Finite Group Databasebeta

← C7×#10237 #13187453

Subgroups — C7×#10237 #13187453

Order 1792.

560 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 154 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
1C2×C7×#477C212yesmaximal
2C7×#2582C211yesmaximal, characteristic
3C7×#1956C211yesmaximal, characteristic
4C2×C7×Q8⋊C8C212yesmaximal
5C7×#1296C211yesmaximal, characteristic
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C4×C7×Q8C2212yes—
8C7×#477—24no—
9C7×#477—24no—
10C2×C7×#107—22no—
11C2×C7×C4⋊C8C2212yes—
12C2×C7×#103—22no—
13C2×C7×#103—22no—
14C2×C4×C7×C8C2211yescharacteristic
15C7×#398—22no—
16C7×#391—22no—
17C7×#389—22no—
18C7×#384C2211yescharacteristic
19C7×Q8⋊C8—24no—
20#10237C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×Q8D412yes—
23C22×C7×C8—21no—
24C22×C7×C8—22no—
25C7×C4⋊Q8—22no—
26C7×C4⋊Q8—21no—
27C7×C4⋊Q8D412yes—
28C4×C7×Q8—22no—
29C4×C7×Q8—22no—
30C4×C7×Q8—22no—
31C4×C7×Q8—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4D412yes—
37C2×C7×C4⋊C4D412yes—
38C2×C42×C7—22no—
39C2×C42×C7C2311yescharacteristic
40C7×#107—28no—
41C7×C4⋊C8—22no—
42C7×C4⋊C8D414yes—
43C7×#103—28no—
44C7×#103—44no—
45C4×C7×C8—22no—
46C7×#95—24no—
47C7×#95—22no—
48C2×#477C2×C712yes—
49#2582C2×C711yescharacteristic
50#1956C2×C711yescharacteristic
51C2×Q8⋊C8C2×C712yes—
52#1296C2×C711yescharacteristic
53C2×C7×Q8—22no—
54C2×C7×Q8—22no—
55C2×C7×Q8—22no—
56C2×C7×Q8—22no—
57C2×C7×Q8—22no—
58C2×C7×Q8—22no—
59C2×C7×Q8—42no—
60C2×C7×Q8—42no—
61C2×C7×Q8—41no—
62C2×C7×Q8Q1614yes—
63C22×C4×C7—22no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—41no—
67C22×C4×C7C2×D411yescharacteristic
68C22×C4×C7C2×D412yes—
69C2×C7×C8—22no—
70C2×C7×C8—24no—
71C2×C7×C8—41no—
72C2×C7×C8—41no—
73C2×C7×C8—42no—
74C2×C7×C8—42no—
75C7×C4⋊C4—24no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—22no—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—42no—
80C7×C4⋊C4—42no—
81C7×C4⋊C4—42no—
82C7×C4⋊C4—42no—
83C7×C4⋊C4—42no—
84C7×C4⋊C4—82no—
85C42×C7—42no—
86C42×C7—42no—
87C42×C7C2×D411yescharacteristic
88C42×C7C2×D412yes—
89C42×C7#4111yescharacteristic
90C2×C4⋊Q8C22×C711yescharacteristic
91C2×C4×Q8C22×C712yes—
92#477—24no—
93#477—24no—
94C2×#107—22no—
95C2×C4⋊C8C22×C712yes—
96C2×#103—22no—
97C2×#103—22no—
98C2×C4×C8C22×C711yescharacteristic
99#398—22no—
100#391—22no—
101#389—22no—
102#384C22×C711yescharacteristic
103Q8⋊C8—24no—
104C23×C7#12011yescharacteristic
105C7×Q8—24no—
106C7×Q8—24no—
107C7×Q8—44no—
108C7×Q8—42no—
109C7×Q8—42no—
110C7×Q8—82no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—24no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—42no—
117C2×C4×C7—42no—
118C2×C4×C7—44no—
119C2×C4×C7—42no—
120C2×C4×C7—42no—
121C2×C4×C7—42no—
122C2×C4×C7—81no—
123C2×C4×C7—81no—
124C2×C4×C7#12012yes—
125C2×C4×C7#12111yescharacteristic
126C2×C4×C7#12112yes—
127C2×C4×C7#12711yescharacteristic
128C2×C4×C7C2×Q1612yes—
129C2×C4×C7#13512yes—
130C2×C4×C7#13712yes—
131C2×C4×C7#13712yes—
132C7×C8—42no—
133C7×C8—44no—
134C22×Q8—21no—
135C22×Q8C7×D412yes—
136C22×C8—21no—
137C22×C8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—21no—
140C4⋊Q8C7×D412yes—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4C7×D412yes—
150C2×C4⋊C4C7×D412yes—
151C2×C42—22no—
152C2×C42C23×C711yescharacteristic
153#107—28no—
154C4⋊C8—22no—
155C4⋊C8C7×D414yes—
156#103—28no—
157#103—44no—
158C4×C8—22no—
159#95—24no—
160#95—22no—
161C22×C7#39211yescharacteristic
162C22×C7#45112yes—
163C22×C7#45212yes—
164C22×C7#45311yescharacteristic
165C22×C7#45411yescharacteristic
166C4×C7—21no—
167C4×C7—21no—
168C4×C7—44no—
169C4×C7—82no—
170C4×C7—82no—
171C4×C7—82no—
172C4×C7#45112yes—
173C4×C7#45212yes—
174C4×C7#46712yes—
175C4×C7#47012yes—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—41no—
185C2×Q8C7×Q1614yes—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—41no—
190C22×C4C2×C7×D411yescharacteristic
191C22×C4C2×C7×D412yes—
192C2×C8—22no—
193C2×C8—24no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—42no—
197C2×C8—42no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—82no—
208C42—42no—
209C42—42no—
210C42C2×C7×D411yescharacteristic
211C42C2×C7×D412yes—
212C42C7×#4111yescharacteristic
213C2×C7#163011yescharacteristic
214C2×C7#163311yescharacteristic
215C2×C7#163612yes—
216C2×C7#200112yes—
217C2×C7#200611yescharacteristic
218C23C7×#12011yescharacteristic
219Q8—24no—
220Q8—24no—
221Q8—44no—
222Q8—42no—
223Q8—42no—
224Q8—82no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4C7×#12012yes—
239C2×C4C7×#12111yescharacteristic
240C2×C4C7×#12112yes—
241C2×C4C7×#12711yescharacteristic
242C2×C4C2×C7×Q1612yes—
243C2×C4C7×#13512yes—
244C2×C4C7×#13712yes—
245C2×C4C7×#13712yes—
246C8—42no—
247C8—44no—
248C7#1023711yescharacteristic, direct product factor, Sylow
249C22C7×#39211yescharacteristic
250C22C7×#45112yes—
251C22C7×#45212yes—
252C22C7×#45311yescharacteristic
253C22C7×#45411yescharacteristic
254C4—21no—
255C4—21no—
256C4—44no—
257C4—82no—
258C4—82no—
259C4—82no—
260C4C7×#45112yes—
261C4C7×#45212yes—
262C4C7×#46712yes—
263C4C7×#47012yes—
264C2C7×#163011yescharacteristic
265C2C7×#163311yescharacteristic
266C2C7×#163612yes—
267C2C7×#200112yes—
268C2C7×#200611yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C7#120
commutator, Frattini, cosocleC2×C42C23×C7
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
20. #10237C7direct (N × H)
248. C7#10237direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4772
C7×#25822
C7×#19562
C2×C7×Q8⋊C82
C7×#12962
#102377