Subgroups — #36334665

Order 21376.

340 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923899	C₂	1	1	yes	maximal, characteristic
2	#24923896	C₂	1	1	yes	maximal, characteristic
3	#24923678	C₂	1	1	yes	maximal, characteristic, complemented
4	#24923677	C₂	1	1	yes	maximal, characteristic, complemented
5	#24923676	C₂	1	1	yes	maximal, characteristic, complemented
6	#24923670	C₂	1	1	yes	maximal, characteristic, complemented
7	#24923668	C₂	1	1	yes	maximal, characteristic, complemented
8	#24923666	C₂	1	1	yes	maximal, characteristic, complemented
9	#24923665	C₂	1	1	yes	maximal, characteristic, complemented
10	#24923310	C₂	1	1	yes	maximal, characteristic
11	#24923308	C₂	1	1	yes	maximal, characteristic
12	#24923306	C₂	1	1	yes	maximal, characteristic
13	#106×D₁₆₇	C₂	1	1	yes	maximal, characteristic
14	#24923226	C₂	1	1	yes	maximal, characteristic
15	#501×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14581933	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₆₇	C₂²	1	1	yes	characteristic
18	#19128832	C₂²	1	1	yes	characteristic
19	#19128831	—	2	1	no	—
20	#19128831	C₂²	1	1	yes	characteristic
21	#19128831	C₂²	1	1	yes	characteristic
22	#19128830	—	2	1	no	—
23	#19128824	C₂²	1	1	yes	characteristic
24	#19128823	C₂²	1	1	yes	characteristic
25	#19128823	C₂²	1	1	yes	characteristic
26	#19128822	—	2	1	no	—
27	#19128822	—	2	1	no	—
28	#19128812	—	2	1	no	—
29	#19128810	—	2	1	no	—
30	#19128810	—	2	1	no	—
31	#19128808	—	2	1	no	—
32	#19128790	C₂²	1	1	yes	characteristic
33	#19128790	C₂²	1	1	yes	characteristic
34	#19128789	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₆₇	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₆₇	C₂²	1	1	yes	characteristic
37	#19128766	C₂²	1	1	yes	characteristic
38	#19128766	C₂²	1	1	yes	characteristic
39	#19128765	C₂²	1	1	yes	characteristic
40	#19128765	C₂²	1	1	yes	characteristic
41	#19128764	C₂²	1	1	yes	characteristic
42	#19128764	C₂²	1	1	yes	characteristic
43	#19128763	C₂²	1	1	yes	characteristic
44	#19128763	C₂²	1	1	yes	characteristic
45	#19128763	C₂²	1	1	yes	characteristic
46	#19128763	C₂²	1	1	yes	characteristic
47	#19128763	C₂²	1	1	yes	characteristic
48	#19128763	C₂²	1	1	yes	characteristic
49	#19128742	C₂²	1	1	yes	characteristic
50	C₄×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₆₇	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
53	#125×C₁₆₇	C₂²	1	1	yes	characteristic
54	#107×C₁₆₇	C₂²	1	1	yes	characteristic
55	#107×C₁₆₇	C₂²	1	1	yes	characteristic
56	#106×C₁₆₇	C₂²	1	1	yes	characteristic
57	#106×C₁₆₇	C₂²	1	1	yes	characteristic
