Finite Group Databasebeta

← #13056596

Subgroups — #13056596

Order 1344.

476 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×#4268C211yesmaximal, characteristic
2C2×C3×C7×C22⋊C4C211yesmaximal, characteristic, complemented
3C2×#66636C211yesmaximal, characteristic, complemented
4C2×#66631C211yesmaximal, characteristic, complemented
5C2×#66631C211yesmaximal, characteristic, complemented
6C2×#66629C211yesmaximal, characteristic, complemented
7#10620670C211yesmaximal, characteristic, complemented
8#89731—31nomaximal, Hall
9C24×C3×C7C2211yescharacteristic
10C23×D21C2211yescharacteristic
11C22×C3×C4×C7C2211yescharacteristic
12C22×C3×C4×C7C2211yescharacteristic
13C2×#4268—22no—
14C2×#4268—22no—
15C2×#4268—22no—
16C22×Dic21C2211yescharacteristic
17C22×Dic21C2211yescharacteristic
18C22×Dic21C2211yescharacteristic
19C3×C7×C22⋊C4—22no—
20#66636—22no—
21#66631—22no—
22#66631—22no—
23#66629—22no—
24C22×C22⋊D7—31no—
25C2×C7×C22⋊C4D311yescharacteristic, complemented
26C2×C22⋊Dic7—31no—
27C2×(C2×D7)⋊C4—31no—
28C2×(C2×D7)⋊C4—31no—
29C2×Dic7⋊C4—31no—
30C7⋊#95—31no—
31#4959—71nomaximal, Hall
32C23×C3×C7—21no—
33C23×C3×C7—21no—
34C23×C3×C7—21no—
35C23×C3×C7—21no—
36C23×C3×C7—21no—
37C23×C3×C7—21no—
38C23×C3×C7D412yes—
39C23×C3×C7C2311yescharacteristic
40C22×D21—21no—
41C22×D21—21no—
42C22×D21—21no—
43C22×D21—21no—
44C22×D21—21no—
45C22×D21—21no—
46C22×D21D412yes—
47C2×C3×C4×C7—21no—
48C2×C3×C4×C7—21no—
49C2×C3×C4×C7—21no—
50C2×C3×C4×C7—21no—
51C2×C3×C4×C7—21no—
52C2×C3×C4×C7—21no—
53#4268—28no—
54C2×Dic21—21no—
55C2×Dic21—21no—
56C2×Dic21—21no—
57C2×Dic21—21no—
58C2×Dic21—21no—
59C2×Dic21—21no—
60C2×Dic21—21no—
61C2×Dic21D412yes—
62C2×Dic21D412yescomplemented
63C24×C7C2×D311yescharacteristic
64C23×D7—31no—
65C22×C4×C7C2×D311yescharacteristic
66C22×C4×C7C2×D311yescharacteristic
67C2×C22⋊D7—62no—
68C2×C22⋊D7—62no—
69C2×C22⋊D7—62no—
70C22×Dic7—31no—
71C22×Dic7—31no—
72C22×Dic7—31no—
73C7×C22⋊C4—22no—
74C22⋊Dic7—62no—
75(C2×D7)⋊C4—62no—
76(C2×D7)⋊C4—62no—
77Dic7⋊C4—62no—
78C22×#67—71no—
79C2×C3×C22⋊C4D711yescharacteristic, complemented
80C2×#217—71no—
81C2×(C2×D3)⋊C4—71no—
82C2×(C2×D3)⋊C4—71no—
83C2×Dic3⋊C4—71no—
84C3⋊#95—71no—
85C22×C3×C7—22no—
86C22×C3×C7—22no—
87C22×C3×C7—22no—
88C22×C3×C7—41no—
89C22×C3×C7—41no—
90C22×C3×C7—41no—
91C22×C3×C7—41no—
92C22×C3×C7C2×D411yescharacteristic
93C22×C3×C7C2×D411yescharacteristic
94C22×C3×C7C2×D411yescharacteristic
95C22×C3×C7C2×D411yescharacteristic
96C22×C3×C7#4111yescharacteristic
97C22×C3×C7#4111yescharacteristic
98C22×C3×C7#4111yescharacteristic
99C2×D21—22no—
100C2×D21—22no—
101C2×D21—22no—
102C2×D21—41no—
103C2×D21—41no—
104C2×D21—41no—
105C2×D21—41no—
106C3×C4×C7—41no—
107C3×C4×C7—41no—
108Dic21—22no—
109Dic21—22no—
110Dic21—41no—
111#394—211noSylow
