Finite Group Databasebeta

← #13245356

Subgroups — #13245356

Order 1792.

774 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 143 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
1#11742834C211yesmaximal, characteristic
2C7×#1800C211yesmaximal, characteristic
3#11729368C211yesmaximal, characteristic
4#11728263C212yesmaximal, complemented
5#11728255C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7#90489—21no—
8C2×#6487—22no—
9#90234—22no—
10#90230—24no—
11D4×Dic14—22no—
12#90191—21no—
13#90174C414yes—
14#90161C2212yes—
15C7×#423C2212yes—
16C2×C7×#100—21no—
17C2×C22⋊(C7⋊C8)—22no—
18C2×C4⋊(C7⋊C8)C2211yescharacteristic
19C2×C4×C7⋊C8C2211yescharacteristic
20C7×#343—21no—
21#89281—22no—
22#89279—24no—
23#89276—22no—
24#8355—71nomaximal, Sylow
25C22×C7×D4D411yescharacteristic
26C23×C4×C7D412yes—
27C22×Dic14—22no—
28C2×C7×M16—22no—
29C4×C7×D4—22no—
30C4×C7×D4—22no—
31C2×C7×C4⋊C4D411yescharacteristic
32C2×C7×C22⋊C4C2×C412yes—
33C2×C42×C7C2311yescharacteristic
34C2×C22⋊Dic7—24no—
35D4×Dic7—24no—
36D4×Dic7—24no—
37#6489—22no—
38C2×C4⋊Dic7—22no—
39#6487—24no—
40#6487—24no—
41#6487—24no—
42#6487—22no—
43#6487C2×C414yes—
44C2×Dic7⋊C4—24no—
45C22×C7⋊C8—21no—
46C22×C7⋊C8—21no—
47#6453—24no—
48#6451—24no—
49#6441—28no—
50#6441—28no—
51#6433—22no—
52#6432D412yes—
53#6432D412yes—
54C4×Dic14—22no—
55C4×Dic14D414yes—
56C7×C4⋊C8—22no—
57C7×C4⋊C8—22no—
58C7×#100—42no—
59C22⋊(C7⋊C8)—44no—
60C4⋊(C7⋊C8)—22no—
61C4×C7⋊C8—22no—
62#3501—71no—
63#1954—71no—
64#1800D711yescharacteristic
65#1550—72no—
66#1493—72no—
67C24×C7C22⋊C412yes—
68C2×C7×D4—22no—
69C2×C7×D4—22no—
70C2×C7×D4SD1612yes—
71C2×C7×D4Q1612yes—
72C22×C4×C7—22no—
73C22×C4×C7—22no—
74C22×C4×C7—42no—
75C22×C4×C7—42no—
76C22×C4×C7C22⋊C412yes—
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C22×Dic7—24no—
81C22×Dic7—24no—
82C2×Dic14—22no—
83C2×Dic14—24no—
84C2×Dic14—82no—
85C2×Dic14—82no—
86C2×Dic14C22⋊C414yes—
87C7×M16—42no—
88C7×M16—42no—
89C2×C7×C8—42no—
90C7×C4⋊C4—22no—
91C7×C22⋊C4—24no—
92C42×C7C2×D412yes—
93C42×C7C2×D412yes—
94C22⋊Dic7—28no—
95C22⋊Dic7—24no—
96C4⋊Dic7—24no—
97C4⋊Dic7—22no—
98C4⋊Dic7—22no—
99C4⋊Dic7—22no—
100C4⋊Dic7C22⋊C414yes—
101Dic7⋊C4—28no—
102Dic7⋊C4—24no—
103Dic7⋊C4—24no—
104C4×Dic7—24no—
105C2×C7⋊C8—22no—
106C2×C7⋊C8—41no—
107C2×C7⋊C8—41no—
108C2×C7⋊C8—42no—
109C2×C7⋊C8—41no—
110#554—144no—
111#554—142no—
112D4×Q8—142no—
113#544—141no—
114#543—74no—
115#533—72no—
116C2×C22⋊Q8—142no—
117#518—141no—
118C2×C4×D4C2×D711yescharacteristic
119#423C2×D712yes—
120C2×C4⋊C8—71no—
121C2×#100—21no—
122C2×C22⋊C8—142no—
123C2×C4×C8—71no—
124#343—21no—
125#333—142no—
126#332—142no—
127Q8⋊C8—144no—
128C23×C7—22no—
129C23×C7—22no—
130C23×C7—42no—
131C23×C7—42no—
132C23×C7#10314yes—
133C23×C7C2×C22⋊C411yescharacteristic
134C7×D4—24no—
135C7×D4—24no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—22no—
139C2×C4×C7—41no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—82no—
144C2×C4×C7C2×C22⋊C412yes—
145C2×C4×C7#12011yescharacteristic
146C2×C4×C7#12011yescharacteristic
147C2×C4×C7#12012yes—
148C2×C4×C7C2×SD1611yescharacteristic
149C2×C4×C7C2×Q1611yescharacteristic
150C2×C4×C7#13712yes—
151C2×Dic7—24no—
152C2×Dic7—24no—
153C2×Dic7—84no—
154C2×Dic7—84no—
155Dic14—42no—
156Dic14—42no—
157Dic14—44no—
158Dic14—84no—
159C7×C8—82no—
160C7⋊C8—42no—
161C7⋊C8—81no—
162C22×Q8—142no—
163C22×D4C22⋊D711yescharacteristic
164C23×C4D2812yes—
165C2×M16—22no—
166C22×C8—141no—
167C22×C8—141no—
