Finite Group Databasebeta

← C7×#12218 #13189434

Subgroups — C7×#12218 #13189434

Order 1792.

456 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2842C211yesmaximal, characteristic
2C7×#2029C211yesmaximal, characteristic
3C7×#1614C212yesmaximal, complemented
4C7×#1352C212yesmaximal, complemented
5C7×#1316C211yesmaximal, characteristic
6C7×#543C2211yescharacteristic
7C7×#484—22no—
8C7×#399—21no—
9C7×#399C2212yes—
10C7×#396—22no—
11C7×#396—22no—
12C7×#379—21no—
13C7×#379—21no—
14C7×#379C2211yescharacteristic
15C7×#342—21no—
16C7×#342C2211yescharacteristic
17C7×#328—22no—
18C7×#323C2212yes—
19#12218C711yesmaximal, characteristic, direct product factor, Sylow
20C25×C7D411yescharacteristic
21C7×C22⋊Q8—22no—
22C7×C22⋊Q8—42no—
23C2×C7×C4⋊C4—22no—
24C2×C7×C4⋊C4—21no—
25C2×C7×C4⋊C4—22no—
26C2×C7×C4⋊C4D412yes—
27C2×C7×C22⋊C4—21no—
28C2×C7×C22⋊C4—21no—
29C2×C7×C22⋊C4—21no—
30C2×C7×C22⋊C4—22no—
31C2×C7×C22⋊C4—21no—
32C2×C7×C22⋊C4—22no—
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—42no—
35C2×C7×C22⋊C4D411yescharacteristic
36C2×C7×C22⋊C4C2311yescharacteristic
37C7×#107—42no—
38C7×#103—42no—
39C7×C22⋊C8—22no—
40C7×#95—21no—
41C7×#95D412yes—
42#2842C2×C711yescharacteristic
43#2029C2×C711yescharacteristic
44#1614C2×C712yescomplemented
45#1352C2×C712yescomplemented
46#1316C2×C711yescharacteristic
47C24×C7—21no—
48C24×C7—21no—
49C24×C7—41no—
50C24×C7—41no—
51C24×C7—82no—
52C24×C7Q1612yes—
53C24×C7C2×D411yescharacteristic
54C2×C7×Q8—22no—
55C22×C4×C7—22no—
56C22×C4×C7—21no—
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—41no—
60C22×C4×C7C2×D412yes—
61C2×C7×C8—42no—
62C7×C4⋊C4—22no—
63C7×C4⋊C4—42no—
64C7×C4⋊C4—42no—
65C7×C4⋊C4—82no—
66C7×C4⋊C4—82no—
67C7×C4⋊C4—81no—
68C7×C22⋊C4—41no—
69C7×C22⋊C4—41no—
70C7×C22⋊C4—41no—
71C7×C22⋊C4—41no—
72C7×C22⋊C4—41no—
73C7×C22⋊C4—41no—
74C7×C22⋊C4—82no—
75C7×C22⋊C4—82no—
76C7×C22⋊C4—82no—
77C7×C22⋊C4—81no—
78C7×C22⋊C4—81no—
79C7×C22⋊C4—84no—
80#543C22×C711yescharacteristic
81#484—22no—
82#399—21no—
83#399C22×C712yes—
84#396—22no—
85#396—22no—
86#379—21no—
87#379—21no—
88#379C22×C711yescharacteristic
89#342—21no—
90#342C22×C711yescharacteristic
91#328—22no—
92#323C22×C712yes—
93C23×C7—21no—
94C23×C7—21no—
95C23×C7—22no—
96C23×C7—42no—
97C23×C7—42no—
98C23×C7—42no—
99C23×C7—42no—
100C23×C7—44no—
101C23×C7—41no—
102C23×C7—41no—
103C23×C7—81no—
104C23×C7—82no—
105C23×C7—82no—
106C23×C7—82no—
107C23×C7—84no—
108C23×C7#12011yescharacteristic
109C23×C7C2×Q1611yescharacteristic
110C23×C7#13711yescharacteristic
111C7×Q8—82no—
112C2×C4×C7—22no—
113C2×C4×C7—42no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—42no—
117C2×C4×C7—42no—
118C2×C4×C7—42no—
119C2×C4×C7—41no—
120C2×C4×C7—42no—
121C2×C4×C7—42no—
122C2×C4×C7—42no—
