Subgroups — #30774086

Order 17152.

470 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863776	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863211	C₂	1	4	yes	maximal
3	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₆₈	—	2	1	no	—
6	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
7	C₂×#14524043	—	2	1	no	—
8	C₂×#14523983	C₂²	1	2	yes	—
9	#16863776	—	2	1	no	—
10	#16863776	—	2	1	no	—
11	#16863776	C₄	1	2	yes	—
12	#16863776	C₄	1	2	yes	—
13	#16863211	—	2	8	no	—
14	#16863172	—	2	1	no	—
15	C₄×C₆₇⋊C₁₆	—	2	1	no	—
16	C₂×#108×C₆₇	—	2	1	no	—
17	C₂×#108×C₆₇	C₂²	1	2	yes	—
18	#423×C₆₇	—	2	1	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₃₄	—	2	2	no	—
21	C₂×Dic₂₆₈	—	2	2	no	—
22	C₂×Dic₂₆₈	—	2	2	no	—
23	C₂×Dic₂₆₈	—	2	4	no	—
24	C₂×Dic₂₆₈	—	2	2	no	—
25	C₂×Dic₂₆₈	—	2	4	no	—
26	C₂×Dic₂₆₈	—	4	2	no	—
27	C₂×Dic₂₆₈	—	4	2	no	—
28	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
29	#14524043	—	2	2	no	—
30	#14524043	—	2	2	no	—
31	#14524043	—	4	2	no	—
32	C₂×C₄⋊Dic₆₇	—	4	2	no	—
33	#14523983	—	2	2	no	—
34	#14523983	D₄	1	4	yes	—
35	C₂×C₆₇⋊C₁₆	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	#108×C₆₇	—	2	4	no	—
41	#108×C₆₇	—	4	2	no	—
42	C₄⋊C₈×C₆₇	—	4	2	no	—
43	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₂×Dic₁₃₄	—	2	4	no	—
48	C₂×Dic₁₃₄	—	4	2	no	—
49	C₂×Dic₁₃₄	—	4	2	no	—
50	C₂×Dic₁₃₄	—	4	2	no	—
51	C₂×Dic₁₃₄	—	4	2	no	—
52	C₂×Dic₁₃₄	—	4	2	no	—
53	C₂²×Dic₆₇	—	4	2	no	—
54	Dic₂₆₈	—	2	8	no	—
55	Dic₂₆₈	—	4	4	no	—
56	Dic₂₆₈	—	4	4	no	—
57	Dic₂₆₈	—	4	4	no	—
58	C₄⋊Dic₆₇	—	4	4	no	—
59	C₄⋊Dic₆₇	—	4	4	no	—
60	C₆₇⋊C₁₆	—	2	4	no	—
61	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	M₁₆×C₆₇	—	4	2	no	—
64	M₁₆×C₆₇	—	4	2	no	—
65	C₂×C₈×C₆₇	—	4	2	no	—
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	D₈	1	2	yes	—
69	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
70	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₆₇	#41	1	2	yes	—
74	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₆₇	D₈	1	2	yes	—
77	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₇	—	2	4	no	—
79	C₂×C₄×C₆₇	#102	1	2	yes	—
80	C₂×C₄×C₆₇	#102	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×Dic₆₇	—	4	4	no	—
88	C₂×Dic₆₇	—	8	2	no	—
89	C₂×Dic₆₇	—	8	2	no	—
90	Dic₁₃₄	—	4	4	no	—
91	Dic₁₃₄	—	4	4	no	—
92	Dic₁₃₄	—	8	2	no	—
93	Dic₁₃₄	—	8	2	no	—
94	C₈×C₆₇	—	8	2	no	—
95	C₈×C₆₇	#102	1	2	yes	—
96	C₈×C₆₇	#102	1	2	yes	—
97	C₈×C₆₇	C₄×D₄	1	2	yes	—
98	C₈×C₆₇	#121	1	2	yes	—
99	C₂²×C₆₇	—	2	2	no	—
100	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
101	C₂²×C₆₇	#486	1	1	yes	characteristic
102	C₂²×C₆₇	#492	1	1	yes	characteristic
103	C₄×C₆₇	—	2	2	no	—
104	C₄×C₆₇	#362	1	4	yes	complemented
105	C₄×C₆₇	#390	1	1	yes	characteristic
106	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
107	C₄×C₆₇	#486	1	1	yes	characteristic
108	C₄×C₆₇	#493	1	1	yes	characteristic
109	Dic₆₇	—	8	4	no	—
110	#13212	—	67	1	no	maximal, Sylow
111	C₂×C₆₇	—	2	1	no	—
112	C₂×C₆₇	—	2	1	no	—
113	C₂×C₆₇	#1957	1	1	yes	characteristic
114	C₂×C₆₇	C₂×#362	1	2	yes	—
115	C₂×#494	—	67	1	no	—
116	C₂×#362	—	67	4	no	—
117	C₄×M₃₂	—	67	1	no	—
118	#1848	D₆₇	1	1	yes	characteristic
119	C₆₇	#13212	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₁₆	—	67	2	no	—
121	C₂²×Q₁₆	—	134	1	no	—
122	C₂×C₄⋊Q₈	—	134	1	no	—
123	C₂×M₃₂	—	67	2	no	—
124	#494	—	67	2	no	—
125	#494	—	67	2	no	—
126	#494	—	134	1	no	—
127	#494	—	134	1	no	—
128	C₂×#108	—	2	1	no	—
129	C₂×#108	C₂×D₆₇	1	2	yes	—
130	#423	—	2	1	no	—
131	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
