Subgroups — #30754908

Order 17152.

486 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	#22568587	C₂	1	1	yes	maximal, characteristic
2	#22560361	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863144	C₂	1	2	yes	maximal
4	C₂×#16863141	C₂	1	2	yes	maximal
5	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	C₂²	1	1	yes	characteristic
7	C₂×#14524043	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	#16863302	—	2	2	no	—
11	#16863302	C₄	1	4	yes	—
12	#16863262	—	2	1	no	—
13	#16863262	—	2	1	no	—
14	#16863144	—	2	4	no	—
15	#16863141	—	2	4	no	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×#97×C₆₇	—	2	1	no	—
18	C₄³×C₆₇	C₂²	1	1	yes	characteristic
19	#336×C₆₇	—	2	1	no	—
20	C₂²×Dic₁₃₄	—	2	1	no	—
21	C₂²×Dic₁₃₄	C₂×C₄	1	2	yes	—
22	C₂²×C₆₇⋊C₈	—	2	2	no	—
23	#14524043	—	2	2	no	—
24	#14524043	—	2	2	no	—
25	#14524043	—	2	2	no	—
26	#14524043	—	2	2	no	—
27	C₂×C₄⋊Dic₆₇	—	2	1	no	—
28	C₂×C₄⋊Dic₆₇	—	2	1	no	—
29	C₂×C₄⋊Dic₆₇	—	2	1	no	—
30	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
31	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
34	C₆₇⋊#95	—	2	4	no	—
35	C₆₇⋊#95	—	2	4	no	—
36	C₂²×C₈×C₆₇	—	2	2	no	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	D₄	1	2	yes	—
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₇	—	2	2	no	—
43	C₄⋊C₈×C₆₇	D₄	1	4	yes	—
44	#97×C₆₇	—	4	2	no	—
45	C₂×Dic₁₃₄	—	2	4	no	—
46	C₂×Dic₁₃₄	—	4	1	no	—
47	C₂×Dic₁₃₄	—	4	1	no	—
48	C₂×Dic₁₃₄	—	4	2	no	—
49	C₂×Dic₁₃₄	—	4	2	no	—
50	C₂×Dic₁₃₄	—	4	4	no	—
51	C₂²×Dic₆₇	—	2	2	no	—
52	C₂²×Dic₆₇	—	2	2	no	—
53	C₂×C₆₇⋊C₈	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	4	4	no	—
55	C₄⋊Dic₆₇	—	2	4	no	—
56	C₄⋊Dic₆₇	—	4	2	no	—
57	C₄⋊Dic₆₇	—	4	2	no	—
58	C₄⋊Dic₆₇	—	4	1	no	—
59	C₄⋊Dic₆₇	—	4	1	no	—
60	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	—	2	4	no	—
66	C₂×C₈×C₆₇	—	4	2	no	—
67	C₂×C₈×C₆₇	—	4	2	no	—
68	C₄²×C₆₇	—	2	2	no	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	—	4	2	no	—
72	C₄²×C₆₇	SD₁₆	1	2	yes	—
73	C₄²×C₆₇	Q₁₆	1	2	yes	—
74	C₄²×C₆₇	C₂×D₄	1	2	yes	—
75	C₄²×C₆₇	#41	1	2	yes	—
76	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	#103	1	4	yes	—
84	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₆₇	#121	1	2	yes	—
86	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
87	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
90	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
91	C₂×C₄×C₆₇	#137	1	2	yes	—
92	C₂×C₄×C₆₇	#137	1	2	yes	—
93	C₂×Dic₆₇	—	4	2	no	—
94	C₂×Dic₆₇	—	4	2	no	—
95	C₂×Dic₆₇	—	4	4	no	—
96	C₂×Dic₆₇	—	4	4	no	—
97	Dic₁₃₄	—	8	2	no	—
98	Dic₁₃₄	—	8	1	no	—
99	Dic₁₃₄	—	8	1	no	—
100	Dic₁₃₄	—	8	4	no	—
101	C₈×C₆₇	—	4	4	no	—
102	C₆₇⋊C₈	—	8	2	no	—
103	C₂²×C₆₇	#390	1	1	yes	characteristic
104	C₂²×C₆₇	#413	1	2	yes	—
105	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
106	C₂²×C₆₇	#417	1	2	yes	—
107	C₂²×C₆₇	#419	1	1	yes	characteristic
108	C₄×C₆₇	—	2	4	no	—
109	C₄×C₆₇	—	2	2	no	—
110	C₄×C₆₇	—	4	2	no	—
111	C₄×C₆₇	#461	1	2	yes	—
112	C₄×C₆₇	#462	1	2	yes	—
113	C₄×C₆₇	#489	1	2	yes	—
114	C₄×C₆₇	#496	1	1	yes	characteristic
115	C₄×C₆₇	#497	1	1	yes	characteristic
116	Dic₆₇	—	8	2	no	—
117	Dic₆₇	—	8	2	no	—
118	#9701	—	67	1	no	maximal, Sylow
119	C₂×C₆₇	#1550	1	2	yes	—
120	C₂×C₆₇	#1559	1	2	yes	—
121	C₂×C₆₇	#1959	1	1	yes	characteristic
122	C₂×C₆₇	#1967	1	1	yes	characteristic
123	C₂×C₆₇	#1981	1	1	yes	characteristic
124	#2862	—	67	1	no	—
125	#1842	—	67	1	no	—
126	#1767	D₆₇	1	1	yes	characteristic
127	C₂×#333	—	67	2	no	—
128	C₂×#332	—	67	2	no	—
129	C₆₇	#9701	1	1	yes	characteristic, complemented, Sylow
