Finite Group Databasebeta

← C7×#13599 #13190815

Subgroups — C7×#13599 #13190815

Order 1792.

312 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3139C211yesmaximal, characteristic
2C2×C7×#356C212yesmaximal
3C7×#1413C214yesmaximal, complemented
4C7×#533C411yescharacteristic
5C2×C7×C4⋊Q8C2211yescharacteristic
6C7×#491C412yes—
7C7×#491C2212yes—
8C7×#487—22no—
9C7×#419—22no—
10C2×C7×#101—22no—
11C4×C7×M16C411yescharacteristic
12C7×#356—22no—
13C7×#356C2214yes—
14#13599C711yesmaximal, characteristic, direct product factor, Sylow
15C22×C7×Q8D412yes—
16C2×C7×M16—21no—
17C2×C7×M16—42no—
18C7×C4⋊Q8—21no—
19C7×C4⋊Q8—41no—
20C7×C4⋊Q8C2×C411yescharacteristic
21C7×C4⋊Q8D412yes—
22C7×C4⋊Q8C2311yescharacteristic
23C7×#125C2×C412yes—
24C7×C22⋊Q8—41no—
25C4×C7×Q8—41no—
26C7×#117—21no—
27C2×C7×C4⋊C4—41no—
28C2×C42×C7C2×C411yescharacteristic
29C7×#103—24no—
30C7×#103—24no—
31C7×#101—24no—
32C7×#101—42no—
33C7×#97C2×C412yes—
34C4×C7×C8—21no—
35#3139C2×C711yescharacteristic
36C2×#356C2×C712yes—
37#1413C2×C714yescomplemented
38C2×C7×Q8—22no—
39C2×C7×Q8—41no—
40C2×C7×Q8—42no—
41C2×C7×Q8—42no—
42C2×C7×Q8C22⋊C412yes—
43C2×C7×Q8C2×D412yes—
44C22×C4×C7—21no—
45C22×C4×C7—22no—
46C22×C4×C7C22⋊C411yescharacteristic
47C7×M16—42no—
48C7×M16—44no—
49C7×M16—82no—
50C2×C7×C8—22no—
51C2×C7×C8—82no—
52C7×C4⋊C4—22no—
53C7×C4⋊C4—41no—
54C7×C4⋊C4—42no—
55C7×C4⋊C4—41no—
56C7×C4⋊C4—42no—
57C7×C4⋊C4—41no—
58C7×C22⋊C4—41no—
59C42×C7—21no—
60C42×C7—41no—
61C42×C7C22⋊C411yescharacteristic
62C42×C7C22×C411yescharacteristic
63#533C4×C711yescharacteristic
64C2×C4⋊Q8C22×C711yescharacteristic
65#491C4×C712yes—
66#491C22×C712yes—
67#487—22no—
68#419—22no—
69C2×#101—22no—
70C4×M16C4×C711yescharacteristic
71#356—22no—
72#356C22×C714yes—
73C23×C7#9911yescharacteristic
74C7×Q8—42no—
75C7×Q8—42no—
76C7×Q8—82no—
77C7×Q8—82no—
78C2×C4×C7—22no—
79C2×C4×C7—21no—
80C2×C4×C7—21no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—42no—
84C2×C4×C7—42no—
85C2×C4×C7—42no—
86C2×C4×C7—42no—
87C2×C4×C7#9911yescharacteristic
88C2×C4×C7C2×C22⋊C411yescharacteristic
89C7×C8—42no—
90C7×C8—84no—
91C22×Q8C7×D412yes—
92C2×M16—21no—
93C2×M16—42no—
94C4⋊Q8—21no—
95C4⋊Q8—41no—
96C4⋊Q8C2×C4×C711yescharacteristic
97C4⋊Q8C7×D412yes—
98C4⋊Q8C23×C711yescharacteristic
99#125C2×C4×C712yes—
100C22⋊Q8—41no—
101C4×Q8—41no—
102#117—21no—
103C2×C4⋊C4—41no—
104C2×C42C2×C4×C711yescharacteristic
105#103—24no—
106#103—24no—
107#101—24no—
108#101—42no—
109#97C2×C4×C712yes—
110C4×C8—21no—
111C22×C7—41no—
112C22×C7#35412yes—
113C22×C7C2×#9911yescharacteristic
114C4×C7—22no—
115C4×C7—21no—
116C4×C7—21no—
117C4×C7—41no—
118C4×C7—42no—
119C4×C7—42no—
120C4×C7—82no—
121C2×Q8—22no—
122C2×Q8—41no—
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8C7×C22⋊C412yes—
126C2×Q8C2×C7×D412yes—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4C7×C22⋊C411yescharacteristic
130M16—42no—
131M16—44no—
132M16—82no—
133C2×C8—22no—
134C2×C8—82no—
135C4⋊C4—22no—
136C4⋊C4—41no—
137C4⋊C4—42no—
138C4⋊C4—41no—
139C4⋊C4—42no—
140C4⋊C4—41no—
141C22⋊C4—41no—
142C42—21no—
143C42—41no—
144C42C7×C22⋊C411yescharacteristic
145C42C22×C4×C711yescharacteristic
146C2×C7—21no—
147C2×C7—22no—
148C2×C7C2×#35411yescharacteristic
149C23C7×#9911yescharacteristic
150Q8—42no—
151Q8—42no—
152Q8—82no—
153Q8—82no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4C7×#9911yescharacteristic
164C2×C4C2×C7×C22⋊C411yescharacteristic
165C8—42no—
166C8—84no—
167C7#1359911yescharacteristic, direct product factor, Sylow
168C22—41no—
169C22C7×#35412yes—
170C22C2×C7×#9911yescharacteristic
171C4—22no—
172C4—21no—
173C4—21no—
174C4—41no—
175C4—42no—
176C4—42no—
177C4—82no—
178C2—21no—
179C2—22no—
180C2C2×C7×#35411yescharacteristic
subgroupG/H
center, socleC2×C7C2×#354
commutatorC42C22×C4×C7
Frattini, cosocleC4⋊Q8C23×C7
prime radicalC23×C7#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C23×C7 ◁ C22×C4×C7 ◁ C2×C7×C4⋊Q8 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1413C2semidirect
14. #13599C7direct (N × H)
37. #1413C2×C7semidirect
167. C7#13599direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31392
C2×C7×#3562
C7×#14132
#135997