Finite Group Databasebeta

← C7×#10170 #13187386

Subgroups — C7×#10170 #13187386

Order 1792.

574 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3206C211yesmaximal, characteristic
2C7×#3152C211yesmaximal, characteristic
3C2×C7×#487C211yesmaximal, characteristic, complemented
4C7×#1997C211yesmaximal, 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×#497—22no—
13C7×#494—21no—
14C7×#492—21no—
15C7×#492—21no—
16C7×#487—21no—
17C7×#487—22no—
18C7×#487—22no—
19C7×#461—22no—
20C7×#448—22no—
21C7×#431C2211yescharacteristic
22C7×#423C2211yescharacteristic
23C2×C7×#103—22no—
24C4×C7×M16C2211yescharacteristic
25C2×C4×C7×C8C2211yescharacteristic
26C7×#343—22no—
27C7×#332—22no—
28C7×D4⋊C8—22no—
29#10170C711yesmaximal, characteristic, direct product factor, Sylow
30C2×C7×#41—21no—
31C22×C7×Q8—21no—
32C22×C7×Q8D412yes—
33C7×#137—44no—
34C7×#137—44no—
35C2×C7×Q16—24no—
36C2×C7×SD16—24no—
37C2×C7×SD16—24no—
38C2×C7×M16—21no—
39C2×C7×M16D412yes—
40C22×C7×C8—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⋊Q8—21no—
48C7×C4⋊Q8D411yescharacteristic
49C7×C4⋊Q8D412yes—
50C7×#127D411yescharacteristic
51C7×#124—21no—
52C7×#121—21no—
53C7×#121—21no—
54C4×C7×D4—21no—
55C4×C7×D4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—21no—
60C2×C42×C7C2311yescharacteristic
61C7×C4⋊C8—21no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8D412yes—
64C7×#103—28no—
65C7×#103—24no—
66C7×C22⋊C8—21no—
67C7×#97—21no—
68C4×C7×C8—21no—
69C4×C7×C8—21no—
70C4×C7×C8—21no—
71C4×C7×C8D412yes—
72#3206C2×C711yescharacteristic
73#3152C2×C711yescharacteristic
74C2×#487C2×C711yescharacteristic, complemented
75#1997C2×C711yescharacteristic, complemented
76#1546C2×C711yescharacteristic, complemented
77#1492C2×C711yescharacteristic
78#1278C2×C711yescharacteristic, complemented
79C7×#41—42no—
80C7×#41—42no—
81C2×C7×Q8—22no—
82C2×C7×Q8—22no—
83C2×C7×Q8—21no—
84C2×C7×Q8—22no—
85C2×C7×Q8—44no—
86C2×C7×Q8—41no—
87C2×C7×Q8—42no—
88C2×C7×Q8—42no—
89C2×C7×D4—21no—
90C2×C7×D4—21no—
91C2×C7×D4—21no—
92C2×C7×D4—41no—
93C22×C4×C7—22no—
94C22×C4×C7—41no—
95C22×C4×C7—41no—
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7#4111yescharacteristic
99C7×Q16—48no—
100C7×SD16—48no—
101C7×SD16—84no—
102C7×M16—22no—
103C7×M16—42no—
104C7×M16SD1614yescomplemented
105C2×C7×C8—22no—
106C2×C7×C8—22no—
107C2×C7×C8—22no—
108C2×C7×C8—42no—
109C2×C7×C8—41no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—22no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C4⋊C4—41no—
120C7×C22⋊C4—41no—
121C7×C22⋊C4—41no—
122C42×C7C2×D411yescharacteristic
123C42×C7C2×D411yescharacteristic
124C42×C7C2×D411yescharacteristic
125C42×C7C2×D411yescharacteristic
126C2×#137—22no—
127#532C22×C711yescharacteristic
128C2×C4⋊Q8C22×C711yescharacteristic
129C2×C4⋊Q8C22×C711yescharacteristic
130#497—22no—
131#494—21no—
132#492—21no—
133#492—21no—
134#487—22no—
135#487—21no—
136#487—22no—
137#461—22no—
138#448—22no—
139#431C22×C711yescharacteristic
140#423C22×C711yescharacteristic
141C2×#103—22no—
142C4×M16C22×C711yescharacteristic
143C2×C4×C8C22×C711yescharacteristic
144#343—22no—
145#332—22no—
146D4⋊C8—22no—
147C23×C7—41no—
148C23×C7#12111yescharacteristic
149C7×Q8—44no—
150C7×Q8—44no—
151C7×Q8—42no—
152C7×Q8—44no—
153C7×Q8—81no—
154C7×Q8—81no—
155C7×D4—42no—
156C7×D4—42no—
157C7×D4—42no—
158C7×D4—82no—
159C2×C4×C7—22no—
160C2×C4×C7—22no—
161C2×C4×C7—41no—
162C2×C4×C7—42no—
163C2×C4×C7—42no—
164C2×C4×C7—82no—
165C2×C4×C7—81no—
166C2×C4×C7—81no—
167C2×C4×C7—82no—
168C2×C4×C7#12011yescharacteristic
169C2×C4×C7#12011yescharacteristic
