Subgroups — #3366×C₁₆₇ #36321726

Order 21376.

396 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#554×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#550×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#490×C₁₆₇	C₂	1	2	yes	maximal, complemented
4	#486×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#478×C₁₆₇	C₂	1	2	yes	maximal, complemented
6	#476×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#459×C₁₆₇	C₂	1	2	yes	maximal, complemented
8	#459×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#446×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#442×C₁₆₇	C₂	1	2	yes	maximal, complemented
11	C₄×D₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₆₇	—	2	1	no	—
13	C₂×D₈×C₁₆₇	C₂²	1	1	yes	characteristic
14	C₂×D₈×C₁₆₇	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
16	#127×C₁₆₇	C₂²	1	1	yes	characteristic
17	#127×C₁₆₇	C₂²	1	2	yes	—
18	#125×C₁₆₇	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₁₆₇	—	2	1	no	—
20	C₂²⋊Q₈×C₁₆₇	—	2	2	no	—
21	#121×C₁₆₇	—	2	1	no	—
22	#121×C₁₆₇	—	2	2	no	—
23	C₄×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₆₇	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₆₇	C₂²	1	2	yes	—
26	C₄×D₄×C₁₆₇	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₆₇	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₆₇	—	2	1	no	—
29	#107×C₁₆₇	C₂²	1	1	yes	characteristic
30	#106×C₁₆₇	C₂²	1	2	yes	—
31	C₄⋊C₈×C₁₆₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₁₆₇	C₂²	1	2	yes	—
33	#102×C₁₆₇	C₂²	1	2	yes	—
34	#102×C₁₆₇	C₂²	1	2	yes	—
35	#102×C₁₆₇	C₂²	1	1	yes	characteristic
36	#102×C₁₆₇	C₂²	1	2	yes	—
37	#102×C₁₆₇	C₂²	1	2	yes	—
38	#102×C₁₆₇	C₂²	1	1	yes	characteristic
39	#97×C₁₆₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
41	#41×C₁₆₇	—	4	4	no	—
42	C₂×Q₈×C₁₆₇	—	2	1	no	—
43	C₂×Q₈×C₁₆₇	D₄	1	1	yes	characteristic
44	C₂×D₄×C₁₆₇	—	2	2	no	—
45	C₂×D₄×C₁₆₇	—	2	1	no	—
46	C₂×D₄×C₁₆₇	—	2	1	no	—
47	C₂×D₄×C₁₆₇	—	2	2	no	—
48	C₂×D₄×C₁₆₇	C₂³	1	2	yes	—
49	C₂×D₄×C₁₆₇	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₆₇	—	2	1	no	—
52	C₂²×C₄×C₁₆₇	—	2	2	no	—
53	C₂²×C₄×C₁₆₇	—	2	1	no	—
54	C₂²×C₄×C₁₆₇	—	2	2	no	—
55	D₈×C₁₆₇	—	2	2	no	—
56	D₈×C₁₆₇	—	2	4	no	—
57	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₁₆₇	C₂³	1	2	yes	—
60	C₄⋊C₄×C₁₆₇	—	2	2	no	—
61	C₄⋊C₄×C₁₆₇	—	2	1	no	—
62	C₄⋊C₄×C₁₆₇	—	2	2	no	—
63	C₄⋊C₄×C₁₆₇	D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₁₆₇	D₄	1	2	yes	—
65	C₄⋊C₄×C₁₆₇	C₂³	1	2	yes	—
66	C₄⋊C₄×C₁₆₇	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₁₆₇	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₁₆₇	—	2	1	no	—
69	C₂²⋊C₄×C₁₆₇	—	2	2	no	—
70	C₂²⋊C₄×C₁₆₇	—	2	1	no	—
71	C₂²⋊C₄×C₁₆₇	—	2	2	no	—
72	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
73	C₄²×C₁₆₇	C₂³	1	2	yes	—
74	C₂³×C₁₆₇	—	2	1	no	—
75	C₂³×C₁₆₇	—	2	2	no	—
76	C₂³×C₁₆₇	—	2	1	no	—
77	Q₈×C₁₆₇	—	2	2	no	—
78	Q₈×C₁₆₇	—	8	1	no	—
79	D₄×C₁₆₇	—	2	1	no	—
80	D₄×C₁₆₇	—	2	2	no	—
81	D₄×C₁₆₇	—	2	1	no	—
82	D₄×C₁₆₇	—	2	2	no	—
83	D₄×C₁₆₇	—	2	1	no	—
