Finite Group Databasebeta

← C7×#9198 #13186414

Subgroups — C7×#9198 #13186414

Order 1792.

578 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 134 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×#2921C211yesmaximal, characteristic, complemented
2C7×#2619C211yesmaximal, characteristic
3C7×#1545C212yesmaximal, complemented
4C7×#1458C211yesmaximal, characteristic, complemented
5C7×#1286C212yesmaximal, complemented
6C2×C7×#127C2211yescharacteristic
7C2×C7×#121—21no—
8C2×C7×#117C2211yescharacteristic
9C7×#433—24no—
10C7×#431C2212yes—
11C7×#423C2212yes—
12C7×#407—21no—
13C4×C7×M16—21no—
14C7×#394—21no—
15C7×#394C412yes—
16C7×#392—21no—
17C7×#392C412yes—
18C7×#390—21no—
19C7×#382C2211yescharacteristic
20C7×#331—24no—
21C7×#324—21no—
22C7×#323—21no—
23#9198C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4—22no—
25C22×C7×D4—21no—
26C22×C7×D4—21no—
27C22×C7×D4C2×C412yes—
28C23×C4×C7D411yescharacteristic
29C2×C7×M16—22no—
30C7×#127—22no—
31C7×#127—22no—
32C7×#121—44no—
33C7×#117—22no—
34C7×#117—22no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4C2×C411yescharacteristic
42C2×C7×C22⋊C4D411yescharacteristic
43C2×C42×C7D411yescharacteristic
44C2×C42×C7C2311yescharacteristic
45C7×C4⋊C8—22no—
46C7×C4⋊C8—22no—
47C7×C4⋊C8D414yescomplemented
48C7×C22⋊C8—24no—
49C7×C22⋊C8—22no—
50C7×#97—22no—
51C4×C7×C8—22no—
52C4×C7×C8—22no—
53C7×#95—21no—
54C7×#95C2×C412yes—
55#2921C2×C711yescharacteristic, complemented
56#2619C2×C711yescharacteristic
57#1545C2×C712yescomplemented
58#1458C2×C711yescharacteristic, complemented
59#1286C2×C712yescomplemented
60C24×C7—22no—
61C24×C7C22⋊C411yescharacteristic
62C2×C7×D4—24no—
63C2×C7×D4—44no—
64C2×C7×D4—41no—
65C2×C7×D4—42no—
66C2×C7×D4—41no—
67C2×C7×D4—42no—
68C2×C7×D4—42no—
69C2×C7×D4—82no—
70C2×C7×D4—82no—
71C2×C7×D4—82no—
72C2×C7×D4—82no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—22no—
76C22×C4×C7—22no—
77C22×C4×C7—22no—
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22×C411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C7×M16—42no—
85C7×M16—42no—
86C2×C7×C8—24no—
87C2×C7×C8—42no—
88C7×C4⋊C4—22no—
89C7×C4⋊C4—22no—
90C7×C4⋊C4—42no—
91C7×C22⋊C4—22no—
92C7×C22⋊C4—22no—
93C7×C22⋊C4—44no—
94C7×C22⋊C4—82no—
95C7×C22⋊C4—82no—
96C42×C7—21no—
97C42×C7—21no—
98C42×C7C2×D412yes—
99C42×C7#4112yes—
100C2×#127C22×C711yescharacteristic
101C2×#121—21no—
102C2×#117C22×C711yescharacteristic
103#433—24no—
104#431C22×C712yes—
105#423C22×C712yes—
106#407—21no—
107C4×M16—21no—
108#394—21no—
109#394C4×C712yes—
110#392—21no—
111#392C4×C712yes—
112#390—21no—
113#382C22×C711yescharacteristic
114#331—24no—
115#324—21no—
116#323—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—41no—
121C23×C7—41no—
122C23×C7—42no—
123C23×C7—42no—
124C23×C7—42no—
125C23×C7—42no—
126C23×C7—42no—
127C23×C7—42no—
128C23×C7—42no—
129C23×C7C2×C22⋊C411yescharacteristic
130C7×D4—84no—
131C7×D4—84no—
132C7×D4—88no—
133C7×D4—82no—
134C7×D4—82no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—42no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
149C2×C4×C7C4×D412yes—
150C2×C4×C7#12011yescharacteristic
151C2×C4×C7#12112yes—
152C2×C4×C7#12311yescharacteristic
153C2×C4×C7#13512yes—
154C2×C4×C7AGL(1,ℤ/8)12yes—
155C7×C8—44no—
156C7×C8—82no—
157C22×D4—22no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4C2×C4×C712yes—
161C23×C4C7×D411yescharacteristic
