Subgroups — #36336901

Order 21376.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24924005	C₂	1	1	yes	maximal, characteristic
2	#24923996	C₂	1	1	yes	maximal, characteristic, complemented
3	#24923627	C₂	1	1	yes	maximal, characteristic, complemented
4	#24923622	C₂	1	1	yes	maximal, characteristic, complemented
5	#24923573	C₂	1	1	yes	maximal, characteristic, complemented
6	#24923571	C₂	1	1	yes	maximal, characteristic, complemented
7	#24923557	C₂	1	1	yes	maximal, characteristic, complemented
8	#24923490	C₂	1	1	yes	maximal, characteristic, complemented
9	#24923490	C₂	1	1	yes	maximal, characteristic, complemented
10	#24923466	C₂	1	1	yes	maximal, characteristic
11	#24923397	C₂	1	1	yes	maximal, characteristic, complemented
12	#24923377	C₂	1	1	yes	maximal, characteristic, complemented
13	#24923366	C₂	1	1	yes	maximal, characteristic, complemented
14	#24923360	C₂	1	1	yes	maximal, characteristic, complemented
15	#490×C₁₆₇	C₂	1	1	yes	maximal, characteristic
16	C₂×Dic₆₆₈	C₂²	1	1	yes	characteristic
17	C₂×#14581938	C₂²	1	1	yes	characteristic
18	C₂×#14581938	C₂²	1	1	yes	characteristic
19	#19128831	—	2	1	no	—
20	#19128831	C₂²	1	1	yes	characteristic
21	#19128823	C₂²	1	1	yes	characteristic
22	#19128822	—	2	1	no	—
23	#19128822	—	2	1	no	—
24	#19128819	—	2	1	no	—
25	#19128819	C₂²	1	1	yes	characteristic
26	#19128819	C₂²	1	1	yes	characteristic
27	#19128813	—	2	1	no	—
28	#19128813	—	2	1	no	—
29	C₄×Dic₃₃₄	C₂²	1	1	yes	characteristic
30	C₄×Dic₃₃₄	C₂²	1	1	yes	characteristic
31	C₄×Dic₃₃₄	C₂²	1	1	yes	characteristic
32	#19128797	—	2	1	no	—
33	#19128797	C₂²	1	1	yes	characteristic
34	#19128797	C₂²	1	1	yes	characteristic
35	D₄×Dic₁₆₇	—	2	1	no	—
36	D₄×Dic₁₆₇	C₂²	1	1	yes	characteristic
37	D₄×Dic₁₆₇	C₂²	1	1	yes	characteristic
38	#19128765	C₂²	1	1	yes	characteristic
39	#19128764	C₂²	1	1	yes	characteristic
40	#19128763	C₂²	1	1	yes	characteristic
41	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
44	#19128757	C₂²	1	1	yes	characteristic
45	#19128757	C₂²	1	1	yes	characteristic
46	#19128755	C₂²	1	1	yes	characteristic
47	#19128755	C₂²	1	1	yes	characteristic
48	#19128753	C₂²	1	1	yes	characteristic
49	#19128753	C₂²	1	1	yes	characteristic
50	#19128742	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₆₇	C₂²	1	1	yes	characteristic
52	#127×C₁₆₇	C₂²	1	1	yes	characteristic
53	#125×C₁₆₇	C₂²	1	1	yes	characteristic
54	#102×C₁₆₇	C₂²	1	1	yes	characteristic
55	#102×C₁₆₇	C₂²	1	1	yes	characteristic
56	#102×C₁₆₇	C₂²	1	1	yes	characteristic
57	#102×C₁₆₇	C₂²	1	1	yes	characteristic
58	#97×C₁₆₇	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₃₄	—	2	1	no	—
60	C₂×Dic₃₃₄	D₄	1	1	yes	characteristic