132	#362	—	134	8	no	—
133	#346	—	134	1	no	—
134	C₄×C₁₆	—	134	1	no	—
135	C₂²×Q₈	—	134	2	no	—
136	C₂×Q₁₆	—	268	2	no	—
137	C₂×Q₁₆	—	268	2	no	—
138	C₂×Q₁₆	—	67	4	no	—
139	C₂×Q₁₆	—	134	2	no	—
140	C₂×Q₁₆	—	134	2	no	—
141	C₂×Q₁₆	—	134	4	no	—
142	C₂×Q₁₆	—	134	2	no	—
143	C₂×Q₁₆	—	134	4	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
147	C₄⋊Q₈	—	268	2	no	—
148	C₄⋊Q₈	—	134	2	no	—
149	C₄⋊Q₈	—	134	2	no	—
150	C₂×C₄⋊C₄	—	268	2	no	—
151	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
152	M₃₂	—	67	4	no	—
153	M₃₂	—	134	2	no	—
154	C₂×C₁₆	—	134	2	no	—
155	#108	—	2	4	no	—
156	#108	—	4	2	no	—
157	C₄⋊C₈	—	4	2	no	—
158	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
159	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
160	C₄×C₈	D₂₆₈	1	1	yes	characteristic
161	C₄×C₈	D₂₆₈	1	1	yes	characteristic
162	C₂×Q₈	—	268	2	no	—
163	C₂×Q₈	—	268	2	no	—
164	C₂×Q₈	—	268	2	no	—
165	C₂×Q₈	—	268	2	no	—
166	C₂×Q₈	—	268	2	no	—
167	C₂×Q₈	—	134	4	no	—
168	C₂²×C₄	—	268	2	no	—
169	C₂²×C₄	#11761615	1	2	yes	—
170	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
171	Q₁₆	—	268	4	no	—
172	Q₁₆	—	268	4	no	—
173	Q₁₆	—	268	4	no	—
174	Q₁₆	—	134	8	no	—
175	M₁₆	—	4	2	no	—
176	M₁₆	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
179	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
180	C₂×C₈	#11761613	1	2	yes	—
181	C₂×C₈	#11761615	1	2	yes	—
182	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
183	C₂×C₈	D₄×D₆₇	1	2	yes	—
184	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
185	C₂×C₈	#11761629	1	2	yes	—
186	C₄⋊C₄	—	268	4	no	—
187	C₄⋊C₄	—	268	4	no	—
188	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
189	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
190	C₄²	#11761613	1	2	yes	—
191	C₁₆	—	134	4	no	—
192	C₂³	C₂×#11761615	1	1	yes	characteristic
193	Q₈	—	268	4	no	—
194	Q₈	—	268	4	no	—
195	Q₈	—	536	2	no	—
196	Q₈	—	536	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	268	4	no	—
199	C₂×C₄	—	536	2	no	—
200	C₂×C₄	—	536	2	no	—
201	C₂×C₄	#14523963	1	2	yes	—
202	C₂×C₄	#14523963	1	2	yes	—
203	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
204	C₂×C₄	#14524032	1	1	yes	characteristic
205	C₂×C₄	#14524039	1	1	yes	characteristic
206	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
207	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
208	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
209	C₈	—	8	2	no	—
210	C₈	#14523963	1	2	yes	—
211	C₈	#14523963	1	2	yes	—
212	C₈	#14524005	1	2	yes	—
213	C₈	#14524018	1	2	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	C₂×#14523963	1	1	yes	characteristic
216	C₂²	#16863744	1	1	yes	characteristic
217	C₂²	#16863770	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	536	4	no	—
220	C₄	#16863211	1	4	yes	complemented
221	C₄	#16863297	1	1	yes	characteristic
222	C₄	C₂×#14523963	1	1	yes	characteristic
223	C₄	#16863744	1	1	yes	characteristic
224	C₄	#16863772	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	#22561014	1	1	yes	characteristic
228	C₂	C₂×#16863211	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523963
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1848	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1848	D₆₇
hypocenter	C₆₇	#13212
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
104. C₄×C₆₇	#362	semidirect
119. C₆₇	#13212	semidirect
220. C₄	#16863211	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863776	2
C₂×#16863211	2
C₄×#14523983	2
C₆₇×#1848	2
#13212	67