Subgroups — #3364×C₁₆₇ #36321724

Order 21376.

336 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#557×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#556×C₁₆₇	C₂	1	1	yes	maximal, characteristic
3	#488×C₁₆₇	C₂	1	2	yes	maximal
4	#487×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#479×C₁₆₇	C₂	1	2	yes	maximal, complemented
6	#474×C₁₆₇	C₂	1	1	yes	maximal, characteristic
7	#464×C₁₆₇	C₂	1	2	yes	maximal
8	#462×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×Q₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×Q₁₆×C₁₆₇	C₂	1	2	yes	maximal, complemented
11	C₄×SD₁₆×C₁₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₁₆×C₁₆₇	C₂²	1	2	yes	—
13	C₂×SD₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₁₆₇	—	2	1	no	—
15	C₄⋊Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
17	#126×C₁₆₇	—	2	2	no	—
18	#125×C₁₆₇	—	2	2	no	—
19	#125×C₁₆₇	C₂²	1	2	yes	—
20	#124×C₁₆₇	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₁₆₇	—	2	1	no	—
22	C₄×Q₈×C₁₆₇	—	2	1	no	—
23	C₄×Q₈×C₁₆₇	C₂²	1	2	yes	—
24	C₄×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₁₆₇	C₂²	1	2	yes	—
26	C₄×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₆₇	C₂²	1	1	yes	characteristic
28	#117×C₁₆₇	—	2	1	no	—
29	#107×C₁₆₇	C₂²	1	2	yes	—
30	#106×C₁₆₇	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₁₆₇	C₂²	1	2	yes	—
32	C₄⋊C₈×C₁₆₇	C₂²	1	1	yes	characteristic
33	#103×C₁₆₇	C₂²	1	2	yes	—
34	#103×C₁₆₇	C₂²	1	2	yes	—
35	#103×C₁₆₇	C₂²	1	2	yes	—
36	#103×C₁₆₇	C₂²	1	1	yes	characteristic
37	#102×C₁₆₇	C₂²	1	1	yes	characteristic
38	#102×C₁₆₇	C₂²	1	2	yes	—
39	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₆₇	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₆₇	—	2	1	no	—
42	C₂×Q₈×C₁₆₇	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₆₇	C₂³	1	2	yes	—
44	C₂×Q₈×C₁₆₇	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₆₇	—	2	1	no	—
47	Q₁₆×C₁₆₇	—	2	4	no	—
48	SD₁₆×C₁₆₇	—	2	2	no	—
49	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₆₇	C₂³	1	2	yes	—
51	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₁₆₇	—	2	1	no	—
53	C₄⋊C₄×C₁₆₇	—	2	2	no	—
54	C₄⋊C₄×C₁₆₇	—	2	2	no	—
55	C₄⋊C₄×C₁₆₇	—	2	1	no	—
56	C₄⋊C₄×C₁₆₇	—	2	2	no	—
57	C₄⋊C₄×C₁₆₇	—	2	1	no	—
58	C₄⋊C₄×C₁₆₇	—	2	2	no	—
59	C₄⋊C₄×C₁₆₇	—	2	1	no	—
60	C₄⋊C₄×C₁₆₇	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₁₆₇	D₄	1	2	yes	—
62	C₄⋊C₄×C₁₆₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₁₆₇	C₂³	1	2	yes	—
64	C₄⋊C₄×C₁₆₇	C₂³	1	1	yes	characteristic
65	C₂²⋊C₄×C₁₆₇	—	2	2	no	—
66	C₂²⋊C₄×C₁₆₇	—	2	1	no	—
67	C₂²⋊C₄×C₁₆₇	—	2	2	no	—
68	C₄²×C₁₆₇	—	2	2	no	—
69	C₄²×C₁₆₇	—	2	2	no	—
70	C₄²×C₁₆₇	—	2	1	no	—