58	#97×C₁₆₇	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₃₄	—	2	1	no	—
60	C₂×Dic₃₃₄	—	2	1	no	—
61	C₂×Dic₃₃₄	—	2	1	no	—
62	C₂×C₄×D₁₆₇	—	2	1	no	—
63	C₂×C₄×D₁₆₇	—	2	1	no	—
64	C₂×C₄×D₁₆₇	C₂³	1	1	yes	characteristic
65	#14581933	Q₈	1	4	yes	complemented
66	C₂×C₁₆₇⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₁₆₇⋊C₈	C₂³	1	1	yes	characteristic
68	C₈×D₁₆₇	Q₈	1	4	yes	complemented
69	C₄⋊Dic₁₆₇	—	2	1	no	—
70	C₄⋊Dic₁₆₇	—	2	1	no	—
71	C₄⋊Dic₁₆₇	—	2	1	no	—
72	C₄⋊Dic₁₆₇	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₆₇	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₆₇	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
81	Dic₁₆₇⋊C₄	—	2	1	no	—
82	Dic₁₆₇⋊C₄	—	2	1	no	—
83	Dic₁₆₇⋊C₄	—	2	1	no	—
84	Dic₁₆₇⋊C₄	D₄	1	1	yes	characteristic
85	Dic₁₆₇⋊C₄	D₄	1	1	yes	characteristic
86	(C₂×D₁₆₇)⋊C₄	—	2	1	no	—
87	(C₂×D₁₆₇)⋊C₄	—	2	1	no	—
88	(C₂×D₁₆₇)⋊C₄	D₄	1	1	yes	characteristic
89	(C₂×D₁₆₇)⋊C₄	D₄	1	1	yes	characteristic
90	C₄×Dic₁₆₇	—	2	1	no	—
91	C₄×Dic₁₆₇	—	2	1	no	—
92	C₄×Dic₁₆₇	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₁₆₇	—	2	1	no	—
94	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₁₆₇	—	2	1	no	—
97	C₄⋊C₄×C₁₆₇	—	2	1	no	—
98	C₄⋊C₄×C₁₆₇	—	2	1	no	—
99	C₄⋊C₄×C₁₆₇	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₆₇	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₆₇	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₆₇	C₂³	1	1	yes	characteristic
103	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
104	C₂²×D₁₆₇	C₂×D₄	1	1	yes	characteristic
105	Dic₃₃₄	—	8	1	no	—
106	Dic₃₃₄	—	8	1	no	—
107	Dic₃₃₄	—	8	1	no	—
108	C₂×Dic₁₆₇	—	2	1	no	—
109	C₂×Dic₁₆₇	—	2	1	no	—
110	C₂×Dic₁₆₇	—	2	1	no	—
111	C₂×Dic₁₆₇	—	2	1	no	—
112	C₂×Dic₁₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×Dic₁₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×Dic₁₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄×D₁₆₇	—	8	1	no	—
116	C₄×D₁₆₇	—	8	1	no	—
117	C₄×D₁₆₇	C₂×Q₈	1	2	yes	—
118	C₄×D₁₆₇	C₂×Q₈	1	2	yes	—
119	C₁₆₇⋊C₈	C₂×Q₈	1	2	yes	—
120	C₁₆₇⋊C₈	C₂×Q₈	1	2	yes	—
121	Q₈×C₁₆₇	—	8	1	no	—
122	C₂×C₄×C₁₆₇	—	2	1	no	—
123	C₂×C₄×C₁₆₇	—	2	1	no	—
124	C₂×C₄×C₁₆₇	—	2	1	no	—
125	C₂×C₄×C₁₆₇	—	2	1	no	—
126	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
129	C₈×C₁₆₇	C₂×Q₈	1	2	yes	—
130	C₈×C₁₆₇	C₂×Q₈	1	2	yes	—
131	C₂²×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₁₆₇	—	2	1	no	—
133	C₂×D₁₆₇	—	2	1	no	—
134	C₂×D₁₆₇	C₄⋊Q₈	1	2	yes	—
135	C₄×C₁₆₇	—	2	1	no	—
136	C₄×C₁₆₇	—	2	1	no	—
137	C₄×C₁₆₇	—	4	1	no	—
138	C₄×C₁₆₇	—	4	1	no	—
139	C₄×C₁₆₇	—	4	1	no	—
140	C₄×C₁₆₇	—	4	1	no	—
