Finite Group Databasebeta

← C7×#10172 #13187388

Subgroups — C7×#10172 #13187388

Order 1792.

562 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3227C211yesmaximal, characteristic
2C2×C7×#498C211yesmaximal, characteristic, complemented
3C7×#3152C211yesmaximal, characteristic
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×#498—22no—
13C7×#498—21no—
14C7×#498—22no—
15C7×#497—22no—
16C7×#494—21no—
17C7×#492—21no—
18C7×#486—21no—
19C7×#481—22no—
20C7×#476—22no—
21C7×#431C2211yescharacteristic
22C2×C7×#106—22no—
23C7×#423C2211yescharacteristic
24C4×C7×M16C2211yescharacteristic
25C2×C4×C7×C8C2211yescharacteristic
26C7×#343—22no—
27C7×#332—22no—
28C7×D4⋊C8—22no—
29#10172C711yesmaximal, 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×#106—24no—
61C7×#106—28no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8D412yes—
65C7×#102—24no—
66C7×C22⋊C8—21no—
67C7×#97—21no—
68C4×C7×C8—21no—
69C4×C7×C8—21no—
70C4×C7×C8—21no—
71C4×C7×C8D412yes—
72#3227C2×C711yescharacteristic
73C2×#498C2×C711yescharacteristic, complemented
74#3152C2×C711yescharacteristic
75#1997C2×C711yescharacteristic, complemented
76#1546C2×C711yescharacteristic, complemented
77#1492C2×C711yescharacteristic
78#1278C2×C711yescharacteristic, complemented
79C7×#41—42no—
80C7×#41—42no—
81C2×C7×Q8—21no—
82C2×C7×Q8—22no—
83C2×C7×Q8—41no—
84C2×C7×Q8—41no—
85C2×C7×Q8—41no—
86C2×C7×Q8—42no—
87C2×C7×Q8—42no—
88C2×C7×Q8—82no—
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×M16—22no—
102C7×M16—42no—
103C7×M16SD1614yes—
104C2×C7×C8—24no—
105C2×C7×C8—22no—
106C2×C7×C8—22no—
107C2×C7×C8—22no—
108C2×C7×C8—41no—
109C7×C4⋊C4—22no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—22no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—22no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C22⋊C4—41no—
120C7×C22⋊C4—41no—
121C42×C7C2×D411yescharacteristic
122C42×C7C2×D411yescharacteristic
123C42×C7C2×D411yescharacteristic
124C42×C7C2×D411yescharacteristic
125C2×#137—22no—
126#532C22×C711yescharacteristic
127C2×C4⋊Q8C22×C711yescharacteristic
128C2×C4⋊Q8C22×C711yescharacteristic
129#498—22no—
130#498—21no—
131#498—22no—
132#497—22no—
133#494—21no—
134#492—21no—
135#486—21no—
136#481—22no—
137#476—22no—
138#431C22×C711yescharacteristic
139C2×#106—22no—
140#423C22×C711yescharacteristic
141C4×M16C22×C711yescharacteristic
142C2×C4×C8C22×C711yescharacteristic
143#343—22no—
144#332—22no—
145D4⋊C8—22no—
146C23×C7—41no—
147C23×C7#12111yescharacteristic
148C7×Q8—44no—
149C7×Q8—42no—
150C7×Q8—82no—
151C7×Q8—82no—
152C7×Q8—81no—
153C7×Q8—81no—
154C7×D4—42no—
155C7×D4—42no—
156C7×D4—42no—
157C7×D4—82no—
158C2×C4×C7—22no—
159C2×C4×C7—22no—
160C2×C4×C7—41no—
161C2×C4×C7—42no—
162C2×C4×C7—42no—
163C2×C4×C7—82no—
164C2×C4×C7—81no—
165C2×C4×C7—81no—
166C2×C4×C7—82no—
167C2×C4×C7#12011yescharacteristic
168C2×C4×C7#12011yescharacteristic
169C2×C4×C7#12011yescharacteristic
170C2×C4×C7#12111yescharacteristic
171C2×C4×C7#12111yescharacteristic
172C2×C4×C7#12711yescharacteristic
173C2×C4×C7C2×SD1612yes—
174C2×C4×C7AGL(1,ℤ/8)12yes—
175C7×C8—24no—
176C7×C8—24no—
177C7×C8—42no—
178C7×C8—81no—
179C2×#41—21no—
180C22×Q8—21no—
181C22×Q8—21no—
182#137—44no—
183#137—44no—
184C2×Q16—24no—
185C2×SD16—24no—