123C2×C4×C7—82no—
124C2×C4×C7—81no—
125C2×C4×C7—82no—
126C7×C8—82no—
127C25C7×D411yescharacteristic
128C22⋊Q8—22no—
129C22⋊Q8—42no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4C7×D412yes—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4C7×D411yescharacteristic
143C2×C22⋊C4C23×C711yescharacteristic
144#107—42no—
145#103—42no—
146C22⋊C8—22no—
147#95—21no—
148#95C7×D412yes—
149C22×C7—22no—
150C22×C7—21no—
151C22×C7—21no—
152C22×C7—41no—
153C22×C7—41no—
154C22×C7—42no—
155C22×C7—42no—
156C22×C7—42no—
157C22×C7—44no—
158C22×C7—42no—
159C22×C7—81no—
160C22×C7—82no—
161C22×C7—82no—
162C22×C7—84no—
163C22×C7—82no—
164C22×C7#45111yescharacteristic
165C22×C7#45511yescharacteristic
166C22×C7B(4,𝔽2)11yescharacteristic
167C4×C7—42no—
168C4×C7—81no—
169C4×C7—82no—
170C4×C7—82no—
171C4×C7—82no—
172C4×C7—161no—
173C24—21no—
174C24—21no—
175C24—41no—
176C24—41no—
177C24—82no—
178C24C7×Q1612yes—
179C24C2×C7×D411yescharacteristic
180C2×Q8—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—41no—
186C22×C4C2×C7×D412yes—
187C2×C8—42no—
188C4⋊C4—22no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C4⋊C4—81no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—82no—
201C22⋊C4—82no—
202C22⋊C4—82no—
203C22⋊C4—81no—
204C22⋊C4—81no—
205C22⋊C4—84no—
206C2×C7—21no—
207C2×C7—21no—
208C2×C7—41no—
209C2×C7—41no—
210C2×C7—82no—
211C2×C7#160211yescharacteristic
212C2×C7#220012yes—
213C23—21no—
214C23—21no—
215C23—22no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—44no—
221C23—41no—
222C23—41no—
223C23—81no—
224C23—82no—
225C23—82no—
226C23—82no—
227C23—84no—
228C23C7×#12011yescharacteristic
229C23C2×C7×Q1611yescharacteristic
230C23C7×#13711yescharacteristic
231Q8—82no—
232C2×C4—22no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—82no—
244C2×C4—81no—
245C2×C4—82no—
246C8—82no—
247C7#1221811yescharacteristic, direct product factor, Sylow
248C22—22no—
249C22—21no—
250C22—21no—
251C22—41no—
252C22—41no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—44no—
257C22—42no—
258C22—81no—
259C22—82no—
260C22—82no—
261C22—84no—
262C22—82no—
263C22C7×#45111yescharacteristic
264C22C7×#45511yescharacteristic
265C22C7×B(4,𝔽2)11yescharacteristic
266C4—42no—
267C4—81no—
268C4—82no—
269C4—82no—
270C4—82no—
271C4—161no—
272C2—21no—
273C2—21no—
274C2—41no—
275C2—41no—
276C2—82no—
277C2C7×#160211yescharacteristic
278C2C7×#220012yes—
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC25×C7D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1614C2semidirect
4. C7×#1352C2semidirect
19. #12218C7direct (N × H)
44. #1614C2×C7semidirect
45. #1352C2×C7semidirect
247. C7#12218direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28422
C7×#20292
C7×#16142
C7×#13522
C7×#13162
#122187