141	C₄×C₁₆₇	C₂²×Q₈	1	1	yes	characteristic
142	C₄×C₁₆₇	C₂²×Q₈	1	1	yes	characteristic
143	Dic₁₆₇	—	2	1	no	—
144	Dic₁₆₇	—	2	1	no	—
145	Dic₁₆₇	—	4	1	no	—
146	Dic₁₆₇	—	4	1	no	—
147	Dic₁₆₇	—	4	1	no	—
148	Dic₁₆₇	—	4	1	no	—
149	Dic₁₆₇	C₄⋊Q₈	1	2	yes	—
150	C₂×C₁₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₂×C₁₆₇	#577	1	1	yes	characteristic
152	C₂×C₁₆₇	#577	1	1	yes	characteristic
153	D₁₆₇	—	2	2	no	—
154	C₁₆₇	#3163	1	1	yes	characteristic, complemented, Sylow
155	#3163	—	167	1	no	maximal, Sylow
156	#544	—	167	1	no	—
157	#544	—	167	1	no	—
158	#501	—	167	1	no	—
159	#501	—	167	1	no	—
160	#501	—	167	1	no	—
161	#501	D₁₆₇	1	1	yes	characteristic, complemented
162	#499	—	167	1	no	—
163	#499	—	167	1	no	—
164	#498	—	167	1	no	—
165	#498	—	167	1	no	—
166	#428	—	167	1	no	—
167	#428	—	167	1	no	—
168	#427	—	167	1	no	—
169	C₂×#106	—	167	1	no	—
170	#405	—	167	1	no	—
171	C₂×M₁₆	—	167	1	no	—
172	C₂²×C₈	—	167	1	no	—
173	C₄⋊Q₈	—	334	1	no	—
174	C₄⋊Q₈	—	334	1	no	—
175	C₄⋊Q₈	—	167	1	no	—
176	C₄⋊Q₈	—	167	1	no	—
177	C₄⋊Q₈	—	167	1	no	—
178	C₄⋊Q₈	C₂×D₁₆₇	1	1	yes	characteristic
179	#125	—	334	1	no	—
180	#125	—	334	1	no	—
181	#125	—	167	1	no	—
182	#125	—	167	1	no	—
183	#125	—	167	1	no	—
184	#125	C₂×D₁₆₇	1	1	yes	characteristic
185	C₂²⋊Q₈	—	334	1	no	—
186	C₂²⋊Q₈	—	334	1	no	—
187	C₂²⋊Q₈	—	334	1	no	—
188	C₂²⋊Q₈	—	334	1	no	—
189	#117	—	167	1	no	—
190	#117	—	167	1	no	—
191	#117	—	167	1	no	—
192	C₂×C₄⋊C₄	—	167	1	no	—
193	C₂×C₄⋊C₄	—	167	1	no	—
194	#107	—	167	1	no	—
195	#107	—	167	1	no	—
196	#107	—	167	1	no	—
197	#107	—	167	1	no	—
198	#107	C₂×D₁₆₇	1	1	yes	characteristic
199	#107	C₂×D₁₆₇	1	1	yes	characteristic
200	#106	—	167	1	no	—
201	#106	—	167	1	no	—
202	#106	—	167	1	no	—
203	#106	—	167	1	no	—
204	#106	—	167	1	no	—
205	#106	—	167	1	no	—
206	#106	—	167	1	no	—
207	#106	—	167	1	no	—
208	#106	C₂×D₁₆₇	1	1	yes	characteristic
209	#106	C₂×D₁₆₇	1	1	yes	characteristic
210	#97	—	167	1	no	—
211	#97	C₂×D₁₆₇	1	1	yes	characteristic
212	C₄×C₈	—	167	1	no	—
213	C₄×C₈	—	167	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	334	1	no	—
216	C₂×Q₈	—	334	1	no	—
217	C₂×Q₈	—	334	1	no	—
218	C₂²×C₄	—	334	1	no	—
219	C₂²×C₄	—	334	1	no	—
220	C₂²×C₄	—	167	1	no	—
221	M₁₆	—	167	4	no	—
222	C₂×C₈	—	167	1	no	—
223	C₂×C₈	—	167	1	no	—
224	C₂×C₈	—	167	4	no	—
225	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
226	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	334	1	no	—
231	C₄⋊C₄	—	334	1	no	—
232	C₄⋊C₄	—	334	1	no	—
233	C₄⋊C₄	—	334	1	no	—
234	C₄⋊C₄	—	334	1	no	—
235	C₄⋊C₄	—	334	1	no	—
236	C₄⋊C₄	—	334	1	no	—
237	C₄⋊C₄	—	334	1	no	—
238	C₄⋊C₄	—	334	1	no	—
