Finite Group Databasebeta

← #28337503

Subgroups — #28337503

Order 13696.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16812134C212yesmaximal, complemented
2C2×#16812036C212yesmaximal, complemented
3#20292528C211yesmaximal, characteristic, complemented
4#20292396C211yesmaximal, characteristic, complemented
5#537×C107C211yesmaximal, characteristic, complemented
6C22×D428C2211yescharacteristic
7#16812134—24no—
8#16812134—24no—
9C2×#13133805—22no—
10C2×#13133803—22no—
11C2×#13133802C2212yes—
12C2×(C2×D107)⋊C4—22no—
13C2×C4×Dic107C2211yescharacteristic
14#16812036—24no—
15C2×#41×C107C2212yes—
16C22⋊Q8×C107—22no—
17#121×C107—22no—
18#121×C107—22no—
19C2×C4⋊C4×C107C2211yescharacteristic
20C23×D107—21no—
21C2×D428—22no—
22C2×D428—22no—
23C2×D428—21no—
24C2×D428D412yescomplemented
25C22×Dic107—21no—
26#13133805—28no—
27#13133803—28no—
28#13133802—22no—
29#13133802D414yescomplemented
30C2×C107⋊C8—22no—
31(C2×D107)⋊C4—44no—
32C4×Dic107—22no—
33#41×C107—22no—
34#41×C107—22no—
35#41×C107—22no—
36#41×C107—22no—
37C2×Q8×C107D412yes—
38C2×D4×C107—22no—
39C2×D4×C107—22no—
40C2×D4×C107D412yes—
41C22×C4×C107—22no—
42C22×C4×C107D412yes—
43C22×C4×C107C2311yescharacteristic
44C4⋊C4×C107—22no—
45C4⋊C4×C107—22no—
46C22⋊C4×C107—24no—
47C23×C107—22no—
48C23×C107C2×D411yescharacteristic
49C22×D107—22no—
50C22×D107—42no—
51C22×D107—42no—
52C22×D107—41no—
53C22×D107—41no—
54Q8×C107—22no—
55Q8×C107—22no—
56D4×C107—22no—
57D4×C107—22no—
58D4×C107—42no—
59D4×C107—42no—
60D4×C107—82no—
61C2×C4×C107—22no—
62C2×C4×C107—22no—
63C2×C4×C107—42no—
64C2×C4×C107—42no—
65C2×C4×C107—42no—
66C2×C4×C107C2×D411yescharacteristic
67C2×C4×C107C2×D412yes—
68C2×C4×C107C2×D412yes—
69C2×C4×C107#4111yescharacteristic
70C2×Dic107—22no—
71C2×Dic107—41no—
72C2×Dic107—41no—
73D428—24no—
74D428—42no—
75C107⋊C8—24no—
76C22×C107—21no—
77C22×C107—21no—
78C22×C107—42no—
79C22×C107—42no—
80C22×C107—42no—
81C22×C107#12011yescharacteristic
82C22×C107#12111yescharacteristic
83C22×C107#12711yescharacteristic
84C2×D107—44no—
85C2×D107—42no—
86C2×D107—82no—
87C2×D107—81no—
88C2×D107—81no—
89C4×C107—42no—
90C4×C107—42no—
91C4×C107#12012yes—
92C4×C107#12112yes—
93Dic107—42no—
94C2×C107—21no—
95C2×C107—21no—
96C2×C107—82no—
97C2×C107#39211yescharacteristic
98C2×C107#45311yescharacteristic
99C2×C107#45411yescharacteristic
100D107—82no—
101#2006—1071nomaximal, Sylow
102C107#200611yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—1072no—
104#537D10711yescharacteristic, complemented
105C2×#104—1072no—
106#390—1071no—
107#343—1071no—
108C2×#41C2×D10712yes—
109C22×D4—1071no—
110AGL(1,ℤ/8)—2144no—
111AGL(1,ℤ/8)—2144no—
112C2×SD16—2142no—
113C2×D8—2142no—
114C2×M16—1072no—
115C22⋊Q8—22no—
116#121—22no—
117#121—22no—
118C2×C4⋊C4C2×D10711yescharacteristic
119C2×C22⋊C4—2142no—
120C2×C42—1071no—
121#104—2144no—
122C24—2141no—
123#41—22no—
124#41—22no—
125#41—22no—
126#41—22no—
127C2×Q8C22⋊D10712yes—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—1072no—
131C2×D4—2142no—
132C2×D4—2142no—
133C2×D4—2141no—
134C2×D4C22⋊D10712yes—
135C22×C4—22no—
136C22×C4—2141no—
137C22×C4C22⋊D10712yes—
138C22×C4C22×D10711yescharacteristic
139SD16—2148no—
140D8—2148no—
141M16—1074no—
142M16—2142no—
143C2×C8—2142no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C22⋊C4—24no—
147C22⋊C4—4284no—
148C42—2142no—
149C23—22no—
150C23—4282no—
151C23—4282no—
152C23—4281no—
153C23—4281no—
154C23—2142no—
155C23C2×C22⋊D10711yescharacteristic
156Q8—22no—
157Q8—22no—
158D4—22no—
159D4—22no—
160D4—42no—
161D4—42no—
162D4—82no—
163D4—4282no—
164D4—2144no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—4281no—
171C2×C4—4281no—
172C2×C4—2142no—
173C2×C4D4×D10711yescharacteristic
174C2×C4D4×D10712yes—
175C2×C4C2×C22⋊D10712yes—
176C2×C4#1313381711yescharacteristic
177C8—2144no—
178C22—21no—
179C22—21no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—8562no—
184C22—8561no—
185C22—8561no—
186C22—4284no—
187C22—4282no—
188C22#1681207811yescharacteristic
189C22#1681208011yescharacteristic
190C22#1681210511yescharacteristic
191C4—42no—
192C4—42no—
193C4—4282no—
194C4#1681207712yes—
195C4#1681208112yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—8562no—
200C2#2029253511yescharacteristic
201C2#2029279511yescharacteristic
202C2#2029280011yescharacteristic
subgroupG/H
centerC22#16812078
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#537×C107C2
socleC22×C107#120
hypercenter#537D107
hypocenterC107#2006
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#16812134C2semidirect
2. C2×#16812036C2semidirect
3. #20292528C2semidirect
4. #20292396C2semidirect
5. #537×C107C2semidirect
24. C2×D428D4semidirect
29. #13133802D4semidirect
102. C107#2006semidirect
104. #537D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121342
C2×#168120362
#202925282
#202923962
#537×C1072
#2006107