Subgroups — #2439×C₁₆₇ #36320799

Order 21376.

508 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#117×C₁₆₇	C₂	1	1	yes	maximal, characteristic
3	#396×C₁₆₇	C₂	1	2	yes	maximal
4	#393×C₁₆₇	C₂	1	2	yes	maximal
5	#391×C₁₆₇	C₂	1	2	yes	maximal, complemented
6	#389×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#389×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#387×C₁₆₇	C₂	1	2	yes	maximal, complemented
9	#382×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#380×C₁₆₇	C₂	1	1	yes	maximal, characteristic
11	C₄×C₂²⋊C₄×C₁₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₁₆₇	C₂²	1	2	yes	—
13	C₂³×C₄×C₁₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₁₆₇	—	2	4	no	—
15	#117×C₁₆₇	—	2	4	no	—
16	C₂×C₄⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₆₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₁₆₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₁₆₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₁₆₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₆₇	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
30	#95×C₁₆₇	C₂²	1	1	yes	characteristic
31	#95×C₁₆₇	C₂²	1	1	yes	characteristic
32	#95×C₁₆₇	C₂²	1	2	yes	—
33	#95×C₁₆₇	C₂²	1	2	yes	—
34	#95×C₁₆₇	C₂²	1	2	yes	—
35	#95×C₁₆₇	C₂²	1	2	yes	—
36	#95×C₁₆₇	C₂²	1	2	yes	—
37	C₂⁴×C₁₆₇	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₁₆₇	—	2	4	no	—
39	C₂×Q₈×C₁₆₇	—	2	4	no	—
40	C₂×Q₈×C₁₆₇	—	4	2	no	—
41	C₂²×C₄×C₁₆₇	—	2	2	no	—
42	C₂²×C₄×C₁₆₇	—	2	2	no	—
43	C₂²×C₄×C₁₆₇	Q₈	1	4	yes	complemented
44	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₆₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₆₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₆₇	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₆₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₁₆₇	—	2	2	no	—
55	C₄⋊C₄×C₁₆₇	—	2	4	no	—
56	C₄⋊C₄×C₁₆₇	—	2	2	no	—
57	C₄⋊C₄×C₁₆₇	—	2	4	no	—
58	C₄⋊C₄×C₁₆₇	—	2	2	no	—
59	C₄⋊C₄×C₁₆₇	—	4	2	no	—
60	C₄⋊C₄×C₁₆₇	D₄	1	4	yes	—
61	C₂²⋊C₄×C₁₆₇	—	2	4	no	—
62	C₂²⋊C₄×C₁₆₇	—	2	2	no	—
63	C₂²⋊C₄×C₁₆₇	—	2	2	no	—
64	C₂²⋊C₄×C₁₆₇	—	4	2	no	—
65	C₄²×C₁₆₇	—	2	4	no	—
66	C₄²×C₁₆₇	—	2	2	no	—
67	C₄²×C₁₆₇	—	2	2	no	—
68	C₂³×C₁₆₇	—	2	1	no	—
69	C₂³×C₁₆₇	—	2	1	no	—
70	C₂³×C₁₆₇	—	2	1	no	—
71	C₂³×C₁₆₇	—	2	1	no	—
72	C₂³×C₁₆₇	—	2	1	no	—
73	C₂³×C₁₆₇	—	2	1	no	—
74	C₂³×C₁₆₇	C₂×Q₈	1	2	yes	—
75	C₂³×C₁₆₇	C₂⁴	1	1	yes	characteristic
76	Q₈×C₁₆₇	—	4	8	no	—
77	C₂×C₄×C₁₆₇	—	2	1	no	—
78	C₂×C₄×C₁₆₇	—	2	1	no	—
79	C₂×C₄×C₁₆₇	—	2	1	no	—
80	C₂×C₄×C₁₆₇	—	2	1	no	—
81	C₂×C₄×C₁₆₇	—	2	2	no	—
82	C₂×C₄×C₁₆₇	—	2	2	no	—
83	C₂×C₄×C₁₆₇	—	2	1	no	—
84	C₂×C₄×C₁₆₇	—	2	1	no	—
85	C₂×C₄×C₁₆₇	—	2	8	no	—
86	C₂×C₄×C₁₆₇	—	2	2	no	—
87	C₂×C₄×C₁₆₇	—	2	2	no	—
88	C₂×C₄×C₁₆₇	—	2	1	no	—
89	C₂×C₄×C₁₆₇	—	2	1	no	—
90	C₂×C₄×C₁₆₇	—	2	1	no	—
91	C₂×C₄×C₁₆₇	—	2	1	no	—
92	C₂×C₄×C₁₆₇	—	2	2	no	—
93	C₂×C₄×C₁₆₇	—	2	2	no	—
