Subgroups — #30767475

Order 17152.

242 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×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#22558968	C₂	1	1	yes	maximal, characteristic
3	#22558282	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1567	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₄	1	2	yes	—
6	#16863420	—	2	1	no	—
7	C₄×C₆₇⋊C₁₆	C₄	1	1	yes	characteristic
8	C₄×C₆₇⋊C₁₆	C₄	1	1	yes	characteristic
9	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	#16863149	—	2	1	no	—
11	#16863149	C₂²	1	2	yes	—
12	#423×C₆₇	—	2	1	no	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#335×C₆₇	—	2	1	no	—
15	#335×C₆₇	C₂²	1	2	yes	—
16	C₂×#11761612	—	4	1	no	—
17	C₂×C₆₇⋊C₁₆	—	2	4	no	—
18	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
20	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
22	C₂×M₁₆×C₆₇	—	4	1	no	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₆₇	—	2	2	no	—
27	C₄⋊C₈×C₆₇	—	4	1	no	—
28	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	#11761612	—	8	2	no	—
33	C₂×C₆₇⋊C₈	—	4	2	no	—
34	C₆₇⋊C₁₆	—	2	2	no	—
35	C₆₇⋊C₁₆	M₁₆	1	4	yes	complemented
36	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
39	M₁₆×C₆₇	—	8	2	no	—
40	C₂×C₈×C₆₇	—	2	2	no	—
41	C₂×C₈×C₆₇	—	2	2	no	—
42	C₂×C₈×C₆₇	—	4	2	no	—
43	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₆₇	D₈	1	1	yes	characteristic
49	C₄²×C₆₇	Q₁₆	1	1	yes	characteristic
50	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂³×C₆₇	#107	1	1	yes	characteristic
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	—	2	1	no	—
54	C₂×C₄×C₆₇	—	2	1	no	—
55	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
58	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
59	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
60	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₈×C₆₇	—	2	1	no	—
64	C₈×C₆₇	—	2	1	no	—
65	C₈×C₆₇	—	8	2	no	—
66	C₈×C₆₇	#108	1	2	yes	—
67	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
68	C₆₇⋊C₈	—	8	2	no	—
69	C₂²×C₆₇	—	2	1	no	—
70	C₂²×C₆₇	—	2	1	no	—
71	C₂²×C₆₇	#367	1	2	yes	—
72	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
73	C₄×C₆₇	—	2	1	no	—
74	C₄×C₆₇	—	2	1	no	—
75	C₄×C₆₇	#367	1	2	yes	—
76	C₄×C₆₇	#368	1	2	yes	—
77	C₄×C₆₇	#423	1	1	yes	characteristic
78	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
79	C₄×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
80	C₄×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
81	#12023	—	67	1	no	maximal, Sylow
82	C₂×C₆₇	—	2	2	no	—
83	C₂×C₆₇	#1567	1	1	yes	characteristic
84	C₂×C₆₇	C₂×#367	1	1	yes	characteristic
85	C₂×C₆₇	C₂×#368	1	1	yes	characteristic
86	C₂×C₄×C₁₆	—	67	1	no	—
87	#1567	—	67	1	no	—
88	#1567	D₆₇	1	1	yes	characteristic
89	#1372	—	67	4	no	—
90	C₆₇	#12023	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₁₆	—	67	2	no	—
92	#423	—	2	1	no	—
93	#423	—	134	1	no	—
94	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
95	C₄×C₁₆	—	67	1	no	—
96	C₄×C₁₆	—	67	1	no	—
97	C₄×C₁₆	—	67	2	no	—
98	#335	—	2	1	no	—
99	#335	—	67	2	no	—
100	#335	—	134	1	no	—
101	#335	C₂×D₆₇	1	2	yes	—
102	C₂×M₁₆	—	4	1	no	—
103	C₂×M₁₆	—	268	1	no	—
104	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
105	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
106	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
107	C₂×C₁₆	—	67	2	no	—
108	C₂×C₁₆	—	67	2	no	—
109	C₂×C₁₆	—	134	4	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	4	1	no	—
112	C₄⋊C₈	—	268	1	no	—
113	C₄⋊C₈	—	134	2	no	—
114	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
115	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
116	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
117	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
118	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
119	C₂²×C₄	#11761613	1	1	yes	characteristic
120	C₂²×C₄	#11761618	1	1	yes	characteristic
121	M₁₆	—	8	2	no	—
122	M₁₆	—	536	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	268	2	no	—
127	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
128	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
129	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
131	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
132	C₄²	#11761613	1	1	yes	characteristic
133	C₄²	#11761618	1	1	yes	characteristic
134	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
135	C₁₆	—	67	4	no	—
136	C₁₆	—	134	2	no	—
137	C₂³	#14523976	1	1	yes	characteristic
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	#14523976	1	1	yes	characteristic
142	C₂×C₄	#14523976	1	1	yes	characteristic
143	C₂×C₄	#14523976	1	1	yes	characteristic
144	C₂×C₄	#14523986	1	2	yes	—
145	C₂×C₄	#14523987	1	2	yes	—
146	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
147	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
148	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	536	2	no	—
153	C₈	#14523978	1	2	yes	—
154	C₈	M₁₆×D₆₇	1	2	yes	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	#16863223	1	2	yes	—
158	C₂²	C₂×#14523976	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#16863223	1	2	yes	—
162	C₄	#16863225	1	2	yes	—
163	C₄	#16863419	1	1	yes	characteristic
164	C₄	C₂×#14523978	1	1	yes	characteristic
165	C₄	C₂×#14523986	1	1	yes	characteristic
166	C₄	C₂×#14523987	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	#22558967	1	1	yes	characteristic
169	C₂	C₂×#16863223	1	1	yes	characteristic
170	C₂	C₂×#16863225	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1567	C₂
socle	C₂²×C₆₇	C₂×#107
prime radical	C₂³×C₆₇	#107
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1567	D₆₇
hypocenter	C₆₇	#12023
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #22558282	C₂	semidirect
35. C₆₇⋊C₁₆	M₁₆	semidirect
90. C₆₇	#12023	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₆₇⋊C₁₆	2
#22558968	2
#22558282	2
C₆₇×#1567	2
#12023	67