Subgroups — #30856254

Order 17152.

1086 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863434	C₂	1	4	yes	maximal
2	#22562616	C₂	1	2	yes	maximal
3	#22560377	C₂	1	8	yes	maximal, complemented
4	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524002	C₂²	1	4	yes	—
6	#16863434	—	2	16	no	—
7	C₂×#14523977	C₂²	1	8	yes	—
8	C₂×#14523975	C₂²	1	8	yes	—
9	#16863266	C₂²	1	4	yes	—
10	#16863262	C₂²	1	4	yes	—
11	#16863256	—	2	4	no	—
12	C₄×C₄⋊Dic₆₇	—	2	4	no	—
13	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#97×C₆₇	C₄	1	8	yes	complemented
16	C₂×#97×C₆₇	C₂²	1	4	yes	—
17	#14524002	—	2	16	no	—
18	C₂×C₄⋊Dic₆₇	—	2	8	no	—
19	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	—
20	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
21	C₂×C₄×Dic₆₇	—	2	4	no	—
22	#14523977	—	2	16	no	—
23	#14523975	—	2	16	no	—
24	C₆₇⋊#95	—	2	8	no	—
25	C₂³×C₄×C₆₇	D₄	1	2	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₆₇	D₄	1	8	yes	—
29	C₂²×C₈×C₆₇	Q₈	1	8	yes	—
30	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₆₇	D₄	1	2	yes	—
33	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
35	#97×C₆₇	C₂×C₄	1	16	yes	—
36	C₂²×Dic₆₇	—	2	8	no	—
37	C₄⋊Dic₆₇	—	2	16	no	—
38	C₄⋊Dic₆₇	—	4	16	no	—
39	C₂²⋊Dic₆₇	—	4	8	no	—
40	C₄×Dic₆₇	—	4	8	no	—
41	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	—	4	4	no	—
43	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
44	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₄⋊C₄	1	32	yes	complemented
52	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
53	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
54	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
55	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
56	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
57	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
58	C₂³×C₆₇	—	2	4	no	—
59	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
60	C₂³×C₆₇	#127	1	2	yes	—
61	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
62	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	—	2	8	no	—
64	C₂×C₄×C₆₇	—	4	8	no	—
65	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
66	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
67	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₆₇	#127	1	2	yes	—
69	C₂×C₄×C₆₇	C₄⋊Q₈	1	8	yes	—
70	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
71	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
73	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
74	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
75	C₂×Dic₆₇	—	4	8	no	—
76	C₂×Dic₆₇	—	4	16	no	—
77	C₈×C₆₇	C₂×C₄⋊C₄	1	16	yes	—
78	C₂²×C₆₇	—	2	4	no	—
79	C₂²×C₆₇	—	2	4	no	—
80	C₂²×C₆₇	—	2	4	no	—
81	C₂²×C₆₇	#384	1	4	yes	—
82	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
84	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
85	C₂²×C₆₇	#575	1	4	yes	—
86	C₄×C₆₇	—	2	8	no	—
87	C₄×C₆₇	#384	1	4	yes	—
88	C₄×C₆₇	C₂²×C₄⋊C₄	1	2	yes	—
89	C₄×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
90	Dic₆₇	—	8	8	no	—
91	#20799	—	67	1	no	maximal, Sylow
92	C₂×C₆₇	—	2	4	no	—
93	C₂×C₆₇	C₂×#384	1	1	yes	characteristic
94	C₂×C₆₇	#2911	1	2	yes	—
95	C₂×C₆₇	#3150	1	2	yes	—
96	C₂×C₆₇	#3162	1	2	yes	—
97	C₂×#427	—	67	4	no	—
98	C₂²×#97	D₆₇	1	1	yes	characteristic
99	#2285	—	67	2	no	—
100	#1847	—	67	8	no	—
101	C₆₇	#20799	1	1	yes	characteristic, complemented, Sylow
102	C₂³×C₈	C₂×D₆₇	1	2	yes	—
103	C₂×#117	—	67	4	no	—
