Subgroups — #30746532

Order 17152.

420 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568851	C₂	1	1	yes	maximal, characteristic
2	#22558592	C₂	1	1	yes	maximal, characteristic
3	#22558592	C₂	1	1	yes	maximal, characteristic
4	#22558492	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558046	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558046	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11761628	C₂²	1	1	yes	characteristic
9	#16863455	—	2	2	no	—
10	#16863449	C₂²	1	1	yes	characteristic
11	#16863449	C₂²	1	1	yes	characteristic
12	#16863447	—	2	1	no	—
13	#16863447	—	2	1	no	—
14	#16863422	C₄	1	2	yes	—
15	#16863362	C₄	1	2	yes	—
16	C₄×#11761612	C₂²	1	1	yes	characteristic
17	#16863128	C₄	1	2	yes	—
18	#16863128	C₄	1	2	yes	—
19	#16863125	C₄	1	2	yes	—
20	#16863125	C₄	1	2	yes	—
21	#16863117	—	2	1	no	—
22	C₈×C₆₇⋊C₈	—	2	1	no	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#322×C₆₇	—	2	2	no	—
27	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
28	C₂×#11761612	C₂×C₄	1	2	yes	—
29	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
31	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
32	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
33	C₄×C₂²⋊D₆₇	—	2	1	no	—
34	C₄×C₂²⋊D₆₇	—	2	1	no	—
35	#14524002	C₂×C₄	1	1	yes	characteristic
36	#14524000	—	2	1	no	—
37	C₄²×D₆₇	—	2	1	no	—
38	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
39	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
40	Dic₆₇⋊C₈	—	2	2	no	—
41	Dic₆₇⋊C₈	—	2	2	no	—
42	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
43	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
44	(C₂×D₆₇)⋊C₈	—	2	2	no	—
45	(C₂×D₆₇)⋊C₈	—	2	2	no	—
46	#14523954	—	2	1	no	—
47	C₄×C₆₇⋊C₈	—	2	1	no	—
48	C₄×C₆₇⋊C₈	—	2	1	no	—
49	C₄×C₆₇⋊C₈	—	2	1	no	—
50	C₂²×C₈×C₆₇	—	2	1	no	—
51	C₂²×C₈×C₆₇	—	2	1	no	—
52	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₆₇	—	2	1	no	—
55	C₄×C₈×C₆₇	—	2	1	no	—
56	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
59	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	D₄	1	2	yes	—
61	#11761628	—	2	4	no	—
62	C₂×Dic₁₃₄	C₄²	1	1	yes	characteristic
63	C₂×D₂₆₈	C₄²	1	1	yes	characteristic
64	C₂×C₂²⋊D₆₇	—	2	1	no	—
65	C₂×C₄×D₆₇	—	2	1	no	—
66	C₂×C₄×D₆₇	—	2	1	no	—
67	C₂×C₄×D₆₇	—	2	1	no	—
68	#11761612	—	2	4	no	—
69	C₂×C₆₇⋊C₈	—	2	2	no	—
70	C₂×C₆₇⋊C₈	—	2	2	no	—
71	C₂×C₆₇⋊C₈	—	2	2	no	—
72	C₄⋊Dic₆₇	C₄²	1	1	yes	characteristic
73	C₄⋊Dic₆₇	C₄²	1	1	yes	characteristic
74	Dic₆₇⋊C₄	—	2	1	no	—
75	Dic₆₇⋊C₄	—	2	1	no	—
76	C₂²⋊Dic₆₇	—	2	1	no	—
77	(C₂×D₆₇)⋊C₄	—	2	1	no	—
78	(C₂×D₆₇)⋊C₄	—	2	1	no	—
79	C₄×Dic₆₇	—	2	1	no	—
80	C₄×Dic₆₇	—	2	1	no	—
81	C₄×Dic₆₇	—	2	1	no	—
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	—	2	4	no	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	4	2	no	—
