Subgroups — #36337164

Order 21376.

350 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923994	C₂	1	1	yes	maximal, characteristic, complemented
2	#24923991	C₂	1	1	yes	maximal, characteristic
3	#24923662	C₂	1	1	yes	maximal, characteristic
4	#24923661	C₂	1	1	yes	maximal, characteristic
5	#24923647	C₂	1	1	yes	maximal, characteristic, complemented
6	#24923493	C₂	1	1	yes	maximal, characteristic
7	#24923492	C₂	1	1	yes	maximal, characteristic
8	#24923478	C₂	1	1	yes	maximal, characteristic, complemented
9	#24923477	C₂	1	1	yes	maximal, characteristic, complemented
10	#24923473	C₂	1	1	yes	maximal, characteristic
11	#24923398	C₂	1	1	yes	maximal, characteristic, complemented
12	#24923374	C₂	1	1	yes	maximal, characteristic, complemented
13	#24923373	C₂	1	1	yes	maximal, characteristic, complemented
14	#24923354	C₂	1	1	yes	maximal, characteristic
15	#462×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₆₆₈	C₂²	1	1	yes	characteristic
17	C₂×#14581940	C₂²	1	1	yes	characteristic
18	C₂×#14581940	C₂²	1	1	yes	characteristic
19	C₂×#14581940	C₂²	1	1	yes	characteristic
20	C₂×#14581939	C₂²	1	1	yes	characteristic
21	C₂×#14581937	C₂²	1	1	yes	characteristic
22	C₂×#14581937	C₂²	1	1	yes	characteristic
23	#19128830	C₂²	1	1	yes	characteristic
24	#19128830	C₂²	1	1	yes	characteristic
25	#19128830	C₂²	1	1	yes	characteristic
26	#19128827	—	2	1	no	—
27	#19128826	—	2	1	no	—
28	#19128823	—	2	1	no	—
29	#19128822	—	2	1	no	—
30	#19128822	—	2	1	no	—
31	#19128821	C₂²	1	1	yes	characteristic
32	#19128820	C₂²	1	1	yes	characteristic
33	#19128810	—	2	1	no	—
34	Q₈×Dic₁₆₇	C₂²	1	1	yes	characteristic
35	Q₈×Dic₁₆₇	C₂²	1	1	yes	characteristic
36	C₄×Dic₃₃₄	C₂²	1	1	yes	characteristic
37	C₄×Dic₃₃₄	C₂²	1	1	yes	characteristic
38	#19128797	—	2	1	no	—
39	#19128797	C₂²	1	1	yes	characteristic
40	#19128794	C₂²	1	1	yes	characteristic
41	#19128789	—	2	1	no	—
42	#19128763	C₂²	1	1	yes	characteristic
43	Dic₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
45	#19128757	C₂²	1	1	yes	characteristic
46	#19128756	C₂²	1	1	yes	characteristic
47	#19128755	C₂²	1	1	yes	characteristic
48	#19128754	C₂²	1	1	yes	characteristic
49	#19128743	C₂²	1	1	yes	characteristic
50	#19128742	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₆₇	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
54	C₄⋊Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₆₇	C₂²	1	1	yes	characteristic
57	#103×C₁₆₇	C₂²	1	1	yes	characteristic
58	#103×C₁₆₇	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₃₄	D₄	1	1	yes	characteristic
60	C₂×Dic₃₃₄	C₂³	1	1	yes	characteristic
