Subgroups — #30774455

Order 17152.

354 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570723	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
3	#22558316	C₂	1	4	yes	maximal
4	C₆₇×#1846	C₂	1	1	yes	maximal, characteristic
5	#16863953	—	2	1	no	—
6	C₂×#14523983	C₂²	1	2	yes	—
7	#16863801	C₄	1	2	yes	complemented
8	#16863798	—	2	1	no	—
9	#16863798	C₄	1	2	yes	complemented
10	#16863794	—	2	1	no	—
11	#16863434	—	2	1	no	—
12	#16863434	C₂²	1	2	yes	—
13	#16863172	—	2	1	no	—
14	C₄×C₆₇⋊C₁₆	—	2	1	no	—
15	C₂×#107×C₆₇	—	2	1	no	—
16	C₂×#106×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#382×C₆₇	—	2	1	no	—
19	#14524043	—	2	2	no	—
20	#14524035	—	2	2	no	—
21	#14524022	—	4	2	no	—
22	C₄×Dic₁₃₄	—	4	2	no	—
23	#14524002	—	2	2	no	—
24	#14523983	—	2	2	no	—
25	#14523983	Q₈	1	4	yes	complemented
26	C₂×C₆₇⋊C₁₆	—	2	2	no	—
27	#14523977	—	2	2	no	—
28	#14523977	C₂×C₄	1	4	yes	—
29	#14523975	—	2	2	no	—
30	#14523975	—	2	2	no	—
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
35	#107×C₆₇	—	4	2	no	—
36	#106×C₆₇	—	2	4	no	—
37	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
41	#95×C₆₇	—	4	2	no	—
42	C₂×Dic₁₃₄	—	4	2	no	—
43	C₄⋊Dic₆₇	—	2	4	no	—
44	C₄⋊Dic₆₇	—	4	2	no	—
45	Dic₆₇⋊C₄	—	4	4	no	—
46	C₂²⋊Dic₆₇	—	4	2	no	—
47	C₄×Dic₆₇	—	4	2	no	—
48	C₆₇⋊C₁₆	—	2	4	no	—
49	C₂²×C₄×C₆₇	—	4	2	no	—
50	C₂²×C₄×C₆₇	D₈	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
55	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
56	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₆₇	#41	1	2	yes	—
60	C₄⋊C₄×C₆₇	—	4	2	no	—
61	C₄⋊C₄×C₆₇	—	4	2	no	—
62	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₆₇	SD₁₆	1	2	yes	—
65	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₆₇	—	4	2	no	—
67	C₂×C₄×C₆₇	—	8	4	no	—
68	C₂×C₄×C₆₇	—	8	2	no	—
69	C₂×C₄×C₆₇	#102	1	2	yes	—
70	C₂×C₄×C₆₇	#102	1	2	yes	—
71	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×Dic₆₇	—	4	4	no	—
78	Dic₁₃₄	—	8	4	no	—
79	C₈×C₆₇	#103	1	2	yes	—
80	C₈×C₆₇	#103	1	2	yes	—
81	C₈×C₆₇	C₄×Q₈	1	2	yes	—
82	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
83	C₂²×C₆₇	—	2	2	no	—
84	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
85	C₂²×C₆₇	#486	1	1	yes	characteristic
86	C₂²×C₆₇	#500	1	1	yes	characteristic
87	C₄×C₆₇	—	2	2	no	—
88	C₄×C₆₇	—	2	2	no	—
89	C₄×C₆₇	—	16	2	no	—
90	C₄×C₆₇	#391	1	1	yes	characteristic
91	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
92	C₄×C₆₇	#487	1	1	yes	characteristic
93	C₄×C₆₇	#498	1	1	yes	characteristic
94	Dic₆₇	—	8	4	no	—
95	#13288	—	67	1	no	maximal, Sylow
96	C₂×C₆₇	—	2	1	no	—
97	C₂×C₆₇	—	2	1	no	—
98	C₂×C₆₇	#1990	1	1	yes	characteristic
99	C₂×C₆₇	#2145	1	2	yes	—
100	#3160	—	67	1	no	—
101	C₄×M₃₂	—	67	1	no	—
102	#1846	D₆₇	1	1	yes	characteristic
103	#1386	—	67	4	no	—
104	C₆₇	#13288	1	1	yes	characteristic, complemented, Sylow
105	#533	—	134	1	no	—
106	C₂×M₃₂	—	67	2	no	—
107	#500	—	67	2	no	—
108	#499	—	67	2	no	—
109	#499	—	134	1	no	—
110	#498	—	134	1	no	—
111	#427	—	67	2	no	—
112	#427	—	134	1	no	—
113	C₂×#107	—	2	1	no	—
114	C₂×#106	C₂×D₆₇	1	2	yes	—
115	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
116	#382	—	2	1	no	—
117	#346	—	134	1	no	—
