Subgroups — #30774396

Order 17152.

334 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863217	C₂	1	1	yes	maximal, characteristic
2	#22560088	C₂	1	1	yes	maximal, characteristic
3	#22558316	C₂	1	2	yes	maximal
4	#22558307	C₂	1	2	yes	maximal
5	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂×#14523979	C₂²	1	1	yes	characteristic
8	#16863434	C₂²	1	1	yes	characteristic
9	#16863351	—	2	1	no	—
10	#16863217	—	2	2	no	—
11	#16863156	—	2	1	no	—
12	C₂×#125×C₆₇	—	2	1	no	—
13	#499×C₆₇	—	2	2	no	—
14	#499×C₆₇	C₄	1	4	yes	complemented
15	C₂×#107×C₆₇	—	2	1	no	—
16	C₂×#106×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#11761612	—	2	1	no	—
19	#14524002	—	2	1	no	—
20	#14523983	—	2	2	no	—
21	#14523983	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	#14523979	—	2	1	no	—
24	#14523979	Q₈	1	2	yes	—
25	#14523977	Q₈	1	2	yes	—
26	#14523975	—	2	1	no	—
27	#14523953	—	4	1	no	—
28	C₈×Dic₆₇	—	4	1	no	—
29	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	#125×C₆₇	—	2	4	no	—
32	#125×C₆₇	—	4	2	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	#107×C₆₇	—	2	4	no	—
37	#106×C₆₇	—	2	2	no	—
38	#106×C₆₇	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₆₇	D₄	1	2	yes	—
41	#11761612	—	2	2	no	—
42	#11761612	—	4	1	no	—
43	C₂×C₆₇⋊C₈	—	4	1	no	—
44	C₄⋊Dic₆₇	—	2	2	no	—
45	C₂²⋊Dic₆₇	—	4	1	no	—
46	C₄×Dic₆₇	—	4	1	no	—
47	C₆₇⋊C₁₆	—	4	2	no	—
48	C₂²×C₄×C₆₇	—	2	1	no	—
49	C₂²×C₄×C₆₇	—	4	2	no	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
55	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
60	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
61	C₄⋊C₄×C₆₇	—	2	4	no	—
62	C₄⋊C₄×C₆₇	—	4	8	no	—
63	C₄⋊C₄×C₆₇	—	4	2	no	—
64	C₄²×C₆₇	—	2	1	no	—
65	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
66	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	4	2	no	—
69	C₂×C₄×C₆₇	—	4	4	no	—
70	C₂×C₄×C₆₇	—	8	4	no	—
71	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×Dic₆₇	—	4	2	no	—
78	C₈×C₆₇	#103	1	2	yes	—
79	C₈×C₆₇	#103	1	2	yes	—
80	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
81	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
82	C₈×C₆₇	#124	1	1	yes	characteristic
83	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
84	C₆₇⋊C₈	—	4	2	no	—
85	C₂²×C₆₇	—	2	2	no	—
86	C₂²×C₆₇	#418	1	1	yes	characteristic
87	C₂²×C₆₇	#476	1	1	yes	characteristic
88	C₂²×C₆₇	#477	1	1	yes	characteristic
89	C₄×C₆₇	—	4	2	no	—
90	C₄×C₆₇	—	8	4	no	—
91	C₄×C₆₇	#391	1	1	yes	characteristic
92	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
93	C₄×C₆₇	#476	1	1	yes	characteristic
94	C₄×C₆₇	#477	1	1	yes	characteristic
95	Dic₆₇	—	8	2	no	—
96	#13273	—	67	1	no	maximal, Sylow
97	C₂×C₆₇	—	2	1	no	—
98	C₂×C₆₇	—	2	1	no	—
99	C₂×C₆₇	#1987	1	1	yes	characteristic
100	C₂×C₆₇	#2231	1	2	yes	—
101	C₂×#499	D₆₇	1	1	yes	characteristic
102	C₂×#364	—	67	1	no	—
103	#1772	—	67	1	no	—
104	#1386	—	67	2	no	—
105	#1383	—	67	2	no	—
106	C₆₇	#13273	1	1	yes	characteristic, complemented, Sylow
107	C₂×#125	—	2	1	no	—
108	C₂×M₃₂	—	67	2	no	—
109	#499	—	2	2	no	—
110	#499	Dic₆₇	1	4	yes	complemented
111	C₂×#108	—	67	1	no	—
112	#427	—	67	1	no	—
113	C₂×#107	—	2	1	no	—
114	C₂×#106	C₂×D₆₇	1	2	yes	—
115	#405	—	134	1	no	—