104	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
105	#427	—	134	16	no	—
106	C₂×#107	—	67	8	no	—
107	C₂×#106	—	67	8	no	—
108	C₂×#97	Dic₆₇	1	8	yes	complemented
109	C₂×#97	C₂×D₆₇	1	4	yes	—
110	#384	—	67	4	no	—
111	#382	—	67	4	no	—
112	#380	—	134	4	no	—
113	C₄×C₄⋊C₄	—	134	4	no	—
114	C₂³×C₄	D₂₆₈	1	2	yes	—
115	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
116	C₂²×C₈	Dic₁₃₄	1	8	yes	—
117	C₂²×C₈	D₂₆₈	1	8	yes	—
118	C₂²×C₈	C₂×Dic₆₇	1	8	yes	—
119	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
120	#117	—	134	16	no	—
121	C₂×C₄⋊C₄	—	67	8	no	—
122	C₂×C₄⋊C₄	—	134	8	no	—
123	C₂×C₂²⋊C₄	—	134	4	no	—
124	C₂×C₄²	—	134	4	no	—
125	C₂×C₄²	Dic₁₃₄	1	4	yes	—
126	C₂×C₄²	D₂₆₈	1	2	yes	—
127	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
128	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
129	#107	—	134	16	no	—
130	#106	—	134	16	no	—
131	#97	C₂×Dic₆₇	1	16	yes	—
132	#95	—	134	8	no	—
133	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
134	C₂²×C₄	—	4	4	no	—
135	C₂²×C₄	—	134	8	no	—
136	C₂²×C₄	C₄⋊Dic₆₇	1	8	yes	—
137	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
138	C₂²×C₄	C₂×D₂₆₈	1	4	yes	—
139	C₂²×C₄	C₂×D₂₆₈	1	4	yes	—
140	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
141	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
142	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
143	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
144	C₂×C₈	C₄⋊Dic₆₇	1	32	yes	complemented
145	C₂×C₈	C₂²×Dic₆₇	1	8	yes	—
146	C₂×C₈	C₂×D₂₆₈	1	8	yes	—
147	C₂×C₈	C₂×Dic₁₃₄	1	8	yes	—
148	C₄⋊C₄	—	268	16	no	—
149	C₄⋊C₄	—	134	16	no	—
150	C₂²⋊C₄	—	268	8	no	—
151	C₄²	—	268	8	no	—
152	C₄²	C₄⋊Dic₆₇	1	8	yes	—
153	C₄²	C₂²×Dic₆₇	1	4	yes	—
154	C₄²	C₂×Dic₁₃₄	1	4	yes	—
155	C₂³	—	2	4	no	—
156	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
157	C₂³	#14524040	1	2	yes	—
158	C₂³	#14524043	1	2	yes	—
159	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	4	8	no	—
162	C₂×C₄	—	268	8	no	—
163	C₂×C₄	—	268	16	no	—
164	C₂×C₄	C₂×C₄⋊Dic₆₇	1	8	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₆₇	1	8	yes	—
167	C₂×C₄	#14524040	1	2	yes	—
168	C₂×C₄	#14524043	1	2	yes	—
169	C₂×C₄	#14524043	1	8	yes	—
170	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
171	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
172	C₂×C₄	C₂²×Dic₁₃₄	1	2	yes	—
173	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
174	C₈	C₂×C₄⋊Dic₆₇	1	16	yes	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	4	no	—
178	C₂²	#16863266	1	4	yes	—
179	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
180	C₂²	C₂×#14524040	1	1	yes	characteristic
181	C₂²	C₂×#14524043	1	1	yes	characteristic
182	C₂²	#16864196	1	4	yes	—
183	C₄	—	2	8	no	—
184	C₄	—	536	8	no	—
185	C₄	#16863266	1	4	yes	—
186	C₄	C₂²×C₄⋊Dic₆₇	1	2	yes	—
187	C₄	C₂×#14524043	1	2	yes	—
188	C₂	—	2	4	no	—
189	C₂	C₂×#16863266	1	1	yes	characteristic
190	C₂	#22568799	1	2	yes	—
191	C₂	#22570655	1	2	yes	—
192	C₂	#22570733	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×#97×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#97	D₆₇
hypocenter	C₆₇	#20799
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #22560377	C₂	semidirect
15. C₂×#97×C₆₇	C₄	semidirect
51. C₂×C₈×C₆₇	C₄⋊C₄	semidirect
101. C₆₇	#20799	semidirect
108. C₂×#97	Dic₆₇	semidirect
144. C₂×C₈	C₄⋊Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863434	2
#22562616	2
#22560377	2
C₂²×#97×C₆₇	2
#20799	67