61	C₂×Dic₃₃₄	C₂³	1	1	yes	characteristic
62	C₂×Dic₃₃₄	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₆₇	—	2	1	no	—
64	C₂²×Dic₁₆₇	—	2	1	no	—
65	Dic₆₆₈	—	2	2	no	—
66	#14581938	—	2	2	no	—
67	#14581938	—	2	2	no	—
68	C₂×C₁₆₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₆₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₆₇	—	2	1	no	—
71	C₄⋊Dic₁₆₇	D₄	1	1	yes	characteristic
72	C₄⋊Dic₁₆₇	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₆₇	C₂³	1	1	yes	characteristic
74	Dic₁₆₇⋊C₄	—	2	1	no	—
75	Dic₁₆₇⋊C₄	—	2	1	no	—
76	Dic₁₆₇⋊C₄	—	2	1	no	—
77	Dic₁₆₇⋊C₄	—	2	1	no	—
78	Dic₁₆₇⋊C₄	—	2	1	no	—
79	Dic₁₆₇⋊C₄	—	2	1	no	—
80	Dic₁₆₇⋊C₄	D₄	1	1	yes	characteristic
81	Dic₁₆₇⋊C₄	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₁₆₇	—	2	1	no	—
83	C₂²⋊Dic₁₆₇	—	2	1	no	—
84	C₂²⋊Dic₁₆₇	—	2	1	no	—
85	C₂²⋊Dic₁₆₇	—	2	1	no	—
86	C₂²⋊Dic₁₆₇	—	2	1	no	—
87	C₂²⋊Dic₁₆₇	—	2	1	no	—
88	C₄×Dic₁₆₇	—	2	1	no	—
89	C₄×Dic₁₆₇	—	2	1	no	—
90	C₄×Dic₁₆₇	—	2	1	no	—
91	C₄×Dic₁₆₇	—	2	1	no	—
92	C₄×Dic₁₆₇	C₂³	1	1	yes	characteristic
93	C₄×Dic₁₆₇	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₁₆₇	—	2	1	no	—
95	C₂×D₄×C₁₆₇	—	2	1	no	—
96	C₂×D₄×C₁₆₇	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₆₇	—	2	1	no	—
101	C₄⋊C₄×C₁₆₇	—	2	1	no	—
102	C₄⋊C₄×C₁₆₇	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₆₇	C₂³	1	1	yes	characteristic
104	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
105	Dic₃₃₄	—	2	1	no	—
106	Dic₃₃₄	—	2	1	no	—
107	Dic₃₃₄	—	2	1	no	—
108	Dic₃₃₄	—	4	1	no	—
109	Dic₃₃₄	—	4	2	no	—
110	Dic₃₃₄	#41	1	2	yes	—
111	C₂×Dic₁₆₇	—	2	1	no	—
112	C₂×Dic₁₆₇	—	2	1	no	—
113	C₂×Dic₁₆₇	—	2	1	no	—
114	C₂×Dic₁₆₇	—	2	1	no	—
115	C₂×Dic₁₆₇	—	8	1	no	—
116	C₂×Dic₁₆₇	—	8	1	no	—
117	C₂×Dic₁₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₆₇	C₂×D₄	1	1	yes	characteristic
121	C₁₆₇⋊C₈	—	2	1	no	—
122	C₁₆₇⋊C₈	—	2	1	no	—
123	C₂³×C₁₆₇	—	2	1	no	—
124	C₂³×C₁₆₇	—	2	1	no	—
125	D₄×C₁₆₇	—	2	1	no	—
126	D₄×C₁₆₇	—	2	1	no	—
127	D₄×C₁₆₇	—	2	1	no	—
128	D₄×C₁₆₇	—	2	1	no	—
129	D₄×C₁₆₇	—	4	2	no	—
130	D₄×C₁₆₇	—	8	1	no	—
131	C₂×C₄×C₁₆₇	—	2	1	no	—
132	C₂×C₄×C₁₆₇	—	2	1	no	—
133	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
136	C₈×C₁₆₇	—	2	1	no	—
137	C₈×C₁₆₇	—	2	1	no	—
138	C₂²×C₁₆₇	—	4	1	no	—
139	C₂²×C₁₆₇	—	4	1	no	—
140	C₂²×C₁₆₇	—	4	1	no	—
141	C₂²×C₁₆₇	—	4	1	no	—
142	C₂²×C₁₆₇	—	4	1	no	—
143	C₂²×C₁₆₇	—	4	1	no	—
144	C₂²×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
145	C₄×C₁₆₇	—	2	1	no	—
146	C₄×C₁₆₇	—	2	1	no	—
147	C₄×C₁₆₇	—	4	1	no	—
148	C₄×C₁₆₇	—	4	1	no	—
149	C₄×C₁₆₇	C₂×#41	1	1	yes	characteristic
150	C₄×C₁₆₇	2₋¹⁺⁴	1	1	yes	characteristic
151	Dic₁₆₇	—	2	2	no	—
152	Dic₁₆₇	—	2	1	no	—
