Subgroups — #30767404

Order 17152.

242 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561965	C₂	1	1	yes	maximal, characteristic
2	#22558492	C₂	1	1	yes	maximal, characteristic
3	#22558278	C₂	1	4	yes	maximal, complemented
4	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	—	2	1	no	—
6	#16863449	C₄	1	1	yes	characteristic
7	#16863449	C₄	1	1	yes	characteristic
8	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
10	#16863117	C₄	1	2	yes	—
11	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₆₇	—	2	1	no	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₆₇	—	2	1	no	—
15	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
16	#14523983	—	4	2	no	—
17	C₂×C₆₇⋊C₁₆	—	2	2	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
19	C₄⋊(C₆₇⋊C₈)	C₈	1	2	yes	—
20	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
21	C₂²⋊(C₆₇⋊C₈)	C₈	1	2	yes	—
22	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₇	—	2	2	no	—
28	C₂×C₁₆×C₆₇	—	4	2	no	—
29	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×C₆₇⋊C₈	—	2	2	no	—
34	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
35	C₆₇⋊C₁₆	—	4	2	no	—
36	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₆₇	—	2	2	no	—
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	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
49	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₁₆×C₆₇	—	4	2	no	—
51	C₂³×C₆₇	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	—	4	1	no	—
54	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#107	1	2	yes	—
58	C₂×C₄×C₆₇	#108	1	2	yes	—
59	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
62	C₈×C₆₇	—	2	1	no	—
63	C₈×C₆₇	—	2	1	no	—
64	C₈×C₆₇	#107	1	2	yes	—
65	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
66	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₆₇⋊C₈	—	2	4	no	—
68	C₆₇⋊C₈	—	4	2	no	—
69	C₂²×C₆₇	—	2	1	no	—
70	C₂²×C₆₇	—	2	1	no	—
71	C₂²×C₆₇	#335	1	2	yes	—
72	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄×C₆₇	—	2	1	no	—
74	C₄×C₆₇	—	2	1	no	—
75	C₄×C₆₇	—	2	1	no	—
76	C₄×C₆₇	#335	1	2	yes	—
77	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
78	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
79	C₄×C₆₇	#504	1	2	yes	—
80	#12005	—	67	1	no	maximal, Sylow
81	C₂×C₆₇	—	2	2	no	—
82	C₂×C₆₇	C₂×#335	1	1	yes	characteristic
83	C₂×C₆₇	#2151	1	1	yes	characteristic
84	C₂×C₆₇	#2153	1	1	yes	characteristic
85	#2150	—	67	1	no	—
86	C₂×C₄⋊C₁₆	D₆₇	1	1	yes	characteristic
87	#1450	—	67	1	no	—
88	#1370	—	67	4	no	—
89	C₆₇	#12005	1	1	yes	characteristic, complemented, Sylow
90	C₂×M₃₂	—	134	1	no	—
91	C₂²×C₁₆	—	2	1	no	—
92	#431	—	67	1	no	—
93	#431	—	67	1	no	—
94	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
95	C₄⋊C₁₆	—	2	1	no	—
96	C₄⋊C₁₆	—	67	2	no	—
97	C₄⋊C₁₆	—	134	1	no	—
98	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
99	#322	—	67	2	no	—
100	C₈²	—	67	2	no	—
101	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
102	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
103	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
104	M₃₂	—	268	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	4	2	no	—
107	C₂×C₁₆	—	134	2	no	—
108	C₄⋊C₈	—	67	2	no	—
109	C₄⋊C₈	—	134	1	no	—
110	C₂²⋊C₈	—	67	2	no	—
111	C₂²⋊C₈	—	134	1	no	—
112	C₄×C₈	—	67	2	no	—
113	C₄×C₈	—	67	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	C₈×D₆₇	1	1	yes	characteristic
120	C₂²×C₄	#11761611	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	67	4	no	—
123	C₂×C₈	—	134	2	no	—
124	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
125	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
126	C₂×C₈	#11761613	1	2	yes	—
127	C₂×C₈	#11761618	1	2	yes	—
128	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
129	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
130	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
131	C₄²	C₈×D₆₇	1	1	yes	characteristic
132	C₄²	#11761611	1	1	yes	characteristic
133	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
134	C₁₆	—	4	2	no	—
135	C₁₆	—	268	2	no	—
136	C₂³	Dic₆₇⋊C₈	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
140	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
141	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
142	C₂×C₄	#14523976	1	2	yes	—
143	C₂×C₄	#14523978	1	2	yes	—
144	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
145	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
146	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	268	2	no	—
150	C₈	—	134	4	no	—
151	C₈	#14523976	1	2	yes	—
152	C₈	C₂×#11761613	1	1	yes	characteristic
153	C₈	C₂×#11761618	1	1	yes	characteristic
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#16863148	1	2	yes	—
157	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#16863148	1	2	yes	—
162	C₄	C₂×#14523976	1	1	yes	characteristic
163	C₄	C₂×#14523978	1	1	yes	characteristic
164	C₄	#16863812	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	C₂×#16863148	1	1	yes	characteristic
167	C₂	#22561967	1	1	yes	characteristic
168	C₂	#22561974	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×C₄⋊C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₈
prime radical	C₂³×C₆₇	C₄⋊C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	#12005
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #22558278	C₂	semidirect
89. C₆₇	#12005	semidirect

Maximal subgroups

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

subgroup	index
#22561965	2
#22558492	2
#22558278	2
C₂×C₄⋊C₁₆×C₆₇	2
#12005	67