Subgroups — #30773594

Order 17152.

324 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561968	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863179	C₂	1	1	yes	maximal, characteristic
3	#22561824	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863171	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560048	C₂	1	1	yes	maximal, characteristic
6	#22558290	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524049	C₂²	1	1	yes	characteristic
9	C₂×#14523983	C₂²	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×#14523969	C₄	1	1	yes	characteristic
12	#16863390	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
15	#16863179	—	2	2	no	—
16	(C₂×D₆₇)⋊C₁₆	—	2	2	no	—
17	#16863171	—	2	2	no	—
18	#16863160	C₄	1	1	yes	characteristic
19	#16863155	C₄	1	1	yes	characteristic
20	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	#405×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
24	#14524049	—	2	1	no	—
25	#14524049	—	2	1	no	—
26	#14524049	—	2	1	no	—
27	#14524049	—	2	1	no	—
28	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
29	C₂×#11761611	—	2	1	no	—
30	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₆₇	—	2	1	no	—
32	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
33	#14523983	—	2	1	no	—
34	#14523983	—	2	1	no	—
35	C₂×C₆₇⋊C₁₆	—	2	1	no	—
36	C₂×C₆₇⋊C₁₆	—	2	1	no	—
37	C₂×C₆₇⋊C₁₆	—	2	1	no	—
38	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
39	#14523969	—	2	2	no	—
40	#14523966	C₈	1	2	yes	—
41	#14523963	C₈	1	2	yes	—
42	C₈×Dic₆₇	—	2	1	no	—
43	C₈×Dic₆₇	—	2	1	no	—
44	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
47	M₃₂×C₆₇	—	2	1	no	—
48	M₃₂×C₆₇	—	2	1	no	—
49	C₂×C₁₆×C₆₇	—	2	1	no	—
50	C₂×C₁₆×C₆₇	—	2	1	no	—
51	C₂×C₁₆×C₆₇	—	2	1	no	—
52	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
53	#97×C₆₇	—	2	1	no	—
54	C₄×C₈×C₆₇	—	2	1	no	—
55	#11761628	—	2	1	no	—
56	#11761628	—	2	1	no	—
57	#11761628	—	2	1	no	—
58	#11761628	—	2	1	no	—
59	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
60	C₂×D₂₆₈	C₂×C₈	1	1	yes	characteristic
61	C₂×C₂²⋊D₆₇	—	2	1	no	—
62	C₂²×Dic₆₇	—	2	1	no	—
63	C₂×C₄×D₆₇	—	2	1	no	—
64	#11761612	—	2	1	no	—
65	#11761612	—	2	1	no	—
66	#11761611	—	4	1	no	—
67	#11761611	—	4	1	no	—
68	C₂×C₆₇⋊C₈	—	2	1	no	—
69	C₂×C₆₇⋊C₈	—	2	1	no	—
70	C₂×C₆₇⋊C₈	—	2	1	no	—
71	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
72	C₈×D₆₇	—	4	1	no	—
73	C₈×D₆₇	—	4	1	no	—
74	C₄×Dic₆₇	—	2	1	no	—
75	C₄×Dic₆₇	M₁₆	1	2	yes	—
76	C₆₇⋊C₁₆	—	2	2	no	—
77	C₆₇⋊C₁₆	—	4	1	no	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	M₁₆×C₆₇	—	2	1	no	—
80	M₁₆×C₆₇	—	2	1	no	—
81	C₂×C₈×C₆₇	—	2	1	no	—
82	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
85	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
86	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
87	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
88	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
89	C₂²⋊C₄×C₆₇	—	2	1	no	—
