Finite Group Databasebeta

← C2×C7×#1405 #13190787

Subgroups — C2×C7×#1405 #13190787

Order 1792.

484 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C2×C7×#456C211yesmaximal, characteristic
2C2×C7×#356C211yesmaximal, characteristic, complemented
3C2×C7×#333C211yesmaximal, characteristic, complemented
4C7×#1405C214yesmaximal, direct product factor
5C2×C7×2−1+4C411yescharacteristic
6C2×C7×#137—21no—
7C2×C7×C4⋊Q8C2211yescharacteristic
8C7×#456—21no—
9C7×#456—21no—
10C7×#456C412yes—
11C7×#456C2212yes—
12C2×C7×C4⋊C8—21no—
13C2×C7×#104—21no—
14C2×C7×#101—21no—
15C2×C7×#101C411yescharacteristic
16C7×#356—21no—
17C7×#356C2212yescomplemented
18C7×#333—21no—
19C7×#333C2212yescomplemented
20C2×#1405C711yesmaximal, characteristic, direct product factor, Sylow
21C7×2−1+4—21no—
22C7×2−1+4—41no—
23C7×2−1+4—42no—
24C7×2−1+4C2×C412yes—
25C2×C7×#41—21no—
26C2×C7×#41—41no—
27C2×C7×#41—41no—
28C22×C7×Q8—41no—
29C22×C7×Q8C2×C411yescharacteristic
30C22×C7×Q8D411yescharacteristic
31C7×#137—22no—
32C7×#137—22no—
33C7×#137—42no—
34C2×C7×Q16—41no—
35C2×C7×SD16—41no—
36C2×C7×M16—21no—
37C2×C7×M16—41no—
38C22×C7×C8—41no—
39C7×C4⋊Q8—41no—
40C7×C4⋊Q8C2×C411yescharacteristic
41C7×C4⋊Q8D411yescharacteristic
42C7×C4⋊Q8D411yescharacteristic
43C7×C4⋊Q8C2311yescharacteristic
44C2×C7×C4⋊C4—41no—
45C2×C42×C7D411yescharacteristic
46C7×C4⋊C8—22no—
47C7×C4⋊C8—41no—
48C7×#104—22no—
49C7×#104—22no—
50C7×#101—21no—
51C7×#101—22no—
52C7×#101—41no—
53C7×#101C2×C412yes—
54C2×#456C2×C711yescharacteristic
55C2×#356C2×C711yescharacteristic, complemented
56C2×#333C2×C711yescharacteristic, complemented
57#1405C2×C714yesdirect product factor
58C7×#41—22no—
59C7×#41—22no—
60C7×#41—41no—
61C7×#41—42no—
62C7×#41—42no—
63C7×#41—42no—
64C7×#41—42no—
65C7×#41—44no—
66C7×#41—42no—
67C7×#41—42no—
68C7×#41—41no—
69C2×C7×Q8—21no—
70C2×C7×Q8—21no—
71C2×C7×Q8—21no—
72C2×C7×Q8—41no—
73C2×C7×Q8—41no—
74C2×C7×Q8—41no—
75C2×C7×Q8—41no—
76C2×C7×Q8—42no—
77C2×C7×Q8—42no—
78C2×C7×Q8—44no—
79C2×C7×Q8—41no—
80C2×C7×Q8C22⋊C411yescharacteristic
81C2×C7×Q8C22⋊C411yescharacteristic
82C2×C7×Q8C22×C411yescharacteristic
83C2×C7×Q8C2×D411yescharacteristic
84C2×C7×D4—21no—
85C2×C7×D4—41no—
86C2×C7×D4—41no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—41no—
91C22×C4×C7—42no—
92C22×C4×C7C22⋊C411yescharacteristic
93C7×Q16—42no—
94C7×Q16—42no—
95C7×SD16—42no—
96C7×SD16—42no—
97C7×M16—22no—
98C7×M16—41no—
99C7×M16—42no—
100C7×M16—81no—
101C2×C7×C8—41no—
102C2×C7×C8—42no—
103C2×C7×C8—82no—
104C2×C7×C8—81no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—41no—
108C7×C4⋊C4—41no—
109C42×C7—21no—
110C42×C7C22⋊C411yescharacteristic
111C42×C7C2×D411yescharacteristic
112C2×2−1+4C4×C711yescharacteristic
113C2×#137—21no—
114C2×C4⋊Q8C22×C711yescharacteristic
115#456—21no—
116#456—21no—
117#456C4×C712yes—
118#456C22×C712yes—
119C2×C4⋊C8—21no—
120C2×#104—21no—
121C2×#101—21no—
122C2×#101C4×C711yescharacteristic
123#356—21no—
124#356C22×C712yescomplemented
125#333—21no—
126#333C22×C712yescomplemented
127C23×C7—41no—
128C23×C7#9911yescharacteristic
129C7×Q8—22no—
130C7×Q8—41no—
131C7×Q8—41no—
132C7×Q8—41no—
133C7×Q8—41no—
134C7×Q8—42no—
135C7×Q8—41no—
136C7×Q8—42no—
137C7×Q8—42no—
138C7×Q8—81no—
139C7×D4—22no—
140C7×D4—41no—
141C7×D4—42no—
142C7×D4—42no—
143C7×D4—42no—
144C7×D4—42no—
145C2×C4×C7—22no—
146C2×C4×C7—22no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—42no—
156C2×C4×C7—42no—
157C2×C4×C7—44no—
158C2×C4×C7—44no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7#9911yescharacteristic
