Subgroups — #30827328

Order 17152.

630 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569160	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#22559117	C₂	1	2	yes	maximal, complemented
4	#22559079	C₂	1	4	yes	maximal, complemented
5	#22559027	C₂	1	2	yes	maximal, complemented
6	#22558637	C₂	1	2	yes	maximal
7	#22558498	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1566	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₆₈	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₃₄	C₂²	1	2	yes	—
11	#16863525	—	2	4	no	—
12	#16863523	C₂²	1	4	yes	—
13	#16863520	C₂²	1	4	yes	—
14	C₄×Dic₂₆₈	C₄	1	1	yes	characteristic
15	C₄×Dic₂₆₈	C₄	1	2	yes	—
16	C₄×Dic₂₆₈	C₄	1	1	yes	characteristic
17	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₆₈	C₂²	1	2	yes	—
19	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
20	#16863449	C₂²	1	1	yes	characteristic
21	C₂×#14523977	C₄	1	1	yes	characteristic
22	#16863419	—	2	2	no	—
23	C₂×#14523968	C₄	1	2	yes	—
24	C₄×#11761612	C₂²	1	1	yes	characteristic
25	M₁₆×Dic₆₇	—	2	2	no	—
26	#16863126	C₂²	1	4	yes	—
27	#16863126	C₂²	1	4	yes	—
28	#16863118	C₂²	1	2	yes	—
29	#16863117	C₂²	1	2	yes	—
30	#423×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#334×C₆₇	C₂²	1	2	yes	—
33	#334×C₆₇	C₂²	1	2	yes	—
34	C₂²×Dic₁₃₄	C₂×C₄	1	2	yes	—
35	C₂×Dic₂₆₈	—	2	4	no	—
36	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
37	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
39	C₂×#11761612	D₄	1	2	yes	—
40	C₄×Dic₁₃₄	—	2	2	no	—
41	C₄×Dic₁₃₄	—	2	2	no	—
42	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₃₄	C₂³	1	2	yes	—
45	C₄×Dic₁₃₄	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
47	C₂×Dic₆₇⋊C₄	—	2	2	no	—
48	C₂×C₄×Dic₆₇	—	2	2	no	—
49	#14523977	C₂×C₄	1	2	yes	—
50	#14523977	C₂×C₄	1	1	yes	characteristic
51	#14523977	C₂×C₄	1	1	yes	characteristic
52	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
53	Dic₆₇⋊C₈	—	2	4	no	—
54	Dic₆₇⋊C₈	—	2	4	no	—
55	#14523968	C₂×C₄	1	2	yes	—
56	#14523968	C₂×C₄	1	2	yes	—
57	#14523968	C₂×C₄	1	2	yes	—
58	#14523968	C₂×C₄	1	2	yes	—
59	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
60	#14523954	D₄	1	2	yes	—
61	#14523953	—	2	4	no	—
62	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
63	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
64	C₈×Dic₆₇	—	2	4	no	—
65	C₂×M₁₆×C₆₇	—	2	2	no	—
66	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₆₇	—	2	2	no	—
70	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
71	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₃₄	—	2	4	no	—
76	C₂×Dic₁₃₄	—	2	2	no	—
77	C₂×Dic₁₃₄	—	4	2	no	—
78	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
80	C₂²×Dic₆₇	—	2	2	no	—
81	C₂²×Dic₆₇	—	2	2	no	—
82	Dic₂₆₈	—	2	4	no	—
83	Dic₂₆₈	—	2	2	no	—
84	Dic₂₆₈	M₁₆	1	4	yes	complemented
85	#11761612	—	2	4	no	—
86	C₂×C₆₇⋊C₈	—	2	2	no	—
87	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
88	C₄⋊Dic₆₇	—	2	2	no	—
89	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
90	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
91	Dic₆₇⋊C₄	—	2	2	no	—
92	Dic₆₇⋊C₄	—	2	2	no	—
93	Dic₆₇⋊C₄	—	2	2	no	—
94	Dic₆₇⋊C₄	—	2	2	no	—
95	C₄×Dic₆₇	—	2	2	no	—
96	C₄×Dic₆₇	—	2	2	no	—
97	C₄×Dic₆₇	—	2	2	no	—
