Subgroups — #30755072

Order 17152.

518 conjugacy classes of subgroups.

The subgroup lattice has 268 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	#22561161	C₂	1	2	yes	maximal
3	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863143	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	—	2	1	no	—
7	C₂²×Dic₂₆₈	—	2	1	no	—
8	C₂×#14524043	C₂²	1	2	yes	—
9	#16863776	—	2	2	no	—
10	#16863776	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
12	#16863158	—	2	4	no	—
13	#16863149	—	2	2	no	—
14	#16863143	—	2	4	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₇	—	2	1	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	#336×C₆₇	—	2	1	no	—
20	C₂²×Dic₁₃₄	—	2	2	no	—
21	C₂×Dic₂₆₈	—	2	4	no	—
22	C₂×Dic₂₆₈	—	2	4	no	—
23	C₂×Dic₂₆₈	—	4	2	no	—
24	C₂×Dic₂₆₈	—	4	2	no	—
25	C₂×Dic₂₆₈	—	4	2	no	—
26	C₂×Dic₂₆₈	—	4	2	no	—
27	C₂²×C₆₇⋊C₈	—	2	2	no	—
28	#14524043	—	2	2	no	—
29	#14524043	—	2	2	no	—
30	#14524043	—	2	2	no	—
31	#14524043	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊Dic₆₇	—	2	2	no	—
33	C₂×C₄⋊Dic₆₇	—	2	2	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
36	C₂²×C₈×C₆₇	—	2	1	no	—
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
40	C₂×C₄²×C₆₇	—	2	1	no	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₇	—	2	1	no	—
43	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
44	C₄×C₈×C₆₇	—	4	2	no	—
45	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₆₇	D₄	1	2	yes	—
47	#95×C₆₇	—	2	1	no	—
48	C₂×Dic₁₃₄	—	2	4	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	C₂×Dic₁₃₄	—	4	2	no	—
54	C₂²×Dic₆₇	—	4	2	no	—
55	Dic₂₆₈	—	4	8	no	—
56	Dic₂₆₈	—	4	8	no	—
57	C₂×C₆₇⋊C₈	—	4	2	no	—
58	C₂×C₆₇⋊C₈	—	4	4	no	—
59	C₄⋊Dic₆₇	—	2	4	no	—
60	C₄⋊Dic₆₇	—	2	4	no	—
61	C₄⋊Dic₆₇	—	4	2	no	—
62	C₄⋊Dic₆₇	—	4	2	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	1	no	—
68	C₂×C₈×C₆₇	—	2	1	no	—
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	4	1	no	—
71	C₂×C₈×C₆₇	—	4	1	no	—
72	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₆₇	D₈	1	2	yes	—
75	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
76	C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₄⋊C₄×C₆₇	Q₁₆	1	4	yes	complemented
78	C₄²×C₆₇	—	4	1	no	—
79	C₄²×C₆₇	—	4	1	no	—
80	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₆₇	#41	1	1	yes	characteristic
84	C₂³×C₆₇	#121	1	1	yes	characteristic
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
98	C₂×C₄×C₆₇	#135	1	2	yes	—
99	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×Dic₆₇	—	4	4	no	—
102	C₂×Dic₆₇	—	8	2	no	—
103	C₂×Dic₆₇	—	8	2	no	—
104	Dic₁₃₄	—	4	4	no	—
105	Dic₁₃₄	—	4	4	no	—
106	Dic₁₃₄	—	8	2	no	—
107	Dic₁₃₄	—	8	2	no	—
108	C₈×C₆₇	—	2	2	no	—
109	C₈×C₆₇	—	4	2	no	—
110	C₈×C₆₇	#102	1	4	yes	—
111	C₆₇⋊C₈	—	8	2	no	—
112	C₂²×C₆₇	#390	1	1	yes	characteristic
113	C₂²×C₆₇	#459	1	1	yes	characteristic
114	C₂²×C₆₇	#460	1	1	yes	characteristic
115	C₂²×C₆₇	#467	1	2	yes	—
116	C₂²×C₆₇	#473	1	2	yes	—
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	—	2	1	no	—
119	C₄×C₆₇	—	4	4	no	—
120	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
121	C₄×C₆₇	#418	1	1	yes	characteristic
122	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
123	C₄×C₆₇	#459	1	1	yes	characteristic
124	C₄×C₆₇	#460	1	1	yes	characteristic
125	C₄×C₆₇	#494	1	1	yes	characteristic
126	C₄×C₆₇	#495	1	1	yes	characteristic
127	Dic₆₇	—	8	4	no	—
128	#9733	—	67	1	no	maximal, Sylow
129	C₂×C₆₇	#1669	1	2	yes	—
130	C₂×C₆₇	#1670	1	2	yes	—
131	C₂×C₆₇	#1969	1	1	yes	characteristic
132	C₂×C₆₇	#1979	1	1	yes	characteristic
133	C₂×C₆₇	#1982	1	1	yes	characteristic
134	C₂×#494	—	67	1	no	—
135	#1988	—	67	2	no	—
136	C₈×C₄⋊C₄	D₆₇	1	1	yes	characteristic
137	C₂×#335	—	67	1	no	—
138	C₂×#332	—	67	2	no	—
139	C₆₇	#9733	1	1	yes	characteristic, complemented, Sylow
140	C₂²×Q₁₆	—	134	1	no	—
141	C₂²×Q₁₆	—	134	1	no	—
142	C₂×C₄⋊Q₈	—	67	2	no	—
143	#494	—	67	4	no	—
144	#494	—	134	2	no	—
145	C₂×C₄⋊C₈	—	67	2	no	—
146	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
147	C₂×C₄×C₈	—	2	1	no	—
148	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