118	C₄×C₁₆	—	134	1	no	—
119	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
121	C₄⋊Q₈	—	134	2	no	—
122	#125	—	134	2	no	—
123	C₂²⋊Q₈	—	268	2	no	—
124	C₄×Q₈	—	268	2	no	—
125	#117	—	134	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
128	M₃₂	—	67	4	no	—
129	M₃₂	—	134	2	no	—
130	C₂×C₁₆	—	134	2	no	—
131	#107	—	4	2	no	—
132	#107	—	67	4	no	—
133	#107	—	134	2	no	—
134	#106	—	2	4	no	—
135	#106	—	134	2	no	—
136	#106	—	134	2	no	—
137	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
138	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
139	C₄×C₈	D₂₆₈	1	1	yes	characteristic
140	C₄×C₈	D₂₆₈	1	1	yes	characteristic
141	#95	—	4	2	no	—
142	C₂×Q₈	—	268	2	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	#11761613	1	2	yes	—
145	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
146	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
147	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
148	C₂×C₈	#11761616	1	2	yes	—
149	C₂×C₈	#11761618	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
151	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
152	C₂×C₈	Q₈×D₆₇	1	2	yes	—
153	C₂×C₈	#11761627	1	2	yes	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	268	4	no	—
157	C₄⋊C₄	—	268	2	no	—
158	C₄⋊C₄	—	134	4	no	—
159	C₂²⋊C₄	—	268	2	no	—
160	C₄²	—	268	2	no	—
161	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
162	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
163	C₄²	#11761615	1	2	yes	—
164	C₁₆	—	134	4	no	—
165	C₂³	C₂×#11761613	1	1	yes	characteristic
166	Q₈	—	536	4	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	8	4	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	—	268	4	no	—
171	C₂×C₄	#14523963	1	2	yes	—
172	C₂×C₄	#14523963	1	2	yes	—
173	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
174	C₂×C₄	#14524030	1	1	yes	characteristic
175	C₂×C₄	#14524041	1	1	yes	characteristic
176	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
177	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
178	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
179	C₈	#14523966	1	2	yes	—
180	C₈	#14523966	1	2	yes	—
181	C₈	#14524008	1	2	yes	—
182	C₈	#14524023	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	C₂×#14523963	1	1	yes	characteristic
185	C₂²	#16863742	1	1	yes	characteristic
186	C₂²	#16863799	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	16	2	no	—
190	C₄	—	536	4	no	—
191	C₄	#16863301	1	1	yes	characteristic
192	C₄	C₂×#14523966	1	1	yes	characteristic
193	C₄	#16863746	1	1	yes	characteristic
194	C₄	#16863792	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	#22561169	1	1	yes	characteristic
198	C₂	#22561944	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523963
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1846	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1846	D₆₇
hypocenter	C₆₇	#13288
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
7. #16863801	C₄	semidirect
9. #16863798	C₄	semidirect
25. #14523983	Q₈	semidirect
104. C₆₇	#13288	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
C₄×#14523983	2
#22558316	2
C₆₇×#1846	2
#13288	67