Subgroups — #30772264

Order 17152.

374 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863221	C₂	1	4	yes	maximal
2	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#22560369	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14523977	—	2	1	no	—
7	C₂×#14523977	C₂²	1	2	yes	—
8	#16863266	—	2	1	no	—
9	#16863221	—	2	8	no	—
10	C₄×C₆₇⋊C₁₆	C₄	1	4	yes	complemented
11	C₂×#107×C₆₇	—	2	1	no	—
12	C₂×#107×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#384×C₆₇	—	2	1	no	—
15	C₂×C₄⋊Dic₆₇	—	2	2	no	—
16	C₂×C₄⋊Dic₆₇	—	4	2	no	—
17	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
18	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
19	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	—
20	#14523977	—	2	4	no	—
21	#14523977	—	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	#107×C₆₇	—	2	4	no	—
28	#107×C₆₇	—	4	2	no	—
29	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₆₇	Q₈	1	2	yes	—
31	C₂²×Dic₆₇	—	4	2	no	—
32	C₄⋊Dic₆₇	—	4	2	no	—
33	C₄⋊Dic₆₇	—	4	2	no	—
34	C₄⋊Dic₆₇	—	8	4	no	—
35	C₆₇⋊C₁₆	C₄⋊C₄	1	8	yes	complemented
36	C₂²×C₄×C₆₇	—	4	2	no	—
37	C₂²×C₄×C₆₇	D₈	1	2	yes	—
38	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₆₇	D₈	1	2	yes	—
41	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
42	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₄⋊C₄×C₆₇	—	4	2	no	—
47	C₄⋊C₄×C₆₇	—	4	2	no	—
48	C₄⋊C₄×C₆₇	—	8	4	no	—
49	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
50	C₄²×C₆₇	Q₁₆	1	2	yes	—
51	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₆₇	—	8	2	no	—
53	C₂×C₄×C₆₇	—	8	4	no	—
54	C₂×C₄×C₆₇	#107	1	4	yes	—
55	C₂×C₄×C₆₇	D₁₆	1	4	yes	—
56	C₂×C₄×C₆₇	Q₃₂	1	4	yes	—
57	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₆₇	—	8	2	no	—
64	C₂×Dic₆₇	—	8	4	no	—
65	C₈×C₆₇	#107	1	4	yes	—
66	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
67	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
68	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
69	C₂²×C₆₇	#493	1	1	yes	characteristic
70	C₂²×C₆₇	#500	1	1	yes	characteristic
71	C₂²×C₆₇	C₂×D₁₆	1	2	yes	—
72	C₂²×C₆₇	C₂×Q₃₂	1	2	yes	—
73	C₄×C₆₇	—	16	2	no	—
74	C₄×C₆₇	#366	1	8	yes	complemented
75	C₄×C₆₇	#384	1	1	yes	characteristic
76	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
77	C₄×C₆₇	#494	1	1	yes	characteristic
78	C₄×C₆₇	#500	1	1	yes	characteristic
79	Dic₆₇	—	16	2	no	—
80	#12905	—	67	1	no	maximal, Sylow
81	C₂×C₆₇	#1845	1	1	yes	characteristic
82	C₂×C₆₇	C₂×#366	1	2	yes	—
83	C₂×C₆₇	#2246	1	1	yes	characteristic
84	C₂×C₆₇	#2247	1	1	yes	characteristic
85	C₂×C₆₇	#2252	1	2	yes	—
86	C₂×#366	—	67	4	no	—
87	C₂×C₄×C₁₆	—	67	1	no	—
88	#1845	—	67	1	no	—
89	#1845	D₆₇	1	1	yes	characteristic
90	C₆₇	#12905	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₁₆	—	67	2	no	—
92	C₂×#107	—	2	1	no	—
93	C₂×#107	—	67	2	no	—
94	C₂×#107	—	134	1	no	—
95	C₂×#107	C₂×D₆₇	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
97	#384	—	2	1	no	—
98	#384	—	134	1	no	—
99	#366	—	134	8	no	—
100	C₄×C₁₆	—	67	4	no	—
101	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	2	2	no	—
104	C₂×C₄⋊C₄	—	4	2	no	—
105	C₂×C₄⋊C₄	—	268	2	no	—
106	C₂×C₄⋊C₄	—	134	2	no	—
107	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
108	C₂×C₁₆	—	67	4	no	—
109	C₂×C₁₆	—	67	4	no	—
110	C₂×C₁₆	—	67	4	no	—
111	#107	—	2	4	no	—
112	#107	—	4	2	no	—
113	#107	—	268	2	no	—
114	#107	—	134	4	no	—
115	C₄×C₈	Dic₁₃₄	1	2	yes	—
116	C₄×C₈	C₄×D₆₇	1	2	yes	—
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	—	268	2	no	—
119	C₂²×C₄	#11761613	1	2	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
121	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
122	C₂×C₈	#11761613	1	2	yes	—
123	C₂×C₈	#11761618	1	2	yes	—
124	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
125	C₂×C₈	D₄×D₆₇	1	2	yes	—
126	C₂×C₈	Q₈×D₆₇	1	2	yes	—
127	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	8	4	no	—
131	C₄⋊C₄	—	268	2	no	—
132	C₄⋊C₄	—	268	2	no	—
133	C₄⋊C₄	—	536	4	no	—
134	C₄²	Dic₆₇⋊C₄	1	2	yes	—
135	C₄²	#11761618	1	2	yes	—
136	C₁₆	—	67	8	no	—
137	C₂³	C₂×#11761613	1	1	yes	characteristic
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	4	no	—
140	C₂×C₄	—	536	2	no	—
141	C₂×C₄	—	536	4	no	—
142	C₂×C₄	#14523976	1	4	yes	—
143	C₂×C₄	#14523984	1	4	yes	—
144	C₂×C₄	#14523989	1	4	yes	—
145	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
146	C₂×C₄	#14524039	1	1	yes	characteristic
147	C₂×C₄	#14524041	1	1	yes	characteristic
148	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
149	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
150	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
151	C₈	#14523976	1	4	yes	—
152	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
153	C₈	#14524042	1	2	yes	—
154	C₂²	C₂×#14523976	1	1	yes	characteristic
155	C₂²	#16863772	1	1	yes	characteristic
156	C₂²	#16863799	1	1	yes	characteristic
157	C₂²	C₂×#14523984	1	2	yes	—
158	C₂²	C₂×#14523989	1	2	yes	—
159	C₄	—	16	2	no	—
160	C₄	—	1072	2	no	—
161	C₄	#16863221	1	8	yes	complemented
162	C₄	#16863265	1	1	yes	characteristic
163	C₄	C₂×#14523976	1	1	yes	characteristic
164	C₄	#16863775	1	1	yes	characteristic
165	C₄	#16863799	1	1	yes	characteristic
166	C₂	#22560368	1	1	yes	characteristic
167	C₂	C₂×#16863221	1	2	yes	—
168	C₂	#22562458	1	1	yes	characteristic
169	C₂	#22562461	1	1	yes	characteristic
170	C₂	#22562482	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1845	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1845	D₆₇
hypocenter	C₆₇	#12905
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
10. C₄×C₆₇⋊C₁₆	C₄	semidirect
35. C₆₇⋊C₁₆	C₄⋊C₄	semidirect
74. C₄×C₆₇	#366	semidirect
90. C₆₇	#12905	semidirect
161. C₄	#16863221	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863221	2
C₂×C₄×C₆₇⋊C₁₆	2
#22560369	2
C₆₇×#1845	2
#12905	67