Subgroups — #36333080

Order 21376.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923823	C₂	1	1	yes	maximal, characteristic
2	#24923400	C₂	1	1	yes	maximal, characteristic, complemented
3	#24923399	C₂	1	1	yes	maximal, characteristic, complemented
4	#24923399	C₂	1	2	yes	maximal, complemented
5	#24923398	C₂	1	2	yes	maximal, complemented
6	#24923393	C₂	1	1	yes	maximal, characteristic, complemented
7	#24923294	C₂	1	2	yes	maximal
8	C₄⋊C₈×D₁₆₇	C₂	1	1	yes	maximal, characteristic
9	#24923223	C₂	1	2	yes	maximal
10	C₄×#14581933	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14581933	C₂²	1	1	yes	characteristic
13	C₂×#14581933	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₁₆₇	C₂²	1	2	yes	—
15	#19128831	C₄	1	1	yes	characteristic
16	#19128830	C₄	1	1	yes	characteristic
17	#19128822	C₄	1	2	yes	—
18	#19128812	C₄	1	1	yes	characteristic
19	#19128810	C₄	1	2	yes	—
20	#19128808	C₄	1	1	yes	characteristic
21	C₄×Dic₃₃₄	C₂²	1	1	yes	characteristic
22	C₄×Dic₃₃₄	C₂²	1	2	yes	—
23	#19128789	C₂²	1	2	yes	—
24	C₄²×D₁₆₇	C₂²	1	1	yes	characteristic
25	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	2	yes	—
27	Dic₁₆₇⋊C₈	C₂²	1	2	yes	—
28	Dic₁₆₇⋊C₈	C₂²	1	2	yes	—
29	Dic₁₆₇⋊C₈	C₂²	1	2	yes	—
30	#19128743	C₂²	1	2	yes	—
31	#19128742	C₂²	1	1	yes	characteristic
32	#19128742	C₂²	1	1	yes	characteristic
33	#19128742	C₂²	1	2	yes	—
34	C₄×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₆₇	C₂²	1	2	yes	—
36	C₄×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₆₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₆₇	C₂²	1	2	yes	—
39	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₆₇	C₂²	1	2	yes	—
41	C₂×Dic₃₃₄	C₂×C₄	1	1	yes	characteristic
42	C₂×Dic₃₃₄	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₆₇	C₂³	1	1	yes	characteristic
46	#14581933	—	2	2	no	—
47	#14581933	Q₈	1	4	yes	complemented
48	C₂×C₁₆₇⋊C₈	C₂³	1	2	yes	—
49	C₂×C₁₆₇⋊C₈	C₂³	1	1	yes	characteristic
50	C₂×C₁₆₇⋊C₈	C₂³	1	1	yes	characteristic
51	C₈×D₁₆₇	—	2	4	no	—
52	C₄⋊Dic₁₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Dic₁₆₇	C₂×C₄	1	2	yes	—
54	Dic₁₆₇⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₁₆₇⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₁₆₇⋊C₄	C₂×C₄	1	2	yes	—
57	Dic₁₆₇⋊C₄	C₂×C₄	1	2	yes	—
58	Dic₁₆₇⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₁₆₇)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₁₆₇)⋊C₄	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₆₇	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₆₇	C₂³	1	2	yes	—
64	C₄×Dic₁₆₇	C₂³	1	1	yes	characteristic
65	C₂×Q₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₆₇	C₂³	1	2	yes	—
