Subgroups — #30772202

Order 17152.

302 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
2	#22560381	C₂	1	1	yes	maximal, characteristic
3	#22558307	C₂	1	4	yes	maximal
4	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₂²	1	2	yes	—
6	C₂×#14523979	—	2	1	no	—
7	C₂×#14523979	C₂²	1	2	yes	—
8	#16863420	—	2	1	no	—
9	#16863172	C₄	1	2	yes	complemented
10	C₄×C₆₇⋊C₁₆	C₄	1	2	yes	complemented
11	C₂×#106×C₆₇	—	2	1	no	—
12	C₂×#106×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#384×C₆₇	—	2	1	no	—
15	C₂×#11761612	—	2	2	no	—
16	#14523983	—	2	2	no	—
17	#14523983	—	2	2	no	—
18	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
19	#14523979	—	2	4	no	—
20	#14523979	—	4	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
22	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
26	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
27	#106×C₆₇	—	2	4	no	—
28	#106×C₆₇	—	4	2	no	—
29	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
33	#11761612	—	4	2	no	—
34	#11761612	—	4	2	no	—
35	C₂×C₆₇⋊C₈	—	4	2	no	—
36	C₆₇⋊C₁₆	—	2	4	no	—
37	C₂²×C₄×C₆₇	—	4	2	no	—
38	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
39	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	D₈	1	2	yes	—
43	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
44	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₆₇	#41	1	2	yes	—
47	C₂×C₈×C₆₇	#41	1	2	yes	—
48	C₄⋊C₄×C₆₇	—	4	2	no	—
49	C₄⋊C₄×C₆₇	—	4	2	no	—
50	C₄⋊C₄×C₆₇	—	8	4	no	—
51	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₆₇	SD₁₆	1	2	yes	—
54	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₄×C₆₇	—	2	4	no	—
56	C₂×C₄×C₆₇	—	8	4	no	—
57	C₂×C₄×C₆₇	—	8	2	no	—
58	C₂×C₄×C₆₇	#106	1	2	yes	—
59	C₂×C₄×C₆₇	#106	1	2	yes	—
60	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₈×C₆₇	#107	1	2	yes	—
67	C₈×C₆₇	#107	1	2	yes	—
68	C₈×C₆₇	#117	1	2	yes	—
69	C₈×C₆₇	#125	1	2	yes	—
70	C₆₇⋊C₈	—	8	2	no	—
71	C₂²×C₆₇	—	2	2	no	—
72	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
73	C₂²×C₆₇	#486	1	1	yes	characteristic
74	C₂²×C₆₇	#487	1	1	yes	characteristic
75	C₄×C₆₇	—	2	2	no	—
76	C₄×C₆₇	—	16	2	no	—
77	C₄×C₆₇	#365	1	4	yes	—
78	C₄×C₆₇	#382	1	1	yes	characteristic
79	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
80	C₄×C₆₇	#486	1	1	yes	characteristic
81	C₄×C₆₇	#487	1	1	yes	characteristic
82	#12886	—	67	1	no	maximal, Sylow
83	C₂×C₆₇	—	2	1	no	—
84	C₂×C₆₇	—	2	1	no	—
85	C₂×C₆₇	#1834	1	1	yes	characteristic
86	C₂×C₆₇	C₂×#365	1	2	yes	—
87	C₄×M₃₂	—	67	1	no	—
88	#1848	—	67	1	no	—
89	#1844	D₆₇	1	1	yes	characteristic
90	#1383	—	67	4	no	—
91	C₆₇	#12886	1	1	yes	characteristic, complemented, Sylow
92	C₂×M₃₂	—	67	2	no	—
93	C₂×#108	—	67	2	no	—
94	C₂×#108	—	134	1	no	—
95	C₂×#106	—	2	1	no	—
96	C₂×#106	C₂×D₆₇	1	2	yes	—
97	#423	—	134	1	no	—
98	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
99	#384	—	2	1	no	—
100	#346	—	67	2	no	—
101	C₄×C₁₆	—	67	2	no	—
102	C₂×M₁₆	—	134	2	no	—
103	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
104	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	4	2	no	—
107	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
108	M₃₂	—	134	2	no	—
109	M₃₂	—	134	2	no	—
110	C₂×C₁₆	—	67	4	no	—
111	#108	—	268	2	no	—
112	#108	—	134	4	no	—
113	#106	—	2	4	no	—
114	#106	—	4	2	no	—
115	C₄⋊C₈	—	268	2	no	—
116	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
117	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
118	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
119	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	#11761616	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
123	M₁₆	—	268	2	no	—
124	M₁₆	—	268	2	no	—
125	C₂×C₈	—	268	2	no	—
126	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
127	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
128	C₂×C₈	#11761613	1	2	yes	—
129	C₂×C₈	#11761618	1	2	yes	—
130	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
131	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
132	C₂×C₈	#11761627	1	2	yes	—
133	C₂×C₈	#11761629	1	2	yes	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	8	4	no	—
137	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
138	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
139	C₄²	#11761615	1	2	yes	—
140	C₁₆	—	134	4	no	—
141	C₂³	C₂×#11761616	1	1	yes	characteristic
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	#14523974	1	2	yes	—
146	C₂×C₄	#14523974	1	2	yes	—
147	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
148	C₂×C₄	#14524030	1	1	yes	characteristic
149	C₂×C₄	#14524032	1	1	yes	characteristic
150	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
151	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
152	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
153	C₈	—	536	2	no	—
154	C₈	#14523976	1	2	yes	—
155	C₈	#14523976	1	2	yes	—
156	C₈	#14524001	1	2	yes	—
157	C₈	#14524034	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	C₂×#14523974	1	1	yes	characteristic
160	C₂²	#16863742	1	1	yes	characteristic
161	C₂²	#16863748	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	16	2	no	—
164	C₄	#16863218	1	4	yes	—
165	C₄	#16863261	1	1	yes	characteristic
166	C₄	C₂×#14523976	1	1	yes	characteristic
167	C₄	#16863744	1	1	yes	characteristic
168	C₄	#16863746	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#22560336	1	1	yes	characteristic
172	C₂	C₂×#16863218	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523974
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1844	C₂
socle	C₂²×C₆₇	C₂×#106
prime radical	C₂³×C₆₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1844	D₆₇
hypocenter	C₆₇	#12886
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1844

Complements

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

N	H ≅ G/N	type
9. #16863172	C₄	semidirect
10. C₄×C₆₇⋊C₁₆	C₄	semidirect
91. C₆₇	#12886	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523983	2
#22560381	2
#22558307	2
C₆₇×#1844	2
#12886	67