61	C₂×Dic₃₃₄	C₂³	1	1	yes	characteristic
62	C₂×D₆₆₈	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₆₇	—	2	1	no	—
64	Dic₆₆₈	—	4	1	no	—
65	#14581940	—	2	2	no	—
66	#14581940	—	2	2	no	—
67	#14581940	—	4	1	no	—
68	#14581939	—	4	1	no	—
69	#14581937	—	2	2	no	—
70	#14581937	—	4	1	no	—
71	C₂×C₁₆₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₁₆₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₆₇	—	2	1	no	—
74	C₄⋊Dic₁₆₇	—	2	1	no	—
75	C₄⋊Dic₁₆₇	D₄	1	1	yes	characteristic
76	C₄⋊Dic₁₆₇	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
85	Dic₁₆₇⋊C₄	D₄	1	1	yes	characteristic
86	Dic₁₆₇⋊C₄	D₄	1	1	yes	characteristic
87	(C₂×D₁₆₇)⋊C₄	—	2	1	no	—
88	(C₂×D₁₆₇)⋊C₄	—	2	1	no	—
89	(C₂×D₁₆₇)⋊C₄	—	2	1	no	—
90	(C₂×D₁₆₇)⋊C₄	—	2	1	no	—
91	(C₂×D₁₆₇)⋊C₄	—	2	1	no	—
92	C₄×Dic₁₆₇	—	2	1	no	—
93	C₄×Dic₁₆₇	—	2	1	no	—
94	C₄×Dic₁₆₇	—	2	1	no	—
95	C₄×Dic₁₆₇	—	2	1	no	—
96	C₄×Dic₁₆₇	C₂³	1	1	yes	characteristic
97	C₄×Dic₁₆₇	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₁₆₇	C₂³	1	1	yes	characteristic
99	C₂×Q₈×C₁₆₇	C₂³	1	1	yes	characteristic
100	C₂×Q₈×C₁₆₇	C₂³	1	1	yes	characteristic
101	Q₁₆×C₁₆₇	—	4	1	no	—
102	Q₁₆×C₁₆₇	—	4	1	no	—
103	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₆₇	—	2	1	no	—
106	C₄⋊C₄×C₁₆₇	—	2	1	no	—
107	C₄⋊C₄×C₁₆₇	—	2	1	no	—
108	C₄⋊C₄×C₁₆₇	C₂³	1	1	yes	characteristic
109	C₄²×C₁₆₇	—	2	1	no	—
110	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
111	C₂²×D₁₆₇	—	2	1	no	—
112	Dic₃₃₄	—	2	2	no	—
113	Dic₃₃₄	—	2	1	no	—
114	Dic₃₃₄	—	2	1	no	—
115	Dic₃₃₄	—	2	1	no	—
116	Dic₃₃₄	—	2	1	no	—
117	D₆₆₈	—	2	1	no	—
118	D₆₆₈	—	2	1	no	—
119	C₂×Dic₁₆₇	—	2	1	no	—
120	C₂×Dic₁₆₇	—	2	1	no	—
121	C₂×Dic₁₆₇	—	2	1	no	—
122	C₂×Dic₁₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₆₇	C₂×D₄	1	1	yes	characteristic
126	C₄×D₁₆₇	—	4	2	no	—
127	C₁₆₇⋊C₈	—	2	1	no	—
128	C₁₆₇⋊C₈	#41	1	2	yes	—
129	Q₈×C₁₆₇	—	2	1	no	—
130	Q₈×C₁₆₇	—	2	1	no	—
131	Q₈×C₁₆₇	—	2	1	no	—
132	Q₈×C₁₆₇	—	2	1	no	—
133	Q₈×C₁₆₇	—	2	1	no	—
134	Q₈×C₁₆₇	—	2	1	no	—
135	C₂×C₄×C₁₆₇	—	2	1	no	—
136	C₂×C₄×C₁₆₇	—	2	1	no	—
137	C₂×C₄×C₁₆₇	—	2	1	no	—
138	C₂×C₄×C₁₆₇	—	2	1	no	—
139	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
142	C₈×C₁₆₇	—	2	1	no	—
143	C₈×C₁₆₇	—	2	1	no	—
144	C₂²×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₆₇	—	4	1	no	—
146	C₂×D₁₆₇	—	4	1	no	—
147	C₂×D₁₆₇	—	4	1	no	—
148	C₄×C₁₆₇	—	2	1	no	—
149	C₄×C₁₆₇	—	2	1	no	—
150	C₄×C₁₆₇	—	4	1	no	—
151	C₄×C₁₆₇	—	4	1	no	—
152	C₄×C₁₆₇	—	4	1	no	—
153	C₄×C₁₆₇	—	4	1	no	—
154	C₄×C₁₆₇	C₂×#41	1	1	yes	characteristic
155	C₄×C₁₆₇	H(5,𝔽₂)	1	1	yes	characteristic
156	Dic₁₆₇	—	2	1	no	—
157	Dic₁₆₇	—	2	1	no	—
158	Dic₁₆₇	—	2	1	no	—
