Subgroups — #30755661

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	#22561171	C₂	1	2	yes	maximal
2	#22561171	C₂	1	2	yes	maximal
3	#22560337	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863118	C₂	1	1	yes	maximal, characteristic
5	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	#16863266	C₂²	1	2	yes	—
9	#16863159	—	2	2	no	—
10	#16863158	—	2	4	no	—
11	#16863158	—	2	4	no	—
12	#16863118	—	2	2	no	—
13	C₂²×Q₁₆×C₆₇	—	2	2	no	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
15	#494×C₆₇	—	2	2	no	—
16	#494×C₆₇	C₄	1	4	yes	complemented
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
19	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	C₂×C₄⋊Dic₆₇	—	2	2	no	—
22	C₂×C₄⋊Dic₆₇	—	2	2	no	—
23	C₄×C₆₇⋊C₈	—	2	1	no	—
24	C₄×C₆₇⋊C₈	—	2	1	no	—
25	C₄×C₆₇⋊C₈	—	2	1	no	—
26	C₄×C₆₇⋊C₈	—	2	1	no	—
27	C₂²×Q₈×C₆₇	—	2	2	no	—
28	C₂×Q₁₆×C₆₇	—	2	8	no	—
29	C₂×Q₁₆×C₆₇	—	4	8	no	—
30	C₂²×C₈×C₆₇	D₄	1	2	yes	—
31	C₄⋊Q₈×C₆₇	—	2	2	no	—
32	C₄⋊Q₈×C₆₇	—	2	2	no	—
33	C₄⋊Q₈×C₆₇	—	2	2	no	—
34	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₆₇	D₄	1	2	yes	—
40	C₂²×Dic₆₇	—	4	2	no	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	2	2	no	—
44	C₂×C₆₇⋊C₈	—	2	4	no	—
45	C₂×C₆₇⋊C₈	Q₁₆	1	4	yes	complemented
46	C₄⋊Dic₆₇	—	4	4	no	—
47	C₄⋊Dic₆₇	—	4	4	no	—
48	C₄⋊Dic₆₇	—	4	4	no	—
49	C₂×Q₈×C₆₇	—	2	4	no	—
50	C₂×Q₈×C₆₇	—	4	2	no	—
51	C₂×Q₈×C₆₇	—	4	8	no	—
52	C₂²×C₄×C₆₇	—	4	2	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	Q₁₆×C₆₇	—	4	16	no	—
57	C₂×C₈×C₆₇	—	2	4	no	—
58	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
59	C₄⋊C₄×C₆₇	—	2	4	no	—
60	C₄⋊C₄×C₆₇	—	2	4	no	—
61	C₄⋊C₄×C₆₇	—	4	2	no	—
62	C₄⋊C₄×C₆₇	—	4	2	no	—
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	Q₈×C₆₇	—	4	8	no	—
69	Q₈×C₆₇	—	8	4	no	—
70	C₂×C₄×C₆₇	—	4	4	no	—
71	C₂×C₄×C₆₇	—	8	4	no	—
72	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
79	C₂×C₄×C₆₇	#135	1	2	yes	—
80	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
81	C₂×C₄×C₆₇	#137	1	4	yes	—
82	C₂×Dic₆₇	—	8	2	no	—
83	C₂×Dic₆₇	—	8	2	no	—
84	C₂×Dic₆₇	—	8	2	no	—
85	C₈×C₆₇	—	2	4	no	—
86	C₆₇⋊C₈	—	2	4	no	—
87	C₆₇⋊C₈	—	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	—	8	4	no	—
94	C₄×C₆₇	#418	1	2	yes	—
95	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
96	C₄×C₆₇	#441	1	2	yes	—
97	C₄×C₆₇	#467	1	2	yes	—
98	C₄×C₆₇	#470	1	2	yes	—
99	C₄×C₆₇	#490	1	2	yes	—
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	D₆₇	1	1	yes	characteristic
108	#1990	—	67	2	no	—
109	#1990	—	67	2	no	—
110	#1834	—	67	1	no	—
111	C₂×#322	—	67	1	no	—
112	C₆₇	#9835	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₁₆	—	2	2	no	—
114	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
115	#494	—	2	2	no	—
116	#494	Dic₆₇	1	4	yes	complemented
117	C₂×C₄×C₈	—	67	1	no	—
118	C₂×C₄×C₈	—	67	1	no	—
119	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
120	#384	—	67	2	no	—
121	#340	—	134	4	no	—
122	#340	—	134	4	no	—
123	#340	—	134	2	no	—
