Finite Group Databasebeta

← C7×#11904 #13189120

Subgroups — C7×#11904 #13189120

Order 1792.

568 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2409C211yesmaximal, characteristic
2C7×#1800C211yesmaximal, characteristic
3C7×#1459C211yesmaximal, characteristic
4C7×#1354C214yesmaximal, complemented
5C23×C7×Q8C411yescharacteristic
6C2×C7×C22⋊Q8—21no—
7C2×C4×C7×D4C2211yescharacteristic
8C7×#423—21no—
9C2×C7×#100—21no—
10C2×C7×C22⋊C8—21no—
11C7×#395C2212yes—
12C7×#393C412yes—
13C7×#391—21no—
14C7×#386—21no—
15C7×#385—21no—
16C7×#380C411yescharacteristic
17C7×#343C2212yes—
18C7×#324C2212yes—
19C7×#323—21no—
20#11904C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×Q8—41no—
23C22×C7×Q8—41no—
24C22×C7×Q8—44no—
25C22×C7×Q8C2×C412yes—
26C22×C7×D4D411yescharacteristic
27C23×C4×C7—21no—
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C2×C7×M16—22no—
31C22×C7×C8—41no—
32C7×C22⋊Q8—44no—
33C4×C7×D4—41no—
34C4×C7×D4—41no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—22no—
37C2×C7×C4⋊C4C2311yescharacteristic
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4C2×C411yescharacteristic
41C2×C7×C22⋊C4D411yescharacteristic
42C2×C42×C7D411yescharacteristic
43C7×C4⋊C8—41no—
44C7×C4⋊C8—41no—
45C7×#100—42no—
46C7×C22⋊C8—22no—
47C7×C22⋊C8—41no—
48C7×#95—22no—
49C7×#95—22no—
50C7×#95C2×C412yes—
51#2409C2×C711yescharacteristic
52#1800C2×C711yescharacteristic
53#1459C2×C711yescharacteristic
54#1354C2×C714yescomplemented
55C24×C7C22⋊C411yescharacteristic
56C24×C7C22⋊C411yescharacteristic
57C2×C7×Q8—24no—
58C2×C7×Q8—42no—
59C2×C7×Q8—44no—
60C2×C7×Q8—42no—
61C2×C7×Q8—48no—
62C2×C7×Q8—44no—
63C2×C7×Q8—44no—
64C2×C7×Q8—41no—
65C2×C7×Q8—41no—
66C2×C7×D4—41no—
67C2×C7×D4—41no—
68C2×C7×D4SD1612yes—
69C2×C7×D4Q1612yes—
70C22×C4×C7—22no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—22no—
76C22×C4×C7—44no—
77C22×C4×C7—42no—
78C22×C4×C7—42no—
79C22×C4×C7—42no—
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22×C411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C7×M16—42no—
86C7×M16—42no—
87C2×C7×C8—42no—
88C2×C7×C8—42no—
89C2×C7×C8—82no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—42no—
92C7×C4⋊C4—44no—
93C7×C22⋊C4—41no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—44no—
96C7×C22⋊C4—42no—
97C42×C7—21no—
98C42×C7—21no—
99C23×Q8C4×C711yescharacteristic
100C2×C22⋊Q8—21no—
101C2×C4×D4C22×C711yescharacteristic
102#423—21no—
103C2×#100—21no—
104C2×C22⋊C8—21no—
105#395C22×C712yes—
106#393C4×C712yes—
107#391—21no—
108#386—21no—
109#385—21no—
110#380C4×C711yescharacteristic
111#343C22×C712yes—
112#324C22×C712yes—
113#323—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—41no—
119C23×C7—41no—
120C23×C7—41no—
121C23×C7—41no—
122C23×C7#10312yes—
123C23×C7#10312yes—
124C23×C7C2×C22⋊C411yescharacteristic
125C7×Q8—44no—
126C7×Q8—44no—
127C7×Q8—44no—
128C7×Q8—44no—
129C7×D4—42no—
130C7×D4—42no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—22no—
134C2×C4×C7—24no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—48no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—81no—
151C2×C4×C7#9911yescharacteristic
152C2×C4×C7#9911yescharacteristic
153C2×C4×C7C2×SD1611yescharacteristic
154C2×C4×C7C2×Q1611yescharacteristic
155C7×C8—82no—