90	C₄²×C₆₇	—	2	1	no	—
91	C₁₆×C₆₇	—	2	2	no	—
92	C₁₆×C₆₇	—	4	1	no	—
93	C₂³×C₆₇	#131	1	1	yes	characteristic
94	C₂²×D₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
102	C₂²⋊D₆₇	—	4	1	no	—
103	C₂²⋊D₆₇	—	4	1	no	—
104	C₂×Dic₆₇	—	2	1	no	—
105	C₂×Dic₆₇	—	2	2	no	—
106	C₂×Dic₆₇	—	4	2	no	—
107	D₂₆₈	—	2	1	no	—
108	D₂₆₈	—	2	1	no	—
109	C₄×D₆₇	—	4	1	no	—
110	C₄×D₆₇	—	4	1	no	—
111	Dic₁₃₄	—	2	1	no	—
112	Dic₁₃₄	—	2	1	no	—
113	C₈×C₆₇	—	4	1	no	—
114	C₈×C₆₇	#121	1	1	yes	characteristic
115	C₈×C₆₇	#123	1	1	yes	characteristic
116	C₈×C₆₇	#124	1	1	yes	characteristic
117	C₈×C₆₇	#126	1	1	yes	characteristic
118	C₆₇⋊C₈	—	2	2	no	—
119	C₆₇⋊C₈	—	4	1	no	—
120	C₂²×C₆₇	—	2	2	no	—
121	C₂²×C₆₇	#433	1	1	yes	characteristic
122	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
123	C₂²×C₆₇	#435	1	1	yes	characteristic
124	C₂×D₆₇	—	4	2	no	—
125	C₂×D₆₇	—	4	1	no	—
126	C₄×C₆₇	—	4	2	no	—
127	C₄×C₆₇	#388	1	1	yes	characteristic
128	C₄×C₆₇	#431	1	1	yes	characteristic
129	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
130	C₄×C₆₇	#436	1	1	yes	characteristic
131	Dic₆₇	—	4	2	no	—
132	Dic₆₇	—	4	1	no	—
133	#13125	—	67	1	no	maximal, Sylow
134	C₂×C₆₇	—	2	1	no	—
135	C₂×C₆₇	—	2	1	no	—
136	C₂×C₆₇	#1923	1	1	yes	characteristic
137	C₂×C₆₇	#2171	1	2	yes	—
138	D₆₇	—	8	1	no	—
139	#2151	—	67	1	no	—
140	C₂×#350	—	67	1	no	—
141	#2113	—	67	1	no	—
142	C₂×#346	—	67	1	no	—
143	#1764	—	67	1	no	—
144	#1379	D₆₇	1	1	yes	characteristic, complemented
145	#1376	—	67	1	no	—
146	C₆₇	#13125	1	1	yes	characteristic, complemented, Sylow
147	C₂×#131	—	67	1	no	—
148	C₂×M₃₂	—	67	1	no	—
149	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
150	C₂²×C₁₆	—	67	1	no	—
151	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
152	C₂×#104	—	67	1	no	—
153	#416	—	67	1	no	—
154	#405	C₂×D₆₇	1	1	yes	characteristic
155	C₂×C₄×C₈	—	67	1	no	—
156	C₄⋊C₁₆	—	134	2	no	—
157	#350	—	134	2	no	—
158	C₂²⋊C₁₆	—	134	2	no	—
159	#346	—	134	2	no	—
160	#341	—	67	1	no	—
161	#339	—	67	1	no	—
162	C₂×#41	—	67	1	no	—
163	#131	—	134	1	no	—
164	#131	—	134	1	no	—
165	#131	—	134	1	no	—
166	#131	—	134	1	no	—
167	C₂×M₁₆	—	67	1	no	—
168	C₂×M₁₆	—	134	1	no	—
169	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
170	C₂²×C₈	—	67	1	no	—
171	C₂²×C₈	—	134	1	no	—
172	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
173	#117	C₄×D₆₇	1	1	yes	characteristic
174	C₂×C₄²	—	67	1	no	—
175	M₃₂	—	2	1	no	—
176	M₃₂	—	2	1	no	—
177	M₃₂	—	134	1	no	—
178	M₃₂	—	134	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	67	2	no	—
183	C₂×C₁₆	—	134	1	no	—
184	C₂×C₁₆	—	134	1	no	—
185	C₂×C₁₆	—	134	1	no	—
186	C₂×C₁₆	D₂₆₈	1	2	yes	—
187	#104	—	134	2	no	—
188	#103	—	67	2	no	—
189	#102	—	67	2	no	—
190	#97	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	134	1	no	—
193	C₄×C₈	—	134	1	no	—
194	#41	—	134	1	no	—
195	#41	—	134	1	no	—
196	#41	—	134	1	no	—
197	#41	—	134	1	no	—
198	C₂×Q₈	—	67	1	no	—
199	C₂×D₄	—	67	1	no	—
200	C₂×D₄	—	134	1	no	—
