Finite Group Databasebeta

← C67×#10101 #30689352

Subgroups — C67×#10101 #30689352

Order 17152.

474 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C67×#3236C211yesmaximal, characteristic
2C67×#3165C211yesmaximal, characteristic
3C67×#2660C211yesmaximal, characteristic
4C67×#1556C211yesmaximal, characteristic
5C67×#1528C211yesmaximal, characteristic
6C67×#1506C211yesmaximal, characteristic
7C67×#1300C211yesmaximal, characteristic
8#533×C67C2211yescharacteristic
9C2×C4⋊Q8×C67C2211yescharacteristic
10C2×#125×C67C2211yescharacteristic
11#501×C67—22no—
12#501×C67—21no—
13#500×C67—21no—
14#498×C67—21no—
15#491×C67—21no—
16#484×C67—21no—
17#483×C67—21no—
18#479×C67—22no—
19#432×C67C2211yescharacteristic
20#432×C67C2211yescharacteristic
21#428×C67—22no—
22C4×M16×C67C2211yescharacteristic
23#398×C67—22no—
24#395×C67—22no—
25#384×C67—21no—
26#382×C67—21no—
27#376×C67C2211yescharacteristic
28#332×C67—22no—
29#330×C67—22no—
30#328×C67—22no—
31C22×Q8×C67—21no—
32C2×M16×C67—21no—
33C4⋊Q8×C67—21no—
34C4⋊Q8×C67—21no—
35C4⋊Q8×C67—21no—
36C4⋊Q8×C67—21no—
37C4⋊Q8×C67D411yescharacteristic
38C4⋊Q8×C67D411yescharacteristic
39#125×C67—21no—
40#125×C67—21no—
41#125×C67—21no—
42#125×C67—21no—
43#125×C67—21no—
44C22⋊Q8×C67—21no—
45C22⋊Q8×C67—21no—
46C4×Q8×C67—21no—
47C4×Q8×C67—21no—
48#117×C67—21no—
49C2×C4⋊C4×C67—21no—
50C2×C4⋊C4×C67—21no—
51C2×C4⋊C4×C67—21no—
52C2×C4⋊C4×C67—24no—
53C2×C4⋊C4×C67D411yescharacteristic
54C2×C4⋊C4×C67D411yescharacteristic
55C2×C4⋊C4×C67D412yes—
56C2×C42×C67—21no—
57C2×C42×C67C2311yescharacteristic
58#107×C67—22no—
59#107×C67—24no—
60#106×C67—22no—
61#106×C67—24no—
62C4⋊C8×C67—21no—
63C4⋊C8×C67D412yes—
64#103×C67—22no—
65#103×C67—22no—
66C22⋊C8×C67—21no—
67C22⋊C8×C67—21no—
68#97×C67—21no—
69#97×C67—21no—
70#97×C67D412yes—
71C4×C8×C67—21no—
72#95×C67—24no—
73#95×C67—21no—
74#95×C67D412yes—
75C2×Q8×C67—21no—
76C2×Q8×C67—21no—
77C2×Q8×C67—41no—
78C2×Q8×C67—41no—
79C2×Q8×C67—82no—
80C22×C4×C67—22no—
81C22×C4×C67—21no—
82C22×C4×C67—22no—
83C22×C4×C67—21no—
84C22×C4×C67C2×D411yescharacteristic
85C22×C4×C67C2×D411yescharacteristic
86C22×C4×C67C2×D411yescharacteristic
87M16×C67—24no—
88C2×C8×C67—22no—
89C2×C8×C67—22no—
90C2×C8×C67—41no—
91C4⋊C4×C67—21no—
92C4⋊C4×C67—21no—
93C4⋊C4×C67—21no—
94C4⋊C4×C67—22no—
95C4⋊C4×C67—21no—
96C4⋊C4×C67—21no—
97C4⋊C4×C67—21no—
98C4⋊C4×C67—21no—
99C4⋊C4×C67—22no—
100C4⋊C4×C67—21no—
101C4⋊C4×C67—41no—
102C4⋊C4×C67—41no—
103C4⋊C4×C67—41no—
104C4⋊C4×C67—41no—
105C4⋊C4×C67—41no—
106C4⋊C4×C67—41no—
107C4⋊C4×C67—41no—
108C4⋊C4×C67—41no—
109C4⋊C4×C67—84no—
110C22⋊C4×C67—41no—
111C42×C67—24no—
112C42×C67—41no—
113C42×C67C2×D411yescharacteristic
114C42×C67C2×D411yescharacteristic
115C42×C67C2×D411yescharacteristic
116C42×C67#4111yescharacteristic
117C23×C67#12011yescharacteristic
118Q8×C67—42no—
119Q8×C67—81no—
120Q8×C67—82no—
121Q8×C67—82no—
122C2×C4×C67—22no—
123C2×C4×C67—22no—
124C2×C4×C67—24no—
125C2×C4×C67—41no—
126C2×C4×C67—42no—
127C2×C4×C67—42no—
128C2×C4×C67—42no—
129C2×C4×C67—44no—
130C2×C4×C67—41no—
131C2×C4×C67—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—42no—
135C2×C4×C67—42no—
136C2×C4×C67#12011yescharacteristic
137C2×C4×C67#12011yescharacteristic
138C2×C4×C67#12111yescharacteristic
139C2×C4×C67#12111yescharacteristic
140C2×C4×C67#12111yescharacteristic
141C2×C4×C67#12711yescharacteristic
142C8×C67—24no—
143C8×C67—42no—
144C8×C67—81no—
145C22×C67—21no—
146C22×C67—21no—
147C22×C67#39211yescharacteristic
148C22×C67#45612yes—
149C4×C67—21no—
150C4×C67—21no—
