Finite Group Databasebeta

← C7×#10174 #13187390

Subgroups — C7×#10174 #13187390

Order 1792.

578 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3226C211yesmaximal, characteristic
2C2×C7×#500C211yesmaximal, characteristic, complemented
3C7×#3152C211yesmaximal, characteristic
4C7×#1966C211yesmaximal, characteristic, complemented
5C7×#1546C211yesmaximal, characteristic, complemented
6C7×#1492C211yesmaximal, characteristic
7C7×#1278C211yesmaximal, characteristic, complemented
8C2×C7×#137—22no—
9C7×#532C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C7×C4⋊Q8C2211yescharacteristic
12C7×#500—22no—
13C7×#500—21no—
14C7×#500—22no—
15C7×#497—22no—
16C7×#494—21no—
17C7×#492—21no—
18C7×#486—21no—
19C7×#480—22no—
20C7×#474—22no—
21C7×#431C2211yescharacteristic
22C2×C7×#107—22no—
23C7×#423C2211yescharacteristic
24C2×C7×#101—22no—
25C4×C7×M16C2211yescharacteristic
26C2×C4×C7×C8C2211yescharacteristic
27C7×#332—22no—
28C7×D4⋊C8—22no—
29#10174C711yesmaximal, characteristic, direct product factor, Sylow
30C2×C7×#41—21no—
31C22×C7×Q8—21no—
32C22×C7×Q8—21no—
33C7×#137—44no—
34C7×#137—44no—
35C2×C7×Q16—24no—
36C2×C7×SD16—24no—
37C2×C7×M16—21no—
38C2×C7×M16D412yes—
39C22×C7×C8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8—21no—
46C7×C4⋊Q8—21no—
47C7×C4⋊Q8D411yescharacteristic
48C7×C4⋊Q8D412yes—
49C7×#127D411yescharacteristic
50C7×#124—21no—
51C7×#121—21no—
52C7×#121—21no—
53C4×C7×D4—21no—
54C4×C7×D4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4D412yes—
59C2×C42×C7C2311yescharacteristic
60C7×#107—24no—
61C7×#107—28no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8D412yes—
65C7×#102—24no—
66C7×#101—44no—
67C7×C22⋊C8—21no—
68C7×#97—21no—
69C4×C7×C8—21no—
70C4×C7×C8—21no—
71C4×C7×C8—21no—
72C4×C7×C8D412yes—
73#3226C2×C711yescharacteristic
74C2×#500C2×C711yescharacteristic, complemented
75#3152C2×C711yescharacteristic
76#1966C2×C711yescharacteristic, complemented
77#1546C2×C711yescharacteristic, complemented
78#1492C2×C711yescharacteristic
79#1278C2×C711yescharacteristic, complemented
80C7×#41—42no—
81C7×#41—42no—
82C2×C7×Q8—21no—
83C2×C7×Q8—22no—
84C2×C7×Q8—41no—
85C2×C7×Q8—41no—
86C2×C7×Q8—41no—
87C2×C7×Q8—42no—
88C2×C7×Q8—42no—
89C2×C7×Q8—42no—
90C2×C7×Q8—42no—
91C2×C7×D4—21no—
92C2×C7×D4—21no—
93C2×C7×D4—21no—
94C2×C7×D4—41no—
95C22×C4×C7—22no—
96C22×C4×C7—41no—
97C22×C4×C7—41no—
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7#4111yescharacteristic
101C7×Q16—48no—
102C7×SD16—48no—
103C7×M16—22no—
104C7×M16—42no—
105C7×M16D814yes—
106C2×C7×C8—24no—
107C2×C7×C8—22no—
108C2×C7×C8—22no—
109C2×C7×C8—22no—
110C2×C7×C8—41no—
111C7×C4⋊C4—22no—
112C7×C4⋊C4—22no—
113C7×C4⋊C4—22no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—22no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—41no—
121C7×C22⋊C4—41no—
122C7×C22⋊C4—41no—
123C42×C7C2×D411yescharacteristic
124C42×C7C2×D411yescharacteristic
125C42×C7C2×D411yescharacteristic
126C42×C7C2×D411yescharacteristic
127C2×#137—22no—
128#532C22×C711yescharacteristic
129C2×C4⋊Q8C22×C711yescharacteristic
130C2×C4⋊Q8C22×C711yescharacteristic
131#500—22no—
132#500—21no—
133#500—22no—
134#497—22no—
135#494—21no—
136#492—21no—
137#486—21no—
138#480—22no—
139#474—22no—
140#431C22×C711yescharacteristic
141C2×#107—22no—
142#423C22×C711yescharacteristic
143C2×#101—22no—
144C4×M16C22×C711yescharacteristic
145C2×C4×C8C22×C711yescharacteristic
146#332—22no—
147D4⋊C8—22no—
148C23×C7—41no—
149C23×C7#12111yescharacteristic
150C7×Q8—44no—
151C7×Q8—42no—
152C7×Q8—82no—
153C7×Q8—81no—
154C7×Q8—81no—
155C7×Q8—81no—
156C7×Q8—81no—
157C7×D4—42no—
158C7×D4—42no—
159C7×D4—42no—
160C7×D4—82no—
161C2×C4×C7—22no—