84	D₄×C₁₆₇	—	4	4	no	—
85	D₄×C₁₆₇	—	4	2	no	—
86	D₄×C₁₆₇	—	8	2	no	—
87	D₄×C₁₆₇	—	8	1	no	—
88	D₄×C₁₆₇	#41	1	2	yes	complemented
89	C₂×C₄×C₁₆₇	—	2	1	no	—
90	C₂×C₄×C₁₆₇	—	2	1	no	—
91	C₂×C₄×C₁₆₇	—	2	2	no	—
92	C₂×C₄×C₁₆₇	—	4	4	no	—
93	C₂×C₄×C₁₆₇	—	4	2	no	—
94	C₂×C₄×C₁₆₇	—	4	2	no	—
95	C₂×C₄×C₁₆₇	—	8	2	no	—
96	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
101	C₈×C₁₆₇	—	2	1	no	—
102	C₈×C₁₆₇	—	2	2	no	—
103	C₈×C₁₆₇	—	2	1	no	—
104	C₂²×C₁₆₇	—	2	2	no	—
105	C₂²×C₁₆₇	—	4	1	no	—
106	C₂²×C₁₆₇	—	4	1	no	—
107	C₂²×C₁₆₇	—	4	2	no	—
108	C₂²×C₁₆₇	—	4	1	no	—
109	C₂²×C₁₆₇	—	4	2	no	—
110	C₂²×C₁₆₇	—	4	1	no	—
111	C₂²×C₁₆₇	—	4	1	no	—
112	C₂²×C₁₆₇	—	4	2	no	—
113	C₂²×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₁₆₇	—	2	2	no	—
115	C₄×C₁₆₇	—	2	2	no	—
116	C₄×C₁₆₇	—	2	1	no	—
117	C₄×C₁₆₇	—	4	1	no	—
118	C₄×C₁₆₇	—	4	1	no	—
119	C₄×C₁₆₇	—	4	2	no	—
120	C₄×C₁₆₇	AGL(1,ℤ/8)	1	2	yes	—
121	C₄×C₁₆₇	C₂×#41	1	1	yes	characteristic
122	C₄×C₁₆₇	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂×C₁₆₇	—	4	2	no	—
124	C₂×C₁₆₇	—	8	2	no	—
125	C₂×C₁₆₇	—	8	1	no	—
126	C₂×C₁₆₇	#548	1	1	yes	characteristic
127	C₂×C₁₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₁₆₇	#576	1	1	yes	characteristic
129	C₁₆₇	#3366	1	1	yes	characteristic, direct product factor, Sylow
130	#3366	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	#554	C₂×C₁₆₇	1	1	yes	characteristic, complemented
132	#550	C₂×C₁₆₇	1	1	yes	characteristic, complemented
133	#490	C₂×C₁₆₇	1	2	yes	complemented
134	#486	C₂×C₁₆₇	1	1	yes	characteristic, complemented
135	#478	C₂×C₁₆₇	1	2	yes	complemented
136	#476	C₂×C₁₆₇	1	1	yes	characteristic, complemented
137	#459	C₂×C₁₆₇	1	2	yes	complemented
138	#459	C₂×C₁₆₇	1	1	yes	characteristic, complemented
139	#446	C₂×C₁₆₇	1	1	yes	characteristic, complemented
140	#442	C₂×C₁₆₇	1	2	yes	complemented
141	C₄×D₈	C₂×C₁₆₇	1	1	yes	characteristic, complemented
142	C₂×#41	—	2	1	no	—
143	C₂×D₈	C₂²×C₁₆₇	1	1	yes	characteristic
144	C₂×D₈	C₂²×C₁₆₇	1	2	yes	—
145	C₄⋊Q₈	C₂²×C₁₆₇	1	1	yes	characteristic
146	#127	C₂²×C₁₆₇	1	1	yes	characteristic
147	#127	C₂²×C₁₆₇	1	2	yes	—
148	#125	C₂²×C₁₆₇	1	2	yes	—
149	C₂²⋊Q₈	—	2	1	no	—
150	C₂²⋊Q₈	—	2	2	no	—
151	#121	—	2	1	no	—
152	#121	—	2	2	no	—
153	C₄×Q₈	C₂²×C₁₆₇	1	1	yes	characteristic
154	C₄×D₄	C₂²×C₁₆₇	1	1	yes	characteristic
155	C₄×D₄	C₂²×C₁₆₇	1	2	yes	—
156	C₄×D₄	C₂²×C₁₆₇	1	1	yes	characteristic
157	C₄×D₄	C₂²×C₁₆₇	1	2	yes	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	#107	C₂²×C₁₆₇	1	1	yes	characteristic
160	#106	C₂²×C₁₆₇	1	2	yes	—
161	C₄⋊C₈	C₂²×C₁₆₇	1	1	yes	characteristic
162	C₄⋊C₈	C₂²×C₁₆₇	1	2	yes	—
163	#102	C₂²×C₁₆₇	1	1	yes	characteristic
164	#102	C₂²×C₁₆₇	1	1	yes	characteristic
165	#102	C₂²×C₁₆₇	1	2	yes	—
166	#102	C₂²×C₁₆₇	1	2	yes	—
167	#102	C₂²×C₁₆₇	1	2	yes	—
168	#102	C₂²×C₁₆₇	1	2	yes	—
169	#97	C₂²×C₁₆₇	1	2	yes	—
170	C₄×C₈	C₂²×C₁₆₇	1	1	yes	characteristic
171	#41	—	4	4	no	—
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	D₄×C₁₆₇	1	1	yes	characteristic
