Finite Group Databasebeta

← #11740895

Subgroups — #11740895

Order 896.

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

normal subgroups only

This group has 346 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
1C7×#541C211yesmaximal, characteristic, complemented
2#90226C211yesmaximal, characteristic
3#89855C212yesmaximal, complemented
4#89838C212yesmaximal, complemented
5#89821C212yesmaximal, complemented
6#89820C212yesmaximal, complemented
7#89799C212yesmaximal, complemented
8#89794C212yesmaximal, complemented
9#89771C211yesmaximal, characteristic, complemented
10C7×C4⋊Q8C2211yescharacteristic
11C7×#126—22no—
12C7×#125—22no—
13C7×#125C2211yescharacteristic
14C7×C22⋊Q8C2212yes—
15C4×C7×Q8C2212yes—
16C7×#117C2211yescharacteristic
17C2×#1143C2212yes—
18C2×#1143C2212yes—
19#6493C2212yes—
20C2×#1136C2211yescharacteristic
21C22×C7⋊C8C2211yescharacteristic
22#6457—22no—
23#6447—22no—
24#6437C2211yescharacteristic
25C4×D28C2212yes—
26#6433C2211yescharacteristic
27#6409C2212yes—
28#6409C2212yes—
29#6383C2212yes—
30#6383C2212yes—
31#6382C2212yes—
32#6381C2212yes—
33C4⋊(C7⋊C8)C2212yes—
34C4⋊(C7⋊C8)C2212yes—
35#3260—71nomaximal, Sylow
36C2×C7×Q8C2312yes—
37C22×C4×C7C2311yescharacteristic
38C2×C22⋊D7—22no—
39C2×D28—22no—
40C2×D28C2312yes—
41C2×C4×D7—22no—
42C7×C4⋊C4—22no—
43C7×C4⋊C4—22no—
44C7×C4⋊C4—22no—
45C7×C4⋊C4—22no—
46C7×C4⋊C4D412yes—
47C7×C4⋊C4C2312yes—
48C7×C22⋊C4—22no—
49C7×C22⋊C4D412yes—
50C42×C7—22no—
51C42×C7C2311yescharacteristic
52C42×C7C2311yescharacteristic
53#1143—42no—
54#1143—42no—
55(C2×D7)⋊C4—22no—
56(C2×D7)⋊C4—22no—
57C4⋊Dic7C2312yes—
58Dic7⋊C4—22no—
59#1136—41no—
60C2×C7⋊C8—41no—
61C2×C7⋊C8C2312yes—
62C2×C7⋊C8C2312yes—
63#541D711yescharacteristic, complemented
64#540—71no—
65#479—72no—
66#478—72no—
67#468—72no—
68#465—72no—
69#461—72no—
70#460—72no—
71#424—71no—
72C23×C7C2×D411yescharacteristic
73C22×D7—22no—
74C7×Q8—22no—
75C7×Q8—22no—
76C2×C4×C7—21no—
77C2×C4×C7—21no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7C2×D412yes—
81C2×C4×C7C2×D411yescharacteristic
82C2×C4×C7C2×D412yes—
83C2×C4×C7C2411yescharacteristic
84C22⋊D7—82no—
85C2×Dic7—22no—
86D28—22no—
87D28—22no—
88D28—82no—
89C4×D7—82no—
90C7⋊C8—22no—
91C7⋊C8—22no—
92C2×SD16—72no—
93C2×SD16—72no—
94C2×M16—71no—
95C22×C8—71no—
96C4⋊Q8C2×D711yescharacteristic
97#127—71no—
98#126—22no—
99#125—22no—
100#125—71no—
101#125C2×D711yescharacteristic
102#123—142no—
103C22⋊Q8C2×D712yes—
104#121—72no—
105#121—142no—
106C4×Q8C2×D712yes—
107C4×D4—72no—
108#117C2×D711yescharacteristic
109#107—72no—
110#106—72no—
111C4⋊C8—72no—
112C4⋊C8—72no—
113#103—72no—
114#103—72no—
115#102—72no—
116#102—72no—
117C22×C7—21no—
118C22×C7—22no—
119C22×C7C22×D411yescharacteristic
120C2×D7—42no—
121C2×D7—42no—
122C2×D7—42no—
123C4×C7—21no—
124C4×C7—22no—
125C4×C7—22no—
126C4×C7—42no—
127C4×C7—42no—
128C4×C7H(5,𝔽2)11yescharacteristic
129C4×C72−1+411yescharacteristic
130Dic7—42no—
131C2×Q8C22×D712yes—
132C2×D4—72no—
133C2×D4—142no—
134C2×D4—142no—
135C22×C4—142no—
136C22×C4C22×D711yescharacteristic
137SD16—282no—
138SD16—282no—
139M16—281no—
140C2×C8—72no—
141C2×C8—72no—
142C2×C8—281no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—72no—
148C4⋊C4—142no—
149C4⋊C4C22⋊D712yes—
150C4⋊C4C22×D712yes—
151C22⋊C4—22no—
152C22⋊C4—142no—
153C22⋊C4—142no—
154C22⋊C4C22⋊D712yes—
155C42—22no—
156C42C22×D711yescharacteristic
157C42C22×D711yescharacteristic
158C2×C7—41no—
159C2×C7#53711yescharacteristic
160C2×C7#57712yes—
161D7—82no—
162C23—142no—
163C23C2×C22⋊D711yescharacteristic
164Q8—22no—
165Q8—22no—
166D4—142no—
167D4—142no—
168D4—562no—
169D4—562no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—142no—
175C2×C4—562no—
176C2×C4C2×C22⋊D712yes—
177C2×C4C2×C22⋊D712yes—
178C2×C4C2×C22⋊D711yescharacteristic
179C2×C4C23×D711yescharacteristic
180C8—142no—
181C8—142no—
182C7#326011yescharacteristic, complemented, Sylow
183C22—21no—
184C22—22no—
185C22—282no—
186C22—282no—
187C22—282no—
188C22C22×C22⋊D711yescharacteristic
189C4—21no—
190C4—22no—
191C4—22no—
192C4—42no—
193C4—42no—
194C4—282no—
195C4#655311yescharacteristic
196C4#655411yescharacteristic
197C2—41no—
198C2—562no—
199C2#9052411yescharacteristic
200C2#9052812yes—
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#541C2
socleC22×C7C22×D4
prime radical#90226C2
hypercenter#541D7
hypocenterC7#3260
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
1. C7×#541C2semidirect
3. #89855C2semidirect
4. #89838C2semidirect
5. #89821C2semidirect
6. #89820C2semidirect
7. #89799C2semidirect
8. #89794C2semidirect
9. #89771C2semidirect
63. #541D7semidirect
182. C7#3260semidirect

Maximal subgroups

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

subgroupindex
C7×#5412
#902262
#898552
#898382
#898212
#898202
#897992
#897942
#897712
#32607