Subgroups — #30775187

Order 17152.

462 conjugacy classes of subgroups.

The subgroup lattice has 266 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	C₂×#16863199	C₂	1	2	yes	maximal
3	C₂×#16863199	C₂	1	2	yes	maximal
4	#22558287	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1942	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	—	2	1	no	—
7	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
8	C₂×#14524043	—	2	1	no	—
9	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	#16863776	—	2	4	no	—
11	#16863199	—	2	4	no	—
12	#16863199	—	2	4	no	—
13	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#389×C₆₇	—	2	1	no	—
17	#340×C₆₇	—	2	1	no	—
18	C₂²×Dic₁₃₄	—	2	2	no	—
19	C₂×Dic₂₆₈	—	2	4	no	—
20	C₂×Dic₂₆₈	—	2	2	no	—
21	C₂×Dic₂₆₈	—	2	2	no	—
22	C₂×Dic₂₆₈	—	2	2	no	—
23	C₂×Dic₂₆₈	—	2	2	no	—
24	C₂×Dic₂₆₈	—	2	2	no	—
25	C₂×Dic₂₆₈	—	4	2	no	—
26	C₂×Dic₂₆₈	—	4	2	no	—
27	C₂×Dic₂₆₈	D₄	1	4	yes	—
28	#14524043	—	2	4	no	—
29	#14524043	—	4	2	no	—
30	C₂×C₄⋊Dic₆₇	—	4	1	no	—
31	C₂×C₄⋊Dic₆₇	—	4	1	no	—
32	C₂×C₆₇⋊C₁₆	—	2	2	no	—
33	C₂×C₆₇⋊C₁₆	—	2	2	no	—
34	C₂×C₆₇⋊C₁₆	—	2	2	no	—
35	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
40	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
41	#107×C₆₇	—	2	1	no	—
42	#107×C₆₇	—	2	2	no	—
43	#107×C₆₇	D₄	1	2	yes	—
44	C₄×C₈×C₆₇	—	2	2	no	—
45	#95×C₆₇	—	4	1	no	—
46	C₂×Dic₁₃₄	—	2	4	no	—
47	C₂×Dic₁₃₄	—	4	2	no	—
48	C₂×Dic₁₃₄	—	4	2	no	—
49	C₂×Dic₁₃₄	—	4	2	no	—
50	C₂×Dic₁₃₄	—	4	2	no	—
51	C₂×Dic₁₃₄	—	4	2	no	—
52	C₂²×Dic₆₇	—	4	2	no	—
53	Dic₂₆₈	—	2	4	no	—
54	Dic₂₆₈	—	2	4	no	—
55	Dic₂₆₈	—	4	8	no	—
56	Dic₂₆₈	—	4	2	no	—
57	Dic₂₆₈	—	4	2	no	—
58	C₄⋊Dic₆₇	—	4	4	no	—
59	C₄⋊Dic₆₇	—	4	4	no	—
60	C₆₇⋊C₁₆	—	4	2	no	—
61	C₂²×C₄×C₆₇	—	4	1	no	—
62	C₂²×C₄×C₆₇	—	4	1	no	—
63	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	—	2	1	no	—
67	C₂×C₈×C₆₇	—	2	1	no	—
68	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
69	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
72	C₂×C₈×C₆₇	#41	1	2	yes	—
73	C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₄⋊C₄×C₆₇	—	4	1	no	—
75	C₄⋊C₄×C₆₇	—	4	1	no	—
76	C₄⋊C₄×C₆₇	—	4	1	no	—
77	C₄⋊C₄×C₆₇	—	8	2	no	—
78	C₄²×C₆₇	—	2	1	no	—
79	C₄²×C₆₇	—	2	1	no	—
80	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	4	2	no	—
86	C₂×C₄×C₆₇	—	8	1	no	—
87	C₂×C₄×C₆₇	—	8	1	no	—
88	C₂×C₄×C₆₇	—	8	1	no	—
89	C₂×C₄×C₆₇	—	8	1	no	—
90	C₂×C₄×C₆₇	—	8	1	no	—
91	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
92	C₂×C₄×C₆₇	Q₃₂	1	2	yes	—
93	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×Dic₆₇	—	4	4	no	—
100	C₂×Dic₆₇	—	8	2	no	—
101	C₂×Dic₆₇	—	8	2	no	—
102	Dic₁₃₄	—	4	4	no	—