90	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂³×C₆₇	#117	1	1	yes	characteristic
97	C₂²×D₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#131	1	4	yes	—
107	C₂²⋊D₆₇	—	4	2	no	—
108	C₂×Dic₆₇	—	2	1	no	—
109	C₂×Dic₆₇	—	2	1	no	—
110	C₂×Dic₆₇	—	2	1	no	—
111	D₂₆₈	—	2	2	no	—
112	C₄×D₆₇	—	4	2	no	—
113	C₄×D₆₇	—	4	2	no	—
114	C₄×D₆₇	—	4	2	no	—
115	Dic₁₃₄	—	2	2	no	—
116	C₈×C₆₇	—	2	4	no	—
117	C₈×C₆₇	—	2	2	no	—
118	C₈×C₆₇	—	2	2	no	—
119	C₈×C₆₇	—	4	2	no	—
120	C₆₇⋊C₈	—	4	2	no	—
121	C₆₇⋊C₈	—	4	2	no	—
122	C₆₇⋊C₈	—	4	2	no	—
123	C₂²×C₆₇	—	2	1	no	—
124	C₂²×C₆₇	—	2	1	no	—
125	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₆₇	#433	1	2	yes	—
127	C₂×D₆₇	—	4	1	no	—
128	C₂×D₆₇	—	4	1	no	—
129	C₂×D₆₇	—	4	1	no	—
130	C₄×C₆₇	—	2	1	no	—
131	C₄×C₆₇	—	2	1	no	—
132	C₄×C₆₇	#405	1	2	yes	—
133	C₄×C₆₇	#408	1	2	yes	—
134	C₄×C₆₇	#443	1	2	yes	—
135	C₄×C₆₇	#444	1	2	yes	—
136	Dic₆₇	—	4	1	no	—
137	Dic₆₇	—	4	1	no	—
138	Dic₆₇	—	4	1	no	—
139	#8189	—	67	1	no	maximal, Sylow
140	C₂×C₆₇	—	2	2	no	—
141	C₂×C₆₇	#1750	1	1	yes	characteristic
142	C₂×C₆₇	#1764	1	1	yes	characteristic
143	C₂×C₆₇	#1765	1	1	yes	characteristic
144	D₆₇	—	8	1	no	—
145	#2922	—	67	1	no	—
146	#1476	—	67	1	no	—
147	#1476	—	67	1	no	—
148	#1450	—	67	1	no	—
149	C₂×#322	D₆₇	1	1	yes	characteristic, complemented
150	#1278	—	67	1	no	—
151	#1278	—	67	1	no	—
152	C₆₇	#8189	1	1	yes	characteristic, complemented, Sylow
153	C₄×#41	—	67	1	no	—
154	#433	—	134	2	no	—
155	#431	—	67	1	no	—
156	#431	—	67	1	no	—
157	#431	—	134	1	no	—
158	#431	—	134	1	no	—
159	#424	—	67	2	no	—
160	#408	—	67	2	no	—
161	C₄×M₁₆	—	67	1	no	—
162	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
165	Q₈⋊C₈	—	67	2	no	—
166	Q₈⋊C₈	—	67	2	no	—
167	D₄⋊C₈	—	67	2	no	—
168	D₄⋊C₈	—	67	2	no	—
169	#322	—	2	2	no	—
170	#322	—	134	1	no	—
171	C₈²	—	134	1	no	—
172	C₂×#41	—	67	1	no	—
173	C₂×M₁₆	—	67	2	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	C₄×D₆₇	1	2	yes	—
177	C₄×Q₈	—	67	1	no	—
178	C₄×Q₈	—	67	1	no	—
179	C₄×D₄	—	67	1	no	—
180	C₄×D₄	—	67	1	no	—
181	C₄×D₄	—	134	1	no	—
182	C₄×D₄	—	134	1	no	—
183	#117	—	67	1	no	—
184	#117	—	134	1	no	—
185	C₂×C₄²	—	134	1	no	—
186	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
187	C₄⋊C₈	—	67	2	no	—
188	C₄⋊C₈	—	67	2	no	—
189	C₄⋊C₈	—	134	2	no	—
190	C₄⋊C₈	—	134	2	no	—
191	C₂²⋊C₈	—	134	1	no	—
192	C₂²⋊C₈	—	134	1	no	—
193	C₂²⋊C₈	—	134	2	no	—
194	C₂²⋊C₈	—	134	2	no	—
195	#97	—	134	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	—	134	1	no	—
199	C₄×C₈	—	134	1	no	—
200	C₄×C₈	—	134	1	no	—
201	C₄×C₈	C₄×D₆₇	1	2	yes	—
202	C₄×C₈	C₄×D₆₇	1	2	yes	—
203	C₄×C₈	D₂₆₈	1	1	yes	characteristic
204	C₄×C₈	D₂₆₈	1	1	yes	characteristic
205	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
206	#41	—	134	4	no	—
207	C₂×Q₈	—	67	1	no	—
208	C₂×D₄	—	67	1	no	—
209	C₂×D₄	—	134	1	no	—