153	Dic₁₆₇	—	2	1	no	—
154	Dic₁₆₇	—	2	1	no	—
155	Dic₁₆₇	—	2	1	no	—
156	Dic₁₆₇	—	4	1	no	—
157	Dic₁₆₇	—	4	1	no	—
158	Dic₁₆₇	—	4	1	no	—
159	C₂×C₁₆₇	—	8	1	no	—
160	C₂×C₁₆₇	—	8	1	no	—
161	C₂×C₁₆₇	#548	1	1	yes	characteristic
162	C₂×C₁₆₇	#575	1	1	yes	characteristic
163	C₂×C₁₆₇	#577	1	1	yes	characteristic
164	C₁₆₇	#3371	1	1	yes	characteristic, complemented, Sylow
165	#3371	—	167	1	no	maximal, Sylow
166	#559	—	167	1	no	—
167	#557	—	167	1	no	—
168	#490	D₁₆₇	1	1	yes	characteristic
169	#489	—	167	1	no	—
170	#488	—	167	1	no	—
171	#479	—	167	1	no	—
172	#478	—	167	1	no	—
173	#476	—	167	1	no	—
174	#464	—	167	1	no	—
175	#464	—	167	1	no	—
176	#460	—	167	1	no	—
177	#446	—	167	1	no	—
178	#441	—	167	1	no	—
179	#440	—	167	1	no	—
180	C₄×SD₁₆	—	167	1	no	—
181	C₂×Q₁₆	—	167	1	no	—
182	C₂×SD₁₆	—	167	1	no	—
183	C₂×SD₁₆	—	167	1	no	—
184	C₄⋊Q₈	—	334	1	no	—
185	C₄⋊Q₈	—	167	1	no	—
186	#127	C₂×D₁₆₇	1	1	yes	characteristic
187	#125	—	334	1	no	—
188	#125	—	334	1	no	—
189	#125	—	167	1	no	—
190	#125	C₂×D₁₆₇	1	1	yes	characteristic
191	#124	—	334	1	no	—
192	#124	—	167	1	no	—
193	#124	—	167	1	no	—
194	#123	—	334	1	no	—
195	#123	—	334	1	no	—
196	C₄×Q₈	—	334	1	no	—
197	C₄×Q₈	—	167	1	no	—
198	C₄×Q₈	—	167	1	no	—
199	C₄×Q₈	—	167	1	no	—
200	C₄×Q₈	—	167	1	no	—
201	C₄×Q₈	—	167	1	no	—
202	C₄×D₄	—	334	1	no	—
203	C₄×D₄	—	167	1	no	—
204	C₄×D₄	—	167	1	no	—
205	#107	—	167	1	no	—
206	#106	—	167	1	no	—
207	#106	—	167	1	no	—
208	C₄⋊C₈	—	167	1	no	—
209	C₄⋊C₈	—	167	1	no	—
210	C₄⋊C₈	—	167	1	no	—
211	#103	—	167	1	no	—
212	#103	—	167	1	no	—
213	#103	—	167	1	no	—
214	#103	—	167	1	no	—
215	#102	—	167	1	no	—
216	#102	—	167	1	no	—
217	#102	C₂×D₁₆₇	1	1	yes	characteristic
218	#102	C₂×D₁₆₇	1	1	yes	characteristic
219	#102	C₂×D₁₆₇	1	1	yes	characteristic
220	#102	C₂×D₁₆₇	1	1	yes	characteristic
221	#97	—	167	1	no	—
222	#97	C₂×D₁₆₇	1	1	yes	characteristic
223	C₄×C₈	—	167	1	no	—
224	C₂×Q₈	—	334	1	no	—
225	C₂×Q₈	—	167	1	no	—
226	C₂×Q₈	—	167	1	no	—
227	C₂×Q₈	—	167	1	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	C₂²×D₁₆₇	1	1	yes	characteristic
231	C₂×D₄	C₂²×D₁₆₇	1	1	yes	characteristic
232	C₂²×C₄	—	334	1	no	—
233	C₂²×C₄	—	334	1	no	—
234	Q₁₆	—	334	2	no	—
235	SD₁₆	—	334	2	no	—
236	SD₁₆	—	334	2	no	—
237	C₂×C₈	—	167	1	no	—
238	C₂×C₈	—	167	1	no	—
239	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	334	1	no	—
244	C₄⋊C₄	—	334	1	no	—
245	C₄⋊C₄	—	334	1	no	—
246	C₄⋊C₄	—	334	1	no	—
247	C₄⋊C₄	—	334	1	no	—
248	C₄⋊C₄	—	334	1	no	—
249	C₄⋊C₄	—	334	1	no	—
250	C₄⋊C₄	—	167	1	no	—
251	C₄⋊C₄	—	167	1	no	—
252	C₄⋊C₄	—	167	1	no	—
253	C₄⋊C₄	—	167	1	no	—
254	C₄⋊C₄	—	167	1	no	—
255	C₄⋊C₄	C₂²×D₁₆₇	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₆₇	1	1	yes	characteristic