103	Dic₁₃₄	—	4	4	no	—
104	Dic₁₃₄	—	8	2	no	—
105	Dic₁₃₄	—	8	2	no	—
106	C₈×C₆₇	—	2	2	no	—
107	C₈×C₆₇	#121	1	2	yes	—
108	C₈×C₆₇	#123	1	2	yes	—
109	C₂²×C₆₇	#447	1	1	yes	characteristic
110	C₂²×C₆₇	#459	1	1	yes	characteristic
111	C₂²×C₆₇	#486	1	1	yes	characteristic
112	C₂²×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
113	C₂²×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
114	C₂²×C₆₇	#510	1	1	yes	characteristic
115	C₂²×C₆₇	#510	1	1	yes	characteristic
116	C₄×C₆₇	—	2	2	no	—
117	C₄×C₆₇	—	2	2	no	—
118	C₄×C₆₇	—	8	2	no	—
119	C₄×C₆₇	—	16	1	no	—
120	C₄×C₆₇	#394	1	1	yes	characteristic
121	C₄×C₆₇	#450	1	1	yes	characteristic
122	C₄×C₆₇	#460	1	1	yes	characteristic
123	C₄×C₆₇	#490	1	1	yes	characteristic
124	Dic₆₇	—	8	4	no	—
125	#13409	—	67	1	no	maximal, Sylow
126	C₂×C₆₇	#2041	1	1	yes	characteristic
127	C₂×C₆₇	#2185	1	1	yes	characteristic
128	C₂×C₆₇	#2188	1	1	yes	characteristic
129	C₂×C₆₇	#2208	1	1	yes	characteristic
130	C₂×C₆₇	#2209	1	1	yes	characteristic
131	C₂×C₆₇	#2241	1	1	yes	characteristic
132	C₂×C₆₇	#2245	1	1	yes	characteristic
133	C₂×#494	—	67	1	no	—
134	C₂×#358	—	67	2	no	—
135	C₂×#358	—	67	2	no	—
136	#1942	D₆₇	1	1	yes	characteristic
137	#1374	—	67	1	no	—
138	C₆₇	#13409	1	1	yes	characteristic, complemented, Sylow
139	C₂²×Q₁₆	—	67	2	no	—
140	C₂²×Q₁₆	—	134	1	no	—
141	C₂×C₄⋊Q₈	—	134	1	no	—
142	C₂²×C₁₆	—	67	2	no	—
143	#494	—	134	4	no	—
144	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
145	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
147	#389	—	2	1	no	—
148	#358	—	134	4	no	—
149	#358	—	134	4	no	—
150	#340	—	2	1	no	—
151	C₂²×Q₈	—	134	2	no	—
152	C₂×Q₁₆	—	268	2	no	—
153	C₂×Q₁₆	—	268	2	no	—
154	C₂×Q₁₆	—	67	4	no	—
155	C₂×Q₁₆	—	134	4	no	—
156	C₂×Q₁₆	—	134	2	no	—
157	C₂×Q₁₆	—	134	2	no	—
158	C₂×Q₁₆	—	134	2	no	—
159	C₂×Q₁₆	—	134	2	no	—
160	C₂×Q₁₆	—	134	2	no	—
161	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
163	C₄⋊Q₈	—	268	2	no	—
164	C₄⋊Q₈	—	134	4	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	4	1	no	—
168	C₂×C₄⋊C₄	—	268	1	no	—
169	C₂×C₄⋊C₄	—	268	1	no	—
170	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
171	C₂×C₁₆	—	134	2	no	—
172	C₂×C₁₆	—	134	2	no	—
173	C₂×C₁₆	—	134	2	no	—
174	#107	—	2	1	no	—
175	#107	—	2	2	no	—
176	#107	D₂₆₈	1	2	yes	—
177	C₄×C₈	—	2	2	no	—
178	#95	—	4	1	no	—
179	C₂×Q₈	—	268	2	no	—
180	C₂×Q₈	—	268	2	no	—
181	C₂×Q₈	—	268	2	no	—
182	C₂×Q₈	—	268	2	no	—
183	C₂×Q₈	—	268	2	no	—
184	C₂×Q₈	—	134	4	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	268	2	no	—
188	C₂²×C₄	#11761613	1	1	yes	characteristic
189	C₂²×C₄	#11761613	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