159	Dic₁₆₇	—	4	1	no	—
160	Dic₁₆₇	—	4	1	no	—
161	Dic₁₆₇	—	4	1	no	—
162	Dic₁₆₇	#137	1	2	yes	—
163	C₂×C₁₆₇	#550	1	1	yes	characteristic
164	C₂×C₁₆₇	C₂×#137	1	1	yes	characteristic
165	C₂×C₁₆₇	#578	1	1	yes	characteristic
166	D₁₆₇	—	8	1	no	—
167	C₁₆₇	#3398	1	1	yes	characteristic, complemented, Sylow
168	#3398	—	167	1	no	maximal, Sylow
169	#557	—	167	1	no	—
170	#556	—	167	1	no	—
171	#497	—	167	1	no	—
172	#497	—	167	1	no	—
173	#494	—	167	1	no	—
174	#464	—	167	1	no	—
175	#464	—	167	1	no	—
176	#462	—	167	1	no	—
177	#462	—	167	1	no	—
178	#462	D₁₆₇	1	1	yes	characteristic, complemented
179	#461	—	167	1	no	—
180	#446	—	167	1	no	—
181	#441	—	167	1	no	—
182	#441	—	167	1	no	—
183	C₄×SD₁₆	—	167	1	no	—
184	C₂×Q₁₆	—	167	1	no	—
185	C₂×Q₁₆	—	167	1	no	—
186	C₂×Q₁₆	—	167	1	no	—
187	C₂×Q₁₆	—	167	1	no	—
188	C₂×Q₁₆	C₂×D₁₆₇	1	1	yes	characteristic
189	C₂×Q₁₆	C₂×D₁₆₇	1	1	yes	characteristic
190	C₂×SD₁₆	—	167	1	no	—
191	C₂×SD₁₆	—	167	1	no	—
192	C₂×SD₁₆	—	167	1	no	—
193	C₄⋊Q₈	—	167	1	no	—
194	C₄⋊Q₈	—	167	1	no	—
195	C₄⋊Q₈	—	167	1	no	—
196	C₄⋊Q₈	C₂×D₁₆₇	1	1	yes	characteristic
197	#126	—	334	1	no	—
198	#126	—	334	1	no	—
199	#125	—	334	1	no	—
200	#125	—	334	1	no	—
201	#125	—	334	1	no	—
202	#124	—	167	1	no	—
203	#124	—	167	1	no	—
204	C₂²⋊Q₈	—	334	1	no	—
205	C₄×Q₈	—	334	1	no	—
206	C₄×Q₈	—	167	1	no	—
207	C₄×Q₈	—	167	1	no	—
208	C₄×Q₈	—	167	1	no	—
209	C₄×Q₈	—	167	1	no	—
210	C₄×Q₈	—	167	1	no	—
211	C₄×Q₈	C₂×D₁₆₇	1	1	yes	characteristic
212	C₄×D₄	—	167	1	no	—
213	#117	—	334	1	no	—
214	#106	—	167	1	no	—
215	C₄⋊C₈	—	167	1	no	—
216	C₄⋊C₈	—	167	1	no	—
217	C₄⋊C₈	C₂×D₁₆₇	1	1	yes	characteristic
218	#103	—	167	1	no	—
219	#103	—	167	1	no	—
220	#103	—	167	1	no	—
221	#103	C₂×D₁₆₇	1	1	yes	characteristic
222	#103	C₂×D₁₆₇	1	1	yes	characteristic
223	#102	—	167	1	no	—
224	#97	—	167	1	no	—
225	#97	—	167	1	no	—
226	C₄×C₈	—	167	1	no	—
227	C₂×Q₈	—	167	1	no	—
228	C₂×Q₈	—	167	1	no	—
229	C₂×Q₈	—	167	1	no	—
230	C₂×Q₈	C₂²×D₁₆₇	1	1	yes	characteristic
231	C₂×Q₈	C₂²×D₁₆₇	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₁₆₇	1	1	yes	characteristic
233	C₂×D₄	—	167	1	no	—
234	C₂²×C₄	—	334	1	no	—
235	Q₁₆	—	4	1	no	—
236	Q₁₆	—	4	1	no	—
237	Q₁₆	—	334	2	no	—
238	Q₁₆	—	334	2	no	—
239	Q₁₆	—	668	1	no	—
240	Q₁₆	—	668	1	no	—
241	SD₁₆	—	334	2	no	—
242	SD₁₆	—	668	1	no	—
243	SD₁₆	—	668	1	no	—
244	C₂×C₈	—	167	1	no	—
245	C₂×C₈	—	167	1	no	—
246	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	334	1	no	—
252	C₄⋊C₄	—	334	1	no	—
253	C₄⋊C₄	—	334	1	no	—
254	C₄⋊C₄	—	334	1	no	—
255	C₄⋊C₄	—	334	1	no	—
256	C₄⋊C₄	—	334	1	no	—
257	C₄⋊C₄	—	334	1	no	—
258	C₄⋊C₄	—	334	1	no	—