71	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
72	C₄²×C₁₆₇	C₂³	1	2	yes	—
73	C₂³×C₁₆₇	—	2	1	no	—
74	Q₈×C₁₆₇	—	2	2	no	—
75	Q₈×C₁₆₇	—	2	1	no	—
76	Q₈×C₁₆₇	—	2	2	no	—
77	Q₈×C₁₆₇	—	2	2	no	—
78	Q₈×C₁₆₇	—	4	2	no	—
79	Q₈×C₁₆₇	#41	1	2	yes	—
80	D₄×C₁₆₇	—	2	1	no	—
81	D₄×C₁₆₇	—	2	1	no	—
82	C₂×C₄×C₁₆₇	—	2	1	no	—
83	C₂×C₄×C₁₆₇	—	2	1	no	—
84	C₂×C₄×C₁₆₇	—	2	2	no	—
85	C₂×C₄×C₁₆₇	—	2	1	no	—
86	C₂×C₄×C₁₆₇	—	2	2	no	—
87	C₂×C₄×C₁₆₇	—	4	2	no	—
88	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
93	C₈×C₁₆₇	—	2	1	no	—
94	C₈×C₁₆₇	—	2	2	no	—
95	C₈×C₁₆₇	—	2	1	no	—
96	C₂²×C₁₆₇	—	4	1	no	—
97	C₂²×C₁₆₇	—	4	1	no	—
98	C₂²×C₁₆₇	—	4	1	no	—
99	C₂²×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
100	C₄×C₁₆₇	—	2	2	no	—
101	C₄×C₁₆₇	—	2	2	no	—
102	C₄×C₁₆₇	—	2	2	no	—
103	C₄×C₁₆₇	—	2	1	no	—
104	C₄×C₁₆₇	—	4	2	no	—
105	C₄×C₁₆₇	—	4	1	no	—
106	C₄×C₁₆₇	—	4	2	no	—
107	C₄×C₁₆₇	—	4	1	no	—
108	C₄×C₁₆₇	#135	1	2	yes	—
109	C₄×C₁₆₇	C₂×#41	1	1	yes	characteristic
110	C₄×C₁₆₇	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×C₁₆₇	—	8	1	no	—
112	C₂×C₁₆₇	#548	1	1	yes	characteristic
113	C₂×C₁₆₇	C₂×#135	1	1	yes	characteristic
114	C₂×C₁₆₇	#578	1	1	yes	characteristic
115	C₁₆₇	#3364	1	1	yes	characteristic, direct product factor, Sylow
116	#3364	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	#557	C₂×C₁₆₇	1	1	yes	characteristic, complemented
118	#556	C₂×C₁₆₇	1	1	yes	characteristic
119	#488	C₂×C₁₆₇	1	2	yes	—
120	#487	C₂×C₁₆₇	1	1	yes	characteristic, complemented
121	#479	C₂×C₁₆₇	1	2	yes	complemented
122	#474	C₂×C₁₆₇	1	1	yes	characteristic
123	#464	C₂×C₁₆₇	1	2	yes	—
124	#462	C₂×C₁₆₇	1	1	yes	characteristic, complemented
125	C₈×Q₈	C₂×C₁₆₇	1	1	yes	characteristic, complemented
126	C₄×Q₁₆	C₂×C₁₆₇	1	2	yes	complemented
127	C₄×SD₁₆	C₂×C₁₆₇	1	1	yes	characteristic
128	C₂×Q₁₆	C₂²×C₁₆₇	1	2	yes	—
129	C₂×SD₁₆	C₂²×C₁₆₇	1	1	yes	characteristic
130	C₄⋊Q₈	—	2	1	no	—
131	C₄⋊Q₈	C₂²×C₁₆₇	1	1	yes	characteristic
132	C₄⋊Q₈	C₂²×C₁₆₇	1	1	yes	characteristic
133	#126	—	2	2	no	—
134	#125	—	2	2	no	—
135	#125	C₂²×C₁₆₇	1	2	yes	—
136	#124	C₂²×C₁₆₇	1	2	yes	—
137	C₂²⋊Q₈	—	2	1	no	—
138	C₄×Q₈	—	2	1	no	—
139	C₄×Q₈	C₂²×C₁₆₇	1	2	yes	—
140	C₄×Q₈	C₂²×C₁₆₇	1	1	yes	characteristic
141	C₄×Q₈	C₂²×C₁₆₇	1	2	yes	—
142	C₄×Q₈	C₂²×C₁₆₇	1	1	yes	characteristic
143	C₄×D₄	C₂²×C₁₆₇	1	1	yes	characteristic
144	#117	—	2	1	no	—
145	#107	C₂²×C₁₆₇	1	2	yes	—
146	#106	C₂²×C₁₆₇	1	1	yes	characteristic
147	C₄⋊C₈	C₂²×C₁₆₇	1	1	yes	characteristic
148	C₄⋊C₈	C₂²×C₁₆₇	1	2	yes	—
149	#103	C₂²×C₁₆₇	1	1	yes	characteristic
150	#103	C₂²×C₁₆₇	1	2	yes	—
151	#103	C₂²×C₁₆₇	1	2	yes	—
152	#103	C₂²×C₁₆₇	1	2	yes	—
153	#102	C₂²×C₁₆₇	1	1	yes	characteristic
154	#102	C₂²×C₁₆₇	1	2	yes	—
