Subgroups — #30755660

Order 17152.

526 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863776	C₂	1	1	yes	maximal, characteristic
2	#22561170	C₂	1	2	yes	maximal
3	#22561170	C₂	1	2	yes	maximal
4	#22560337	C₂	1	1	yes	maximal, characteristic
5	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	—	2	2	no	—
7	C₂×#14524043	C₂²	1	2	yes	—
8	#16863776	—	2	2	no	—
9	#16863776	C₄	1	4	yes	complemented
10	#16863266	C₂²	1	2	yes	—
11	#16863159	—	2	2	no	—
12	#16863159	—	2	4	no	—
13	#16863159	—	2	4	no	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#322×C₆₇	—	2	2	no	—
18	C₂²×Dic₁₃₄	—	2	2	no	—
19	C₂×Dic₂₆₈	—	2	8	no	—
20	C₂×Dic₂₆₈	—	4	8	no	—
21	#14524043	—	2	2	no	—
22	#14524043	—	2	2	no	—
23	#14524043	—	2	2	no	—
24	#14524043	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	C₂×C₄⋊Dic₆₇	—	2	2	no	—
28	C₂×C₄⋊Dic₆₇	—	2	2	no	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	D₄	1	2	yes	—
31	C₂²×C₈×C₆₇	D₄	1	2	yes	—
32	C₂²×C₈×C₆₇	D₄	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₆₇	—	2	1	no	—
35	C₄×C₈×C₆₇	—	2	1	no	—
36	C₄×C₈×C₆₇	—	2	1	no	—
37	C₄×C₈×C₆₇	—	2	1	no	—
38	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₆₇	D₄	1	2	yes	—
40	C₂×Dic₁₃₄	—	2	4	no	—
41	C₂×Dic₁₃₄	—	4	8	no	—
42	C₂×Dic₁₃₄	—	4	2	no	—
43	C₂²×Dic₆₇	—	4	2	no	—
44	C₂²×Dic₆₇	—	4	2	no	—
45	Dic₂₆₈	—	4	16	no	—
46	C₄⋊Dic₆₇	—	2	4	no	—
47	C₄⋊Dic₆₇	—	2	4	no	—
48	C₄⋊Dic₆₇	—	4	2	no	—
49	C₄⋊Dic₆₇	—	4	2	no	—
50	C₄⋊Dic₆₇	—	4	4	no	—
51	C₄⋊Dic₆₇	—	4	4	no	—
52	C₄⋊Dic₆₇	—	4	4	no	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	—	2	4	no	—
59	C₂×C₈×C₆₇	—	2	2	no	—
60	C₂×C₈×C₆₇	—	2	4	no	—
61	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
62	C₂×C₈×C₆₇	Q₁₆	1	4	yes	complemented
63	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₆₇	#41	1	1	yes	characteristic
66	C₄²×C₆₇	#41	1	1	yes	characteristic
67	C₂³×C₆₇	#127	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
73	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
75	C₂×C₄×C₆₇	#135	1	2	yes	—
76	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
77	C₂×C₄×C₆₇	#137	1	4	yes	—
78	C₂×Dic₆₇	—	4	4	no	—
79	C₂×Dic₆₇	—	8	4	no	—
80	C₂×Dic₆₇	—	8	2	no	—
81	C₂×Dic₆₇	—	8	2	no	—
82	C₂×Dic₆₇	—	8	2	no	—
83	Dic₁₃₄	—	4	8	no	—
84	Dic₁₃₄	—	8	4	no	—
85	C₈×C₆₇	—	2	4	no	—
86	C₈×C₆₇	—	2	4	no	—
87	C₈×C₆₇	—	4	2	no	—
88	C₂²×C₆₇	#390	1	1	yes	characteristic
89	C₂²×C₆₇	#494	1	1	yes	characteristic
90	C₂²×C₆₇	#495	1	1	yes	characteristic
91	C₂²×C₆₇	#496	1	2	yes	—
92	C₂²×C₆₇	#497	1	2	yes	—
93	C₄×C₆₇	#418	1	2	yes	—
94	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
95	C₄×C₆₇	#441	1	2	yes	—
96	C₄×C₆₇	#467	1	2	yes	—
97	C₄×C₆₇	#470	1	2	yes	—
98	C₄×C₆₇	#490	1	2	yes	—
99	Dic₆₇	—	8	4	no	—
100	Dic₆₇	—	16	2	no	—
101	#9835	—	67	1	no	maximal, Sylow
102	C₂×C₆₇	#1720	1	2	yes	—
103	C₂×C₆₇	#1723	1	2	yes	—
104	C₂×C₆₇	#1961	1	1	yes	characteristic
105	C₂×C₆₇	#1969	1	1	yes	characteristic
106	C₂×C₆₇	#1971	1	1	yes	characteristic
107	C₂×#494	—	67	1	no	—
108	#1990	—	67	2	no	—
109	#1990	—	67	2	no	—
110	#1834	—	67	1	no	—
111	C₂×#322	D₆₇	1	1	yes	characteristic
112	C₆₇	#9835	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₁₆	—	134	2	no	—
