Finite Group Databasebeta

← C7×#10359 #13187575

Subgroups — C7×#10359 #13187575

Order 1792.

400 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2844C211yesmaximal, characteristic
2C7×#1993C212yesmaximal
3C7×#1960C211yesmaximal, characteristic
4C7×#1295C212yesmaximal
5C7×#1295C211yesmaximal, characteristic
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C7×#103—21no—
8C2×C7×#103—21no—
9C2×C7×#97C2212yes—
10C2×C7×#97C2211yescharacteristic
11C7×#400—22no—
12C7×#400—21no—
13C7×#395—22no—
14C7×#395—21no—
15C7×#382C2212yes—
16C7×#382C2211yescharacteristic
17C7×#340—22no—
18C7×#340—22no—
19#10359C711yesmaximal, characteristic, direct product factor, Sylow
20C22×C7×Q8—21no—
21C22×C7×Q8—21no—
22C22×C7×C8—22no—
23C22×C7×C8—21no—
24C7×C4⋊Q8—22no—
25C7×C4⋊Q8—21no—
26C7×C4⋊Q8Q812yescomplemented
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4D412yes—
31C2×C7×C4⋊C4D412yes—
32C2×C7×C4⋊C4D411yescharacteristic
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C42×C7C2311yescharacteristic
35C7×#103—42no—
36C7×#103—42no—
37C7×#97—22no—
38C7×#97—22no—
39C7×#97—22no—
40C7×#95—22no—
41C7×#95—21no—
42C7×#95—21no—
43C7×#95—22no—
44C7×#95—22no—
45C7×#95—22no—
46#2844C2×C711yescharacteristic
47#1993C2×C712yes—
48#1960C2×C711yescharacteristic
49#1295C2×C711yescharacteristic
50#1295C2×C712yes—
51C2×C7×Q8—22no—
52C2×C7×Q8—41no—
53C2×C7×Q8—41no—
54C2×C7×Q8—41no—
55C2×C7×Q8—41no—
56C2×C7×Q8—41no—
57C2×C7×Q8—81no—
58C2×C7×Q8—82no—
59C22×C4×C7—21no—
60C22×C4×C7—22no—
61C22×C4×C7—21no—
62C22×C4×C7—22no—
63C22×C4×C7—41no—
64C22×C4×C7C2×D412yes—
65C22×C4×C7C2×D411yescharacteristic
66C2×C7×C8—22no—
67C2×C7×C8—24no—
68C2×C7×C8—41no—
69C2×C7×C8—42no—
70C2×C7×C8—42no—
71C2×C7×C8—41no—
72C7×C4⋊C4—24no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4—41no—
75C7×C4⋊C4—42no—
76C7×C4⋊C4—42no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—81no—
79C7×C4⋊C4—82no—
80C42×C7C2×Q811yescharacteristic
81C42×C7#4112yes—
82C42×C7#4111yescharacteristic
83C2×C4⋊Q8C22×C711yescharacteristic
84C2×#103—21no—
85C2×#103—21no—
86C2×#97C22×C711yescharacteristic
87C2×#97C22×C712yes—
88#400—22no—
89#400—21no—
90#395—22no—
91#395—21no—
92#382C22×C711yescharacteristic
93#382C22×C712yes—
94#340—22no—
95#340—22no—
96C23×C7#12011yescharacteristic
97C7×Q8—42no—
98C7×Q8—42no—
99C7×Q8—81no—
100C7×Q8—81no—
101C7×Q8—84no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—42no—
107C2×C4×C7—41no—
108C2×C4×C7—41no—
109C2×C4×C7—42no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—42no—
113C2×C4×C7—42no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—42no—
117C2×C4×C7—81no—
118C2×C4×C7—81no—
119C2×C4×C7C22⋊Q812yes—
120C2×C4×C7C22⋊Q811yescharacteristic
121C2×C4×C7#12412yes—
122C2×C4×C7#12411yescharacteristic
123C7×C8—42no—
124C7×C8—44no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×C8—22no—
128C22×C8—21no—
129C4⋊Q8—22no—
130C4⋊Q8—21no—
131C4⋊Q8C7×Q812yescomplemented
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4C7×D412yes—
136C2×C4⋊C4C7×D412yes—
137C2×C4⋊C4C7×D411yescharacteristic
138C2×C4⋊C4C7×D411yescharacteristic
139C2×C42C23×C711yescharacteristic
140#103—42no—
141#103—42no—
142#97—22no—
143#97—22no—
144#97—22no—
145#95—22no—
146#95—21no—
147#95—21no—
148#95—22no—
149#95—22no—
150#95—22no—
151C22×C7#39311yescharacteristic
152C22×C7#45311yescharacteristic
153C22×C7#45411yescharacteristic
154C22×C7#45512yes—
155C22×C7#45612yes—
156C4×C7—21no—
157C4×C7—21no—
158C4×C7—22no—
159C4×C7—22no—
160C4×C7—81no—
161C4×C7—82no—
162C4×C7—81no—
163C4×C7—82no—
164C4×C7—82no—
165C2×Q8—22no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—81no—
172C2×Q8—82no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—41no—
178C22×C4C2×C7×D412yes—
179C22×C4C2×C7×D411yescharacteristic
180C2×C8—22no—
181C2×C8—24no—
182C2×C8—41no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—41no—
186C4⋊C4—24no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—81no—
193C4⋊C4—82no—
194C42C2×C7×Q811yescharacteristic
195C42C7×#4112yes—
196C42C7×#4111yescharacteristic
197C2×C7#165911yescharacteristic
198C2×C7#166011yescharacteristic
199C2×C7#166412yes—
200C2×C7#202711yescharacteristic
201C2×C7#202812yes—
202C23C7×#12011yescharacteristic
203Q8—42no—
204Q8—42no—
205Q8—81no—
206Q8—81no—
207Q8—84no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4C7×C22⋊Q812yes—
226C2×C4C7×C22⋊Q811yescharacteristic
227C2×C4C7×#12412yes—
228C2×C4C7×#12411yescharacteristic
229C8—42no—
230C8—44no—
231C7#1035911yescharacteristic, direct product factor, Sylow
232C22C7×#39311yescharacteristic
233C22C7×#45311yescharacteristic
234C22C7×#45411yescharacteristic
235C22C7×#45512yes—
236C22C7×#45612yes—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—82no—
242C4—81no—
243C4—82no—
244C4—81no—
245C4—82no—
246C2C7×#165911yescharacteristic
247C2C7×#166011yescharacteristic
248C2C7×#166412yes—
249C2C7×#202711yescharacteristic
250C2C7×#202812yes—
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
19. #10359C7direct (N × H)
26. C7×C4⋊Q8Q8semidirect
131. C4⋊Q8C7×Q8semidirect
231. C7#10359direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28442
C7×#19932
C7×#19602
C7×#12952
#103597