239	C₄⋊C₄	—	334	1	no	—
240	C₄⋊C₄	—	334	1	no	—
241	C₄⋊C₄	—	167	1	no	—
242	C₄⋊C₄	—	167	1	no	—
243	C₄⋊C₄	—	167	1	no	—
244	C₄⋊C₄	—	167	1	no	—
245	C₄⋊C₄	—	167	1	no	—
246	C₄⋊C₄	—	167	1	no	—
247	C₄⋊C₄	C₂²×D₁₆₇	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₁₆₇	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₆₇	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₁₆₇	1	1	yes	characteristic
251	C₂²⋊C₄	—	334	1	no	—
252	C₂²⋊C₄	—	334	1	no	—
253	C₂²⋊C₄	—	167	1	no	—
254	C₂²⋊C₄	—	167	1	no	—
255	C₄²	—	334	1	no	—
256	C₄²	—	334	1	no	—
257	C₄²	—	167	1	no	—
258	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
259	C₂³	—	167	1	no	—
260	Q₈	—	8	1	no	—
261	Q₈	—	1336	1	no	—
262	Q₈	—	1336	1	no	—
263	Q₈	—	1336	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	1336	1	no	—
269	C₂×C₄	—	1336	1	no	—
270	C₂×C₄	—	334	1	no	—
271	C₂×C₄	—	334	1	no	—
272	C₂×C₄	—	334	1	no	—
273	C₂×C₄	—	334	1	no	—
274	C₂×C₄	—	167	1	no	—
275	C₂×C₄	—	167	1	no	—
276	C₂×C₄	—	167	1	no	—
277	C₂×C₄	—	167	2	no	—
278	C₂×C₄	—	167	2	no	—
279	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
280	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
281	C₂×C₄	C₂³×D₁₆₇	1	1	yes	characteristic
282	C₈	—	167	2	no	—
283	C₈	—	167	2	no	—
284	C₈	Q₈×D₁₆₇	1	2	yes	—
285	C₈	Q₈×D₁₆₇	1	2	yes	—
286	C₂²	—	334	1	no	—
287	C₂²	—	334	1	no	—
288	C₂²	—	167	2	no	—
289	C₂²	C₂×D₄×D₁₆₇	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	334	1	no	—
297	C₄	—	334	1	no	—
298	C₄	—	668	1	no	—
299	C₄	—	668	1	no	—
300	C₄	—	668	1	no	—
301	C₄	—	668	1	no	—
302	C₄	—	167	2	no	—
303	C₄	C₂×Q₈×D₁₆₇	1	1	yes	characteristic
304	C₄	C₂×Q₈×D₁₆₇	1	1	yes	characteristic
305	C₂	—	334	2	no	—
306	C₂	C₄⋊Q₈×D₁₆₇	1	1	yes	characteristic
307	C₂	#24924078	1	1	yes	characteristic
308	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	#501×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂²×D₄
prime radical	C₂²×D₁₆₇	C₂×D₄
hypercenter	#501	D₁₆₇
hypocenter	C₁₆₇	#3163
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
3. #24923678	C₂	semidirect
4. #24923677	C₂	semidirect
5. #24923676	C₂	semidirect
6. #24923670	C₂	semidirect
7. #24923668	C₂	semidirect
8. #24923666	C₂	semidirect
9. #24923665	C₂	semidirect
15. #501×C₁₆₇	C₂	semidirect
65. #14581933	Q₈	semidirect
68. C₈×D₁₆₇	Q₈	semidirect
154. C₁₆₇	#3163	semidirect
161. #501	D₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
#24923899	2
#24923896	2
#24923678	2
#24923677	2
#24923676	2
#24923670	2
#24923668	2
#24923666	2
#24923665	2
#24923310	2
#24923308	2
#24923306	2
#106×D₁₆₇	2
#24923226	2
#501×C₁₆₇	2
#3163	167