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7#6712yes—
119C23×C7C22×D311yescharacteristic
120C22×D7—32no—
121C22×D7—61no—
122C22×D7—61no—
123C22×D7—61no—
124C22×D7—61no—
125C22×D7—61no—
126C22×D7—61no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C22⋊D7—68no—
134C2×Dic7—32no—
135C2×Dic7—32no—
136C2×Dic7—61no—
137C2×Dic7—61no—
138C2×Dic7—61no—
139C2×Dic7—61no—
140C2×Dic7—61no—
141C2×Dic7—61no—
142C2×Dic7—61no—
143C24×C3C2×D711yescharacteristic
144C23×D3—71no—
145C22×C3×C4C2×D711yescharacteristic
146C22×C3×C4C2×D711yescharacteristic
147C2×#67—142no—
148C2×#67—142no—
149C2×#67—142no—
150C22×Dic3—71no—
151C22×Dic3—71no—
152C22×Dic3—71no—
153C3×C22⋊C4—22no—
154#217—142no—
155(C2×D3)⋊C4—142no—
156(C2×D3)⋊C4—142no—
157Dic3⋊C4—142no—
158C2×C3×C7—42no—
159C2×C3×C7#12011yescharacteristic
160C2×C3×C7#12111yescharacteristic
161C2×C3×C7#12111yescharacteristic
162C2×C3×C7#12111yescharacteristic
163C2×C3×C7#12311yescharacteristic
164C2×C3×C7#12311yescharacteristic
165C2×C3×C7#12411yescharacteristic
166D21—42no—
167C22×D4—211no—
168C2×C4⋊C4—211no—
169C2×C22⋊C4—211no—
170C2×C22⋊C4—211no—
171C2×C22⋊C4—211no—
172C2×C22⋊C4D2111yescharacteristic, complemented
173#95—211no—
174C22×C7—22no—
175C22×C7—22no—
176C22×C7—22no—
177C22×C7—41no—
178C22×C7—41no—
179C22×C7—41no—
180C22×C7—41no—
181C22×C7#23511yescharacteristic
182C22×C7#23511yescharacteristic
183C22×C7D3×D411yescharacteristic
184C22×C7D3×D411yescharacteristic
185C22×C7D3×D411yescharacteristic
186C22×C7#23711yescharacteristic
187C22×C7C2×#6711yescharacteristic
188C2×D7—62no—
189C2×D7—62no—
190C2×D7—62no—
191C2×D7—121no—
192C2×D7—121no—
193C2×D7—121no—
194C2×D7—121no—
195C4×C7—41no—
196C4×C7—41no—
197Dic7—62no—
198Dic7—62no—
199Dic7—121no—
200C23×C3—21no—
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—21no—
204C23×C3—21no—
205C23×C3—21no—
206C23×C3C22⋊D712yes—
207C23×C3C22×D711yescharacteristic
208C22×D3—72no—
209C22×D3—141no—
210C22×D3—141no—
211C22×D3—141no—
212C22×D3—141no—
213C22×D3—141no—
214C22×D3—141no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221#67—148no—
222C2×Dic3—72no—
223C2×Dic3—72no—
224C2×Dic3—141no—
225C2×Dic3—141no—
226C2×Dic3—141no—
227C2×Dic3—141no—
228C2×Dic3—141no—
229C2×Dic3—141no—
230C2×Dic3—141no—
231C3×C7#39411yescharacteristic, complemented, Hall
232C24—211no—
233C24C2×D2111yescharacteristic
234C2×D4—422no—
235C2×D4—422no—
236C2×D4—422no—
237C22×C4—211no—
238C22×C4—211no—
239C22×C4—211no—
240C22×C4C2×D2111yescharacteristic
241C22×C4C2×D2111yescharacteristic
242C4⋊C4—422no—
243C22⋊C4—22no—
244C22⋊C4—422no—
245C22⋊C4—422no—
246C22⋊C4—422no—
247C2×C7—42no—
248C2×C7#88311yescharacteristic
249C2×C7#88411yescharacteristic
250C2×C7#88411yescharacteristic
251C2×C7#88511yescharacteristic
252C2×C7#92911yescharacteristic
253C2×C7#93711yescharacteristic