114	C₂×C₄⋊Q₈	—	67	2	no	—
115	#494	—	67	4	no	—
116	#494	—	134	2	no	—
117	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
120	#384	—	67	2	no	—
121	#340	—	134	2	no	—
122	#340	—	134	4	no	—
123	#340	—	134	4	no	—
124	#322	—	2	2	no	—
125	C₂²×Q₈	—	134	2	no	—
126	C₂×Q₁₆	—	268	8	no	—
127	C₂×Q₁₆	—	134	8	no	—
128	C₂²×C₈	D₂₆₈	1	2	yes	—
129	C₂²×C₈	D₂₆₈	1	2	yes	—
130	C₂²×C₈	D₂₆₈	1	2	yes	—
131	C₄⋊Q₈	—	67	4	no	—
132	C₄⋊Q₈	—	134	2	no	—
133	C₄⋊Q₈	—	134	2	no	—
134	C₄⋊Q₈	—	134	2	no	—
135	C₂×C₄⋊C₄	—	134	2	no	—
136	C₂×C₄⋊C₄	—	134	2	no	—
137	C₂×C₄⋊C₄	—	134	2	no	—
138	C₂×C₄⋊C₄	—	134	2	no	—
139	C₂×C₄⋊C₄	—	134	2	no	—
140	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	C₄×D₆₇	1	2	yes	—
146	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
147	C₂×Q₈	—	268	2	no	—
148	C₂×Q₈	—	268	8	no	—
149	C₂×Q₈	—	134	4	no	—
150	C₂²×C₄	—	268	2	no	—
151	C₂²×C₄	—	268	2	no	—
152	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
155	Q₁₆	—	268	16	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
162	C₂×C₈	Dic₂₆₈	1	4	yes	complemented
163	C₄⋊C₄	—	268	2	no	—
164	C₄⋊C₄	—	268	2	no	—
165	C₄⋊C₄	—	268	4	no	—
166	C₄⋊C₄	—	268	4	no	—
167	C₄⋊C₄	—	268	4	no	—
168	C₄⋊C₄	—	134	4	no	—
169	C₄⋊C₄	—	134	4	no	—
170	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
171	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
172	C₄²	#11761628	1	1	yes	characteristic
173	C₄²	#11761628	1	1	yes	characteristic
174	C₂³	#14524040	1	1	yes	characteristic
175	Q₈	—	268	8	no	—
176	Q₈	—	536	4	no	—
177	C₂×C₄	—	268	4	no	—
178	C₂×C₄	—	536	4	no	—
179	C₂×C₄	—	536	2	no	—
180	C₂×C₄	—	536	2	no	—
181	C₂×C₄	—	536	2	no	—
182	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
183	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
184	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
185	C₂×C₄	#14524017	1	1	yes	characteristic
186	C₂×C₄	#14524017	1	1	yes	characteristic
187	C₂×C₄	#14524031	1	1	yes	characteristic
188	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
189	C₂×C₄	#14524065	1	2	yes	—
190	C₂×C₄	#14524071	1	4	yes	—
191	C₂×C₄	#14524076	1	4	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	2	4	no	—
194	C₈	—	4	2	no	—
195	C₂²	#16863298	1	1	yes	characteristic
196	C₂²	#16863776	1	1	yes	characteristic
197	C₂²	#16863780	1	1	yes	characteristic
198	C₂²	#16863785	1	2	yes	—
199	C₂²	#16863790	1	2	yes	—
200	C₄	—	536	4	no	—
201	C₄	—	1072	2	no	—
202	C₄	#16863402	1	2	yes	—
203	C₄	C₄×Dic₂₆₈	1	2	yes	—
204	C₄	#16863504	1	2	yes	—
205	C₄	#16863637	1	2	yes	—
206	C₄	#16863654	1	2	yes	—
207	C₄	#16863761	1	2	yes	—
208	C₂	#22559869	1	2	yes	—
209	C₂	#22559886	1	2	yes	—
210	C₂	#22561033	1	1	yes	characteristic
211	C₂	#22561070	1	1	yes	characteristic
212	C₂	#22561080	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524040
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#9835
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
9. #16863776	C₄	semidirect
62. C₂×C₈×C₆₇	Q₁₆	semidirect
112. C₆₇	#9835	semidirect
162. C₂×C₈	Dic₂₆₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863776	2
#22561170	2
#22560337	2
C₂×#322×C₆₇	2
#9835	67