98	C₄×Dic₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
102	M₁₆×C₆₇	—	4	4	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	4	no	—
105	C₂×C₈×C₆₇	—	4	1	no	—
106	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₆₇	#41	1	1	yes	characteristic
113	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	4	1	no	—
118	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#137	1	4	yes	—
122	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₆₇	—	2	2	no	—
126	C₂×Dic₆₇	—	2	2	no	—
127	C₂×Dic₆₇	—	2	2	no	—
128	C₂×Dic₆₇	—	2	2	no	—
129	C₂×Dic₆₇	—	4	4	no	—
130	C₂×Dic₆₇	—	8	2	no	—
131	Dic₁₃₄	—	2	2	no	—
132	Dic₁₃₄	—	2	2	no	—
133	Dic₁₃₄	—	2	2	no	—
134	Dic₁₃₄	—	4	4	no	—
135	Dic₁₃₄	C₂×M₁₆	1	4	yes	—
136	C₈×C₆₇	—	2	1	no	—
137	C₈×C₆₇	—	2	1	no	—
138	C₈×C₆₇	—	2	1	no	—
139	C₈×C₆₇	—	4	4	no	—
140	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
141	C₆₇⋊C₈	—	2	4	no	—
142	C₆₇⋊C₈	—	4	2	no	—
143	C₂²×C₆₇	—	2	1	no	—
144	C₂²×C₆₇	—	2	1	no	—
145	C₂²×C₆₇	#441	1	2	yes	—
146	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	—	2	1	no	—
150	C₄×C₆₇	#441	1	2	yes	—
151	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
152	C₄×C₆₇	#568	1	1	yes	characteristic
153	C₄×C₆₇	C₂×#137	1	2	yes	—
154	Dic₆₇	—	2	4	no	—
155	Dic₆₇	—	4	2	no	—
156	Dic₆₇	—	4	2	no	—
157	Dic₆₇	—	4	2	no	—
158	#18042	—	67	1	no	maximal, Sylow
159	C₂×C₆₇	—	2	2	no	—
160	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
161	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
162	C₂×C₆₇	#2957	1	1	yes	characteristic
163	Q₈×M₁₆	—	67	2	no	—
164	C₂×C₄×Q₁₆	—	67	1	no	—
165	#1593	—	67	2	no	—
166	#1587	—	67	4	no	—
167	#1579	—	67	2	no	—
168	#1566	D₆₇	1	1	yes	characteristic
169	#1487	—	67	2	no	—
170	#1451	—	67	1	no	—
171	C₆₇	#18042	1	1	yes	characteristic, complemented, Sylow
172	C₂²×Q₁₆	—	67	1	no	—
173	C₂×C₄×Q₈	—	67	2	no	—
174	#446	—	67	4	no	—
175	#446	—	134	4	no	—
176	C₈×Q₈	—	67	4	no	—
177	C₄×Q₁₆	—	67	1	no	—
178	C₄×Q₁₆	—	67	2	no	—
179	C₄×Q₁₆	—	67	1	no	—
180	C₄×Q₁₆	—	67	1	no	—
181	C₄×Q₁₆	—	67	2	no	—
182	C₄×Q₁₆	—	67	1	no	—
183	#431	—	67	1	no	—
184	C₂×#107	—	67	1	no	—
185	#423	—	134	2	no	—
186	#423	C₂×D₆₇	1	2	yes	—
187	C₂×#103	—	67	2	no	—
188	C₄×M₁₆	—	67	1	no	—
189	C₄×M₁₆	—	134	2	no	—
190	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
191	#334	C₂×D₆₇	1	2	yes	—
192	#334	C₂×D₆₇	1	2	yes	—
193	Q₈⋊C₈	—	67	4	no	—
194	Q₈⋊C₈	—	67	4	no	—
195	#322	—	67	2	no	—
196	#322	—	67	2	no	—
197	C₂²×Q₈	—	67	2	no	—
198	C₂×Q₁₆	—	67	1	no	—
199	C₂×Q₁₆	—	67	2	no	—
200	C₂×Q₁₆	—	67	1	no	—
201	C₂×Q₁₆	—	134	4	no	—
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	—	67	2	no	—
204	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
205	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
206	C₄×Q₈	—	67	2	no	—
207	C₄×Q₈	—	67	2	no	—
208	C₄×Q₈	—	67	2	no	—
209	C₄×Q₈	—	67	2	no	—
210	C₄×Q₈	—	134	2	no	—
211	C₄×Q₈	—	134	2	no	—
212	C₂×C₄⋊C₄	—	67	2	no	—
213	C₂×C₄⋊C₄	—	134	2	no	—
214	C₂×C₄²	—	134	2	no	—
215	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
216	#107	—	67	2	no	—
217	#107	—	67	1	no	—
218	#107	—	67	1	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	134	4	no	—
221	C₄⋊C₈	—	134	4	no	—
222	C₄⋊C₈	—	134	1	no	—
223	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
224	#103	—	67	2	no	—
