Finite Group Databasebeta

← #28338147

Subgroups — #28338147

Order 13696.

230 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 55 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#20293197C211yesmaximal, characteristic
2#101×D107C212yesmaximal, complemented
3#20292425C212yesmaximal, complemented
4#20292424C211yesmaximal, characteristic, complemented
5#355×C107C211yesmaximal, characteristic, complemented
6C2×#13133817C411yescharacteristic
7C2×Q8×D107C2211yescharacteristic
8M16×D107—22no—
9#16812109C411yescharacteristic
10#16812107C2211yescharacteristic
11#16812096C411yescharacteristic
12#16812091—22no—
13#16812086—22no—
14#16812066C411yescharacteristic
15#16812028C2212yes—
16#16812027—22no—
17#124×C107C2211yescharacteristic
18#101×C107C2212yes—
19#13133817—42no—
20C2×Dic214C2×C411yescharacteristic
21C2×Dic214C2×C411yescharacteristic
22C2×Dic214D412yes—
23Q8×D107—42no—
24C2×C22⋊D107—21no—
25C22×Dic107—21no—
26C2×C4×D107C2×C411yescharacteristic
27C2×C4×D107D412yes—
28#13133802—22no—
29#13133801—42no—
30C8×D107—42no—
31C4⋊Dic107—22no—
32Dic107⋊C4—22no—
33Dic107⋊C4—21no—
34C22⋊Dic107—22no—
35(C2×D107)⋊C4—21no—
36C4×Dic107C2×C411yescharacteristic
37C2×Q8×C107C2311yescharacteristic
38C2×D4×C107C2×C411yescharacteristic
39M16×C107—22no—
40C22⋊C4×C107—22no—
41C42×C107C2×C411yescharacteristic
42C23×C107—21no—
43C22×D107C22⋊C411yescharacteristic
44Q8×C107—41no—
45D4×C107—41no—
46C2×C4×C107—21no—
47C2×C4×C107C22×C411yescharacteristic
48C2×C4×C107C2×D412yes—
49C22⋊D107—42no—
50C2×Dic107—21no—
51C2×Dic107—21no—
52C2×Dic107—42no—
53C2×Dic107C22⋊C412yes—
54C2×Dic107C22⋊C411yescharacteristic
55C4×D107—22no—
56C4×D107—22no—
57C4×D107—22no—
58Dic214—41no—
59Dic214—42no—
60Dic214—41no—
61C8×C107—42no—
62C107⋊C8—42no—
63C22×C107—42no—
64C22×C107—41no—
65C22×C107C2×C22⋊C411yescharacteristic
66C2×D107—41no—
67C2×D107#9912yes—
68C4×C107—22no—
69C4×C107—21no—
70C4×C107—41no—
71Dic107—21no—
72Dic107—22no—
73Dic107—41no—
74Dic107—41no—
75C2×C107—21no—
76C2×C107—81no—
77C2×C107C2×#9911yescharacteristic
78D107—22no—
79#2133—1071nomaximal, Sylow
80C107#213311yescharacteristic, complemented, Sylow
81#536—1071no—
82C2×#101—1072no—
83#356—1072no—
84#355—1071no—
85#355D10711yescharacteristic, complemented
86C2×#41—1071no—
87C22×Q8—1071no—
88C2×M16—2142no—
89C4⋊Q8—1071no—
90C4⋊Q8—1071no—
91#124—1071no—
92#124C2×D10711yescharacteristic
93#123—2142no—
94C22⋊Q8—2142no—
95#117—1071no—
96#101—1072no—
97#101—2142no—
98#101C2×D10712yes—
99#41—4282no—
100C2×Q8—1071no—
101C2×Q8—1072no—
102C2×Q8—1071no—
103C2×Q8—4282no—
104C2×Q8C22×D10711yescharacteristic
105C2×D4—2141no—
106C2×D4C4×D10711yescharacteristic
107C22×C4—1071no—
108C22×C4—1072no—
109C22×C4—2141no—
110M16—22no—
111M16—4282no—
112M16—2142no—
113C2×C8—4282no—
114C4⋊C4—2141no—
115C4⋊C4—2142no—
116C4⋊C4—2142no—
117C22⋊C4—22no—
118C22⋊C4—2141no—
119C22⋊C4—2142no—
120C42—1071no—
121C42C4×D10711yescharacteristic
122C23—21no—
123C23—1071no—
124Q8—41no—
125Q8—4281no—
126Q8—4282no—
127Q8—4281no—
128D4—41no—
129D4—4282no—
130C2×C4—21no—
131C2×C4—1072no—
132C2×C4—1071no—
133C2×C4—4282no—
134C2×C4—2142no—
135C2×C4—2142no—
136C2×C4—2142no—
137C2×C4—2141no—
138C2×C4—2141no—
139C2×C4C2×C4×D10711yescharacteristic
140C2×C4D4×D10712yes—
141C8—42no—
142C8—4282no—
143C22—42no—
144C22—41no—
145C22—1072no—
146C22—4281no—
147C22C22⋊C4×D10711yescharacteristic
148C4—22no—
149C4—21no—
150C4—41no—
151C4—4281no—
152C4—4281no—
153C4—2141no—
154C4—2142no—
155C2—21no—
156C2—81no—
157C2—2142no—
158C2#99×D10711yescharacteristic
subgroupG/H
centerC2#99×D107
commutatorC2×C4×C107C22×C4
FrattiniC2×Q8C22×D107
Fitting#355×C107C2
socleC2×C107C2×#99
prime radicalC2×#13133817C4
cosocleC2×Q8×C107C23
hypercenter#355D107
hypocenterC107#2133
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #101×D107C2semidirect
3. #20292425C2semidirect
4. #20292424C2semidirect
5. #355×C107C2semidirect
80. C107#2133semidirect
85. #355D107semidirect

Maximal subgroups

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

subgroupindex
#202931972
#101×D1072
#202924252
#202924242
#355×C1072
#2133107