Finite Group Databasebeta

← #11741076

Subgroups — #11741076

Order 896.

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

normal subgroups only

This group has 340 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
1C7×#554C211yesmaximal, characteristic
2Q8×Dic14C211yesmaximal, characteristic, complemented
3#89863C211yesmaximal, characteristic, complemented
4#89850C211yesmaximal, characteristic, complemented
5#89837C212yesmaximal, complemented
6#89835C212yesmaximal, complemented
7#89796C211yesmaximal, characteristic, complemented
8#89790C212yesmaximal
9C4×#1142C211yesmaximal, characteristic
10#89780C211yesmaximal, characteristic
11#89779C212yesmaximal
12C7×C4⋊Q8C2211yescharacteristic
13C7×#125C2212yes—
14C7×C22⋊Q8—21no—
15C7×C22⋊Q8—22no—
16C4×C7×Q8C2211yescharacteristic
17C4×C7×D4C2212yes—
18C4×C7×D4C2211yescharacteristic
19C2×C7×C4⋊C4—21no—
20Q8×Dic7—21no—
21C2×#1142C2211yescharacteristic
22#6451—21no—
23#6451—22no—
24#6432C2212yes—
25#6432C2211yescharacteristic
26C4×Dic14C2211yescharacteristic
27C4×Dic14C2212yes—
28#6406C2211yescharacteristic
29#6406C2212yes—
30#6384C2212yes—
31#6384C2211yescharacteristic
32#6382C2212yes—
33#6382C2211yescharacteristic
34#6381C2212yes—
35#6381C2212yes—
36#6381C2212yes—
37C4⋊(C7⋊C8)C2211yescharacteristic
38C4⋊(C7⋊C8)C2212yes—
39#6377C2212yes—
40C4×C7⋊C8C2211yescharacteristic
41#3385—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×Q8D411yescharacteristic
44C2×C7×D4C2311yescharacteristic
45C22×C4×C7—21no—
46C22×C4×C7—22no—
47C2×Dic14—21no—
48C2×Dic14—22no—
49C2×Dic14C2311yescharacteristic
50C7×C4⋊C4—22no—
51C7×C4⋊C4—21no—
52C7×C4⋊C4—22no—
53C7×C4⋊C4D411yescharacteristic
54C7×C4⋊C4D412yes—
55C7×C4⋊C4C2311yescharacteristic
56C7×C4⋊C4C2312yes—
57C7×C22⋊C4—21no—
58C7×C22⋊C4—22no—
59C42×C7C2312yes—
60C42×C7C2311yescharacteristic
61#1142Q814yescomplemented
62C4⋊Dic7—22no—
63C4⋊Dic7—21no—
64C4⋊Dic7C2311yescharacteristic
65C4⋊Dic7C2312yes—
66Dic7⋊C4—21no—
67Dic7⋊C4—22no—
68C4×Dic7—22no—
69C4×Dic7—21no—
70C2×C7⋊C8C2311yescharacteristic
71C2×C7⋊C8C2312yes—
72C2×C7⋊C8C2311yescharacteristic
73Q82—71no—
74#554D711yescharacteristic
75#501—72no—
76#500—71no—
77#479—72no—
78#477—71no—
79#476—71no—
80#474—72no—
81#446—71no—
82#440—72no—
83C4×SD16—71no—
84C23×C7—21no—
85C7×Q8—22no—
86C7×Q8—81no—
87C7×D4—21no—
88C7×D4C2×Q812yes—
89C2×C4×C7—21no—
90C2×C4×C7—22no—
91C2×C4×C7—42no—
92C2×C4×C7—82no—
93C2×C4×C7C2×D412yes—
94C2×C4×C7C2×D411yescharacteristic
95C2×C4×C7C2×D412yes—
96C2×C4×C7C2×D411yescharacteristic
97C2×C4×C7C2411yescharacteristic
98C2×Dic7—21no—
99C2×Dic7—22no—
100C2×Dic7—21no—
101Dic14—21no—
102Dic14—82no—
103Dic14—81no—
104Dic14C2×Q812yes—
105C7⋊C8—21no—
106C7⋊C8—22no—
107C7⋊C8C2×Q812yes—
108C2×SD16—71no—
109C4⋊Q8—72no—
110C4⋊Q8—71no—
111C4⋊Q8—141no—
112C4⋊Q8—142no—