116	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
117	#364	—	134	2	no	—
118	#339	—	134	1	no	—
119	C₂×M₁₆	—	134	1	no	—
120	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
122	#125	—	2	4	no	—
123	#125	—	4	2	no	—
124	#117	—	134	1	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	4	2	no	—
127	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
128	M₃₂	—	134	2	no	—
129	M₃₂	—	134	2	no	—
130	C₂×C₁₆	—	134	2	no	—
131	#108	—	67	2	no	—
132	#108	—	134	1	no	—
133	#107	—	2	4	no	—
134	#107	—	67	2	no	—
135	#106	—	2	2	no	—
136	#106	—	134	1	no	—
137	#106	C₂×Dic₆₇	1	4	yes	—
138	#97	—	268	1	no	—
139	C₄×C₈	—	268	1	no	—
140	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
141	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
145	M₁₆	—	268	1	no	—
146	M₁₆	—	134	2	no	—
147	C₂×C₈	—	268	1	no	—
148	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
149	C₂×C₈	#11761615	1	1	yes	characteristic
150	C₂×C₈	#11761615	1	1	yes	characteristic
151	C₂×C₈	#11761618	1	2	yes	—
152	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
153	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
154	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
155	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
156	C₂×C₈	#11761629	1	1	yes	characteristic
157	C₂×C₈	#11761629	1	1	yes	characteristic
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	8	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	134	2	no	—
162	C₂²⋊C₄	—	268	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	268	1	no	—
165	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
166	C₁₆	—	268	2	no	—
167	C₂³	#14524069	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	8	4	no	—
172	C₂×C₄	—	268	2	no	—
173	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
174	C₂×C₄	#14524019	1	1	yes	characteristic
175	C₂×C₄	#14524019	1	1	yes	characteristic
176	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
177	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
178	C₂×C₄	#14524069	1	1	yes	characteristic
179	C₈	—	268	2	no	—
180	C₈	#14523967	1	2	yes	—
181	C₈	#14523967	1	2	yes	—
182	C₈	Q₈×Dic₆₇	1	1	yes	characteristic
183	C₈	Q₈×Dic₆₇	1	1	yes	characteristic
184	C₈	#14524032	1	1	yes	characteristic
185	C₈	#14524041	1	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	#16863401	1	1	yes	characteristic
188	C₂²	#16863683	1	1	yes	characteristic
189	C₂²	#16863688	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	8	4	no	—
192	C₄	—	536	2	no	—
193	C₄	#16863303	1	1	yes	characteristic
194	C₄	C₂×#14523967	1	1	yes	characteristic
195	C₄	#16863683	1	1	yes	characteristic
196	C₄	#16863688	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#22561158	1	1	yes	characteristic
200	C₂	#22562382	1	2	yes	—

	subgroup	G/H
center	C₂²	#16863683
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#499×C₆₇	C₂
socle	C₂²×C₆₇	#476
prime radical	C₂³×C₆₇	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#499	D₆₇
hypocenter	C₆₇	#13273
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
14. #499×C₆₇	C₄	semidirect
106. C₆₇	#13273	semidirect
110. #499	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863217	2
#22560088	2
#22558316	2
#22558307	2
C₂×#499×C₆₇	2
#13273	67