259	C₄⋊C₄	—	334	1	no	—
260	C₄⋊C₄	—	334	1	no	—
261	C₄⋊C₄	—	167	1	no	—
262	C₄⋊C₄	—	167	1	no	—
263	C₄⋊C₄	—	167	1	no	—
264	C₄⋊C₄	—	167	1	no	—
265	C₄⋊C₄	C₂²×D₁₆₇	1	1	yes	characteristic
266	C₂²⋊C₄	—	334	1	no	—
267	C₂²⋊C₄	—	334	1	no	—
268	C₂²⋊C₄	—	334	1	no	—
269	C₂²⋊C₄	—	334	1	no	—
270	C₂²⋊C₄	—	334	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	334	1	no	—
273	C₄²	—	334	1	no	—
274	C₄²	—	334	1	no	—
275	C₄²	—	334	1	no	—
276	C₄²	—	167	1	no	—
277	C₄²	—	167	1	no	—
278	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
279	C₂³	—	334	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	334	2	no	—
287	Q₈	—	334	1	no	—
288	Q₈	—	334	1	no	—
289	Q₈	—	334	1	no	—
290	Q₈	—	334	1	no	—
291	D₄	—	334	1	no	—
292	D₄	—	334	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	334	1	no	—
298	C₂×C₄	—	334	1	no	—
299	C₂×C₄	—	334	1	no	—
300	C₂×C₄	—	668	2	no	—
301	C₂×C₄	—	167	1	no	—
302	C₂×C₄	—	167	1	no	—
303	C₂×C₄	—	167	1	no	—
304	C₂×C₄	—	167	1	no	—
305	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
306	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₆₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	334	1	no	—
311	C₈	—	167	2	no	—
312	C₂²	—	668	1	no	—
313	C₂²	—	668	1	no	—
314	C₂²	—	668	1	no	—
315	C₂²	C₂×D₄×D₁₆₇	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	334	1	no	—
323	C₄	—	334	1	no	—
324	C₄	—	334	1	no	—
325	C₄	—	668	1	no	—
326	C₄	—	668	1	no	—
327	C₄	—	668	1	no	—
328	C₄	—	167	2	no	—
329	C₄	#41×D₁₆₇	1	1	yes	characteristic
330	C₄	#19128877	1	1	yes	characteristic
331	C₂	—	1336	1	no	—
332	C₂	#24923937	1	1	yes	characteristic
333	C₂	#137×D₁₆₇	1	1	yes	characteristic
334	C₂	#24924082	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₆₇
commutator, cosocle	C₂×C₄×C₁₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₆₇
Fitting	#462×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂²×D₄
prime radical	C₂×D₆₆₈	C₂³
hypercenter	#462	D₁₆₇
hypocenter	C₁₆₇	#3398
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

Complements

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

N	H ≅ G/N	type
1. #24923994	C₂	semidirect
5. #24923647	C₂	semidirect
8. #24923478	C₂	semidirect
9. #24923477	C₂	semidirect
11. #24923398	C₂	semidirect
12. #24923374	C₂	semidirect
13. #24923373	C₂	semidirect
15. #462×C₁₆₇	C₂	semidirect
167. C₁₆₇	#3398	semidirect
178. #462	D₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
#24923994	2
#24923991	2
#24923662	2
#24923661	2
#24923647	2
#24923493	2
#24923492	2
#24923478	2
#24923477	2
#24923473	2
#24923398	2
#24923374	2
#24923373	2
#24923354	2
#462×C₁₆₇	2
#3398	167