Finite Group Databasebeta

← #11737815

Subgroups — #11737815

Order 896.

348 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 117 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#90404C211yesmaximal, characteristic
2#90261C211yesmaximal, characteristic, complemented
3C7×#460C211yesmaximal, characteristic
4#89938C212yesmaximal, complemented
5SD16×Dic7C212yesmaximal, complemented
6#89842C211yesmaximal, characteristic
7C4×#1144C211yesmaximal, characteristic, complemented
8#89623C211yesmaximal, characteristic, complemented
9#89602C211yesmaximal, characteristic, complemented
10#89540C212yesmaximal, complemented
11#89535C212yesmaximal, complemented
12C2×C7×SD16C2212yes—
13C7×#127C2211yescharacteristic
14C4×C7×Q8C2211yescharacteristic
15Q8×Dic7C2212yes—
16C2×#1144C2211yescharacteristic
17#6499—21no—
18#6499C2212yes—
19#6498—22no—
20D4×Dic7—21no—
21D4×Dic7C2212yes—
22#6453—22no—
23#6453C2212yes—
24#6432—21no—
25#6432C2211yescharacteristic
26C4×Dic14—21no—
27C4×Dic14C2211yescharacteristic
28C4×Dic14C2211yescharacteristic
29C7×C4⋊C8C2211yescharacteristic
30C7×#102C2212yes—
31#6409C2211yescharacteristic
32#6406C2212yes—
33#6406C2212yes—
34#6391C2212yes—
35#6390C2212yes—
36Dic7⋊C8C2212yes—
37C8×Dic7C2212yes—
38#6384C2211yescharacteristic
39#6381C2211yescharacteristic
40C4×C7⋊C8C2211yescharacteristic
41#3365—71nomaximal, Sylow
42C2×C7×Q8C2311yescharacteristic
43C2×C7×D4—21no—
44C2×C7×D4—21no—
45C2×C7×D4C2312yes—
46C22×Dic7—22no—
47C2×Dic14—21no—
48C2×Dic14D411yescharacteristic
49C2×Dic14C2311yescharacteristic
50C7×SD16—24no—
51C2×C7×C8C2312yes—
52C7×C4⋊C4—21no—
53C7×C4⋊C4C2311yescharacteristic
54C42×C7—21no—
55C42×C7C2311yescharacteristic
56C22⋊Dic7—22no—
57C22⋊Dic7—22no—
58C22⋊Dic7—22no—
59#1144—22no—
60C4⋊Dic7—21no—
61C4⋊Dic7—22no—
62C4⋊Dic7—22no—
63C4⋊Dic7D411yescharacteristic
64C4⋊Dic7C2312yes—
65C4⋊Dic7C2311yescharacteristic
66Dic7⋊C4—21no—
67Dic7⋊C4—22no—
68Dic7⋊C4D412yes—
69C4×Dic7—21no—
70C4×Dic7—22no—
71C4×Dic7C2312yes—
72C2×C7⋊C8C2311yescharacteristic
73C2×C7⋊C8C2311yescharacteristic
74#559—71no—
75#557—71no—
76#488—72no—
77#486—71no—
78#478—72no—
79#475—71no—
80#464—72no—
81#460D711yescharacteristic
82C8×Q8—71no—
83C4×Q16—71no—
84C4×SD16—72no—
85C23×C7—22no—
86C7×Q8—21no—
87C7×Q8#4112yes—
88C7×D4—22no—
89C7×D4—22no—
90C7×D4—42no—
91C7×D4—81no—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7C2×D411yescharacteristic
95C2×C4×C7C2×D411yescharacteristic
96C2×C4×C7C2411yescharacteristic
97C2×Dic7—22no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—82no—
101C2×Dic7C2×D412yes—
102C2×Dic7C2×D412yes—
103Dic14—22no—
104Dic14—21no—
105Dic14—21no—
106Dic14—42no—
107C7×C8—22no—
108C7⋊C8—21no—
109C7⋊C8—21no—
110C2×Q16—71no—
111C2×SD16C2×D712yes—
112C4⋊Q8—71no—
113C4⋊Q8—141no—
114#127C2×D711yescharacteristic