151C4×C67—21no—
152C4×C67—21no—
153C4×C67—21no—
154C4×C67—44no—
155C4×C67—81no—
156C4×C67—81no—
157C4×C67—81no—
158C4×C67—82no—
159C4×C67—81no—
160C4×C67#45612yes—
161#10101C6711yesmaximal, characteristic, direct product factor, Sylow
162C2×C67—22no—
163C2×C67#200511yescharacteristic
164C2×C67#200511yescharacteristic
165C2×C67#200811yescharacteristic
166#3236C2×C6711yescharacteristic
167#3165C2×C6711yescharacteristic
168#2660C2×C6711yescharacteristic
169#1556C2×C6711yescharacteristic
170#1528C2×C6711yescharacteristic
171#1506C2×C6711yescharacteristic
172#1300C2×C6711yescharacteristic
173C67#1010111yescharacteristic, direct product factor, Sylow
174#533C22×C6711yescharacteristic
175C2×C4⋊Q8C22×C6711yescharacteristic
176C2×#125C22×C6711yescharacteristic
177#501—21no—
178#501—22no—
179#500—21no—
180#498—21no—
181#491—21no—
182#484—21no—
183#483—21no—
184#479—22no—
185#432C22×C6711yescharacteristic
186#432C22×C6711yescharacteristic
187#428—22no—
188C4×M16C22×C6711yescharacteristic
189#398—22no—
190#395—22no—
191#384—21no—
192#382—21no—
193#376C22×C6711yescharacteristic
194#332—22no—
195#330—22no—
196#328—22no—
197C22×Q8—21no—
198C2×M16—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8D4×C6711yescharacteristic
204C4⋊Q8D4×C6711yescharacteristic
205#125—21no—
206#125—21no—
207#125—21no—
208#125—21no—
209#125—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C4×Q8—21no—
213C4×Q8—21no—
214#117—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4D4×C6711yescharacteristic
220C2×C4⋊C4D4×C6711yescharacteristic
221C2×C4⋊C4D4×C6712yes—
222C2×C42—21no—
223C2×C42C23×C6711yescharacteristic
224#107—24no—
225#107—22no—
226#106—24no—
227#106—22no—
228C4⋊C8—21no—
229C4⋊C8D4×C6712yes—
230#103—22no—
231#103—22no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234#97—21no—
235#97—21no—
236#97D4×C6712yes—
237C4×C8—21no—
238#95—24no—
239#95—21no—
240#95D4×C6712yes—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—82no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4C2×D4×C6711yescharacteristic
251C22×C4C2×D4×C6711yescharacteristic
252C22×C4C2×D4×C6711yescharacteristic
253M16—24no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—41no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—84no—
276C22⋊C4—41no—
277C42—24no—
278C42—41no—
279C42C2×D4×C6711yescharacteristic
280C42C2×D4×C6711yescharacteristic
281C42C2×D4×C6711yescharacteristic
282C42#41×C6711yescharacteristic
283C23#120×C6711yescharacteristic
284Q8—42no—
285Q8—81no—
286Q8—82no—
287Q8—82no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4#120×C6711yescharacteristic
303C2×C4#120×C6711yescharacteristic
304C2×C4#121×C6711yescharacteristic
305C2×C4#121×C6711yescharacteristic
306C2×C4#121×C6711yescharacteristic
307C2×C4#127×C6711yescharacteristic
308C8—24no—
309C8—42no—
310C8—81no—
311C22—21no—
312C22—21no—
313C22#392×C6711yescharacteristic
314C22#456×C6712yes—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—44no—
321C4—81no—
322C4—81no—
323C4—82no—
324C4—81no—
325C4—81no—
326C4#456×C6712yes—
327C2—22no—
328C2C67×#200511yescharacteristic
329C2C67×#200511yescharacteristic
330C2C67×#200811yescharacteristic
subgroupG/H
center, socleC22×C67#392
commutator, Frattini, cosocleC2×C42C23×C67
prime radicalC23×C67#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
161. #10101C67direct (N × H)
173. C67#10101direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#32362
C67×#31652
C67×#26602
C67×#15562
C67×#15282
C67×#15062
C67×#13002
#1010167