Subgroups — #30749498

Order 17152.

378 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560361	C₂	1	1	yes	maximal, characteristic
2	#22560326	C₂	1	1	yes	maximal, characteristic
3	#22558044	C₂	1	4	yes	maximal
4	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	#16863151	—	2	2	no	—
9	C₂×#97×C₆₇	C₄	1	2	yes	—
10	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
11	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
13	C₄³×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₆₇⋊C₈	—	2	2	no	—
15	C₂²×C₆₇⋊C₈	—	2	2	no	—
16	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
17	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
18	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
19	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₆₇	D₄	1	2	yes	—
24	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
25	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
26	#97×C₆₇	—	2	4	no	—
27	#97×C₆₇	—	2	2	no	—
28	#97×C₆₇	C₈	1	4	yes	complemented
29	C₄×C₈×C₆₇	—	2	2	no	—
30	C₄×C₈×C₆₇	—	2	2	no	—
31	C₂×C₆₇⋊C₈	—	4	2	no	—
32	C₂×C₆₇⋊C₈	—	4	2	no	—
33	C₂×C₆₇⋊C₈	—	4	4	no	—
34	C₂×C₆₇⋊C₈	—	4	2	no	—
35	C₂×C₆₇⋊C₈	—	4	2	no	—
36	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
37	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₆₇	—	2	2	no	—
42	C₂×C₈×C₆₇	—	2	2	no	—
43	C₂×C₈×C₆₇	—	2	2	no	—
44	C₂×C₈×C₆₇	—	2	2	no	—
45	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
46	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
47	C₄²×C₆₇	—	2	2	no	—
48	C₄²×C₆₇	—	2	2	no	—
49	C₄²×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	—	4	2	no	—
51	C₄²×C₆₇	C₂×C₈	1	2	yes	—
52	C₄²×C₆₇	M₁₆	1	2	yes	—
53	C₄²×C₆₇	#41	1	2	yes	—
54	C₄²×C₆₇	#41	1	2	yes	—
55	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
63	C₂×C₄×C₆₇	#117	1	2	yes	—
64	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#125	1	2	yes	—
67	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₆₇	#137	1	2	yes	—
72	C₈×C₆₇	—	2	4	no	—
73	C₈×C₆₇	—	2	4	no	—
74	C₆₇⋊C₈	—	8	2	no	—
75	C₆₇⋊C₈	—	8	2	no	—
76	C₂²×C₆₇	#382	1	1	yes	characteristic
77	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₂²×C₆₇	#423	1	1	yes	characteristic
79	C₂²×C₆₇	#428	1	2	yes	—
80	C₂²×C₆₇	#430	1	2	yes	—
81	C₄×C₆₇	—	2	4	no	—
82	C₄×C₆₇	—	2	2	no	—
83	C₄×C₆₇	—	4	2	no	—
84	C₄×C₆₇	#431	1	2	yes	—
85	C₄×C₆₇	#432	1	2	yes	—
86	C₄×C₆₇	#489	1	2	yes	—
87	C₄×C₆₇	#490	1	1	yes	characteristic
88	C₄×C₆₇	#491	1	1	yes	characteristic
89	#8786	—	67	1	no	maximal, Sylow
90	C₂×C₆₇	#1565	1	2	yes	—
91	C₂×C₆₇	#1569	1	2	yes	—
92	C₂×C₆₇	#1831	1	1	yes	characteristic
93	C₂×C₆₇	#1836	1	1	yes	characteristic
94	C₂×C₆₇	#1838	1	1	yes	characteristic
95	#1842	—	67	1	no	—
96	#1831	—	67	1	no	—
97	C₄×#97	D₆₇	1	1	yes	characteristic
98	#1277	—	67	4	no	—
99	C₆₇	#8786	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄⋊C₈	—	67	2	no	—
101	C₂×C₄⋊C₈	—	67	2	no	—
102	C₂×C₄⋊C₈	—	134	2	no	—
103	C₂×#97	Dic₆₇	1	2	yes	—
104	C₂×#97	Dic₆₇	1	1	yes	characteristic
105	C₂×#97	Dic₆₇	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
107	C₄³	C₂×D₆₇	1	1	yes	characteristic
108	#336	—	134	2	no	—
109	C₂²×C₈	—	134	2	no	—
110	C₂²×C₈	—	134	2	no	—
111	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
112	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
113	C₂×C₄²	Dic₁₃₄	1	2	yes	—
114	C₂×C₄²	D₂₆₈	1	2	yes	—
115	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
116	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
117	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
118	C₄⋊C₈	—	268	4	no	—
119	C₄⋊C₈	—	134	4	no	—
120	C₄⋊C₈	—	134	4	no	—
121	#97	—	2	4	no	—
122	#97	—	2	2	no	—
123	#97	C₆₇⋊C₈	1	4	yes	complemented
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
127	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
128	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
129	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
130	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	268	2	no	—
136	C₂×C₈	—	268	2	no	—
137	C₂×C₈	—	268	4	no	—
138	C₂×C₈	—	268	2	no	—
139	C₂×C₈	—	268	2	no	—
140	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
141	C₂×C₈	#11761612	1	4	yes	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
147	C₄²	#11761612	1	2	yes	—
148	C₄²	#11761628	1	2	yes	—
149	C₄²	#11761628	1	2	yes	—
150	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
158	C₂×C₄	#14524002	1	2	yes	—
159	C₂×C₄	#14524031	1	1	yes	characteristic
160	C₂×C₄	#14524031	1	1	yes	characteristic
161	C₂×C₄	#14524035	1	2	yes	—
162	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
163	C₂×C₄	C₂×#11761612	1	2	yes	—
164	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
165	C₂×C₄	#14524071	1	2	yes	—
166	C₂×C₄	#14524076	1	2	yes	—
167	C₈	—	2	4	no	—
168	C₈	—	2	4	no	—
169	C₈	—	536	2	no	—
170	C₈	—	536	2	no	—
171	C₂²	#16863262	1	1	yes	characteristic
172	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
173	C₂²	#16863420	1	1	yes	characteristic
174	C₂²	#16863438	1	2	yes	—
175	C₂²	#16863445	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	2	no	—
179	C₄	#16863449	1	2	yes	—
180	C₄	#16863453	1	2	yes	—
181	C₄	#16863756	1	2	yes	—
182	C₄	#16863761	1	1	yes	characteristic
183	C₄	#16863765	1	1	yes	characteristic
184	C₂	#22558961	1	2	yes	—
185	C₂	#22558976	1	2	yes	—
186	C₂	#22560326	1	1	yes	characteristic
187	C₂	#22560343	1	1	yes	characteristic
188	C₂	#22560350	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#97×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#97	D₆₇
hypocenter	C₆₇	#8786
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
28. #97×C₆₇	C₈	semidirect
99. C₆₇	#8786	semidirect
123. #97	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22560361	2
#22560326	2
#22558044	2
C₄×#97×C₆₇	2
#8786	67