Finite Group Databasebeta

← C7×#12288 #13189504

Subgroups — C7×#12288 #13189504

Order 1792.

354 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3245C211yesmaximal, characteristic
2C7×#2232C212yesmaximal, complemented
3C7×#2086C211yesmaximal, characteristic, complemented
4C7×#1347C212yesmaximal, complemented
5C7×#1322C211yesmaximal, characteristic, complemented
6C7×#532—21no—
7C7×#474—22no—
8C7×#466—21no—
9C7×#466C2212yes—
10C7×#459—22no—
11C2×C7×#107C2211yescharacteristic
12C2×C7×C4⋊C8C2211yescharacteristic
13C7×#389—21no—
14C7×#366—22no—
15C7×#357—22no—
16C7×C22⋊C16C2212yes—
17C7×#340—21no—
18C7×#340C2211yescharacteristic
19#12288C711yesmaximal, characteristic, direct product factor, Sylow
20C2×C7×#41—41no—
21C2×C7×D8—21no—
22C2×C7×D8D412yes—
23C22×C7×C8D411yescharacteristic
24C22×C7×C8C2311yescharacteristic
25C7×C4⋊Q8—21no—
26C7×#127—21no—
27C7×#124—41no—
28C7×#121—22no—
29C7×#121—41no—
30C4×C7×D4—22no—
31C4×C7×D4—41no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—41no—
35C2×C42×C7D411yescharacteristic
36C2×C7×C16—22no—
37C7×#107—21no—
38C7×#107—22no—
39C7×#107D412yes—
40C7×C4⋊C8—22no—
41C7×#102—22no—
42C7×#102—22no—
43C7×#95—41no—
44#3245C2×C711yescharacteristic
45#2232C2×C712yescomplemented
46#2086C2×C711yescharacteristic, complemented
47#1347C2×C712yescomplemented
48#1322C2×C711yescharacteristic, complemented
49C7×#41—82no—
50C7×#41—82no—
51C2×C7×Q8—41no—
52C2×C7×D4—22no—
53C2×C7×D4—41no—
54C2×C7×D4—41no—
55C2×C7×D4—42no—
56C22×C4×C7—42no—
57C22×C4×C7—41no—
58C22×C4×C7—41no—
59C22×C4×C7D811yescharacteristic
60C22×C4×C7D811yescharacteristic
61C22×C4×C7C2×D411yescharacteristic
62C7×D8—42no—
63C7×D8—81no—
64C2×C7×C8—21no—
65C2×C7×C8—41no—
66C2×C7×C8—41no—
67C2×C7×C8C2×D412yes—
68C7×C4⋊C4—22no—
69C7×C4⋊C4—22no—
70C7×C4⋊C4—41no—
71C7×C4⋊C4—41no—
72C7×C4⋊C4—41no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4—82no—
75C7×C22⋊C4—42no—
76C7×C22⋊C4—42no—
77C42×C7—21no—
78C42×C7—21no—
79C7×C16—42no—
80#532—21no—
81#474—22no—
82#466—21no—
83#466C22×C712yes—
84#459—22no—
85C2×#107C22×C711yescharacteristic
86C2×C4⋊C8C22×C711yescharacteristic
87#389—21no—
88#366—22no—
89#357—22no—
90C22⋊C16C22×C712yes—
91#340—21no—
92#340C22×C711yescharacteristic
93C23×C7—42no—
94C23×C7C2×D811yescharacteristic
95C7×Q8—82no—
96C7×D4—42no—
97C7×D4—42no—
98C7×D4—82no—
99C7×D4—82no—
100C7×D4—82no—
101C7×D4—82no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—41no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7—84no—
111C2×C4×C7—81no—
112C2×C4×C7—81no—
113C2×C4×C7—81no—
114C2×C4×C7—81no—
115C2×C4×C7—81no—
116C2×C4×C7D1612yes—
117C2×C4×C7#12011yescharacteristic
118C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
119C7×C8—22no—
120C7×C8—41no—
121C2×#41—41no—
122C2×D8—21no—
123C2×D8C7×D412yes—
124C22×C8C7×D411yescharacteristic
125C22×C8C23×C711yescharacteristic
126C4⋊Q8—21no—
127#127—21no—
128#124—41no—
129#121—22no—
130#121—41no—
131C4×D4—22no—
132C4×D4—41no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—41no—
136C2×C42C7×D411yescharacteristic
137C2×C16—22no—
138#107—21no—
139#107—22no—
140#107C7×D412yes—
141C4⋊C8—22no—
142#102—22no—
143#102—22no—
144#95—41no—
145C22×C7—21no—
146C22×C7—21no—
147C22×C7—82no—
148C22×C7—82no—
149C22×C7—82no—
150C22×C7#44711yescharacteristic
151C22×C7C2×D1611yescharacteristic
152C22×C7#50911yescharacteristic
153C4×C7—21no—
154C4×C7—22no—
155C4×C7—41no—
156C4×C7—82no—
157C4×C7—82no—
158C4×C7—161no—
159C4×C7#45611yescharacteristic
160C4×C7#45811yescharacteristic
161#41—82no—
162#41—82no—
163C2×Q8—41no—
164C2×D4—22no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—42no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4C7×D811yescharacteristic
172C22×C4C7×D811yescharacteristic
173C22×C4C2×C7×D411yescharacteristic
174D8—42no—
175D8—81no—
176C2×C8—21no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8C2×C7×D412yes—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—82no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C42—21no—
190C42—21no—
191C16—42no—
192C2×C7—21no—
193C2×C7—21no—
194C2×C7—162no—
195C2×C7#161011yescharacteristic
196C2×C7#218411yescharacteristic
197C2×C7#219111yescharacteristic
198C23—42no—
199C23C2×C7×D811yescharacteristic
200Q8—82no—
201D4—42no—
202D4—42no—
203D4—82no—
204D4—82no—
205D4—82no—
206D4—82no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—84no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4C7×D1612yes—
222C2×C4C7×#12011yescharacteristic
223C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
224C8—22no—
225C8—41no—
226C7#1228811yescharacteristic, direct product factor, Sylow
227C22—21no—
228C22—21no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22C7×#44711yescharacteristic
233C22C2×C7×D1611yescharacteristic
234C22C7×#50911yescharacteristic
235C4—21no—
236C4—22no—
237C4—41no—
238C4—82no—
239C4—82no—
240C4—161no—
241C4C7×#45611yescharacteristic
242C4C7×#45811yescharacteristic
243C2—21no—
244C2—21no—
245C2—162no—
246C2C7×#161011yescharacteristic
247C2C7×#218411yescharacteristic
248C2C7×#219111yescharacteristic
subgroupG/H
center, socleC22×C7#447
commutator, Frattini, cosocleC22×C8C23×C7
prime radicalC7×#3245C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×C4⋊C8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2232C2semidirect
3. C7×#2086C2semidirect
4. C7×#1347C2semidirect
5. C7×#1322C2semidirect
19. #12288C7direct (N × H)
45. #2232C2×C7semidirect
46. #2086C2×C7semidirect
47. #1347C2×C7semidirect
48. #1322C2×C7semidirect
226. C7#12288direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32452
C7×#22322
C7×#20862
C7×#13472
C7×#13222
#122887