191	Q₁₆	—	268	8	no	—
192	Q₁₆	—	268	2	no	—
193	Q₁₆	—	268	2	no	—
194	Q₁₆	—	134	4	no	—
195	Q₁₆	—	134	4	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	#11761615	1	2	yes	—
199	C₂×C₈	D₄×D₆₇	1	2	yes	—
200	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
201	C₂×C₈	#11761628	1	1	yes	characteristic
202	C₂×C₈	#11761629	1	2	yes	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	1	no	—
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₄⋊C₄	—	8	2	no	—
208	C₄⋊C₄	—	268	4	no	—
209	C₄⋊C₄	—	268	4	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₁₆	—	268	2	no	—
213	C₂³	C₂×#11761613	1	1	yes	characteristic
214	Q₈	—	268	4	no	—
215	Q₈	—	268	4	no	—
216	Q₈	—	536	2	no	—
217	Q₈	—	536	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	1	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	8	1	no	—
226	C₂×C₄	—	8	1	no	—
227	C₂×C₄	—	8	1	no	—
228	C₂×C₄	—	268	4	no	—
229	C₂×C₄	—	536	2	no	—
230	C₂×C₄	—	536	2	no	—
231	C₂×C₄	#14523987	1	2	yes	—
232	C₂×C₄	#14523989	1	2	yes	—
233	C₂×C₄	#14524011	1	1	yes	characteristic
234	C₂×C₄	#14524017	1	1	yes	characteristic
235	C₂×C₄	#14524032	1	1	yes	characteristic
236	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
237	C₂×C₄	#14524068	1	1	yes	characteristic
238	C₂×C₄	#14524068	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	#14524018	1	2	yes	—
241	C₈	#14524028	1	2	yes	—
242	C₂²	#16863529	1	1	yes	characteristic
243	C₂²	#16863591	1	1	yes	characteristic
244	C₂²	#16863744	1	1	yes	characteristic
245	C₂²	C₂×#14523987	1	1	yes	characteristic
246	C₂²	C₂×#14523989	1	1	yes	characteristic
247	C₂²	#16863836	1	1	yes	characteristic
248	C₂²	#16863836	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	8	2	no	—
252	C₄	—	16	1	no	—
253	C₄	—	536	4	no	—
254	C₄	#16863313	1	1	yes	characteristic
255	C₄	#16863546	1	1	yes	characteristic
256	C₄	#16863596	1	1	yes	characteristic
257	C₄	#16863762	1	1	yes	characteristic
258	C₂	#22561443	1	1	yes	characteristic
259	C₂	#22562125	1	1	yes	characteristic
260	C₂	#22562142	1	1	yes	characteristic
261	C₂	#22562254	1	1	yes	characteristic
262	C₂	#22562259	1	1	yes	characteristic
263	C₂	#22562438	1	1	yes	characteristic
264	C₂	#22562456	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1942	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	#1942	D₆₇
hypocenter	C₆₇	#13409
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1942

Complements

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

N	H ≅ G/N	type
138. C₆₇	#13409	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
C₂×#16863199	2
#22558287	2
C₆₇×#1942	2
#13409	67