94	C₂×C₄×C₁₆₇	—	2	2	no	—
95	C₂×C₄×C₁₆₇	—	2	2	no	—
96	C₂×C₄×C₁₆₇	—	2	2	no	—
97	C₂×C₄×C₁₆₇	—	2	2	no	—
98	C₂×C₄×C₁₆₇	—	2	1	no	—
99	C₂×C₄×C₁₆₇	—	2	2	no	—
100	C₂×C₄×C₁₆₇	—	2	1	no	—
101	C₂×C₄×C₁₆₇	C₂×D₄	1	4	yes	—
102	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₆₇	C₂×Q₈	1	2	yes	—
104	C₂×C₄×C₁₆₇	C₂×Q₈	1	2	yes	—
105	C₂×C₄×C₁₆₇	#41	1	2	yes	—
106	C₂×C₄×C₁₆₇	#41	1	2	yes	—
107	C₂×C₄×C₁₆₇	#41	1	2	yes	—
108	C₂²×C₁₆₇	—	2	2	no	—
109	C₂²×C₁₆₇	—	2	2	no	—
110	C₂²×C₁₆₇	—	2	2	no	—
111	C₂²×C₁₆₇	—	4	1	no	—
112	C₂²×C₁₆₇	—	4	1	no	—
113	C₂²×C₁₆₇	—	4	1	no	—
114	C₂²×C₁₆₇	—	4	1	no	—
115	C₂²×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
116	C₂²×C₁₆₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂²×C₁₆₇	C₂×#41	1	1	yes	characteristic
118	C₂²×C₁₆₇	C₂×#41	1	1	yes	characteristic
119	C₂²×C₁₆₇	C₂×#41	1	1	yes	characteristic
120	C₂²×C₁₆₇	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂²×C₁₆₇	2₋¹⁺⁴	1	1	yes	characteristic
122	C₄×C₁₆₇	—	2	2	no	—
123	C₄×C₁₆₇	—	2	4	no	—
124	C₄×C₁₆₇	—	2	2	no	—
125	C₄×C₁₆₇	—	2	2	no	—
126	C₄×C₁₆₇	—	4	2	no	—
127	C₄×C₁₆₇	—	4	2	no	—
128	C₄×C₁₆₇	—	4	1	no	—
129	C₄×C₁₆₇	—	4	2	no	—
130	C₄×C₁₆₇	—	4	1	no	—
131	C₄×C₁₆₇	C₂²⋊Q₈	1	4	yes	—
132	C₂×C₁₆₇	—	4	2	no	—
133	C₂×C₁₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₁₆₇	#533	1	1	yes	characteristic
135	C₂×C₁₆₇	#544	1	1	yes	characteristic
136	C₂×C₁₆₇	#546	1	1	yes	characteristic
137	C₂×C₁₆₇	#548	1	1	yes	characteristic
138	C₂×C₁₆₇	#555	1	1	yes	characteristic
139	C₂×C₁₆₇	#556	1	1	yes	characteristic
140	C₁₆₇	#2439	1	1	yes	characteristic, direct product factor, Sylow
141	#2439	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂×C₄⋊Q₈	C₂×C₁₆₇	1	1	yes	characteristic, complemented
143	C₂×#117	C₂×C₁₆₇	1	1	yes	characteristic
144	#396	C₂×C₁₆₇	1	2	yes	—
145	#393	C₂×C₁₆₇	1	2	yes	—
146	#391	C₂×C₁₆₇	1	2	yes	complemented
147	#389	C₂×C₁₆₇	1	1	yes	characteristic, complemented
148	#389	C₂×C₁₆₇	1	1	yes	characteristic, complemented
149	#387	C₂×C₁₆₇	1	2	yes	complemented
150	#382	C₂×C₁₆₇	1	1	yes	characteristic, complemented
151	#380	C₂×C₁₆₇	1	1	yes	characteristic
152	C₄×C₂²⋊C₄	C₂×C₁₆₇	1	1	yes	characteristic
153	C₂²×Q₈	C₂²×C₁₆₇	1	2	yes	—
154	C₂³×C₄	C₂²×C₁₆₇	1	1	yes	characteristic
155	C₄⋊Q₈	—	2	4	no	—
156	#117	—	2	4	no	—
157	C₂×C₄⋊C₄	C₂²×C₁₆₇	1	1	yes	characteristic
158	C₂×C₄⋊C₄	C₂²×C₁₆₇	1	1	yes	characteristic
159	C₂×C₄⋊C₄	C₂²×C₁₆₇	1	2	yes	—
160	C₂×C₄⋊C₄	C₂²×C₁₆₇	1	1	yes	characteristic
161	C₂×C₄⋊C₄	C₂²×C₁₆₇	1	2	yes	—
162	C₂×C₄⋊C₄	C₂²×C₁₆₇	1	2	yes	—
163	C₂×C₄⋊C₄	C₂²×C₁₆₇	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	C₂²×C₁₆₇	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₁₆₇	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	C₂²×C₁₆₇	1	1	yes	characteristic
167	C₂×C₂²⋊C₄	C₂²×C₁₆₇	1	2	yes	—
168	C₂×C₄²	C₂²×C₁₆₇	1	2	yes	—
169	C₂×C₄²	C₂²×C₁₆₇	1	1	yes	characteristic
