Subgroups — #30754723

Order 17152.

482 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
2	#22561145	C₂	1	2	yes	maximal
3	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863128	C₂	1	2	yes	maximal
5	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₃₄	C₂²	1	2	yes	—
8	C₄×Dic₂₆₈	—	2	2	no	—
9	C₄×Dic₂₆₈	—	2	2	no	—
10	C₂×#14523977	C₄	1	1	yes	characteristic
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×#14523968	C₄	1	2	yes	—
13	#16863150	—	2	4	no	—
14	#16863149	—	2	2	no	—
15	#16863128	—	2	4	no	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#322×C₆₇	—	2	2	no	—
20	C₂²×Dic₁₃₄	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₆₈	—	2	4	no	—
22	C₂×Dic₂₆₈	—	2	2	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₄×Dic₁₃₄	—	2	2	no	—
25	C₄×Dic₁₃₄	—	2	2	no	—
26	C₄×Dic₁₃₄	—	2	2	no	—
27	C₄×Dic₁₃₄	—	2	2	no	—
28	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₆₇⋊C₄	—	2	2	no	—
30	C₂×C₄×Dic₆₇	—	2	2	no	—
31	#14523977	—	2	2	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
34	#14523968	—	2	4	no	—
35	C₂²×C₈×C₆₇	—	2	1	no	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	D₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₆₇	—	2	1	no	—
41	C₄×C₈×C₆₇	—	2	1	no	—
42	C₄×C₈×C₆₇	—	2	1	no	—
43	C₄×C₈×C₆₇	D₄	1	2	yes	—
44	C₄×C₈×C₆₇	D₄	1	2	yes	—
45	C₄×C₈×C₆₇	D₄	1	2	yes	—
46	C₂×Dic₁₃₄	—	2	2	no	—
47	C₂×Dic₁₃₄	—	2	2	no	—
48	C₂×Dic₁₃₄	—	2	2	no	—
49	C₂×Dic₁₃₄	—	2	2	no	—
50	C₂×Dic₁₃₄	—	2	2	no	—
51	C₂×Dic₁₃₄	M₁₆	1	4	yes	—
52	C₂²×Dic₆₇	—	2	2	no	—
53	C₂²×Dic₆₇	—	2	2	no	—
54	Dic₂₆₈	—	2	8	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	4	4	no	—
57	C₄⋊Dic₆₇	—	2	2	no	—
58	C₄⋊Dic₆₇	—	2	2	no	—
59	Dic₆₇⋊C₄	—	4	2	no	—
60	Dic₆₇⋊C₄	—	4	2	no	—
61	C₄×Dic₆₇	—	4	2	no	—
62	C₄×Dic₆₇	—	4	2	no	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
66	C₂×C₈×C₆₇	—	2	2	no	—
67	C₂×C₈×C₆₇	—	2	2	no	—
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	4	2	no	—
72	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₆₇	Q₁₆	1	4	yes	complemented
75	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₆₇	#41	1	1	yes	characteristic
79	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
84	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
86	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₆₇	#131	1	2	yes	—
89	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₆₇	#135	1	2	yes	—
91	C₂×Dic₆₇	—	2	4	no	—
92	C₂×Dic₆₇	—	4	2	no	—
93	C₂×Dic₆₇	—	4	2	no	—
94	C₂×Dic₆₇	—	4	2	no	—
95	C₂×Dic₆₇	—	4	4	no	—
96	Dic₁₃₄	—	2	4	no	—
97	Dic₁₃₄	—	2	4	no	—
98	Dic₁₃₄	—	4	2	no	—
99	Dic₁₃₄	—	4	2	no	—
100	C₈×C₆₇	—	2	2	no	—
101	C₈×C₆₇	—	2	2	no	—
102	C₈×C₆₇	—	2	4	no	—
103	C₈×C₆₇	—	4	2	no	—
104	C₆₇⋊C₈	—	8	2	no	—
105	C₂²×C₆₇	#390	1	1	yes	characteristic
106	C₂²×C₆₇	#435	1	2	yes	—
107	C₂²×C₆₇	C₄×Q₁₆	1	2	yes	—
108	C₂²×C₆₇	#444	1	2	yes	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	#407	1	1	yes	characteristic
112	C₄×C₆₇	#408	1	1	yes	characteristic
113	C₄×C₆₇	#435	1	2	yes	—
114	C₄×C₆₇	#467	1	2	yes	—
115	C₄×C₆₇	#494	1	1	yes	characteristic
116	C₄×C₆₇	#495	1	1	yes	characteristic
117	Dic₆₇	—	4	4	no	—
118	Dic₆₇	—	8	2	no	—
119	#9662	—	67	1	no	maximal, Sylow
120	C₂×C₆₇	#1584	1	4	yes	—
121	C₂×C₆₇	#1955	1	1	yes	characteristic
122	C₂×C₆₇	#1969	1	1	yes	characteristic
123	C₂×C₆₇	#1975	1	1	yes	characteristic
124	C₂×C₄×Q₁₆	—	67	1	no	—
125	#1984	—	67	2	no	—
126	C₂×#335	—	67	1	no	—
127	C₂×Q₈⋊C₈	—	67	2	no	—
128	C₂×#322	D₆₇	1	1	yes	characteristic
129	C₆₇	#9662	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₁₆	—	67	1	no	—
131	C₂×C₄×Q₈	—	67	2	no	—
132	C₄×Q₁₆	—	134	2	no	—
133	C₄×Q₁₆	—	134	2	no	—
134	C₂×#107	—	67	1	no	—