149	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
150	#340	—	134	4	no	—
151	#336	—	2	1	no	—
152	#335	—	134	2	no	—
153	#332	—	134	4	no	—
154	C₂²×Q₈	—	134	2	no	—
155	C₂×Q₁₆	—	268	2	no	—
156	C₂×Q₁₆	—	268	2	no	—
157	C₂×Q₁₆	—	268	2	no	—
158	C₂×Q₁₆	—	268	2	no	—
159	C₂×Q₁₆	—	134	4	no	—
160	C₂×Q₁₆	—	134	4	no	—
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	134	2	no	—
163	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
164	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
165	C₄⋊Q₈	—	67	4	no	—
166	C₄⋊Q₈	—	134	2	no	—
167	C₄⋊Q₈	—	134	2	no	—
168	C₄⋊Q₈	—	134	2	no	—
169	C₂×C₄⋊C₄	—	134	2	no	—
170	C₂×C₄⋊C₄	—	134	2	no	—
171	C₂×C₄⋊C₄	D₂₆₈	1	2	yes	—
172	C₂×C₄²	—	2	1	no	—
173	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	134	2	no	—
176	C₄⋊C₈	—	134	2	no	—
177	C₄⋊C₈	D₂₆₈	1	2	yes	—
178	C₄×C₈	—	4	2	no	—
179	C₄×C₈	C₄×D₆₇	1	2	yes	—
180	C₄×C₈	D₂₆₈	1	2	yes	—
181	#95	—	2	1	no	—
182	C₂×Q₈	—	268	2	no	—
183	C₂×Q₈	—	268	2	no	—
184	C₂×Q₈	—	268	2	no	—
185	C₂×Q₈	—	268	2	no	—
186	C₂×Q₈	—	268	2	no	—
187	C₂×Q₈	—	134	4	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	268	2	no	—
190	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
192	C₂²×C₄	#11761628	1	1	yes	characteristic
193	Q₁₆	—	268	8	no	—
194	Q₁₆	—	268	8	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	268	2	no	—
201	C₂×C₈	—	268	4	no	—
202	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
203	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
204	C₂×C₈	#11761613	1	2	yes	—
205	C₂×C₈	#11761615	1	2	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	268	2	no	—
208	C₄⋊C₄	—	268	2	no	—
209	C₄⋊C₄	—	134	4	no	—
210	C₄⋊C₄	—	134	4	no	—
211	C₄⋊C₄	Dic₂₆₈	1	4	yes	complemented
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
215	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
216	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
217	C₄²	#11761628	1	1	yes	characteristic
218	C₂³	#14524017	1	1	yes	characteristic
219	Q₈	—	268	4	no	—
220	Q₈	—	268	4	no	—
221	Q₈	—	536	2	no	—
222	Q₈	—	536	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	268	4	no	—
228	C₂×C₄	—	536	2	no	—
229	C₂×C₄	—	536	2	no	—
230	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
231	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
232	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
233	C₂×C₄	#14524017	1	1	yes	characteristic
234	C₂×C₄	#14524031	1	1	yes	characteristic
235	C₂×C₄	#14524040	1	1	yes	characteristic
236	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
237	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
238	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
239	C₂×C₄	#14524065	1	2	yes	—
240	C₂×C₄	#14524068	1	1	yes	characteristic
241	C₂×C₄	#14524068	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	536	2	no	—
245	C₈	#14523963	1	4	yes	—
246	C₂²	#16863298	1	1	yes	characteristic
247	C₂²	#16863591	1	1	yes	characteristic
248	C₂²	#16863596	1	1	yes	characteristic
249	C₂²	#16863637	1	2	yes	—
250	C₂²	#16863671	1	2	yes	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	536	4	no	—
255	C₄	C₂×#14523963	1	1	yes	characteristic
256	C₄	#16863400	1	1	yes	characteristic
257	C₄	C₄×Dic₂₆₈	1	2	yes	—
258	C₄	#16863591	1	1	yes	characteristic
259	C₄	#16863596	1	1	yes	characteristic
260	C₄	#16863776	1	1	yes	characteristic
261	C₄	#16863780	1	1	yes	characteristic
262	C₂	#22559574	1	2	yes	—
263	C₂	#22559579	1	2	yes	—
264	C₂	#22561070	1	1	yes	characteristic
265	C₂	#22561124	1	1	yes	characteristic
266	C₂	#22561136	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524017
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₈×C₄⋊C₄×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₈×C₄⋊C₄	D₆₇
hypocenter	C₆₇	#9733
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
10. #16863776	C₄	semidirect
77. C₄⋊C₄×C₆₇	Q₁₆	semidirect
139. C₆₇	#9733	semidirect
211. C₄⋊C₄	Dic₂₆₈	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
#22561161	2
C₂×#16863149	2
C₂×#16863143	2
C₈×C₄⋊C₄×C₆₇	2
#9733	67