Subgroups — #30772207

Order 17152.

286 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	#22560381	C₂	1	1	yes	maximal, characteristic
2	#22560373	C₂	1	1	yes	maximal, characteristic
3	#22558308	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523979	—	2	1	no	—
6	C₂×#14523979	C₂²	1	2	yes	—
7	C₂×#14523977	—	2	1	no	—
8	C₂×#14523975	C₂²	1	2	yes	—
9	#16863420	—	2	1	no	—
10	#16863262	—	2	1	no	—
11	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#346×C₆₇	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₆₇	C₄	1	2	yes	complemented
15	C₂×#11761612	—	2	2	no	—
16	C₂×C₄⋊Dic₆₇	—	2	2	no	—
17	#14523979	—	2	4	no	—
18	#14523979	—	4	2	no	—
19	#14523977	—	4	2	no	—
20	#14523975	—	2	4	no	—
21	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
22	C₆₇⋊#95	—	4	2	no	—
23	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
26	M₃₂×C₆₇	—	2	2	no	—
27	M₃₂×C₆₇	—	2	2	no	—
28	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
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	C₂²×Dic₆₇	—	4	2	no	—
34	#11761612	—	4	2	no	—
35	#11761612	—	4	2	no	—
36	C₂×C₆₇⋊C₈	—	4	2	no	—
37	C₄⋊Dic₆₇	—	4	2	no	—
38	C₄⋊Dic₆₇	—	4	2	no	—
39	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
40	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	D₈	1	2	yes	—
44	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
45	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₆₇	#41	1	2	yes	—
48	C₂×C₈×C₆₇	#41	1	2	yes	—
49	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₆₇	SD₁₆	1	2	yes	—
52	C₁₆×C₆₇	—	2	4	no	—
53	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₄×C₆₇	—	4	2	no	—
55	C₂×C₄×C₆₇	#106	1	2	yes	—
56	C₂×C₄×C₆₇	#106	1	2	yes	—
57	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
59	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₆₇	—	8	4	no	—
64	C₂×Dic₆₇	—	8	2	no	—
65	C₈×C₆₇	#107	1	2	yes	—
66	C₈×C₆₇	#107	1	2	yes	—
67	C₈×C₆₇	#117	1	2	yes	—
68	C₈×C₆₇	#125	1	2	yes	—
69	C₆₇⋊C₈	—	8	2	no	—
70	C₂²×C₆₇	—	2	2	no	—
71	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
72	C₂²×C₆₇	#486	1	1	yes	characteristic
73	C₂²×C₆₇	#487	1	1	yes	characteristic
74	C₄×C₆₇	—	2	2	no	—
75	C₄×C₆₇	—	2	2	no	—
76	C₄×C₆₇	#382	1	1	yes	characteristic
77	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
78	C₄×C₆₇	#486	1	1	yes	characteristic
79	C₄×C₆₇	#487	1	1	yes	characteristic
80	Dic₆₇	—	16	2	no	—
81	#12887	—	67	1	no	maximal, Sylow
82	C₂×C₆₇	—	2	1	no	—
83	C₂×C₆₇	—	2	1	no	—
84	C₂×C₆₇	#1834	1	1	yes	characteristic
85	C₂×C₆₇	#2155	1	2	yes	—
86	C₄×M₃₂	D₆₇	1	1	yes	characteristic
87	#1848	—	67	1	no	—
88	#1846	—	67	1	no	—
89	#1383	—	67	4	no	—
90	C₆₇	#12887	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
92	C₂×#108	—	67	2	no	—
93	C₂×#108	—	134	1	no	—
94	C₂×#107	—	134	1	no	—
95	C₂×#106	—	67	2	no	—
96	#423	—	134	1	no	—
97	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
98	#382	—	134	1	no	—
99	#346	Dic₆₇	1	2	yes	complemented
100	C₄×C₁₆	Dic₆₇	1	2	yes	complemented
101	C₂×M₁₆	—	134	2	no	—
102	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	134	2	no	—
105	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	2	no	—
108	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
109	#108	—	268	2	no	—
110	#108	—	134	4	no	—
111	#107	—	268	2	no	—
112	#106	—	134	4	no	—
113	C₄⋊C₈	—	268	2	no	—
114	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
115	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
118	#95	—	268	2	no	—
119	C₂²×C₄	—	268	2	no	—
120	C₂²×C₄	#11761617	1	2	yes	—
121	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
122	M₁₆	—	268	2	no	—
123	M₁₆	—	268	2	no	—
124	C₂×C₈	—	268	2	no	—
125	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
126	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
127	C₂×C₈	D₅₃₆	1	2	yes	—
128	C₂×C₈	Dic₂₆₈	1	2	yes	—
129	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
131	C₂×C₈	#11761628	1	2	yes	—
132	C₂×C₈	#11761628	1	2	yes	—
133	C₄⋊C₄	—	268	2	no	—
134	C₄⋊C₄	—	268	2	no	—
135	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
136	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
137	C₄²	#11761617	1	2	yes	—
138	C₁₆	—	2	4	no	—
139	C₂³	C₂×#11761617	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	536	4	no	—
142	C₂×C₄	—	536	2	no	—
143	C₂×C₄	#14523975	1	2	yes	—
144	C₂×C₄	#14523975	1	2	yes	—
145	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
146	C₂×C₄	#14524031	1	1	yes	characteristic
147	C₂×C₄	#14524031	1	1	yes	characteristic
148	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
149	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
150	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
151	C₈	—	536	2	no	—
152	C₈	#14523977	1	2	yes	—
153	C₈	#14523977	1	2	yes	—
154	C₈	#14524002	1	2	yes	—
155	C₈	#14524035	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×#14523975	1	1	yes	characteristic
158	C₂²	#16863743	1	1	yes	characteristic
159	C₂²	#16863747	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	1072	2	no	—
163	C₄	#16863262	1	1	yes	characteristic
164	C₄	C₂×#14523977	1	1	yes	characteristic
165	C₄	#16863743	1	1	yes	characteristic
166	C₄	#16863747	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#22560337	1	1	yes	characteristic
170	C₂	#22561982	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523975
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₄×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#106
prime radical	C₂³×C₆₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₄×M₃₂	D₆₇
hypocenter	C₆₇	#12887
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
13. #346×C₆₇	C₄	semidirect
14. C₄×C₁₆×C₆₇	C₄	semidirect
90. C₆₇	#12887	semidirect
99. #346	Dic₆₇	semidirect
100. C₄×C₁₆	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22560381	2
#22560373	2
#22558308	2
C₄×M₃₂×C₆₇	2
#12887	67