Subgroups — #30765609

Order 17152.

386 conjugacy classes of subgroups.

The subgroup lattice has 242 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	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
3	#22558178	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₆₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₃₄	—	2	1	no	—
7	C₂×#14523983	—	2	1	no	—
8	C₄×Dic₂₆₈	—	2	1	no	—
9	C₄×Dic₂₆₈	—	2	1	no	—
10	C₄×Dic₂₆₈	C₄	1	2	yes	—
11	C₄×Dic₂₆₈	C₂²	1	2	yes	—
12	C₂×#14523977	C₄	1	1	yes	characteristic
13	C₂×#14523968	—	2	1	no	—
14	#16863172	C₂²	1	2	yes	—
15	C₄×C₆₇⋊C₁₆	—	2	1	no	—
16	#431×C₆₇	—	2	1	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#335×C₆₇	—	2	1	no	—
19	#334×C₆₇	C₂²	1	2	yes	—
20	C₂²×Dic₁₃₄	—	2	1	no	—
21	C₂×Dic₂₆₈	—	2	1	no	—
22	C₂×Dic₂₆₈	—	2	2	no	—
23	C₂×Dic₂₆₈	C₈	1	4	yes	complemented
24	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₃₄	—	2	2	no	—
26	C₄×Dic₁₃₄	—	2	2	no	—
27	C₄×Dic₁₃₄	—	4	1	no	—
28	C₄×Dic₁₃₄	—	4	1	no	—
29	C₂×C₄⋊Dic₆₇	—	2	1	no	—
30	C₂×Dic₆₇⋊C₄	—	4	1	no	—
31	C₂×C₄×Dic₆₇	—	4	1	no	—
32	#14523983	—	2	4	no	—
33	C₂×C₆₇⋊C₁₆	—	2	2	no	—
34	#14523977	—	2	1	no	—
35	#14523977	C₂×C₄	1	2	yes	—
36	#14523968	—	2	2	no	—
37	#14523968	—	2	2	no	—
38	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
41	C₄⋊C₈×C₆₇	—	2	2	no	—
42	C₂²⋊C₈×C₆₇	—	4	1	no	—
43	C₄×C₈×C₆₇	—	4	1	no	—
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	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₃₄	—	2	4	no	—
49	C₂×Dic₁₃₄	—	2	2	no	—
50	C₂×Dic₁₃₄	—	4	1	no	—
51	C₂×Dic₁₃₄	—	4	2	no	—
52	C₂²×Dic₆₇	—	4	1	no	—
53	C₂²×Dic₆₇	—	4	1	no	—
54	Dic₂₆₈	—	2	4	no	—
55	Dic₂₆₈	—	2	2	no	—
56	Dic₂₆₈	C₂×C₈	1	4	yes	—
57	C₄⋊Dic₆₇	—	2	2	no	—
58	C₄⋊Dic₆₇	—	4	1	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	C₆₇⋊C₁₆	—	2	4	no	—
64	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	2	no	—
68	C₂×C₈×C₆₇	—	4	2	no	—
69	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
73	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₆₇	D₈	1	1	yes	characteristic
77	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
78	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂³×C₆₇	#102	1	1	yes	characteristic
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	4	1	no	—
82	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
83	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
84	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
86	C₂×C₄×C₆₇	#104	1	2	yes	—
87	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×Dic₆₇	—	4	4	no	—
91	C₂×Dic₆₇	—	4	2	no	—
92	C₂×Dic₆₇	—	4	2	no	—
93	C₂×Dic₆₇	—	8	2	no	—
94	Dic₁₃₄	—	2	4	no	—
95	Dic₁₃₄	—	4	4	no	—
96	Dic₁₃₄	—	4	2	no	—
97	C₈×C₆₇	—	2	1	no	—
98	C₈×C₆₇	—	2	1	no	—
99	C₈×C₆₇	—	8	2	no	—
100	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
101	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
102	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂²×C₆₇	—	2	1	no	—
104	C₂²×C₆₇	—	2	1	no	—
105	C₂²×C₆₇	D₄⋊C₈	1	2	yes	—
106	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
107	C₄×C₆₇	—	2	1	no	—
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	D₄⋊C₈	1	2	yes	—
111	C₄×C₆₇	#362	1	2	yes	—
112	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
113	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
114	Dic₆₇	—	4	4	no	—
115	Dic₆₇	—	8	2	no	—
116	#11600	—	67	1	no	maximal, Sylow
117	C₂×C₆₇	—	2	2	no	—
118	C₂×C₆₇	C₂×D₄⋊C₈	1	1	yes	characteristic
119	C₂×C₆₇	#2145	1	1	yes	characteristic
120	C₂×C₆₇	C₂×#362	1	1	yes	characteristic
121	C₂×C₄×Q₁₆	—	67	1	no	—
122	C₄×M₃₂	—	67	1	no	—
123	#1564	D₆₇	1	1	yes	characteristic
124	#1334	—	67	4	no	—
125	C₆₇	#11600	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₁₆	—	67	1	no	—
127	C₂×C₄×Q₈	—	134	1	no	—
128	C₂×M₃₂	—	134	1	no	—
129	C₄×Q₁₆	—	67	2	no	—
130	C₄×Q₁₆	—	67	2	no	—
131	C₄×Q₁₆	—	134	1	no	—