254C2×C7#93911yescharacteristic
255D7—122no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—22no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3#115711yescharacteristic
264C22×C3#115711yescharacteristic
265C22×C3D4×D711yescharacteristic
266C22×C3D4×D711yescharacteristic
267C22×C3D4×D711yescharacteristic
268C22×C3#115911yescharacteristic
269C22×C3C2×C22⋊D711yescharacteristic
270C2×D3—142no—
271C2×D3—142no—
272C2×D3—142no—
273C2×D3—281no—
274C2×D3—281no—
275C2×D3—281no—
276C2×D3—281no—
277C3×C4—41no—
278C3×C4—41no—
279Dic3—142no—
280Dic3—142no—
281Dic3—281no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—212no—
289C23—421no—
290C23—421no—
291C23—421no—
292C23—421no—
293C23—421no—
294C23—421no—
295C23#426812yes—
296C23C22×D2111yescharacteristic
297D4—428no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—212no—
305C2×C4—212no—
306C2×C4—421no—
307C2×C4—421no—
308C2×C4—421no—
309C2×C4—421no—
310C2×C4—421no—
311C2×C4—421no—
312C2×C4—421no—
313C7#495911yescharacteristic, complemented, Sylow
314C2×C3—42no—
315C2×C3#644611yescharacteristic
316C2×C3#644711yescharacteristic
317C2×C3#644711yescharacteristic
318C2×C3#644811yescharacteristic
319C2×C3#649211yescharacteristic
320C2×C3#650011yescharacteristic
321C2×C3#650211yescharacteristic
322D3—282no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—422no—
331C22—422no—
332C22—422no—
333C22—841no—
334C22—841no—
335C22—841no—
336C22—841no—
337C22#6672811yescharacteristic
338C22#6672811yescharacteristic
339C22D4×D2111yescharacteristic
340C22D4×D2111yescharacteristic
341C22D4×D2111yescharacteristic
342C22#6673011yescharacteristic
343C22C2×#426811yescharacteristic
344C4—41no—
345C4—41no—
346C4—422no—
347C4—422no—
348C4—841no—
349C3#8973111yescharacteristic, complemented, Sylow
350C2—42no—
351C2—842no—
352C2#1062136611yescharacteristic
353C2#1062136711yescharacteristic
354C2#1062136711yescharacteristic
355C2#1062136811yescharacteristic
356C2#1062141211yescharacteristic
357C2#1062142011yescharacteristic
358C2#1062142211yescharacteristic
subgroupG/H
center, FrattiniC23C22×D21
commutator, socle, cosocleC23×C3×C7C23
FittingC2×C3×C7×C22⋊C4C2
prime radicalC22×#4268C2
hypercenterC2×C22⋊C4D21
hypocenterC3×C7#394
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C7×C22⋊C4C2semidirect
3. C2×#66636C2semidirect
4. C2×#66631C2semidirect
5. C2×#66631C2semidirect
6. C2×#66629C2semidirect
7. #10620670C2semidirect
25. C2×C7×C22⋊C4D3semidirect
62. C2×Dic21D4semidirect
79. C2×C3×C22⋊C4D7semidirect
172. C2×C22⋊C4D21semidirect
231. C3×C7#394semidirect
313. C7#4959semidirect
349. C3#89731semidirect

Maximal subgroups

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

subgroupindex
C22×#42682
C2×C3×C7×C22⋊C42
C2×#666362
C2×#666312
C2×#666292
#106206702
#897313
#49597