Finite Group Databasebeta

← C7×#10037 #13187253

Subgroups — C7×#10037 #13187253

Order 1792.

476 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3162C211yesmaximal, characteristic
2C7×#3104C212yesmaximal
3C7×#1985C211yesmaximal, characteristic
4C7×#1481C212yesmaximal
5C7×#1275C211yesmaximal, characteristic
6C7×#533C2212yes—
7C2×C4×C7×Q8C2211yescharacteristic
8C7×#501—22no—
9C7×#501—22no—
10C7×#484—22no—
11C7×#477—22no—
12C7×#462—22no—
13C7×#451—22no—
14C4×C7×Q16—24no—
15C7×#431C2212yes—
16C7×#427—22no—
17C2×C7×C4⋊C8C2211yescharacteristic
18C2×C7×#97C2211yescharacteristic
19C7×#336—22no—
20C7×Q8⋊C8—24no—
21#10037C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×Q8D411yescharacteristic
23C2×C7×Q16—24no—
24C22×C7×C8—21no—
25C22×C7×C8D412yes—
26C7×C4⋊Q8D412yes—
27C7×C4⋊Q8D412yes—
28C7×#125—22no—
29C7×C22⋊Q8—22no—
30C7×C22⋊Q8—22no—
31C4×C7×Q8—22no—
32C4×C7×Q8—22no—
33C4×C7×Q8—22no—
34C4×C7×Q8—22no—
35C7×#117—22no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C42×C7—21no—
39C2×C42×C7C2311yescharacteristic
40C7×#107—24no—
41C7×#107—22no—
42C7×#107—22no—
43C7×#106—24no—
44C7×C4⋊C8—22no—
45C7×C4⋊C8—22no—
46C7×#103—24no—
47C7×#103—24no—
48C7×C22⋊C8—22no—
49C7×#97—22no—
50C4×C7×C8D414yes—
51#3162C2×C711yescharacteristic
52#3104C2×C712yes—
53#1985C2×C711yescharacteristic
54#1481C2×C712yes—
55#1275C2×C711yescharacteristic
56C2×C7×Q8—22no—
57C2×C7×Q8—22no—
58C2×C7×Q8—22no—
59C2×C7×Q8—42no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7C2×D411yescharacteristic
63C22×C4×C7C2×D411yescharacteristic
64C22×C4×C7#4111yescharacteristic
65C7×Q16—48no—
66C2×C7×C8—22no—
67C2×C7×C8—22no—
68C2×C7×C8—24no—
69C2×C7×C8—41no—
70C2×C7×C8—42no—
71C2×C7×C8Q1614yescomplemented
72C7×C4⋊C4—22no—
73C7×C4⋊C4—22no—
74C7×C4⋊C4—22no—
75C7×C4⋊C4—22no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—42no—
80C7×C22⋊C4—42no—
81C42×C7—41no—
82C42×C7—41no—
83C42×C7—42no—
84C42×C7C2×D412yes—
85C42×C7C2×D412yes—
86#533C22×C712yes—
87C2×C4×Q8C22×C711yescharacteristic
88#501—22no—
89#501—22no—
90#484—22no—
91#477—22no—
92#462—22no—
93#451—22no—
94C4×Q16—24no—
95#431C22×C712yes—
96#427—22no—
97C2×C4⋊C8C22×C711yescharacteristic
98C2×#97C22×C711yescharacteristic
99#336—22no—
100Q8⋊C8—24no—
101C23×C7#12111yescharacteristic
102C7×Q8—24no—
103C7×Q8—42no—
104C7×Q8—44no—
105C7×Q8—44no—
106C2×C4×C7—22no—
107C2×C4×C7—41no—
108C2×C4×C7—41no—
109C2×C4×C7—41no—
110C2×C4×C7—41no—
111C2×C4×C7—42no—
112C2×C4×C7—41no—
113C2×C4×C7—42no—
114C2×C4×C7—42no—
115C2×C4×C7#12012yes—
116C2×C4×C7#12011yescharacteristic
117C2×C4×C7#12112yes—
118C2×C4×C7#12711yescharacteristic
119C2×C4×C7C2×Q1612yes—
120C2×C4×C7#13712yes—
121C7×C8—24no—
122C7×C8—44no—
123C7×C8—81no—
124C22×Q8C7×D411yescharacteristic
125C2×Q16—24no—
126C22×C8—21no—
127C22×C8C7×D412yes—
128C4⋊Q8C7×D412yes—
129C4⋊Q8C7×D412yes—
130#125—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137#117—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4C7×D411yescharacteristic
140C2×C42—21no—
141C2×C42C23×C711yescharacteristic
142#107—22no—
143#107—22no—
144#107—24no—
145#106—24no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148#103—24no—
149#103—24no—
150C22⋊C8—22no—
151#97—22no—
152C4×C8C7×D414yes—
153C22×C7—21no—
154C22×C7—21no—
155C22×C7#39211yescharacteristic
156C22×C7#46212yes—
157C4×C7—22no—
158C4×C7—22no—
159C4×C7—42no—
160C4×C7—82no—
161C4×C7—81no—
162C4×C7—82no—
163C4×C7#45112yes—
164C4×C7#49411yescharacteristic
165C4×C7#49711yescharacteristic
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—42no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4C2×C7×D411yescharacteristic
173C22×C4C2×C7×D411yescharacteristic
174C22×C4C7×#4111yescharacteristic
175Q16—48no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—24no—
179C2×C8—41no—
180C2×C8—42no—
181C2×C8C7×Q1614yescomplemented
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C22⋊C4—42no—
191C42—41no—
192C42—41no—
193C42—42no—
194C42C2×C7×D412yes—
195C42C2×C7×D412yes—
196C2×C7—22no—
197C2×C7#201011yescharacteristic
198C2×C7#201611yescharacteristic
199C2×C7#201711yescharacteristic
200C23C7×#12111yescharacteristic
201Q8—24no—
202Q8—42no—
203Q8—44no—
204Q8—44no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4C7×#12012yes—
215C2×C4C7×#12011yescharacteristic
216C2×C4C7×#12112yes—
217C2×C4C7×#12711yescharacteristic
218C2×C4C2×C7×Q1612yes—
219C2×C4C7×#13712yes—
220C8—24no—
221C8—44no—
222C8—81no—
223C7#1003711yescharacteristic, direct product factor, Sylow
224C22—21no—
225C22—21no—
226C22C7×#39211yescharacteristic
227C22C7×#46212yes—
228C4—22no—
229C4—22no—
230C4—42no—
231C4—82no—
232C4—81no—
233C4—82no—
234C4C7×#45112yes—
235C4C7×#49411yescharacteristic
236C4C7×#49711yescharacteristic
237C2—22no—
238C2C7×#201011yescharacteristic
239C2C7×#201611yescharacteristic
240C2C7×#201711yescharacteristic
subgroupG/H
centerC2×C4×C7#127
commutator, Frattini, cosocleC2×C42C23×C7
socleC22×C7#392
prime radicalC23×C7#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
21. #10037C7direct (N × H)
71. C2×C7×C8Q16semidirect
181. C2×C8C7×Q16semidirect
223. C7#10037direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31622
C7×#31042
C7×#19852
C7×#14812
C7×#12752
#100377