132	C₄×Q₁₆	—	134	1	no	—
133	#431	—	2	1	no	—
134	C₂×#107	—	67	1	no	—
135	C₂×#103	—	134	1	no	—
136	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
137	#346	—	67	2	no	—
138	C₄×C₁₆	—	134	1	no	—
139	#335	—	2	1	no	—
140	#334	C₂×D₆₇	1	2	yes	—
141	C₂²×Q₈	—	134	1	no	—
142	C₂×Q₁₆	—	67	4	no	—
143	C₂×Q₁₆	—	67	2	no	—
144	C₂×Q₁₆	—	134	1	no	—
145	C₂×Q₁₆	—	134	2	no	—
146	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
147	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
148	C₄×Q₈	—	268	1	no	—
149	C₄×Q₈	—	268	1	no	—
150	C₄×Q₈	—	134	2	no	—
151	C₄×Q₈	—	134	2	no	—
152	C₂×C₄⋊C₄	—	268	1	no	—
153	C₂×C₄⋊C₄	—	134	1	no	—
154	C₂×C₄²	—	268	1	no	—
155	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
156	M₃₂	—	134	4	no	—
157	C₂×C₁₆	—	134	2	no	—
158	#107	—	67	2	no	—
159	#107	—	134	1	no	—
160	C₄⋊C₈	—	2	2	no	—
161	#103	—	134	2	no	—
162	#103	—	134	2	no	—
163	C₂²⋊C₈	—	4	1	no	—
164	C₄×C₈	—	4	1	no	—
165	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
166	C₄×C₈	D₂₆₈	1	1	yes	characteristic
167	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
169	C₂×Q₈	—	268	1	no	—
170	C₂×Q₈	—	268	2	no	—
171	C₂×Q₈	—	134	4	no	—
172	C₂×Q₈	—	134	2	no	—
173	C₂²×C₄	—	268	1	no	—
174	C₂²×C₄	—	268	1	no	—
175	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	#11761613	1	1	yes	characteristic
177	C₂²×C₄	#11761615	1	1	yes	characteristic
178	Q₁₆	—	67	4	no	—
179	Q₁₆	—	134	4	no	—
180	Q₁₆	—	134	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
184	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
185	C₂×C₈	C₈×D₆₇	1	2	yes	—
186	C₂×C₈	#11761611	1	2	yes	—
187	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
188	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
189	C₄⋊C₄	—	268	2	no	—
190	C₄⋊C₄	—	268	1	no	—
191	C₄⋊C₄	—	268	2	no	—
192	C₄⋊C₄	—	134	2	no	—
193	C₄²	—	268	2	no	—
194	C₄²	—	268	2	no	—
195	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
196	C₄²	#11761613	1	1	yes	characteristic
197	C₄²	#11761615	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
199	C₁₆	—	134	4	no	—
200	C₂³	#14523963	1	1	yes	characteristic
201	Q₈	—	268	4	no	—
202	Q₈	—	268	2	no	—
203	Q₈	—	134	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	268	4	no	—
207	C₂×C₄	—	268	2	no	—
208	C₂×C₄	—	268	2	no	—
209	C₂×C₄	—	536	2	no	—
210	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
211	C₂×C₄	#14523963	1	1	yes	characteristic
212	C₂×C₄	#14523963	1	1	yes	characteristic
213	C₂×C₄	#14523963	1	1	yes	characteristic
214	C₂×C₄	#14523969	1	2	yes	—
215	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
216	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
217	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	8	2	no	—
221	C₈	(C₂×D₆₇)⋊C₈	1	2	yes	—
222	C₈	C₂×C₈×D₆₇	1	1	yes	characteristic
223	C₈	C₂×#11761611	1	1	yes	characteristic
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	#16863123	1	2	yes	—
227	C₂²	C₂×#14523963	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	268	4	no	—
232	C₄	—	536	2	no	—
233	C₄	#16863123	1	2	yes	—
234	C₄	#16863211	1	2	yes	—
235	C₄	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
236	C₄	C₂×#14523969	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	C₂×#16863123	1	1	yes	characteristic
239	C₂	#22561944	1	1	yes	characteristic
240	C₂	C₂×#16863211	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14523963
commutator	C₈×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1564	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	#102
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1564	D₆₇
hypocenter	C₆₇	#11600
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #22558178	C₂	semidirect
23. C₂×Dic₂₆₈	C₈	semidirect
125. C₆₇	#11600	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
C₄×#14523983	2
#22558178	2
C₆₇×#1564	2
#11600	67