257	C₂²⋊C₄	—	334	1	no	—
258	C₂²⋊C₄	—	334	1	no	—
259	C₂²⋊C₄	—	334	1	no	—
260	C₂²⋊C₄	—	334	1	no	—
261	C₂²⋊C₄	—	334	1	no	—
262	C₂²⋊C₄	—	334	1	no	—
263	C₄²	—	334	1	no	—
264	C₄²	—	334	1	no	—
265	C₄²	—	334	1	no	—
266	C₄²	—	334	1	no	—
267	C₄²	—	167	1	no	—
268	C₄²	—	167	1	no	—
269	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	Q₈	—	334	1	no	—
273	Q₈	—	334	1	no	—
274	Q₈	—	334	1	no	—
275	Q₈	—	668	1	no	—
276	Q₈	—	668	2	no	—
277	Q₈	—	167	2	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	2	1	no	—
282	D₄	—	4	2	no	—
283	D₄	—	8	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	1336	1	no	—
287	C₂×C₄	—	1336	1	no	—
288	C₂×C₄	—	334	1	no	—
289	C₂×C₄	—	334	1	no	—
290	C₂×C₄	—	334	1	no	—
291	C₂×C₄	—	334	1	no	—
292	C₂×C₄	—	167	1	no	—
293	C₂×C₄	—	167	1	no	—
294	C₂×C₄	—	167	1	no	—
295	C₂×C₄	—	167	1	no	—
296	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
297	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₆₇	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	334	1	no	—
302	C₈	—	334	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	C₂×D₄×D₁₆₇	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	334	2	no	—
315	C₄	—	334	1	no	—
316	C₄	—	334	1	no	—
317	C₄	—	334	1	no	—
318	C₄	—	334	1	no	—
319	C₄	—	668	1	no	—
320	C₄	—	668	1	no	—
321	C₄	—	668	1	no	—
322	C₄	#41×D₁₆₇	1	1	yes	characteristic
323	C₄	#19128879	1	1	yes	characteristic
324	C₂	—	8	1	no	—
325	C₂	—	8	1	no	—
326	C₂	#24923927	1	1	yes	characteristic
327	C₂	#24924065	1	1	yes	characteristic
328	C₂	#24924079	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₆₇
commutator, cosocle	C₂×C₄×C₁₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₆₇
Fitting	#490×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂²×D₄
prime radical	#127×C₁₆₇	C₂²
hypercenter	#490	D₁₆₇
hypocenter	C₁₆₇	#3371
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #490

Complements

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

N	H ≅ G/N	type
2. #24923996	C₂	semidirect
3. #24923627	C₂	semidirect
4. #24923622	C₂	semidirect
5. #24923573	C₂	semidirect
6. #24923571	C₂	semidirect
7. #24923557	C₂	semidirect
8. #24923490	C₂	semidirect
9. #24923490	C₂	semidirect
11. #24923397	C₂	semidirect
12. #24923377	C₂	semidirect
13. #24923366	C₂	semidirect
14. #24923360	C₂	semidirect
164. C₁₆₇	#3371	semidirect

Maximal subgroups

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

subgroup	index
#24924005	2
#24923996	2
#24923627	2
#24923622	2
#24923573	2
#24923571	2
#24923557	2
#24923490	2
#24923466	2
#24923397	2
#24923377	2
#24923366	2
#24923360	2
#490×C₁₆₇	2
#3371	167