67	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₆₇	C₂×C₄	1	2	yes	—
70	C₄⋊C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
72	C₄²×C₁₆₇	C₂³	1	2	yes	—
73	C₂²×D₁₆₇	C₂²×C₄	1	1	yes	characteristic
74	Dic₃₃₄	—	2	2	no	—
75	Dic₃₃₄	—	4	2	no	—
76	C₂×Dic₁₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₆₇	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₆₇	C₂²×C₄	1	2	yes	—
80	C₂×Dic₁₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₄×D₁₆₇	—	2	2	no	—
82	C₄×D₁₆₇	—	4	1	no	—
83	C₄×D₁₆₇	C₂×Q₈	1	2	yes	—
84	C₄×D₁₆₇	#41	1	2	yes	—
85	C₁₆₇⋊C₈	—	2	2	no	—
86	C₁₆₇⋊C₈	—	2	1	no	—
87	C₁₆₇⋊C₈	C₂×Q₈	1	2	yes	—
88	Q₈×C₁₆₇	—	2	2	no	—
89	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₆₇	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₆₇	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
94	C₈×C₁₆₇	—	2	2	no	—
95	C₈×C₁₆₇	—	2	1	no	—
96	C₈×C₁₆₇	C₂×Q₈	1	2	yes	—
97	C₂²×C₁₆₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₁₆₇	—	2	1	no	—
99	C₂×D₁₆₇	—	2	1	no	—
100	C₂×D₁₆₇	C₄×Q₈	1	2	yes	—
101	C₄×C₁₆₇	—	2	2	no	—
102	C₄×C₁₆₇	—	2	1	no	—
103	C₄×C₁₆₇	—	2	2	no	—
104	C₄×C₁₆₇	#131	1	2	yes	—
105	C₄×C₁₆₇	C₂²×Q₈	1	1	yes	characteristic
106	C₄×C₁₆₇	C₂×#41	1	1	yes	characteristic
107	Dic₁₆₇	—	2	1	no	—
108	Dic₁₆₇	—	2	2	no	—
109	Dic₁₆₇	—	2	1	no	—
110	Dic₁₆₇	—	2	2	no	—
111	Dic₁₆₇	C₄×Q₈	1	2	yes	—
112	C₂×C₁₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
113	C₂×C₁₆₇	C₂×#131	1	1	yes	characteristic
114	C₂×C₁₆₇	#568	1	1	yes	characteristic
115	D₁₆₇	—	2	2	no	—
116	C₁₆₇	#2962	1	1	yes	characteristic, complemented, Sylow
117	#2962	—	167	1	no	maximal, Sylow
118	#533	—	167	1	no	—
119	#446	—	167	1	no	—
120	#446	—	167	2	no	—
121	#446	—	167	1	no	—
122	#446	—	167	2	no	—
123	C₈×Q₈	—	167	1	no	—
124	C₈×Q₈	D₁₆₇	1	1	yes	characteristic, complemented
125	#424	—	167	2	no	—
126	C₂×C₄⋊C₈	—	167	1	no	—
127	#405	—	167	2	no	—
128	C₄×M₁₆	—	167	1	no	—
129	C₂×M₁₆	—	167	1	no	—
130	C₂×M₁₆	—	167	1	no	—
131	C₂²×C₈	—	167	2	no	—
132	C₄⋊Q₈	—	167	1	no	—
133	C₄⋊Q₈	—	167	1	no	—
134	#125	—	167	2	no	—
135	C₂²⋊Q₈	—	167	1	no	—
136	C₂²⋊Q₈	—	167	2	no	—
137	C₂²⋊Q₈	—	167	1	no	—
138	C₄×Q₈	—	167	1	no	—
139	C₄×Q₈	—	167	2	no	—
140	C₄×Q₈	C₂×D₁₆₇	1	1	yes	characteristic
141	#117	—	167	2	no	—
142	C₂×C₄²	—	167	1	no	—
143	C₄⋊C₈	—	167	2	no	—
144	C₄⋊C₈	—	167	2	no	—
145	C₄⋊C₈	—	167	2	no	—
146	C₄⋊C₈	—	167	1	no	—
147	C₄⋊C₈	—	167	2	no	—
148	C₄⋊C₈	C₂×D₁₆₇	1	1	yes	characteristic
149	C₄⋊C₈	C₂×D₁₆₇	1	2	yes	—
150	#97	—	167	1	no	—
151	#97	—	167	1	no	—
152	#97	—	167	2	no	—
153	#97	—	167	2	no	—
154	C₄×C₈	—	167	2	no	—
155	C₄×C₈	—	167	1	no	—
156	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
157	C₄×C₈	C₂×D₁₆₇	1	2	yes	—