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	1	no	—
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	C₂³×C₁₆₇	1	1	yes	characteristic
179	C₂×D₄	C₂³×C₁₆₇	1	2	yes	—
180	C₂×D₄	C₂³×C₁₆₇	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	D₈	—	2	2	no	—
186	D₈	—	2	4	no	—
187	C₂×C₈	C₂³×C₁₆₇	1	1	yes	characteristic
188	C₂×C₈	C₂³×C₁₆₇	1	1	yes	characteristic
189	C₂×C₈	C₂³×C₁₆₇	1	2	yes	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	D₄×C₁₆₇	1	1	yes	characteristic
194	C₄⋊C₄	D₄×C₁₆₇	1	2	yes	—
195	C₄⋊C₄	C₂³×C₁₆₇	1	2	yes	—
196	C₄⋊C₄	C₂³×C₁₆₇	1	1	yes	characteristic
197	C₄⋊C₄	C₂³×C₁₆₇	1	1	yes	characteristic
198	C₂²⋊C₄	—	2	1	no	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₄²	C₂³×C₁₆₇	1	1	yes	characteristic
203	C₄²	C₂³×C₁₆₇	1	2	yes	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	1	no	—
207	Q₈	—	2	2	no	—
208	Q₈	—	8	1	no	—
209	D₄	—	2	1	no	—
210	D₄	—	2	2	no	—
211	D₄	—	2	1	no	—
212	D₄	—	2	1	no	—
213	D₄	—	2	2	no	—
214	D₄	—	4	4	no	—
215	D₄	—	4	2	no	—
216	D₄	—	8	2	no	—
217	D₄	—	8	1	no	—
218	D₄	#41×C₁₆₇	1	2	yes	complemented
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	C₂×D₄×C₁₆₇	1	1	yes	characteristic
227	C₂×C₄	C₂×D₄×C₁₆₇	1	2	yes	—
228	C₂×C₄	C₂×D₄×C₁₆₇	1	2	yes	—
229	C₂×C₄	C₂×D₄×C₁₆₇	1	1	yes	characteristic
230	C₂×C₄	C₂⁴×C₁₆₇	1	1	yes	characteristic
231	C₈	—	2	1	no	—
232	C₈	—	2	2	no	—
233	C₈	—	2	1	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	2	no	—
243	C₂²	C₂²×D₄×C₁₆₇	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	1	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	4	2	no	—
250	C₄	AGL(1,ℤ/8)×C₁₆₇	1	2	yes	—
251	C₄	C₂×#41×C₁₆₇	1	1	yes	characteristic
252	C₄	2₋¹⁺⁴×C₁₆₇	1	1	yes	characteristic
253	C₂	—	4	2	no	—
254	C₂	—	8	2	no	—
255	C₂	—	8	1	no	—
256	C₂	#548×C₁₆₇	1	1	yes	characteristic
257	C₂	C₂×AGL(1,ℤ/8)×C₁₆₇	1	1	yes	characteristic
258	C₂	#576×C₁₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆₇ ◁ C₄×Q₈×C₁₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #554×C₁₆₇	C₂	semidirect
2. #550×C₁₆₇	C₂	semidirect
3. #490×C₁₆₇	C₂	semidirect
4. #486×C₁₆₇	C₂	semidirect
5. #478×C₁₆₇	C₂	semidirect
6. #476×C₁₆₇	C₂	semidirect
7. #459×C₁₆₇	C₂	semidirect
8. #459×C₁₆₇	C₂	semidirect
9. #446×C₁₆₇	C₂	semidirect
10. #442×C₁₆₇	C₂	semidirect
11. C₄×D₈×C₁₆₇	C₂	semidirect
88. D₄×C₁₆₇	#41	semidirect
129. C₁₆₇	#3366	direct (N × H)
130. #3366	C₁₆₇	direct (N × H)
131. #554	C₂×C₁₆₇	semidirect
132. #550	C₂×C₁₆₇	semidirect
133. #490	C₂×C₁₆₇	semidirect
134. #486	C₂×C₁₆₇	semidirect
135. #478	C₂×C₁₆₇	semidirect
136. #476	C₂×C₁₆₇	semidirect
137. #459	C₂×C₁₆₇	semidirect
138. #459	C₂×C₁₆₇	semidirect
139. #446	C₂×C₁₆₇	semidirect
140. #442	C₂×C₁₆₇	semidirect
141. C₄×D₈	C₂×C₁₆₇	semidirect
218. D₄	#41×C₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
#554×C₁₆₇	2
#550×C₁₆₇	2
#490×C₁₆₇	2
#486×C₁₆₇	2
#478×C₁₆₇	2
#476×C₁₆₇	2
#459×C₁₆₇	2
#446×C₁₆₇	2
#442×C₁₆₇	2
C₄×D₈×C₁₆₇	2
#3366	167

Subgroups — #3366×C167 #36321726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3366×C₁₆₇ #36321726