170	C₂×C₄²	C₂²×C₁₆₇	1	1	yes	characteristic
171	#95	C₂²×C₁₆₇	1	1	yes	characteristic
172	#95	C₂²×C₁₆₇	1	1	yes	characteristic
173	#95	C₂²×C₁₆₇	1	2	yes	—
174	#95	C₂²×C₁₆₇	1	2	yes	—
175	#95	C₂²×C₁₆₇	1	2	yes	—
176	#95	C₂²×C₁₆₇	1	2	yes	—
177	#95	C₂²×C₁₆₇	1	2	yes	—
178	C₂⁴	C₂³×C₁₆₇	1	1	yes	characteristic
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	Q₈×C₁₆₇	1	4	yes	complemented
185	C₂²×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₁₆₇	1	2	yes	—
187	C₂²×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₁₆₇	1	2	yes	—
189	C₂²×C₄	C₂³×C₁₆₇	1	2	yes	—
190	C₂²×C₄	C₂³×C₁₆₇	1	2	yes	—
191	C₂²×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	D₄×C₁₆₇	1	4	yes	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₂²⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	4	2	no	—
206	C₄²	—	2	4	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	C₂×Q₈×C₁₆₇	1	2	yes	—
216	C₂³	C₂⁴×C₁₆₇	1	1	yes	characteristic
217	Q₈	—	4	8	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	C₂×D₄×C₁₆₇	1	4	yes	—
243	C₂×C₄	C₂×D₄×C₁₆₇	1	2	yes	—
244	C₂×C₄	C₂×Q₈×C₁₆₇	1	2	yes	—
245	C₂×C₄	C₂×Q₈×C₁₆₇	1	2	yes	—
246	C₂×C₄	#41×C₁₆₇	1	2	yes	—
247	C₂×C₄	#41×C₁₆₇	1	2	yes	—
248	C₂×C₄	#41×C₁₆₇	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	C₂²×D₄×C₁₆₇	1	1	yes	characteristic
257	C₂²	C₂²×Q₈×C₁₆₇	1	1	yes	characteristic
258	C₂²	C₂×#41×C₁₆₇	1	1	yes	characteristic
259	C₂²	C₂×#41×C₁₆₇	1	1	yes	characteristic
260	C₂²	C₂×#41×C₁₆₇	1	1	yes	characteristic
261	C₂²	H(5,𝔽₂)×C₁₆₇	1	1	yes	characteristic
262	C₂²	2₋¹⁺⁴×C₁₆₇	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	2	no	—
271	C₄	—	4	1	no	—
272	C₄	C₂²⋊Q₈×C₁₆₇	1	4	yes	—
273	C₂	—	4	2	no	—
274	C₂	C₂×C₂²⋊Q₈×C₁₆₇	1	1	yes	characteristic
275	C₂	#533×C₁₆₇	1	1	yes	characteristic
276	C₂	#544×C₁₆₇	1	1	yes	characteristic
277	C₂	#546×C₁₆₇	1	1	yes	characteristic
278	C₂	#548×C₁₆₇	1	1	yes	characteristic
279	C₂	#555×C₁₆₇	1	1	yes	characteristic
280	C₂	#556×C₁₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₆₇
prime radical	C₂⁴×C₁₆₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Q₈×C₁₆₇	C₂	semidirect
5. #391×C₁₆₇	C₂	semidirect
6. #389×C₁₆₇	C₂	semidirect
7. #389×C₁₆₇	C₂	semidirect
8. #387×C₁₆₇	C₂	semidirect
9. #382×C₁₆₇	C₂	semidirect
43. C₂²×C₄×C₁₆₇	Q₈	semidirect
140. C₁₆₇	#2439	direct (N × H)
141. #2439	C₁₆₇	direct (N × H)
142. C₂×C₄⋊Q₈	C₂×C₁₆₇	semidirect
146. #391	C₂×C₁₆₇	semidirect
147. #389	C₂×C₁₆₇	semidirect
148. #389	C₂×C₁₆₇	semidirect
149. #387	C₂×C₁₆₇	semidirect
150. #382	C₂×C₁₆₇	semidirect
184. C₂²×C₄	Q₈×C₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×C₁₆₇	2
C₂×#117×C₁₆₇	2
#396×C₁₆₇	2
#393×C₁₆₇	2
#391×C₁₆₇	2
#389×C₁₆₇	2
#387×C₁₆₇	2
#382×C₁₆₇	2
#380×C₁₆₇	2
C₄×C₂²⋊C₄×C₁₆₇	2
#2439	167

Subgroups — #2439×C167 #36320799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2439×C₁₆₇ #36320799