Finite Group Databasebeta

← #356×D107 #28338140

Subgroups — #356×D107 #28338140

Order 13696.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C4⋊Q8×D107C211yesmaximal, characteristic
2#101×D107C212yesmaximal
3#20292426C211yesmaximal, characteristic, complemented
4#20292425C212yesmaximal, complemented
5#356×C107C211yesmaximal, characteristic, complemented
6C2×Q8×D107C411yescharacteristic
7C2×Q8×D107C2211yescharacteristic
8M16×D107—22no—
9#16812109C411yescharacteristic
10#16812108—22no—
11#16812107C411yescharacteristic
12#16812107C2211yescharacteristic
13C4⋊C4×D107—22no—
14C42×D107C411yescharacteristic
15#16812028C2212yes—
16#16812027—22no—
17C4⋊Q8×C107C2211yescharacteristic
18#101×C107C2212yes—
19C2×Dic214—21no—
20C2×Dic214C2×C411yescharacteristic
21C2×Dic214C2×C411yescharacteristic
22C2×Dic214D412yes—
23Q8×D107—41no—
24Q8×D107—41no—
25Q8×D107—41no—
26Q8×D107—41no—
27C2×C4×D107—21no—
28C2×C4×D107—21no—
29C2×C4×D107C2×C411yescharacteristic
30C2×C4×D107D412yes—
31#13133802—22no—
32#13133801—42no—
33C8×D107—42no—
34C4⋊Dic107—22no—
35Dic107⋊C4—22no—
36Dic107⋊C4—22no—
37C4×Dic107—21no—
38C4×Dic107C2×C411yescharacteristic
39C2×Q8×C107C2×C411yescharacteristic
40C2×Q8×C107C2311yescharacteristic
41M16×C107—22no—
42C4⋊C4×C107—22no—
43C42×C107C2×C411yescharacteristic
44C22×D107C22⋊C411yescharacteristic
45Q8×C107—41no—
46Q8×C107—41no—
47C2×C4×C107—21no—
48C2×C4×C107—21no—
49C2×C4×C107C22×C411yescharacteristic
50C2×C4×C107C2×D412yes—
51C2×Dic107—21no—
52C2×Dic107—21no—
53C2×Dic107C22⋊C411yescharacteristic
54C2×Dic107C22⋊C412yes—
55C4×D107—24no—
56C4×D107—22no—
57C4×D107—22no—
58C4×D107—21no—
59C4×D107—21no—
60C4×D107—41no—
61C4×D107—41no—
62Dic214—41no—
63Dic214—42no—
64Dic214—42no—
65Dic214—41no—
66C8×C107—42no—
67C107⋊C8—42no—
68C22×C107C2×C22⋊C411yescharacteristic
69C2×D107—22no—
70C2×D107#9911yescharacteristic
71C2×D107#9911yescharacteristic
72C4×C107—22no—
73C4×C107—22no—
74C4×C107—21no—
75C4×C107—41no—
76Dic107—22no—
77Dic107—21no—
78Dic107—22no—
79Dic107—41no—
80C2×C107—21no—
81C2×C107C2×#9911yescharacteristic
82D107—21no—
83D107#35612yesdirect product factor
84C2×#356—1071nomaximal, Sylow
85C107C2×#35611yescharacteristic, complemented, Sylow
86C2×C4⋊Q8—1071no—
87C2×#101—1072no—
88#356—1071no—
89#356—1072no—
90#356D10711yescharacteristic, direct product factor
91C22×Q8—1071no—
92C22×Q8—1071no—
93C2×M16—2142no—
94C4⋊Q8—1071no—
95C4⋊Q8—1071no—
96C4⋊Q8—1071no—
97C4⋊Q8—2142no—
98C4⋊Q8C2×D10711yescharacteristic
99C2×C4⋊C4—2142no—
100C2×C42—1071no—
101#101—1072no—
102#101—2142no—
103#101C2×D10712yes—
104C2×Q8—1071no—
105C2×Q8—1072no—
106C2×Q8—1071no—
107C2×Q8—4281no—
108C2×Q8—4281no—
109C2×Q8—4281no—
110C2×Q8—4281no—
111C2×Q8—2141no—
112C2×Q8C4×D10711yescharacteristic
113C2×Q8C22×D10711yescharacteristic
114C22×C4—1071no—
115C22×C4—1072no—
116C22×C4—2141no—
117C22×C4—2141no—
118M16—22no—
119M16—4282no—
120M16—2142no—
121C2×C8—4282no—
122C4⋊C4—22no—
123C4⋊C4—2142no—
124C4⋊C4—2142no—
125C4⋊C4—2142no—
126C42—1071no—
127C42—2141no—
128C42C4×D10711yescharacteristic
129C23—1071no—
130Q8—41no—
131Q8—41no—
132Q8—4281no—
133Q8—4282no—
134Q8—4281no—
135Q8—4282no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—1071no—
139C2×C4—1072no—
140C2×C4—4281no—
141C2×C4—4281no—
142C2×C4—2142no—
143C2×C4—2142no—
144C2×C4—2141no—
145C2×C4—2141no—
146C2×C4—2144no—
147C2×C4—2141no—
148C2×C4—2141no—
149C2×C4C2×C4×D10711yescharacteristic
150C2×C4D4×D10712yes—
151C8—42no—
152C8—4282no—
153C22—1071no—
154C22—1071no—
155C22—2142no—
156C22C22⋊C4×D10711yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—4281no—
162C4—2142no—
163C4—2141no—
164C4—2142no—
165C2—21no—
166C2—1072no—
167C2—2141no—
168C2#99×D10711yescharacteristic
subgroupG/H
centerC2#99×D107
commutatorC2×C4×C107C22×C4
FrattiniC2×Q8C22×D107
Fitting#356×C107C2
socleC2×C107C2×#99
prime radicalC22×D107C22⋊C4
cosocleC2×Q8×C107C23
hypercenter#356D107
hypocenterC107C2×#356
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #20292426C2semidirect
4. #20292425C2semidirect
5. #356×C107C2semidirect
83. D107#356direct (N × H)
85. C107C2×#356semidirect
90. #356D107direct (N × H)

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1072
#101×D1072
#202924262
#202924252
#356×C1072
C2×#356107