×Q812yes—
91C71⋊C8—21no—
92C71⋊C8—22no—
93C71⋊C8C2×Q812yes—
94C22×C71C22×D411yescharacteristic
95C4×C71—22no—
96C4×C71—22no—
97C4×C71—22no—
98C4×C71—21no—
99C4×C71—42no—
100C4×C71—41no—
101C4×C71#13712yes—
102C4×C71C22×Q811yescharacteristic
103C4×C712−1+411yescharacteristic
104Dic71—22no—
105Dic71—41no—
106Dic71—42no—
107C2×C71D4×Q811yescharacteristic
108C2×C71C2×#13711yescharacteristic
109C2×C71#57711yescharacteristic
110#3387—711nomaximal, Sylow
111C71#338711yescharacteristic, complemented, Sylow
112Q82—711no—
113#557D7111yescharacteristic
114#501—712no—
115#498—711no—
116#479—712no—
117#477—711no—
118#475—711no—
119#475—712no—
120#446—711no—
121#441—712no—
122C4×Q16—711no—
123C2×Q16—711no—
124C4⋊Q8—21no—
125C4⋊Q8—712no—
126C4⋊Q8—711no—
127C4⋊Q8—1421no—
128C4⋊Q8—1422no—
129C4⋊Q8C2×D7111yescharacteristic
130#125—22no—
131#125C2×D7112yes—
132C4×Q8—21no—
133C4×Q8—711no—
134C4×Q8—712no—
135C4×Q8—1421no—
136C4×Q8C2×D7111yescharacteristic
137C4×Q8C2×D7112yes—
138C4×Q8C2×D7111yescharacteristic
139#107—711no—
140#107—712no—
141#106—712no—
142#106—712no—
143#106—712no—
144C4⋊C8—711no—
145C4⋊C8—712no—
146#103—711no—
147#103—712no—
148#103—711no—
149#103—712no—
150#97—712no—
151C4×C8—711no—
152C2×Q8—21no—
153C2×Q8—711no—
154C2×Q8—1422no—
155C2×Q8—1421no—
156C2×Q8C22⋊D7111yescharacteristic
157C2×Q8C22×D7111yescharacteristic
158Q16—714no—
159C2×C8—711no—
160C2×C8—711no—
161C2×C8—712no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—711no—
168C4⋊C4—712no—
169C4⋊C4—1422no—
170C4⋊C4—1421no—
171C4⋊C4—1422no—
172C4⋊C4—1421no—
173C4⋊C4C22⋊D7111yescharacteristic
174C4⋊C4C22⋊D7112yes—
175C4⋊C4C22×D7112yes—
176C4⋊C4C22×D7111yescharacteristic
177C42—22no—
178C42—21no—
179C42—1422no—
180C42—1421no—
181C42C22×D7111yescharacteristic
182C42C22×D7112yes—
183Q8—21no—
184Q8—22no—
185Q8—42no—
186Q8—5682no—
187Q8—5681no—
188Q8—712no—
189Q8—1421no—
190Q8Q8×D7112yes—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—1422no—
195C2×C4—1421no—
196C2×C4—1421no—
197C2×C4C2×C22⋊D7112yes—
198C2×C4C2×C22⋊D7112yes—
199C2×C4C2×C22⋊D7111yescharacteristic
200C2×C4C2×C22⋊D7111yescharacteristic
201C2×C4C23×D7111yescharacteristic
202C8—712no—
203C8—1421no—
204C8—1422no—
205C22C22×C22⋊D7111yescharacteristic
206C4—22no—
207C4—22no—
208C4—22no—
209C4—21no—
210C4—42no—
211C4—41no—
212C4—2842no—
213C4—2841no—
214C4—1422no—
215C4#1455615812yes—
216C4C2×Q8×D7111yescharacteristic
217C4#1455617611yescharacteristic
218C2Q8×C22⋊D7111yescharacteristic
219C2C2×#1455615811yescharacteristic
220C2#1798680711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#557×C71C2
socleC22×C71C22×D4
hypercenter#557D71
hypocenterC71#3387
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
44. #11765855Q8semidirect
111. C71#3387semidirect