155	C₄×C₈	C₂²×C₁₆₇	1	1	yes	characteristic
156	C₄×C₈	C₂²×C₁₆₇	1	2	yes	—
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	D₄×C₁₆₇	1	1	yes	characteristic
159	C₂×Q₈	C₂³×C₁₆₇	1	2	yes	—
160	C₂×Q₈	C₂³×C₁₆₇	1	1	yes	characteristic
161	C₂×D₄	C₂³×C₁₆₇	1	1	yes	characteristic
162	C₂²×C₄	—	2	1	no	—
163	Q₁₆	—	2	4	no	—
164	SD₁₆	—	2	2	no	—
165	C₂×C₈	C₂³×C₁₆₇	1	1	yes	characteristic
166	C₂×C₈	C₂³×C₁₆₇	1	2	yes	—
167	C₂×C₈	C₂³×C₁₆₇	1	1	yes	characteristic
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	D₄×C₁₆₇	1	1	yes	characteristic
177	C₄⋊C₄	D₄×C₁₆₇	1	2	yes	—
178	C₄⋊C₄	C₂³×C₁₆₇	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₁₆₇	1	2	yes	—
180	C₄⋊C₄	C₂³×C₁₆₇	1	1	yes	characteristic
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	1	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	1	no	—
187	C₄²	C₂³×C₁₆₇	1	1	yes	characteristic
188	C₄²	C₂³×C₁₆₇	1	2	yes	—
189	C₂³	—	2	1	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	2	1	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	4	2	no	—
195	Q₈	#41×C₁₆₇	1	2	yes	—
196	D₄	—	2	1	no	—
197	D₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	C₂×D₄×C₁₆₇	1	2	yes	—
205	C₂×C₄	C₂×D₄×C₁₆₇	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×C₁₆₇	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₁₆₇	1	1	yes	characteristic
208	C₂×C₄	C₂⁴×C₁₆₇	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	2	no	—
211	C₈	—	2	1	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	—	4	1	no	—
215	C₂²	C₂²×D₄×C₁₆₇	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	1	no	—
224	C₄	#135×C₁₆₇	1	2	yes	—
225	C₄	C₂×#41×C₁₆₇	1	1	yes	characteristic
226	C₄	2₋¹⁺⁴×C₁₆₇	1	1	yes	characteristic
227	C₂	—	8	1	no	—
228	C₂	#548×C₁₆₇	1	1	yes	characteristic
229	C₂	C₂×#135×C₁₆₇	1	1	yes	characteristic
230	C₂	#578×C₁₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₆₇
prime radical	C₂×D₄×C₁₆₇	C₂³
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. #557×C₁₆₇	C₂	semidirect
4. #487×C₁₆₇	C₂	semidirect
5. #479×C₁₆₇	C₂	semidirect
8. #462×C₁₆₇	C₂	semidirect
9. C₈×Q₈×C₁₆₇	C₂	semidirect
10. C₄×Q₁₆×C₁₆₇	C₂	semidirect
115. C₁₆₇	#3364	direct (N × H)
116. #3364	C₁₆₇	direct (N × H)
117. #557	C₂×C₁₆₇	semidirect
120. #487	C₂×C₁₆₇	semidirect
121. #479	C₂×C₁₆₇	semidirect
124. #462	C₂×C₁₆₇	semidirect
125. C₈×Q₈	C₂×C₁₆₇	semidirect
126. C₄×Q₁₆	C₂×C₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
#557×C₁₆₇	2
#556×C₁₆₇	2
#488×C₁₆₇	2
#487×C₁₆₇	2
#479×C₁₆₇	2
#474×C₁₆₇	2
#464×C₁₆₇	2
#462×C₁₆₇	2
C₈×Q₈×C₁₆₇	2
C₄×Q₁₆×C₁₆₇	2
C₄×SD₁₆×C₁₆₇	2
#3364	167

Subgroups — #3364×C167 #36321724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3364×C₁₆₇ #36321724