186C2×M16—21no—
187C2×M16C7×D412yes—
188C22×C8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8C7×D412yes—
197C4⋊Q8C7×D411yescharacteristic
198#127C7×D411yescharacteristic
199#124—21no—
200#121—21no—
201#121—21no—
202C4×D4—21no—
203C4×D4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C7×D412yes—
208C2×C42C23×C711yescharacteristic
209#106—28no—
210#106—24no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8C7×D412yes—
214#102—24no—
215C22⋊C8—21no—
216#97—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8C7×D412yes—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—81no—
224C22×C7—81no—
225C22×C7—81no—
226C22×C7#39211yescharacteristic
227C22×C7#46012yes—
228C4×C7—21no—
229C4×C7—21no—
230C4×C7—81no—
231C4×C7—82no—
232C4×C7—82no—
233C4×C7#45012yes—
234C4×C7#45612yes—
235C4×C7#47112yes—
236C4×C7#49211yescharacteristic
237C4×C7#49611yescharacteristic
238#41—42no—
239#41—42no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—82no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—41no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4C2×C7×D411yescharacteristic
256C22×C4C2×C7×D411yescharacteristic
257C22×C4C7×#4111yescharacteristic
258Q16—48no—
259SD16—48no—
260M16—22no—
261M16—42no—
262M16C7×SD1614yes—
263C2×C8—24no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—41no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—21no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C42C2×C7×D411yescharacteristic
281C42C2×C7×D411yescharacteristic
282C42C2×C7×D411yescharacteristic
283C42C2×C7×D411yescharacteristic
284C2×C7—22no—
285C2×C7—161no—
286C2×C7#200511yescharacteristic
287C2×C7#201411yescharacteristic
288C2×C7#201911yescharacteristic
289C23—41no—
290C23C7×#12111yescharacteristic
291Q8—44no—
292Q8—42no—
293Q8—82no—
294Q8—82no—
295Q8—81no—
296Q8—81no—
297D4—42no—
298D4—42no—
299D4—42no—
300D4—82no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—82no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—82no—
310C2×C4C7×#12011yescharacteristic
311C2×C4C7×#12011yescharacteristic
312C2×C4C7×#12011yescharacteristic
313C2×C4C7×#12111yescharacteristic
314C2×C4C7×#12111yescharacteristic
315C2×C4C7×#12711yescharacteristic
316C2×C4C2×C7×SD1612yes—
317C2×C4C7×AGL(1,ℤ/8)12yes—
318C8—24no—
319C8—24no—
320C8—42no—
321C8—81no—
322C7#1017211yescharacteristic, direct product factor, Sylow
323C22—21no—
324C22—21no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22C7×#39211yescharacteristic
329C22C7×#46012yes—
330C4—21no—
331C4—21no—
332C4—81no—
333C4—82no—
334C4—82no—
335C4C7×#45012yes—
336C4C7×#45612yes—
337C4C7×#47112yes—
338C4C7×#49211yescharacteristic
339C4C7×#49611yescharacteristic
340C2—22no—
341C2—161no—
342C2C7×#200511yescharacteristic
343C2C7×#201411yescharacteristic
344C2C7×#201911yescharacteristic
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×#498C2semidirect
4. C7×#1997C2semidirect
5. C7×#1546C2semidirect
7. C7×#1278C2semidirect
29. #10172C7direct (N × H)
73. C2×#498C2×C7semidirect
75. #1997C2×C7semidirect
76. #1546C2×C7semidirect
78. #1278C2×C7semidirect
322. C7#10172direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32272
C2×C7×#4982
C7×#31522
C7×#19972
C7×#15462
C7×#14922
C7×#12782
#101727