Finite Group Databasebeta

← #28346266

Subgroups — #28346266

Order 13696.

376 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 153 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#20293184C211yesmaximal, characteristic
2#20292934C214yesmaximal, complemented
3#20292922C212yesmaximal, complemented
4#20292912C212yesmaximal, complemented
5#20292667C212yesmaximal
6#20292651C211yesmaximal, characteristic
7#20292636C212yesmaximal
8C2×C4×Q8×C107C211yesmaximal, characteristic
9C2×#13133802C2212yes—
10#16812109C2211yescharacteristic
11#16812109C2211yescharacteristic
12#16812101C2212yes—
13#16812088—21no—
14#16812088—21no—
15C4×Dic214—21no—
16C4×Dic214—21no—
17#16812068C2212yes—
18#16812042C2214yes—
19#16812040C2214yes—
20C4⋊(C107⋊C8)C2212yes—
21C4⋊(C107⋊C8)C2212yes—
22#16812033C2214yes—
23#16812033C2214yes—
24C22×Q8×C107C2211yescharacteristic
25C4×Q8×C107—21no—
26C4×Q8×C107—21no—
27C4×Q8×C107C2212yes—
28C4×Q8×C107C2212yes—
29C2×C4⋊C4×C107—21no—
30C2×C4⋊C4×C107C2211yescharacteristic
31C2×C42×C107—21no—
32C2×C42×C107C2211yescharacteristic
33C2×Dic214—21no—
34C2×Dic214—21no—
35#13133802—24no—
36C2×C107⋊C8C2314yes—
37C4⋊Dic107—21no—
38C4⋊Dic107—21no—
39C4⋊Dic107C2312yes—
40C4⋊Dic107C2312yes—
41Dic107⋊C4—22no—
42Dic107⋊C4—22no—
43C22⋊Dic107—22no—
44C4×Dic107—22no—
45C2×Q8×C107—21no—
46C2×Q8×C107—22no—
47C2×Q8×C107Q814yescomplemented
48C2×Q8×C107C2312yes—
49C22×C4×C107—21no—
50C22×C4×C107—21no—
51C22×C4×C107D411yescharacteristic
52C22×C4×C107D411yescharacteristic
53C22×C4×C107C2311yescharacteristic
54C4⋊C4×C107—22no—
55C4⋊C4×C107—22no—
56C4⋊C4×C107—21no—
57C4⋊C4×C107C2312yes—
58C42×C107—22no—
59C42×C107—22no—
60C42×C107D411yescharacteristic
61C42×C107D411yescharacteristic
62C42×C107C2311yescharacteristic
63C42×C107C2311yescharacteristic
64C23×C107C2×D411yescharacteristic
65Q8×C107—24no—
66Q8×C107—22no—
67Q8×C107C2×Q814yes—
68C2×C4×C107—24no—
69C2×C4×C107—22no—
70C2×C4×C107—22no—
71C2×C4×C107—22no—
72C2×C4×C107—41no—
73C2×C4×C107—42no—
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107C2×D411yescharacteristic
76C2×C4×C107C2×D411yescharacteristic
77C2×C4×C107C2×D411yescharacteristic
78C2×C4×C107C2×D411yescharacteristic
79C2×C4×C107C2×Q812yes—
80C2×C4×C107#4112yes—
81C2×C4×C107C2411yescharacteristic
82C2×Dic107—22no—
83C2×Dic107—22no—
84Dic214—42no—
85Dic214—42no—
86C107⋊C8—24no—
87C22×C107—21no—
88C22×C107—21no—
89C22×C107C22⋊Q812yes—
90C22×C107C22×D411yescharacteristic
91C4×C107—24no—
92C4×C107—21no—
93C4×C107—21no—
94C4×C107—21no—
95C4×C107—42no—
96C4×C107C22⋊Q812yes—
97C4×C107#13712yes—
98C4×C107C22×Q811yescharacteristic
99C4×C107C2×#4111yescharacteristic
100Dic107—42no—
101Dic107—42no—
102C2×C107—22no—
103C2×C107C2×C22⋊Q811yescharacteristic
104C2×C107C2×#13711yescharacteristic
105C2×C107#57511yescharacteristic
106#3078—1071nomaximal, Sylow