170C2×C4×C7#12011yescharacteristic
171C2×C4×C7#12111yescharacteristic
172C2×C4×C7#12111yescharacteristic
173C2×C4×C7#12711yescharacteristic
174C2×C4×C7C2×SD1612yes—
175C2×C4×C7AGL(1,ℤ/8)12yes—
176C7×C8—24no—
177C7×C8—42no—
178C7×C8—42no—
179C7×C8—81no—
180C2×#41—21no—
181C22×Q8—21no—
182C22×Q8C7×D412yes—
183#137—44no—
184#137—44no—
185C2×Q16—24no—
186C2×SD16—24no—
187C2×SD16—24no—
188C2×M16—21no—
189C2×M16C7×D412yes—
190C22×C8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8C7×D412yes—
199C4⋊Q8C7×D411yescharacteristic
200#127C7×D411yescharacteristic
201#124—21no—
202#121—21no—
203#121—21no—
204C4×D4—21no—
205C4×D4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C42C23×C711yescharacteristic
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8C7×D412yes—
214#103—28no—
215#103—24no—
216C22⋊C8—21no—
217#97—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8C7×D412yes—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7—81no—
225C22×C7—81no—
226C22×C7—81no—
227C22×C7#39211yescharacteristic
228C22×C7#46012yes—
229C4×C7—21no—
230C4×C7—21no—
231C4×C7—82no—
232C4×C7—81no—
233C4×C7—82no—
234C4×C7#45012yes—
235C4×C7#45512yes—
236C4×C7#47312yes—
237C4×C7#49211yescharacteristic
238C4×C7#49611yescharacteristic
239#41—42no—
240#41—42no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—21no—
244C2×Q8—22no—
245C2×Q8—44no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—41no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4C2×C7×D411yescharacteristic
257C22×C4C2×C7×D411yescharacteristic
258C22×C4C7×#4111yescharacteristic
259Q16—48no—
260SD16—48no—
261SD16—84no—
262M16—22no—
263M16—42no—
264M16C7×SD1614yescomplemented
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—42no—
269C2×C8—41no—
270C4⋊C4—22no—
271C4⋊C4—21no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C42C2×C7×D411yescharacteristic
283C42C2×C7×D411yescharacteristic
284C42C2×C7×D411yescharacteristic
285C42C2×C7×D411yescharacteristic
286C2×C7—22no—
287C2×C7—161no—
288C2×C7#200411yescharacteristic
289C2×C7#201411yescharacteristic
290C2×C7#202011yescharacteristic
291C23—41no—
292C23C7×#12111yescharacteristic
293Q8—44no—
294Q8—44no—
295Q8—42no—
296Q8—44no—
297Q8—81no—
298Q8—81no—
299D4—42no—
300D4—42no—
301D4—42no—
302D4—82no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4C7×#12011yescharacteristic
313C2×C4C7×#12011yescharacteristic
314C2×C4C7×#12011yescharacteristic
315C2×C4C7×#12111yescharacteristic
316C2×C4C7×#12111yescharacteristic
317C2×C4C7×#12711yescharacteristic
318C2×C4C2×C7×SD1612yes—
319C2×C4C7×AGL(1,ℤ/8)12yes—
320C8—24no—
321C8—42no—
322C8—42no—
323C8—81no—
324C7#1017011yescharacteristic, direct product factor, Sylow
325C22—21no—
326C22—21no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22C7×#39211yescharacteristic
331C22C7×#46012yes—
332C4—21no—
333C4—21no—
334C4—82no—
335C4—81no—
336C4—82no—
337C4C7×#45012yes—
338C4C7×#45512yes—
339C4C7×#47312yes—
340C4C7×#49211yescharacteristic
341C4C7×#49611yescharacteristic
342C2—22no—
343C2—161no—
344C2C7×#200411yescharacteristic
345C2C7×#201411yescharacteristic
346C2C7×#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
3. C2×C7×#487C2semidirect
4. C7×#1997C2semidirect
5. C7×#1546C2semidirect
7. C7×#1278C2semidirect
29. #10170C7direct (N × H)
74. C2×#487C2×C7semidirect
75. #1997C2×C7semidirect
76. #1546C2×C7semidirect
78. #1278C2×C7semidirect
104. C7×M16SD16semidirect
264. M16C7×SD16semidirect
324. C7#10170direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32062
C7×#31522
C2×C7×#4872
C7×#19972
C7×#15462
C7×#14922
C7×#12782
#101707