135	C₂×C₄⋊C₈	—	67	2	no	—
136	C₂×#103	—	67	2	no	—
137	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
140	#336	—	134	4	no	—
141	#335	—	134	2	no	—
142	Q₈⋊C₈	—	134	4	no	—
143	#322	—	2	2	no	—
144	C₂²×Q₈	—	67	2	no	—
145	C₂×Q₁₆	—	134	4	no	—
146	C₂×Q₁₆	—	134	2	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	134	2	no	—
149	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
150	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
151	C₂²×C₈	D₂₆₈	1	2	yes	—
152	C₄×Q₈	—	134	2	no	—
153	C₄×Q₈	—	134	2	no	—
154	C₄×Q₈	—	134	2	no	—
155	C₄×Q₈	—	134	2	no	—
156	C₂×C₄⋊C₄	—	67	2	no	—
157	C₂×C₄⋊C₄	—	134	2	no	—
158	C₂×C₄²	—	134	2	no	—
159	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
160	#107	—	134	2	no	—
161	C₄⋊C₈	—	134	2	no	—
162	C₄⋊C₈	—	134	2	no	—
163	#103	—	134	4	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	D₂₆₈	1	2	yes	—
168	C₄×C₈	D₂₆₈	1	2	yes	—
169	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
170	C₂×Q₈	—	67	4	no	—
171	C₂×Q₈	—	134	2	no	—
172	C₂×Q₈	—	134	2	no	—
173	C₂×Q₈	—	134	2	no	—
174	C₂×Q₈	—	134	2	no	—
175	C₂×Q₈	—	134	2	no	—
176	C₂²×C₄	—	134	2	no	—
177	C₂²×C₄	—	134	2	no	—
178	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
180	C₂²×C₄	#11761628	1	1	yes	characteristic
181	Q₁₆	—	134	8	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	268	2	no	—
189	C₂×C₈	—	268	4	no	—
190	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
191	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
192	C₂×C₈	Dic₂₆₈	1	4	yes	complemented
193	C₄⋊C₄	—	268	2	no	—
194	C₄⋊C₄	—	268	2	no	—
195	C₄⋊C₄	—	134	2	no	—
196	C₄⋊C₄	—	134	2	no	—
197	C₄²	—	268	2	no	—
198	C₄²	—	268	2	no	—
199	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
200	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
201	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
202	C₄²	#11761628	1	1	yes	characteristic
203	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
204	Q₈	—	268	2	no	—
205	Q₈	—	268	2	no	—
206	Q₈	—	134	4	no	—
207	Q₈	—	134	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	268	2	no	—
210	C₂×C₄	—	268	2	no	—
211	C₂×C₄	—	268	4	no	—
212	C₂×C₄	—	268	2	no	—
213	C₂×C₄	—	134	4	no	—
214	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
215	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
216	C₂×C₄	#14524017	1	1	yes	characteristic
217	C₂×C₄	#14524017	1	1	yes	characteristic
218	C₂×C₄	#14524031	1	1	yes	characteristic
219	C₂×C₄	#14524040	1	1	yes	characteristic
220	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
221	C₂×C₄	#14524051	1	2	yes	—
222	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
223	C₂×C₄	#14524065	1	2	yes	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	4	no	—
227	C₈	—	4	2	no	—
228	C₈	—	536	2	no	—
229	C₂²	#16863298	1	1	yes	characteristic
230	C₂²	#16863469	1	2	yes	—
231	C₂²	C₄×Dic₂₆₈	1	2	yes	—
232	C₂²	#16863519	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	268	4	no	—
236	C₄	—	536	2	no	—
237	C₄	#16863359	1	1	yes	characteristic
238	C₄	#16863364	1	1	yes	characteristic
239	C₄	#16863469	1	2	yes	—
240	C₄	#16863637	1	2	yes	—
241	C₄	#16863776	1	1	yes	characteristic
242	C₄	#16863780	1	1	yes	characteristic
243	C₂	#22559061	1	4	yes	—
244	C₂	#22561007	1	1	yes	characteristic
245	C₂	#22561070	1	1	yes	characteristic
246	C₂	#22561102	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#9662
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
74. C₂×C₈×C₆₇	Q₁₆	semidirect
129. C₆₇	#9662	semidirect
192. C₂×C₈	Dic₂₆₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₆₈	2
#22561145	2
C₂×#16863149	2
C₂×#16863128	2
C₂×#322×C₆₇	2
#9662	67