162C2×C4×C7—22no—
163C2×C4×C7—41no—
164C2×C4×C7—42no—
165C2×C4×C7—42no—
166C2×C4×C7—44no—
167C2×C4×C7—82no—
168C2×C4×C7—81no—
169C2×C4×C7—81no—
170C2×C4×C7#12011yescharacteristic
171C2×C4×C7#12011yescharacteristic
172C2×C4×C7#12011yescharacteristic
173C2×C4×C7#12111yescharacteristic
174C2×C4×C7#12111yescharacteristic
175C2×C4×C7#12711yescharacteristic
176C2×C4×C7C2×D812yes—
177C2×C4×C7AGL(1,ℤ/8)12yes—
178C7×C8—24no—
179C7×C8—24no—
180C7×C8—42no—
181C7×C8—81no—
182C2×#41—21no—
183C22×Q8—21no—
184C22×Q8—21no—
185#137—44no—
186#137—44no—
187C2×Q16—24no—
188C2×SD16—24no—
189C2×M16—21no—
190C2×M16C7×D412yes—
191C22×C8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8C7×D412yes—
200C4⋊Q8C7×D411yescharacteristic
201#127C7×D411yescharacteristic
202#124—21no—
203#121—21no—
204#121—21no—
205C4×D4—21no—
206C4×D4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4C7×D412yes—
211C2×C42C23×C711yescharacteristic
212#107—28no—
213#107—24no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8C7×D412yes—
217#102—24no—
218#101—44no—
219C22⋊C8—21no—
220#97—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8C7×D412yes—
225C22×C7—21no—
226C22×C7—21no—
227C22×C7—81no—
228C22×C7—81no—
229C22×C7—81no—
230C22×C7#39211yescharacteristic
231C22×C7#45912yes—
232C4×C7—21no—
233C4×C7—21no—
234C4×C7—82no—
235C4×C7—81no—
236C4×C7—82no—
237C4×C7#44712yes—
238C4×C7#45612yes—
239C4×C7#47312yes—
240C4×C7#49311yescharacteristic
241C4×C7#49611yescharacteristic
242#41—42no—
243#41—42no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—41no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4C2×C7×D411yescharacteristic
261C22×C4C2×C7×D411yescharacteristic
262C22×C4C7×#4111yescharacteristic
263Q16—48no—
264SD16—48no—
265M16—22no—
266M16—42no—
267M16C7×D814yes—
268C2×C8—24no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—41no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C42C2×C7×D411yescharacteristic
286C42C2×C7×D411yescharacteristic
287C42C2×C7×D411yescharacteristic
288C42C2×C7×D411yescharacteristic
289C2×C7—22no—
290C2×C7—161no—
291C2×C7#200511yescharacteristic
292C2×C7#201111yescharacteristic
293C2×C7#202011yescharacteristic
294C23—41no—
295C23C7×#12111yescharacteristic
296Q8—44no—
297Q8—42no—
298Q8—81no—
299Q8—81no—
300Q8—81no—
301Q8—81no—
302Q8—82no—
303D4—42no—
304D4—42no—
305D4—42no—
306D4—82no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—82no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4C7×#12011yescharacteristic
317C2×C4C7×#12011yescharacteristic
318C2×C4C7×#12011yescharacteristic
319C2×C4C7×#12111yescharacteristic
320C2×C4C7×#12111yescharacteristic
321C2×C4C7×#12711yescharacteristic
322C2×C4C2×C7×D812yes—
323C2×C4C7×AGL(1,ℤ/8)12yes—
324C8—24no—
325C8—24no—
326C8—42no—
327C8—81no—
328C7#1017411yescharacteristic, direct product factor, Sylow
329C22—21no—
330C22—21no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22C7×#39211yescharacteristic
335C22C7×#45912yes—
336C4—21no—
337C4—21no—
338C4—82no—
339C4—81no—
340C4—82no—
341C4C7×#44712yes—
342C4C7×#45612yes—
343C4C7×#47312yes—
344C4C7×#49311yescharacteristic
345C4C7×#49611yescharacteristic
346C2—22no—
347C2—161no—
348C2C7×#200511yescharacteristic
349C2C7×#201111yescharacteristic
350C2C7×#202011yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#500C2semidirect
4. C7×#1966C2semidirect
5. C7×#1546C2semidirect
7. C7×#1278C2semidirect
29. #10174C7direct (N × H)
74. C2×#500C2×C7semidirect
76. #1966C2×C7semidirect
77. #1546C2×C7semidirect
79. #1278C2×C7semidirect
328. C7#10174direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32262
C2×C7×#5002
C7×#31522
C7×#19662
C7×#15462
C7×#14922
C7×#12782
#101747