Subgroups — #30864619

Order 17152.

854 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	C₄⋊Q₈×Dic₆₇	C₂	1	2	yes	maximal
2	#22561171	C₂	1	2	yes	maximal
3	#22560948	C₂	1	4	yes	maximal
4	#22560366	C₂	1	1	yes	maximal, characteristic
5	#22560108	C₂	1	4	yes	maximal
6	C₄×#14523953	C₂	1	1	yes	maximal, characteristic
7	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₆₇	—	2	2	no	—
9	C₂×#14523975	C₂²	1	4	yes	—
10	C₂×#14523953	C₂²	1	2	yes	—
11	C₂×#14523953	C₂²	1	2	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#16863303	—	2	2	no	—
14	#16863295	—	2	2	no	—
15	#16863295	C₂²	1	4	yes	—
16	#16863266	C₂²	1	2	yes	—
17	C₄⋊C₄×Dic₆₇	—	2	2	no	—
18	C₄⋊C₄×Dic₆₇	C₂²	1	4	yes	—
19	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
20	#16863158	C₂²	1	8	yes	—
21	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	#500×C₆₇	C₄	1	8	yes	complemented
23	C₂×#107×C₆₇	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
26	Q₈×Dic₆₇	—	4	8	no	—
27	C₂×C₄⋊Dic₆₇	—	2	4	no	—
28	C₂×C₄⋊Dic₆₇	—	2	2	no	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
31	C₂×C₄×Dic₆₇	—	2	4	no	—
32	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
33	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
34	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
36	#14523975	—	2	8	no	—
37	#14523953	—	2	4	no	—
38	#14523953	Q₈	1	8	yes	complemented
39	C₄×C₆₇⋊C₈	—	2	2	no	—
40	C₆₇⋊#95	—	2	4	no	—
41	C₆₇⋊#95	—	2	4	no	—
42	C₂²×Q₈×C₆₇	—	2	2	no	—
43	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₆₇	—	2	4	no	—
45	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
46	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	#107×C₆₇	C₂×C₄	1	16	yes	—
50	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
51	C₂²×Dic₆₇	—	2	4	no	—
52	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
53	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
54	C₂×C₆₇⋊C₈	—	2	2	no	—
55	C₂×C₆₇⋊C₈	—	2	2	no	—
56	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
57	C₄⋊Dic₆₇	—	2	8	no	—
58	C₄⋊Dic₆₇	—	4	8	no	—
59	C₄⋊Dic₆₇	—	4	4	no	—
60	C₄⋊Dic₆₇	—	4	4	no	—
61	C₄×Dic₆₇	—	2	4	no	—
62	C₄×Dic₆₇	—	2	4	no	—
63	C₄×Dic₆₇	—	2	2	no	—
64	C₄×Dic₆₇	—	4	8	no	—
65	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
66	C₂×Q₈×C₆₇	—	2	8	no	—
67	C₂×Q₈×C₆₇	—	4	4	no	—
68	C₂×Q₈×C₆₇	—	4	4	no	—
69	C₂²×C₄×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
75	C₂×C₈×C₆₇	#41	1	4	yes	—
76	C₄⋊C₄×C₆₇	—	2	8	no	—
77	C₄⋊C₄×C₆₇	—	2	4	no	—
78	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
79	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₆₇	#41	1	2	yes	—
81	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
82	Q₈×C₆₇	—	4	16	no	—
83	C₂×C₄×C₆₇	—	2	8	no	—
84	C₂×C₄×C₆₇	—	4	4	no	—
85	C₂×C₄×C₆₇	—	4	4	no	—
86	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
89	C₂×C₄×C₆₇	#137	1	4	yes	—
90	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
92	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
94	C₂×Dic₆₇	—	2	2	no	—
95	C₂×Dic₆₇	—	2	2	no	—
96	C₂×Dic₆₇	—	2	2	no	—
97	C₂×Dic₆₇	—	2	2	no	—
98	C₂×Dic₆₇	—	2	2	no	—
99	C₂×Dic₆₇	—	4	4	no	—
100	C₂×Dic₆₇	—	4	4	no	—
101	C₂×Dic₆₇	—	4	4	no	—
102	C₂×Dic₆₇	C₄⋊Q₈	1	4	yes	—
103	C₈×C₆₇	C₄×Q₈	1	8	yes	complemented
104	C₆₇⋊C₈	—	2	4	no	—
105	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
106	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
107	C₂²×C₆₇	#532	1	1	yes	characteristic
108	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
109	C₂²×C₆₇	C₂×#137	1	2	yes	—
110	C₄×C₆₇	—	4	8	no	—
111	C₄×C₆₇	#441	1	4	yes	—
112	C₄×C₆₇	#442	1	4	yes	—
113	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
114	C₄×C₆₇	#533	1	2	yes	—
115	Dic₆₇	—	2	4	no	—
116	Dic₆₇	—	4	2	no	—
117	Dic₆₇	—	8	4	no	—
118	#21726	—	67	1	no	maximal, Sylow
119	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
121	C₂×C₆₇	C₂×#442	1	1	yes	characteristic
122	C₂×C₆₇	#3165	1	2	yes	—
123	C₂×C₆₇	#3182	1	2	yes	—
124	C₂×#500	D₆₇	1	1	yes	characteristic
125	C₄×C₄⋊Q₈	—	67	2	no	—
126	#1990	—	67	2	no	—
127	#1943	—	67	4	no	—
128	#1844	—	67	1	no	—
129	#1776	—	67	4	no	—
130	C₄×#97	—	67	1	no	—