124	#322	—	134	2	no	—
125	C₂²×Q₈	—	2	2	no	—
126	C₂×Q₁₆	—	2	8	no	—
127	C₂×Q₁₆	—	4	8	no	—
128	C₂²×C₈	—	67	2	no	—
129	C₂²×C₈	—	67	2	no	—
130	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	C₂×Dic₆₇	1	4	yes	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	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	—	134	1	no	—
142	C₄×C₈	—	134	1	no	—
143	C₄×C₈	—	134	1	no	—
144	C₄×C₈	—	134	1	no	—
145	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
146	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	4	8	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	—	268	2	no	—
152	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
153	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
155	Q₁₆	—	4	16	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	67	4	no	—
158	C₂×C₈	—	134	2	no	—
159	C₂×C₈	—	134	4	no	—
160	C₂×C₈	—	134	2	no	—
161	C₂×C₈	—	134	2	no	—
162	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	268	4	no	—
168	C₄⋊C₄	—	268	4	no	—
169	C₄⋊C₄	—	268	4	no	—
170	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
171	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
172	C₄²	#11761627	1	1	yes	characteristic
173	C₄²	#11761627	1	1	yes	characteristic
174	C₂³	#14524039	1	1	yes	characteristic
175	Q₈	—	4	8	no	—
176	Q₈	—	8	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	8	4	no	—
179	C₂×C₄	—	536	2	no	—
180	C₂×C₄	—	536	2	no	—
181	C₂×C₄	—	536	2	no	—
182	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
183	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
184	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
185	C₂×C₄	#14524014	1	1	yes	characteristic
186	C₂×C₄	#14524014	1	1	yes	characteristic
187	C₂×C₄	#14524030	1	1	yes	characteristic
188	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
189	C₂×C₄	#14524066	1	2	yes	—
190	C₂×C₄	#14524069	1	4	yes	—
191	C₂×C₄	#14524075	1	4	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	268	2	no	—
194	C₈	—	134	4	no	—
195	C₂²	#16863299	1	1	yes	characteristic
196	C₂²	#16863774	1	1	yes	characteristic
197	C₂²	#16863781	1	1	yes	characteristic
198	C₂²	#16863783	1	2	yes	—
199	C₂²	#16863789	1	2	yes	—
200	C₄	—	8	4	no	—
201	C₄	—	1072	2	no	—
202	C₄	#16863401	1	2	yes	—
203	C₄	Q₁₆×Dic₆₇	1	2	yes	—
204	C₄	#16863503	1	2	yes	—
205	C₄	#16863634	1	2	yes	—
206	C₄	#16863651	1	2	yes	—
207	C₄	#16863760	1	2	yes	—
208	C₂	#22559872	1	2	yes	—
209	C₂	#22559884	1	2	yes	—
210	C₂	#22561034	1	1	yes	characteristic
211	C₂	#22561071	1	1	yes	characteristic
212	C₂	#22561081	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#494×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#494	D₆₇
hypocenter	C₆₇	#9835
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
16. #494×C₆₇	C₄	semidirect
45. C₂×C₆₇⋊C₈	Q₁₆	semidirect
112. C₆₇	#9835	semidirect
116. #494	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22561171	2
#22560337	2
C₂×#16863118	2
C₂×#494×C₆₇	2
#9835	67