225	#103	—	67	2	no	—
226	#103	—	67	2	no	—
227	#103	—	67	2	no	—
228	C₂²⋊C₈	—	134	1	no	—
229	#97	—	67	2	no	—
230	#97	—	134	4	no	—
231	C₄×C₈	—	67	2	no	—
232	C₄×C₈	—	67	2	no	—
233	C₄×C₈	—	134	4	no	—
234	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
235	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
238	C₂×Q₈	—	268	2	no	—
239	C₂×Q₈	—	67	2	no	—
240	C₂×Q₈	—	67	2	no	—
241	C₂×Q₈	—	134	4	no	—
242	C₂×Q₈	—	134	2	no	—
243	C₂²×C₄	—	134	2	no	—
244	C₂²×C₄	—	134	2	no	—
245	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
246	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
247	C₂²×C₄	#11761629	1	1	yes	characteristic
248	Q₁₆	—	67	4	no	—
249	Q₁₆	—	134	4	no	—
250	Q₁₆	—	134	2	no	—
251	M₁₆	—	4	4	no	—
252	M₁₆	—	134	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	67	4	no	—
257	C₂×C₈	—	134	2	no	—
258	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
259	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
260	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
261	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
262	C₄⋊C₄	—	67	2	no	—
263	C₄⋊C₄	—	67	2	no	—
264	C₄⋊C₄	—	134	2	no	—
265	C₄⋊C₄	—	134	2	no	—
266	C₄⋊C₄	—	134	2	no	—
267	C₄⋊C₄	—	134	2	no	—
268	C₄⋊C₄	—	134	2	no	—
269	C₄²	—	134	2	no	—
270	C₄²	—	134	2	no	—
271	C₄²	—	134	2	no	—
272	C₄²	—	134	2	no	—
273	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
274	C₄²	D₄×D₆₇	1	1	yes	characteristic
275	C₄²	#11761629	1	1	yes	characteristic
276	C₄²	C₂³×D₆₇	1	1	yes	characteristic
277	C₂³	#14524005	1	1	yes	characteristic
278	Q₈	—	268	4	no	—
279	Q₈	—	67	4	no	—
280	Q₈	—	134	2	no	—
281	Q₈	—	134	2	no	—
282	Q₈	—	134	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	268	4	no	—
287	C₂×C₄	—	536	2	no	—
288	C₂×C₄	—	134	2	no	—
289	C₂×C₄	—	134	2	no	—
290	C₂×C₄	—	134	2	no	—
291	C₂×C₄	—	134	2	no	—
292	C₂×C₄	#14524005	1	1	yes	characteristic
293	C₂×C₄	#14524005	1	1	yes	characteristic
294	C₂×C₄	#14524005	1	1	yes	characteristic
295	C₂×C₄	#14524072	1	4	yes	—
296	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
298	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	4	4	no	—
303	C₈	—	268	2	no	—
304	C₈	—	134	4	no	—
305	C₈	M₁₆×D₆₇	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#16863502	1	2	yes	—
309	C₂²	C₂×#14524005	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	268	2	no	—
314	C₄	—	268	2	no	—
315	C₄	—	268	2	no	—
316	C₄	—	134	4	no	—
317	C₄	#16863502	1	2	yes	—
318	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
319	C₄	#16864162	1	1	yes	characteristic
320	C₄	C₂×#14524072	1	2	yes	—
321	C₂	—	2	2	no	—
322	C₂	#22568944	1	1	yes	characteristic
323	C₂	C₂×#16863502	1	1	yes	characteristic
324	C₂	#22569125	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1566	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1566	D₆₇
hypocenter	C₆₇	#18042
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. #22559117	C₂	semidirect
4. #22559079	C₂	semidirect
5. #22559027	C₂	semidirect
84. Dic₂₆₈	M₁₆	semidirect
171. C₆₇	#18042	semidirect

Maximal subgroups

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

subgroup	index
#22569160	2
C₂×C₄×Dic₂₆₈	2
#22559117	2
#22559079	2
#22559027	2
#22558637	2
#22558498	2
C₆₇×#1566	2
#18042	67