131	C₆₇	#21726	1	1	yes	characteristic, complemented, Sylow
132	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
133	C₂×C₄×Q₈	—	134	2	no	—
134	#500	Dic₆₇	1	8	yes	complemented
135	C₂×#107	C₂×D₆₇	1	4	yes	—
136	C₂×#106	—	67	4	no	—
137	C₂×#97	—	67	2	no	—
138	C₂×#97	—	67	2	no	—
139	C₂×C₄×C₈	—	67	1	no	—
140	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
141	#391	—	134	2	no	—
142	#389	—	67	4	no	—
143	#389	—	134	2	no	—
144	#384	—	67	2	no	—
145	C₄×C₄⋊C₄	—	67	4	no	—
146	C₄×C₄⋊C₄	—	134	2	no	—
147	C₄³	—	67	1	no	—
148	#340	—	67	8	no	—
149	C₂²×Q₈	—	2	2	no	—
150	C₂²×C₈	—	67	2	no	—
151	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
152	C₄⋊Q₈	—	2	4	no	—
153	C₄⋊Q₈	C₂×Dic₆₇	1	8	yes	—
154	C₄×Q₈	—	268	8	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	67	4	no	—
157	C₂×C₄⋊C₄	—	134	4	no	—
158	C₂×C₄⋊C₄	—	134	2	no	—
159	C₂×C₄⋊C₄	—	134	2	no	—
160	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
161	C₂×C₄²	—	67	1	no	—
162	C₂×C₄²	—	67	2	no	—
163	C₂×C₄²	—	67	2	no	—
164	C₂×C₄²	—	67	1	no	—
165	C₂×C₄²	—	134	4	no	—
166	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
167	#107	C₂×Dic₆₇	1	16	yes	—
168	#106	—	134	8	no	—
169	#97	—	67	8	no	—
170	#97	—	134	4	no	—
171	C₄×C₈	—	134	2	no	—
172	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
173	#95	—	134	4	no	—
174	#95	—	134	4	no	—
175	C₂×Q₈	—	2	8	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂×Q₈	—	4	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	67	2	no	—
180	C₂²×C₄	—	67	2	no	—
181	C₂²×C₄	—	134	4	no	—
182	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
183	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
184	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
185	C₂×C₈	—	67	4	no	—
186	C₂×C₈	—	134	2	no	—
187	C₂×C₈	—	134	2	no	—
188	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
189	C₂×C₈	Q₈×D₆₇	1	4	yes	—
190	C₂×C₈	#11761629	1	4	yes	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	268	8	no	—
194	C₄⋊C₄	—	268	4	no	—
195	C₄⋊C₄	—	268	4	no	—
196	C₄⋊C₄	—	134	8	no	—
197	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
198	C₄²	—	268	8	no	—
199	C₄²	—	67	4	no	—
200	C₄²	—	134	4	no	—
201	C₄²	—	134	4	no	—
202	C₄²	—	134	2	no	—
203	C₄²	C₂²×Dic₆₇	1	2	yes	—
204	C₄²	#11761627	1	2	yes	—
205	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
206	Q₈	—	4	16	no	—
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	268	4	no	—
211	C₂×C₄	—	268	4	no	—
212	C₂×C₄	—	268	4	no	—
213	C₂×C₄	—	67	4	no	—
214	C₂×C₄	—	134	2	no	—
215	C₂×C₄	—	134	2	no	—
216	C₂×C₄	—	134	2	no	—
217	C₂×C₄	—	134	2	no	—
218	C₂×C₄	—	134	2	no	—
219	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
220	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
221	C₂×C₄	#14524067	1	4	yes	—
222	C₂×C₄	#14524075	1	4	yes	—
223	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
224	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
225	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
226	C₂×C₄	C₂×#11761629	1	2	yes	—
227	C₈	—	134	4	no	—
228	C₈	Q₈×Dic₆₇	1	8	yes	complemented
229	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
230	C₂²	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
231	C₂²	#16863947	1	1	yes	characteristic
232	C₂²	C₂×#14524067	1	2	yes	—
233	C₂²	C₂×#14524075	1	2	yes	—
234	C₄	—	4	8	no	—
235	C₄	—	268	2	no	—
236	C₄	—	536	4	no	—
237	C₄	—	134	4	no	—
238	C₄	#16863503	1	4	yes	—
239	C₄	#16863508	1	4	yes	—
240	C₄	C₂×Q₈×Dic₆₇	1	2	yes	—
241	C₄	#16863951	1	2	yes	—
242	C₂	C₄⋊Q₈×Dic₆₇	1	1	yes	characteristic
243	C₂	C₂×#16863503	1	1	yes	characteristic
244	C₂	C₂×#16863508	1	1	yes	characteristic
245	C₂	#22570752	1	2	yes	—
246	C₂	#22570929	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	#21726
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
22. #500×C₆₇	C₄	semidirect
38. #14523953	Q₈	semidirect
103. C₈×C₆₇	C₄×Q₈	semidirect
131. C₆₇	#21726	semidirect
134. #500	Dic₆₇	semidirect
228. C₈	Q₈×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×Dic₆₇	2
#22561171	2
#22560948	2
#22560366	2
#22560108	2
C₄×#14523953	2
C₂×#500×C₆₇	2
#21726	67