113C4⋊Q8C2×D711yescharacteristic
114#125C2×D712yes—
115C22⋊Q8—21no—
116C22⋊Q8—22no—
117C4×Q8—71no—
118C4×Q8—72no—
119C4×Q8—141no—
120C4×Q8C2×D711yescharacteristic
121C4×D4C2×D711yescharacteristic
122C4×D4C2×D712yes—
123C2×C4⋊C4—21no—
124#107—72no—
125#107—72no—
126#107—72no—
127#106—71no—
128#106—72no—
129C4⋊C8—72no—
130C4⋊C8—71no—
131#103—71no—
132#103—72no—
133#102—71no—
134#102—72no—
135#97—72no—
136C4×C8—71no—
137C22×C7—22no—
138C22×C7—41no—
139C22×C7—41no—
140C22×C7C22×D411yescharacteristic
141C4×C7—22no—
142C4×C7—22no—
143C4×C7—21no—
144C4×C7—41no—
145C4×C7—42no—
146C4×C7#13712yes—
147C4×C7C22×Q811yescharacteristic
148C4×C72−1+411yescharacteristic
149Dic7—22no—
150Dic7—41no—
151Dic7—42no—
152C2×Q8—21no—
153C2×Q8—71no—
154C2×Q8—141no—
155C2×Q8—142no—
156C2×Q8C22⋊D711yescharacteristic
157C2×D4C22×D711yescharacteristic
158C22×C4—21no—
159C22×C4—22no—
160SD16—74no—
161C2×C8—71no—
162C2×C8—71no—
163C2×C8—72no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—71no—
168C4⋊C4—72no—
169C4⋊C4—141no—
170C4⋊C4—142no—
171C4⋊C4—142no—
172C4⋊C4—141no—
173C4⋊C4C22⋊D711yescharacteristic
174C4⋊C4C22⋊D712yes—
175C4⋊C4C22×D711yescharacteristic
176C4⋊C4C22×D712yes—
177C22⋊C4—21no—
178C22⋊C4—22no—
179C42—142no—
180C42—141no—
181C42C22×D711yescharacteristic
182C42C22×D712yes—
183C2×C7—42no—
184C2×C7D4×Q811yescharacteristic
185C2×C7C2×#13711yescharacteristic
186C2×C7#57611yescharacteristic
187C23—21no—
188Q8—22no—
189Q8—72no—
190Q8—81no—
191Q8—141no—
192Q8—562no—
193Q8—561no—
194D4—21no—
195D4Q8×D712yes—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—42no—
199C2×C4—82no—
200C2×C4—141no—
201C2×C4—142no—
202C2×C4—141no—
203C2×C4C2×C22⋊D712yes—
204C2×C4C2×C22⋊D711yescharacteristic
205C2×C4C2×C22⋊D712yes—
206C2×C4C2×C22⋊D711yescharacteristic
207C2×C4C23×D711yescharacteristic
208C8—72no—
209C8—141no—
210C8—142no—
211C7#338511yescharacteristic, complemented, Sylow
212C22—22no—
213C22—41no—
214C22—41no—
215C22C22×C22⋊D711yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—41no—
220C4—42no—
221C4—142no—
222C4—281no—
223C4—282no—
224C4#651412yes—
225C4C2×Q8×D711yescharacteristic
226C4#655111yescharacteristic
227C2—42no—
228C2Q8×C22⋊D711yescharacteristic
229C2C2×#651411yescharacteristic
230C2#9052611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#554C2
socleC22×C7C22×D4
prime radicalC2×C7×D4C23
hypercenter#554D7
hypocenterC7#3385
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
2. Q8×Dic14C2semidirect
3. #89863C2semidirect
4. #89850C2semidirect
5. #89837C2semidirect
6. #89835C2semidirect
7. #89796C2semidirect
61. #1142Q8semidirect
211. C7#3385semidirect

Maximal subgroups

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

subgroupindex
C7×#5542
Q8×Dic142
#898632
#898502
#898372
#898352
#897962
#897902
C4×#11422
#897802
#897792
#33857