210	C₂²×C₄	—	134	1	no	—
211	C₂²×C₄	—	134	1	no	—
212	C₂²×C₄	—	134	1	no	—
213	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
214	C₂²×C₄	#11761628	1	1	yes	characteristic
215	C₂²×C₄	#11761628	1	1	yes	characteristic
216	M₁₆	—	134	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	134	2	no	—
223	C₂×C₈	—	134	2	no	—
224	C₂×C₈	—	134	2	no	—
225	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
226	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
227	C₄⋊C₄	—	67	1	no	—
228	C₄⋊C₄	—	67	1	no	—
229	C₄⋊C₄	—	134	1	no	—
230	C₄⋊C₄	—	134	1	no	—
231	C₂²⋊C₄	—	134	1	no	—
232	C₂²⋊C₄	—	134	1	no	—
233	C₂²⋊C₄	—	134	1	no	—
234	C₄²	—	134	1	no	—
235	C₄²	—	134	1	no	—
236	C₄²	—	134	1	no	—
237	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
238	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
239	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
240	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
241	C₂³	—	134	1	no	—
242	C₂³	#14524000	1	1	yes	characteristic
243	Q₈	—	134	2	no	—
244	D₄	—	268	2	no	—
245	D₄	—	134	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	268	2	no	—
249	C₂×C₄	—	268	2	no	—
250	C₂×C₄	—	268	2	no	—
251	C₂×C₄	—	134	1	no	—
252	C₂×C₄	—	134	1	no	—
253	C₂×C₄	—	134	1	no	—
254	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
255	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
257	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
258	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
259	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
260	C₂×C₄	#14524051	1	4	yes	—
261	C₈	—	2	4	no	—
262	C₈	—	2	2	no	—
263	C₈	—	2	2	no	—
264	C₈	—	4	2	no	—
265	C₈	—	268	2	no	—
266	C₈	—	268	2	no	—
267	C₈	—	268	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	268	1	no	—
271	C₂²	—	268	1	no	—
272	C₂²	—	268	1	no	—
273	C₂²	C₄×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
274	C₂²	#16863458	1	2	yes	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	268	1	no	—
278	C₄	—	268	1	no	—
279	C₄	—	268	1	no	—
280	C₄	#16863353	1	2	yes	—
281	C₄	#16863364	1	2	yes	—
282	C₄	#16863514	1	2	yes	—
283	C₄	#16863519	1	2	yes	—
284	C₂	—	2	2	no	—
285	C₂	—	536	1	no	—
286	C₂	#22559978	1	1	yes	characteristic
287	C₂	#22560046	1	1	yes	characteristic
288	C₂	#22560053	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle	C₂²×C₆₇	C₄×C₂²⋊C₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#8189
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. #22558492	C₂	semidirect
5. #22558046	C₂	semidirect
6. #22558046	C₂	semidirect
7. C₂×#322×C₆₇	C₂	semidirect
149. C₂×#322	D₆₇	semidirect
152. C₆₇	#8189	semidirect

Maximal subgroups

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

subgroup	index
#22568851	2
#22558592	2
#22558492	2
#22558046	2
C₂×#322×C₆₇	2
#8189	67