107C107#307811yescharacteristic, complemented, Sylow
108#533—1071no—
109C2×C4×Q8D10711yescharacteristic
110#479—1074no—
111#477—1072no—
112#475—1072no—
113#428—1072no—
114#423—1071no—
115#419—1072no—
116C22×Q8C2×D10711yescharacteristic
117C2×M16—1072no—
118C4⋊Q8—1071no—
119C4⋊Q8—1071no—
120#125—1072no—
121C22⋊Q8—2141no—
122C22⋊Q8—2141no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8—2141no—
126C4×Q8—2141no—
127C4×Q8C2×D10712yes—
128C4×Q8C2×D10712yes—
129#117—1072no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4C2×D10711yescharacteristic
132C2×C42—21no—
133C2×C42C2×D10711yescharacteristic
134#107—1074no—
135#106—1074no—
136C4⋊C8—1072no—
137C4⋊C8—1072no—
138#103—1074no—
139#103—1074no—
140C2×Q8—21no—
141C2×Q8—22no—
142C2×Q8—2141no—
143C2×Q8—2141no—
144C2×Q8Dic21414yescomplemented
145C2×Q8C22×D10712yes—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4C22⋊D10711yescharacteristic
149C22×C4C22⋊D10711yescharacteristic
150C22×C4C22×D10711yescharacteristic
151M16—2144no—
152C2×C8—1074no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—1072no—
157C4⋊C4—1072no—
158C4⋊C4—2142no—
159C4⋊C4—2142no—
160C4⋊C4—2141no—
161C4⋊C4—2141no—
162C4⋊C4C22×D10712yes—
163C22⋊C4—2142no—
164C42—22no—
165C42—22no—
166C42—2142no—
167C42C22⋊D10711yescharacteristic
168C42C22⋊D10711yescharacteristic
169C42C22×D10711yescharacteristic
170C42C22×D10711yescharacteristic
171C23C2×C22⋊D10711yescharacteristic
172Q8—24no—
173Q8—22no—
174Q8—4282no—
175Q8—4282no—
176Q8C2×Dic21414yes—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—2142no—
184C2×C4—2142no—
185C2×C4C2×C22⋊D10711yescharacteristic
186C2×C4C2×C22⋊D10711yescharacteristic
187C2×C4C2×C22⋊D10711yescharacteristic
188C2×C4C2×C22⋊D10711yescharacteristic
189C2×C4C2×C22⋊D10711yescharacteristic
190C2×C4C2×Dic21412yes—
191C2×C4#1313381812yes—
192C2×C4C23×D10711yescharacteristic
193C8—2144no—
194C22—21no—
195C22—21no—
196C22#1681208812yes—
197C22C22×C22⋊D10711yescharacteristic
198C4—24no—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—42no—
203C4—4282no—
204C4—4282no—
205C4#1681208812yes—
206C4#1681214012yes—
207C4C22×Dic21411yescharacteristic
208C4C2×#1313381811yescharacteristic
209C2—22no—
210C2C2×#1681208811yescharacteristic
211C2C2×#1681214011yescharacteristic
212C2#2029342511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×C4×Q8×C107C2
socleC22×C107C22×D4
prime radicalC23×C107C2×D4
hypercenterC2×C4×Q8D107
hypocenterC107#3078
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #20292934C2semidirect
3. #20292922C2semidirect
4. #20292912C2semidirect
47. C2×Q8×C107Q8semidirect
107. C107#3078semidirect
144. C2×Q8Dic214semidirect

Maximal subgroups

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

subgroupindex
#202931842
#202929342
#202929222
#202929122
#202926672
#202926512
#202926362
C2×C4×Q8×C1072
#3078107