130	C₂×C₄⋊Q₈	—	67	1	no	—
131	C₂×C₄⋊Q₈	—	67	1	no	—
132	C₂×C₄⋊C₈	—	67	2	no	—
133	C₂×C₄⋊C₈	—	134	2	no	—
134	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
135	C₂×#97	—	2	1	no	—
136	#391	—	67	4	no	—
137	#391	—	134	2	no	—
138	#382	—	134	1	no	—
139	#382	—	134	1	no	—
140	C₄³	C₂×D₆₇	1	1	yes	characteristic
141	#336	—	2	1	no	—
142	#333	—	134	4	no	—
143	#332	—	134	4	no	—
144	C₂²×Q₈	—	67	2	no	—
145	C₂²×Q₈	—	134	1	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	134	2	no	—
148	C₄⋊Q₈	—	134	2	no	—
149	C₄⋊Q₈	—	134	2	no	—
150	C₄⋊Q₈	—	134	2	no	—
151	C₄⋊Q₈	—	134	2	no	—
152	C₂×C₄⋊C₄	—	67	2	no	—
153	C₂×C₄⋊C₄	—	134	1	no	—
154	C₂×C₄⋊C₄	—	134	1	no	—
155	C₂×C₄⋊C₄	—	134	1	no	—
156	C₂×C₄²	C₄×D₆₇	1	2	yes	—
157	C₂×C₄²	D₂₆₈	1	2	yes	—
158	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
160	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	268	4	no	—
163	C₄⋊C₈	—	134	2	no	—
164	C₄⋊C₈	—	134	2	no	—
165	C₄⋊C₈	D₂₆₈	1	4	yes	—
166	#97	—	4	2	no	—
167	#95	—	134	4	no	—
168	#95	—	134	4	no	—
169	C₂×Q₈	—	268	4	no	—
170	C₂×Q₈	—	268	1	no	—
171	C₂×Q₈	—	268	1	no	—
172	C₂×Q₈	—	268	2	no	—
173	C₂×Q₈	—	268	2	no	—
174	C₂×Q₈	—	134	4	no	—
175	C₂²×C₄	—	134	2	no	—
176	C₂²×C₄	—	134	2	no	—
177	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
178	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
179	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	268	2	no	—
186	C₂×C₈	—	268	4	no	—
187	C₄⋊C₄	—	268	2	no	—
188	C₄⋊C₄	—	268	2	no	—
189	C₄⋊C₄	—	268	1	no	—
190	C₄⋊C₄	—	268	1	no	—
191	C₄⋊C₄	—	134	4	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	#11761616	1	2	yes	—
197	C₄²	#11761618	1	2	yes	—
198	C₄²	C₂×D₂₆₈	1	2	yes	—
199	C₄²	#11761628	1	2	yes	—
200	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
201	Q₈	—	536	4	no	—
202	Q₈	—	536	2	no	—
203	Q₈	—	536	1	no	—
204	Q₈	—	536	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	268	2	no	—
212	C₂×C₄	—	268	2	no	—
213	C₂×C₄	—	268	4	no	—
214	C₂×C₄	—	268	4	no	—
215	C₂×C₄	#14523966	1	4	yes	—
216	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
217	C₂×C₄	#14524017	1	2	yes	—
218	C₂×C₄	#14524031	1	1	yes	characteristic
219	C₂×C₄	#14524040	1	1	yes	characteristic
220	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
221	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
222	C₂×C₄	#14524071	1	2	yes	—
223	C₂×C₄	#14524073	1	2	yes	—
224	C₂×C₄	#14524076	1	2	yes	—
225	C₈	—	4	4	no	—
226	C₈	—	536	2	no	—
227	C₂²	#16863298	1	1	yes	characteristic
228	C₂²	#16863378	1	2	yes	—
229	C₂²	C₂×#14523966	1	1	yes	characteristic
230	C₂²	#16863398	1	2	yes	—
231	C₂²	#16863404	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	4	no	—
234	C₄	—	4	2	no	—
235	C₄	—	536	2	no	—
236	C₄	—	536	2	no	—
237	C₄	#16863601	1	2	yes	—
238	C₄	#16863606	1	2	yes	—
239	C₄	#16863756	1	2	yes	—
240	C₄	#16863785	1	1	yes	characteristic
241	C₄	#16863790	1	1	yes	characteristic
242	C₂	#22558900	1	2	yes	—
243	C₂	#22558940	1	2	yes	—
244	C₂	#22561025	1	1	yes	characteristic
245	C₂	#22561058	1	1	yes	characteristic
246	C₂	#22561132	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₆₇)⋊C₄
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1767	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1767	D₆₇
hypocenter	C₆₇	#9701
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
129. C₆₇	#9701	semidirect

Maximal subgroups

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

subgroup	index
#22568587	2
#22560361	2
C₂×#16863144	2
C₂×#16863141	2
C₆₇×#1767	2
#9701	67