162C2×M16—22no—
163#127—22no—
164#127—22no—
165#121—44no—
166#117—22no—
167#117—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4C2×C4×C711yescharacteristic
170C2×C4⋊C4C7×D411yescharacteristic
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4C2×C4×C711yescharacteristic
175C2×C22⋊C4C7×D411yescharacteristic
176C2×C42C7×D411yescharacteristic
177C2×C42C23×C711yescharacteristic
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8C7×D414yescomplemented
181C22⋊C8—24no—
182C22⋊C8—22no—
183#97—22no—
184C4×C8—22no—
185C4×C8—22no—
186#95—21no—
187#95C2×C4×C712yes—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—42no—
191C22×C7—41no—
192C22×C7—41no—
193C22×C7—41no—
194C22×C7—81no—
195C22×C7—82no—
196C22×C7—82no—
197C22×C7—82no—
198C22×C7—82no—
199C22×C7—82no—
200C22×C7—82no—
201C22×C7—82no—
202C22×C7#38611yescharacteristic
203C22×C7#41611yescharacteristic
204C22×C7#41811yescharacteristic
205C4×C7—22no—
206C4×C7—22no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—82no—
211C4×C7#44912yes—
212C4×C7#48212yes—
213C24—22no—
214C24C7×C22⋊C411yescharacteristic
215C2×D4—24no—
216C2×D4—44no—
217C2×D4—41no—
218C2×D4—42no—
219C2×D4—41no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—82no—
223C2×D4—82no—
224C2×D4—82no—
225C2×D4—82no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4C7×C22⋊C411yescharacteristic
232C22×C4C7×C22⋊C411yescharacteristic
233C22×C4C7×C22⋊C411yescharacteristic
234C22×C4C22×C4×C711yescharacteristic
235C22×C4C2×C7×D411yescharacteristic
236C22×C4C2×C7×D411yescharacteristic
237M16—42no—
238M16—42no—
239C2×C8—24no—
240C2×C8—42no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—44no—
247C22⋊C4—82no—
248C22⋊C4—82no—
249C42—21no—
250C42—21no—
251C42C2×C7×D412yes—
252C42C7×#4112yes—
253C2×C7—21no—
254C2×C7—21no—
255C2×C7—81no—
256C2×C7—162no—
257C2×C7#187511yescharacteristic
258C2×C7#191011yescharacteristic
259C2×C7#191411yescharacteristic
260C23—21no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23C2×C7×C22⋊C411yescharacteristic
273D4—84no—
274D4—84no—
275D4—88no—
276D4—82no—
277D4—82no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4C4×C7×D412yes—
293C2×C4C7×#12011yescharacteristic
294C2×C4C7×#12112yes—
295C2×C4C7×#12311yescharacteristic
296C2×C4C7×#13512yes—
297C2×C4C7×AGL(1,ℤ/8)12yes—
298C8—44no—
299C8—82no—
300C7#919811yescharacteristic, direct product factor, Sylow
301C22—21no—
302C22—21no—
303C22—42no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22—82no—
312C22—82no—
313C22—82no—
314C22—82no—
315C22C7×#38611yescharacteristic
316C22C7×#41611yescharacteristic
317C22C7×#41811yescharacteristic
318C4—22no—
319C4—22no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—82no—
324C4C7×#44912yes—
325C4C7×#48212yes—
326C2—21no—
327C2—21no—
328C2—81no—
329C2—162no—
330C2C7×#187511yescharacteristic
331C2C7×#191011yescharacteristic
332C2C7×#191411yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#2619C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2921C2semidirect
3. C7×#1545C2semidirect
4. C7×#1458C2semidirect
5. C7×#1286C2semidirect
23. #9198C7direct (N × H)
47. C7×C4⋊C8D4semidirect
55. #2921C2×C7semidirect
57. #1545C2×C7semidirect
58. #1458C2×C7semidirect
59. #1286C2×C7semidirect
180. C4⋊C8C7×D4semidirect
300. C7#9198direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#29212
C7×#26192
C7×#15452
C7×#14582
C7×#12862
#91987