156C7×C8—82no—
157C22×Q8—21no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—44no—
161C22×Q8C2×C4×C712yes—
162C22×D4C7×D411yescharacteristic
163C23×C4—21no—
164C23×C4C2×C4×C711yescharacteristic
165C23×C4C7×D411yescharacteristic
166C2×M16—22no—
167C22×C8—41no—
168C22⋊Q8—44no—
169C4×D4—41no—
170C4×D4—41no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C23×C711yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4C2×C4×C711yescharacteristic
177C2×C22⋊C4C7×D411yescharacteristic
178C2×C42C7×D411yescharacteristic
179C4⋊C8—41no—
180C4⋊C8—41no—
181#100—42no—
182C22⋊C8—22no—
183C22⋊C8—41no—
184#95—22no—
185#95—22no—
186#95C2×C4×C712yes—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—22no—
190C22×C7—22no—
191C22×C7—42no—
192C22×C7—42no—
193C22×C7—42no—
194C22×C7—41no—
195C22×C7—41no—
196C22×C7—41no—
197C22×C7—41no—
198C22×C7—81no—
199C22×C7C2×#9911yescharacteristic
200C22×C7C2×#10311yescharacteristic
201C22×C7#42111yescharacteristic
202C4×C7—21no—
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—41no—
206C4×C7—44no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—82no—
210C4×C7AΣL(1,𝔽16)12yes—
211C24C7×C22⋊C411yescharacteristic
212C24C7×C22⋊C411yescharacteristic
213C2×Q8—24no—
214C2×Q8—44no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—48no—
219C2×Q8—44no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4C7×SD1612yes—
225C2×D4C7×Q1612yes—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4C7×C22⋊C411yescharacteristic
237C22×C4C7×C22⋊C411yescharacteristic
238C22×C4C22×C4×C711yescharacteristic
239C22×C4C2×C7×D411yescharacteristic
240C22×C4C2×C7×D411yescharacteristic
241M16—42no—
242M16—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—82no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—44no—
252C22⋊C4—42no—
253C42—21no—
254C42—21no—
255C2×C7—21no—
256C2×C7—21no—
257C2×C7—42no—
258C2×C7—42no—
259C2×C7#152011yescharacteristic
260C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
261C2×C7#213511yescharacteristic
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23C7×#10312yes—
271C23C7×#10312yes—
272C23C2×C7×C22⋊C411yescharacteristic
273Q8—44no—
274Q8—44no—
275Q8—44no—
276Q8—44no—
277D4—42no—
278D4—42no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—48no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4C7×#9911yescharacteristic
300C2×C4C7×#9911yescharacteristic
301C2×C4C2×C7×SD1611yescharacteristic
302C2×C4C2×C7×Q1611yescharacteristic
303C8—82no—
304C8—82no—
305C7#1190411yescharacteristic, direct product factor, Sylow
306C22—21no—
307C22—21no—
308C22—22no—
309C22—22no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×C7×#9911yescharacteristic
319C22C2×C7×#10311yescharacteristic
320C22C7×#42111yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—44no—
326C4—41no—
327C4—41no—
328C4—82no—
329C4C7×AΣL(1,𝔽16)12yes—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2C7×#152011yescharacteristic
335C2C2×C7×AΣL(1,𝔽16)11yescharacteristic
336C2C7×#213511yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1354C2semidirect
20. #11904C7direct (N × H)
54. #1354C2×C7semidirect
305. C7#11904direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24092
C7×#18002
C7×#14592
C7×#13542
#119047