158	C₂×Q₈	—	167	1	no	—
159	C₂×Q₈	—	167	2	no	—
160	C₂×Q₈	C₄×D₁₆₇	1	1	yes	characteristic
161	C₂²×C₄	—	167	1	no	—
162	C₂²×C₄	—	167	1	no	—
163	C₂²×C₄	—	167	1	no	—
164	M₁₆	—	334	2	no	—
165	M₁₆	—	167	4	no	—
166	C₂×C₈	—	334	4	no	—
167	C₂×C₈	—	167	1	no	—
168	C₂×C₈	—	167	2	no	—
169	C₂×C₈	—	167	1	no	—
170	C₂×C₈	C₂²×D₁₆₇	1	2	yes	—
171	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
173	C₄⋊C₄	—	167	2	no	—
174	C₄⋊C₄	—	167	2	no	—
175	C₄⋊C₄	—	167	2	no	—
176	C₄⋊C₄	—	167	2	no	—
177	C₄⋊C₄	—	167	2	no	—
178	C₄⋊C₄	—	167	2	no	—
179	C₄⋊C₄	—	167	1	no	—
180	C₄⋊C₄	C₄×D₁₆₇	1	2	yes	—
181	C₄⋊C₄	C₄×D₁₆₇	1	1	yes	characteristic
182	C₂²⋊C₄	—	167	2	no	—
183	C₂²⋊C₄	—	167	2	no	—
184	C₄²	—	167	1	no	—
185	C₄²	—	167	1	no	—
186	C₄²	—	167	2	no	—
187	C₄²	—	167	1	no	—
188	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
189	C₄²	C₂²×D₁₆₇	1	2	yes	—
190	C₂³	—	167	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	334	2	no	—
193	Q₈	—	668	2	no	—
194	C₂×C₄	—	334	2	no	—
195	C₂×C₄	—	668	1	no	—
196	C₂×C₄	—	167	1	no	—
197	C₂×C₄	—	167	1	no	—
198	C₂×C₄	—	167	2	no	—
199	C₂×C₄	—	167	2	no	—
200	C₂×C₄	—	167	1	no	—
201	C₂×C₄	—	167	2	no	—
202	C₂×C₄	—	167	2	no	—
203	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₁₆₇	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₁₆₇	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₁₆₇	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	334	2	no	—
211	C₈	—	334	1	no	—
212	C₈	—	167	2	no	—
213	C₈	Q₈×D₁₆₇	1	2	yes	—
214	C₂²	—	334	1	no	—
215	C₂²	—	334	1	no	—
216	C₂²	—	167	2	no	—
217	C₂²	C₂²×C₄×D₁₆₇	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	334	1	no	—
222	C₄	—	334	2	no	—
223	C₄	—	334	1	no	—
224	C₄	—	334	2	no	—
225	C₄	—	167	2	no	—
226	C₄	#19128838	1	2	yes	—
227	C₄	C₂×Q₈×D₁₆₇	1	1	yes	characteristic
228	C₄	#41×D₁₆₇	1	1	yes	characteristic
229	C₂	—	334	2	no	—
230	C₂	C₄×Q₈×D₁₆₇	1	1	yes	characteristic
231	C₂	C₂×#19128838	1	1	yes	characteristic
232	C₂	#24924034	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₆₇
commutator, socle	C₂²×C₁₆₇	C₂³×C₄
Fitting	C₈×Q₈×C₁₆₇	C₂
prime radical	C₂²×D₁₆₇	C₂²×C₄
cosocle	C₂×C₄×C₁₆₇	C₂⁴
hypercenter	C₈×Q₈	D₁₆₇
hypocenter	C₁₆₇	#2962
LCS	G ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #24923400	C₂	semidirect
3. #24923399	C₂	semidirect
4. #24923399	C₂	semidirect
5. #24923398	C₂	semidirect
6. #24923393	C₂	semidirect
11. C₈×Q₈×C₁₆₇	C₂	semidirect
47. #14581933	Q₈	semidirect
116. C₁₆₇	#2962	semidirect
124. C₈×Q₈	D₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
#24923823	2
#24923400	2
#24923399	2
#24923398	2
#24923393	2
#24923294	2
C₄⋊C₈×D₁₆₇	2
#24923223	2
C₄×#14581933	2
C₈×Q₈×C₁₆₇	2
#2962	167