115#125—72no—
116#125—142no—
117#124—72no—
118#124—141no—
119#123—142no—
120C4×Q8—71no—
121C4×Q8—72no—
122C4×Q8—71no—
123C4×Q8—141no—
124C4×Q8C2×D711yescharacteristic
125C4×D4—72no—
126C4×D4—141no—
127#107—71no—
128#106—72no—
129C4⋊C8—72no—
130C4⋊C8C2×D711yescharacteristic
131#103—71no—
132#103—72no—
133#103—71no—
134#102—72no—
135#102—72no—
136#102C2×D712yes—
137C4×C8—72no—
138C4×C8—71no—
139C22×C7—42no—
140C22×C7—42no—
141C22×C7—42no—
142C22×C7C22×D411yescharacteristic
143C4×C7—22no—
144C4×C7—21no—
145C4×C7—41no—
146C4×C7#13512yes—
147C4×C7C2×#4111yescharacteristic
148C4×C72−1+411yescharacteristic
149Dic7—22no—
150Dic7—22no—
151Dic7—41no—
152Dic7—42no—
153Dic7—41no—
154C2×Q8—71no—
155C2×Q8—71no—
156C2×Q8—141no—
157C2×Q8C22×D711yescharacteristic
158C2×D4—21no—
159C2×D4—21no—
160C2×D4C22×D712yes—
161C22×C4—142no—
162Q16—142no—
163SD16—24no—
164C2×C8—71no—
165C2×C8—71no—
166C2×C8C22×D712yes—
167C4⋊C4—21no—
168C4⋊C4—72no—
169C4⋊C4—71no—
170C4⋊C4—72no—
171C4⋊C4—71no—
172C4⋊C4—142no—
173C4⋊C4—142no—
174C4⋊C4—142no—
175C4⋊C4—141no—
176C4⋊C4—141no—
177C4⋊C4C22×D711yescharacteristic
178C22⋊C4—142no—
179C22⋊C4—142no—
180C22⋊C4—142no—
181C42—21no—
182C42—72no—
183C42—142no—
184C42—141no—
185C42C22×D711yescharacteristic
186C2×C7—82no—
187C2×C7#54811yescharacteristic
188C2×C7C2×#13511yescharacteristic
189C2×C7#57711yescharacteristic
190C23—22no—
191Q8—21no—
192Q8—142no—
193Q8—141no—
194Q8—141no—
195Q8—282no—
196Q8#115912yes—
197D4—22no—
198D4—22no—
199D4—42no—
200D4—81no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—72no—
204C2×C4—72no—
205C2×C4—141no—
206C2×C4—141no—
207C2×C4—142no—
208C2×C4—562no—
209C2×C4D4×D711yescharacteristic
210C2×C4D4×D711yescharacteristic
211C2×C4C23×D711yescharacteristic
212C8—22no—
213C8—141no—
214C8—141no—
215C7#336511yescharacteristic, complemented, Sylow
216C22—42no—
217C22—42no—
218C22—42no—
219C22C2×D4×D711yescharacteristic
220C4—21no—
221C4—22no—
222C4—41no—
223C4—142no—
224C4—142no—
225C4—282no—
226C4—281no—
227C4—281no—
228C4#647912yes—
229C4C2×#115911yescharacteristic
230C4#655411yescharacteristic
231C2—82no—
232C2#9028911yescharacteristic
233C2C2×#647911yescharacteristic
234C2#9046611yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#460C2
socleC22×C7C22×D4
prime radicalC7×#127C22
hypercenter#460D7
hypocenterC7#3365
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #90261C2semidirect
4. #89938C2semidirect
5. SD16×Dic7C2semidirect
7. C4×#1144C2semidirect
8. #89623C2semidirect
9. #89602C2semidirect
10. #89540C2semidirect
11. #89535C2semidirect
215. C7#3365semidirect

Maximal subgroups

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

subgroupindex
#904042
#902612
C7×#4602
#899382
SD16×Dic72
#898422
C4×#11442
#896232
#896022
#895402
#895352
#33657