165C2×C4×C7C2×C22⋊C411yescharacteristic
166C7×C8—42no—
167C7×C8—82no—
168C7×C8—82no—
1692−1+4—21no—
1702−1+4—41no—
1712−1+4—42no—
1722−1+4C2×C4×C712yes—
173C2×#41—21no—
174C2×#41—41no—
175C2×#41—41no—
176C22×Q8—41no—
177C22×Q8C2×C4×C711yescharacteristic
178C22×Q8C7×D411yescharacteristic
179#137—22no—
180#137—22no—
181#137—42no—
182C2×Q16—41no—
183C2×SD16—41no—
184C2×M16—21no—
185C2×M16—41no—
186C22×C8—41no—
187C4⋊Q8—41no—
188C4⋊Q8C2×C4×C711yescharacteristic
189C4⋊Q8C7×D411yescharacteristic
190C4⋊Q8C7×D411yescharacteristic
191C4⋊Q8C23×C711yescharacteristic
192C2×C4⋊C4—41no—
193C2×C42C7×D411yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—41no—
196#104—22no—
197#104—22no—
198#101—21no—
199#101—22no—
200#101—41no—
201#101C2×C4×C712yes—
202C22×C7—22no—
203C22×C7—42no—
204C22×C7—82no—
205C22×C7AΣL(1,𝔽16)11yescharacteristic
206C22×C7AΣL(1,𝔽16)11yescharacteristic
207C22×C7C2×#9911yescharacteristic
208C4×C7—21no—
209C4×C7—22no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—42no—
218#41—22no—
219#41—22no—
220#41—41no—
221#41—42no—
222#41—42no—
223#41—42no—
224#41—42no—
225#41—44no—
226#41—42no—
227#41—42no—
228#41—41no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—44no—
239C2×Q8—41no—
240C2×Q8C7×C22⋊C411yescharacteristic
241C2×Q8C7×C22⋊C411yescharacteristic
242C2×Q8C22×C4×C711yescharacteristic
243C2×Q8C2×C7×D411yescharacteristic
244C2×D4—21no—
245C2×D4—41no—
246C2×D4—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4C7×C22⋊C411yescharacteristic
253Q16—42no—
254Q16—42no—
255SD16—42no—
256SD16—42no—
257M16—22no—
258M16—41no—
259M16—42no—
260M16—81no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—82no—
264C2×C8—81no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C42—21no—
270C42C7×C22⋊C411yescharacteristic
271C42C2×C7×D411yescharacteristic
272C2×C7—21no—
273C2×C7—21no—
274C2×C7—82no—
275C2×C7#140512yesdirect product factor
276C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
277C23—41no—
278C23C7×#9911yescharacteristic
279Q8—22no—
280Q8—41no—
281Q8—41no—
282Q8—41no—
283Q8—41no—
284Q8—42no—
285Q8—41no—
286Q8—42no—
287Q8—42no—
288Q8—81no—
289D4—22no—
290D4—41no—
291D4—42no—
292D4—42no—
293D4—42no—
294D4—42no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4C7×#9911yescharacteristic
315C2×C4C2×C7×C22⋊C411yescharacteristic
316C8—42no—
317C8—82no—
318C8—82no—
319C7C2×#140511yescharacteristic, direct product factor, Sylow
320C22—22no—
321C22—42no—
322C22—82no—
323C22C7×AΣL(1,𝔽16)11yescharacteristic
324C22C7×AΣL(1,𝔽16)11yescharacteristic
325C22C2×C7×#9911yescharacteristic
326C4—21no—
327C4—22no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—42no—
336C2—21no—
337C2—21no—
338C2—82no—
339C2C7×#140512yesdirect product factor
340C2C2×C7×AΣL(1,𝔽16)11yescharacteristic
subgroupG/H
center, socleC22×C7AΣL(1,𝔽16)
commutatorC2×Q8C22×C4×C7
Frattini, cosocleC4⋊Q8C23×C7
prime radicalC2×C7×2−1+4C4
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C22×C4×C7 ◁ C2×C7×C4⋊Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#356C2semidirect
3. C2×C7×#333C2semidirect
4. C7×#1405C2direct (N × H)
17. C7×#356C22semidirect
19. C7×#333C22semidirect
20. C2×#1405C7direct (N × H)
55. C2×#356C2×C7semidirect
56. C2×#333C2×C7semidirect
57. #1405C2×C7direct (N × H)
124. #356C22×C7semidirect
126. #333C22×C7semidirect
275. C2×C7#1405direct (N × H)
319. C7C2×#1405direct (N × H)
339. C2C7×#1405direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4562
C2×C7×#3562
C2×C7×#3332
C7×#14052
C2×#14057