168C4⋊Q8—72no—
169C4⋊Q8—72no—
170C4⋊Q8—144no—
171#125—144no—
172#125—142no—
173C22⋊Q8—74no—
174C22⋊Q8—148no—
175C22⋊Q8—148no—
176C22⋊Q8—144no—
177C22⋊Q8—144no—
178C22⋊Q8—144no—
179C22⋊Q8—142no—
180C4×Q8—74no—
181C4×Q8—142no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—144no—
185C4×D4—144no—
186#117—142no—
187C2×C4⋊C4—144no—
188C2×C4⋊C4—142no—
189C2×C4⋊C4C22⋊D711yescharacteristic
190C2×C22⋊C4—144no—
191C2×C22⋊C4C4×D712yes—
192C2×C42C22×D711yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—142no—
196#100—42no—
197C22⋊C8—284no—
198C4×C8—142no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—24no—
202C22×C7—44no—
203C22×C7—42no—
204C22×C7—42no—
205C22×C7—82no—
206C22×C7#37911yescharacteristic
207C22×C7C2×#10311yescharacteristic
208C22×C7#41911yescharacteristic
209C4×C7—21no—
210C4×C7—22no—
211C4×C7—42no—
212C4×C7#44812yes—
213C4×C7#45112yes—
214C4×C7#45311yescharacteristic
215C4×C7#45511yescharacteristic
216Dic7—44no—
217Dic7—44no—
218C24(C2×D7)⋊C412yes—
219C2×Q8—74no—
220C2×Q8—144no—
221C2×Q8—142no—
222C2×Q8—562no—
223C2×Q8—562no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4#114212yes—
227C2×D4#114412yes—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—144no—
233C22×C4—144no—
234C22×C4(C2×D7)⋊C412yes—
235C22×C4C2×C4×D711yescharacteristic
236C22×C4C2×D2811yescharacteristic
237C22×C4C2×C22⋊D711yescharacteristic
238M16—42no—
239M16—42no—
240C2×C8—42no—
241C2×C8—142no—
242C2×C8—281no—
243C2×C8—281no—
244C2×C8—282no—
245C2×C8—281no—
246C4⋊C4—22no—
247C4⋊C4—74no—
248C4⋊C4—148no—
249C4⋊C4—144no—
250C4⋊C4—144no—
251C4⋊C4—144no—
252C4⋊C4—142no—
253C4⋊C4—142no—
254C4⋊C4—142no—
255C22⋊C4—24no—
256C22⋊C4—148no—
257C22⋊C4—144no—
258C42—144no—
259C42D4×D712yes—
260C42D4×D712yes—
261C2×C7—21no—
262C2×C7—21no—
263C2×C7—44no—
264C2×C7#178711yescharacteristic
265C2×C7#179111yescharacteristic
266C2×C7#179211yescharacteristic
267C23—22no—
268C23—22no—
269C23—42no—
270C23—42no—
271C23#638414yes—
272C23C2×(C2×D7)⋊C411yescharacteristic
273Q8—284no—
274Q8—282no—
275Q8—282no—
276Q8—564no—
277D4—24no—
278D4—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—82no—
287C2×C4—144no—
288C2×C4—144no—
289C2×C4—564no—
290C2×C4—564no—
291C2×C4D7×C22⋊C412yes—
292C2×C4#644511yescharacteristic
293C2×C4#644511yescharacteristic
294C2×C4C2×#114211yescharacteristic
295C2×C4#650012yes—
296C2×C4C2×#114411yescharacteristic
297C2×C4#651412yes—
298C8—82no—
299C8—282no—
300C8—561no—
301C7#835511yescharacteristic, complemented, Sylow
302C22—21no—
303C22—21no—
304C22—24no—
305C22—44no—
306C22—42no—
307C22—42no—
308C22—82no—
309C22#8972711yescharacteristic
310C22C2×#638411yescharacteristic
311C22#8977611yescharacteristic
312C4—22no—
313C4—21no—
314C4—42no—
315C4—284no—
316C4—284no—
317C4#8951811yescharacteristic
318C4#8951911yescharacteristic
319C4#8981212yes—
320C4#8982212yes—
321C2—21no—
322C2—21no—
323C2—44no—
324C2#1172727911yescharacteristic
325C2#1172784011yescharacteristic
326C2#1172789711yescharacteristic
subgroupG/H
centerC22#89727
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1800C2
socleC22×C7#379
prime radicalC22×C7×D4D4
cosocleC2×C42×C7C23
hypercenter#1800D7
hypocenterC7#8355
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
4. #11728263C2semidirect
5. #11728255C2semidirect
301. C7#8355semidirect

Maximal subgroups

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

subgroupindex
#117428342
C7×#18002
#117293682
#117282632
#117282552
#83557