Subgroups — #30856448

Order 17152.

758 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568842	C₂	1	2	yes	maximal
2	#22560384	C₂	1	2	yes	maximal, complemented
3	#22560350	C₂	1	2	yes	maximal, complemented
4	#22560251	C₂	1	4	yes	maximal
5	C₄×#14523979	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#16863453	—	2	4	no	—
8	#16863449	—	2	4	no	—
9	C₂×#14523979	C₂²	1	8	yes	—
10	#16863420	C₂²	1	4	yes	—
11	#16863360	C₂²	1	4	yes	—
12	C₄×#11761612	C₂²	1	4	yes	—
13	#16863161	C₂²	1	8	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₆₇	C₄	1	8	yes	—
17	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×#11761612	C₂³	1	8	yes	—
21	#14523979	—	2	16	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
23	C₂²⋊(C₆₇⋊C₈)	—	2	8	no	—
24	#14523954	—	2	4	no	—
25	C₄×C₆₇⋊C₈	—	2	4	no	—
26	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
27	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₆₇	—	2	8	no	—
29	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
30	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₆₇	D₄	1	2	yes	—
34	C₂×C₄²×C₆₇	D₄	1	2	yes	—
35	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	#97×C₆₇	C₂×C₄	1	4	yes	—
39	#97×C₆₇	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
41	#11761612	—	2	16	no	—
42	C₂×C₆₇⋊C₈	—	2	8	no	—
43	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	—	4	4	no	—
46	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
54	M₁₆×C₆₇	—	2	16	no	—
55	C₂×C₈×C₆₇	—	2	4	no	—
56	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₆₇	#41	1	8	yes	—
59	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
60	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
61	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
62	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
63	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₆₇	C₂×D₄	1	2	yes	—
65	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂³×C₆₇	—	2	2	no	—
67	C₂³×C₆₇	—	4	2	no	—
68	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
69	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	2	4	no	—
72	C₂×C₄×C₆₇	—	2	4	no	—
73	C₂×C₄×C₆₇	—	2	8	no	—
74	C₂×C₄×C₆₇	—	2	4	no	—
75	C₂×C₄×C₆₇	—	4	4	no	—
76	C₂×C₄×C₆₇	—	4	2	no	—
77	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
84	C₈×C₆₇	—	2	4	no	—
85	C₈×C₆₇	#117	1	8	yes	—
86	C₆₇⋊C₈	—	4	8	no	—
87	C₂²×C₆₇	—	2	2	no	—
88	C₂²×C₆₇	—	2	4	no	—
89	C₂²×C₆₇	—	2	2	no	—
90	C₂²×C₆₇	—	4	4	no	—
91	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₆₇	#533	1	2	yes	—
93	C₄×C₆₇	—	2	4	no	—
94	C₄×C₆₇	—	2	2	no	—
95	C₄×C₆₇	—	2	2	no	—
96	C₄×C₆₇	#430	1	4	yes	—
97	C₄×C₆₇	C₂×#117	1	2	yes	—
98	C₄×C₆₇	#532	1	2	yes	—
99	#20827	—	67	1	no	maximal, Sylow
100	C₂×C₆₇	—	2	2	no	—
101	C₂×C₆₇	—	4	2	no	—
102	C₂×C₆₇	#2285	1	1	yes	characteristic
103	C₂×C₆₇	C₂×#430	1	2	yes	—
104	#2920	—	67	2	no	—
105	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
106	#1849	—	67	2	no	—
107	#1838	—	67	2	no	—
108	#1811	—	67	4	no	—
109	C₄×#108	—	67	4	no	—
110	C₆₇	#20827	1	1	yes	characteristic, complemented, Sylow
111	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
112	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
113	#432	—	134	4	no	—
114	#431	—	134	4	no	—
115	C₂×#108	—	67	8	no	—
116	#423	—	67	4	no	—
117	#407	—	67	4	no	—
118	C₄×M₁₆	—	67	4	no	—
119	C₄×M₁₆	Dic₆₇	1	8	yes	—
120	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
121	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
123	#341	—	67	8	no	—
124	C₂³×C₄	Dic₁₃₄	1	2	yes	—
125	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
126	C₂×M₁₆	—	2	8	no	—
127	C₂×M₁₆	—	67	8	no	—
128	C₂×M₁₆	C₂×Dic₆₇	1	8	yes	—
129	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
130	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
131	C₂×C₄²	Dic₁₃₄	1	2	yes	—
132	C₂×C₄²	D₂₆₈	1	2	yes	—
133	C₂×C₄²	D₂₆₈	1	2	yes	—
134	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
135	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
136	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
137	#108	—	134	16	no	—
138	C₄⋊C₈	—	134	8	no	—
139	C₂²⋊C₈	—	134	8	no	—
140	#97	—	134	4	no	—
141	#97	C₂×Dic₆₇	1	4	yes	—
142	#97	C₂×Dic₆₇	1	4	yes	—
143	C₄×C₈	—	134	4	no	—
144	C₄×C₈	C₂×Dic₆₇	1	8	yes	—
145	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	4	4	no	—
148	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
149	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
150	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
151	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
152	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
153	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
154	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
155	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
156	M₁₆	—	2	16	no	—
157	M₁₆	—	134	16	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	134	8	no	—
160	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
161	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
162	C₂×C₈	#11761628	1	8	yes	—
163	C₄²	C₄⋊Dic₆₇	1	4	yes	—
164	C₄²	C₄⋊Dic₆₇	1	4	yes	—
165	C₄²	C₂²×Dic₆₇	1	2	yes	—
166	C₄²	C₂²×Dic₆₇	1	2	yes	—
167	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
168	C₄²	C₂×D₂₆₈	1	2	yes	—
169	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
170	C₂³	—	2	2	no	—
171	C₂³	—	4	2	no	—
172	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
173	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
185	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
186	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
187	C₂×C₄	C₂×#11761628	1	4	yes	—
188	C₈	—	2	4	no	—
189	C₈	—	268	8	no	—
190	C₈	#14524002	1	8	yes	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	4	4	no	—
195	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
196	C₂²	#16863953	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	2	4	no	—
199	C₄	—	2	2	no	—
200	C₄	#16863445	1	4	yes	—
201	C₄	C₂×#14524002	1	2	yes	—
202	C₄	#16863946	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	4	2	no	—
205	C₂	#22562616	1	1	yes	characteristic
206	C₂	C₂×#16863445	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	#20827
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22560384	C₂	semidirect
3. #22560350	C₂	semidirect
5. C₄×#14523979	C₂	semidirect
110. C₆₇	#20827	semidirect

Maximal subgroups

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

subgroup	index
#22568842	2
#22560384	2
#22560350	2
#22560251	2
C₄×#14523979	2
C₂×C₄×M₁₆×C₆₇	2
#20827	67