201	C₂²×C₄	—	134	1	no	—
202	C₂²×C₄	—	134	1	no	—
203	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
204	M₁₆	—	2	1	no	—
205	M₁₆	—	2	1	no	—
206	M₁₆	—	268	1	no	—
207	M₁₆	—	268	1	no	—
208	M₁₆	—	134	1	no	—
209	M₁₆	—	134	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	268	1	no	—
212	C₂×C₈	—	268	1	no	—
213	C₂×C₈	—	67	2	no	—
214	C₂×C₈	—	134	1	no	—
215	C₂×C₈	—	134	1	no	—
216	C₂×C₈	—	134	1	no	—
217	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
218	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
219	C₂×C₈	#11761627	1	1	yes	characteristic
220	C₂×C₈	#11761627	1	1	yes	characteristic
221	C₂×C₈	#11761628	1	1	yes	characteristic
222	C₂×C₈	#11761629	1	1	yes	characteristic
223	C₄⋊C₄	C₈×D₆₇	1	2	yes	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	67	2	no	—
227	C₄²	—	134	1	no	—
228	C₁₆	—	2	2	no	—
229	C₁₆	—	4	1	no	—
230	C₁₆	—	268	1	no	—
231	C₁₆	—	134	2	no	—
232	C₂³	—	134	1	no	—
233	C₂³	#14524051	1	1	yes	characteristic
234	Q₈	—	134	1	no	—
235	Q₈	—	134	1	no	—
236	D₄	—	268	1	no	—
237	D₄	—	268	1	no	—
238	D₄	—	134	1	no	—
239	D₄	—	134	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	268	2	no	—
242	C₂×C₄	—	268	1	no	—
243	C₂×C₄	—	268	1	no	—
244	C₂×C₄	—	134	1	no	—
245	C₂×C₄	—	134	2	no	—
246	C₂×C₄	#14524001	1	1	yes	characteristic
247	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
248	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
249	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
250	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
251	C₂×C₄	#14524049	1	1	yes	characteristic
252	C₈	—	4	1	no	—
253	C₈	—	268	1	no	—
254	C₈	—	134	2	no	—
255	C₈	#14524018	1	1	yes	characteristic
256	C₈	#14524027	1	1	yes	characteristic
257	C₈	#14524029	1	1	yes	characteristic
258	C₈	#14524038	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	268	2	no	—
261	C₂²	—	268	1	no	—
262	C₂²	#16863456	1	1	yes	characteristic
263	C₂²	(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic
264	C₂²	#16863469	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	268	2	no	—
267	C₄	—	268	1	no	—
268	C₄	#16863288	1	1	yes	characteristic
269	C₄	#16863448	1	1	yes	characteristic
270	C₄	C₈×D₂₆₈	1	1	yes	characteristic
271	C₄	#16863471	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	536	1	no	—
275	C₂	#22560816	1	1	yes	characteristic
276	C₂	#22562048	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1379	C₂
socle	C₂²×C₆₇	C₈×D₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1379	D₆₇
hypocenter	C₆₇	#13125
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
1. #22561968	C₂	semidirect
4. C₂×#16863171	C₂	semidirect
6. #22558290	C₂	semidirect
7. C₆₇×#1379	C₂	semidirect
144. #1379	D₆₇	semidirect
146. C₆₇	#13125	semidirect

Maximal subgroups

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

subgroup	index
#22561968	2
C₂×#16863179	2
#22561824	2
C₂×#16863171	2
#22560048	2
#22558290	2
C₆₇×#1379	2
#13125	67