Subgroups — #30827290

Order 17152.

638 conjugacy classes of subgroups.

The subgroup lattice has 338 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	#22559105	C₂	1	2	yes	maximal, complemented
4	#22559058	C₂	1	4	yes	maximal, complemented
5	#22559015	C₂	1	2	yes	maximal, complemented
6	#22558637	C₂	1	2	yes	maximal
7	C₈×#11761612	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1567	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	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
30	#423×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#335×C₆₇	C₂²	1	2	yes	—
33	#335×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	1	yes	characteristic
40	C₂×#11761612	D₄	1	1	yes	characteristic
41	C₄×Dic₁₃₄	—	2	2	no	—
42	C₄×Dic₁₃₄	—	2	2	no	—
43	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
45	C₄×Dic₁₃₄	C₂³	1	2	yes	—
46	C₄×Dic₁₃₄	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
48	C₂×Dic₆₇⋊C₄	—	2	2	no	—
49	C₂×C₄×Dic₆₇	—	2	2	no	—
50	#14523977	C₂×C₄	1	2	yes	—
51	#14523977	C₂×C₄	1	1	yes	characteristic
52	#14523977	C₂×C₄	1	1	yes	characteristic
53	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
54	Dic₆₇⋊C₈	—	2	4	no	—
55	Dic₆₇⋊C₈	—	2	4	no	—
56	#14523968	C₂×C₄	1	2	yes	—
57	#14523968	C₂×C₄	1	2	yes	—
58	#14523968	C₂×C₄	1	2	yes	—
59	#14523968	C₂×C₄	1	2	yes	—
60	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
61	#14523954	D₄	1	2	yes	—
62	#14523953	—	2	4	no	—
63	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
64	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
65	C₈×Dic₆₇	—	2	4	no	—
66	C₂×M₁₆×C₆₇	—	2	2	no	—
67	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₆₇	—	2	2	no	—
71	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
72	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×Dic₁₃₄	—	2	4	no	—
77	C₂×Dic₁₃₄	—	2	2	no	—
78	C₂×Dic₁₃₄	—	4	2	no	—
79	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
80	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
81	C₂²×Dic₆₇	—	2	2	no	—
82	C₂²×Dic₆₇	—	2	2	no	—
83	Dic₂₆₈	—	2	4	no	—
84	Dic₂₆₈	—	2	2	no	—
85	Dic₂₆₈	M₁₆	1	4	yes	complemented
86	#11761612	—	2	2	no	—
87	#11761612	Q₁₆	1	4	yes	complemented
88	C₂×C₆₇⋊C₈	—	2	2	no	—
89	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
90	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
91	C₄⋊Dic₆₇	—	2	2	no	—
92	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
93	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
94	Dic₆₇⋊C₄	—	2	2	no	—
95	Dic₆₇⋊C₄	—	2	2	no	—
96	Dic₆₇⋊C₄	—	2	2	no	—
97	Dic₆₇⋊C₄	—	2	2	no	—
98	C₄×Dic₆₇	—	2	2	no	—
99	C₄×Dic₆₇	—	2	2	no	—
100	C₄×Dic₆₇	—	2	2	no	—
101	C₄×Dic₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
105	M₁₆×C₆₇	—	4	4	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	4	no	—
108	C₂×C₈×C₆₇	—	4	1	no	—
109	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₇	#41	1	1	yes	characteristic
116	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
117	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	4	1	no	—
121	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
125	C₂×C₄×C₆₇	#135	1	2	yes	—
126	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
129	C₂×Dic₆₇	—	2	2	no	—
130	C₂×Dic₆₇	—	2	2	no	—
131	C₂×Dic₆₇	—	2	2	no	—
132	C₂×Dic₆₇	—	2	2	no	—
133	C₂×Dic₆₇	—	4	4	no	—
134	C₂×Dic₆₇	—	8	2	no	—
135	Dic₁₃₄	—	2	2	no	—
136	Dic₁₃₄	—	2	2	no	—
137	Dic₁₃₄	—	2	2	no	—
138	Dic₁₃₄	—	4	4	no	—
139	Dic₁₃₄	C₂×M₁₆	1	4	yes	—
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	—	4	4	no	—
144	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
145	C₆₇⋊C₈	—	2	2	no	—
146	C₆₇⋊C₈	—	4	2	no	—
147	C₆₇⋊C₈	C₂×Q₁₆	1	4	yes	complemented
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	C₄×Q₁₆	1	2	yes	—
151	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
156	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
157	C₄×C₆₇	#568	1	1	yes	characteristic
158	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
159	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
160	Dic₆₇	—	2	4	no	—
161	Dic₆₇	—	4	2	no	—
162	Dic₆₇	—	4	2	no	—
163	Dic₆₇	—	4	2	no	—
164	M₁₆×Q₁₆	—	67	1	no	maximal, Sylow
165	C₂×C₆₇	—	2	2	no	—
166	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
167	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
168	C₂×C₆₇	#2957	1	1	yes	characteristic
169	Q₈×M₁₆	—	67	2	no	—
170	C₂×C₄×Q₁₆	—	67	1	no	—
171	#1591	—	67	2	no	—
172	#1584	—	67	4	no	—
173	C₈×Q₁₆	—	67	2	no	—
174	#1567	D₆₇	1	1	yes	characteristic
175	#1487	—	67	2	no	—
176	C₈×M₁₆	—	67	1	no	—
177	C₆₇	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	—	67	1	no	—
179	C₂×C₄×Q₈	—	67	2	no	—
180	#446	—	67	4	no	—
181	#446	—	134	4	no	—
182	C₈×Q₈	—	67	4	no	—
183	C₄×Q₁₆	—	67	1	no	—
184	C₄×Q₁₆	—	67	2	no	—
185	C₄×Q₁₆	—	67	1	no	—
186	C₄×Q₁₆	—	67	1	no	—
187	C₄×Q₁₆	—	67	2	no	—
188	C₄×Q₁₆	—	67	1	no	—
189	#431	—	67	1	no	—
190	C₂×#107	—	67	1	no	—
191	#423	—	134	2	no	—
192	#423	C₂×D₆₇	1	2	yes	—
193	C₂×#103	—	67	2	no	—
194	C₄×M₁₆	—	67	1	no	—
195	C₄×M₁₆	—	134	2	no	—
196	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
197	#335	C₂×D₆₇	1	2	yes	—
198	#335	C₂×D₆₇	1	2	yes	—
199	Q₈⋊C₈	—	67	4	no	—
200	Q₈⋊C₈	—	67	4	no	—
201	#322	—	67	2	no	—
202	C₈²	—	67	2	no	—
203	C₂²×Q₈	—	67	2	no	—
204	C₂×Q₁₆	—	67	1	no	—
205	C₂×Q₁₆	—	67	2	no	—
206	C₂×Q₁₆	—	67	1	no	—
207	C₂×Q₁₆	—	134	4	no	—
208	C₂×M₁₆	—	2	2	no	—
209	C₂×M₁₆	—	67	1	no	—
210	C₂×M₁₆	—	67	1	no	—
211	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
212	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
213	C₄×Q₈	—	67	2	no	—
214	C₄×Q₈	—	67	2	no	—
215	C₄×Q₈	—	67	2	no	—
216	C₄×Q₈	—	67	2	no	—
217	C₄×Q₈	—	134	2	no	—
218	C₄×Q₈	—	134	2	no	—
219	C₂×C₄⋊C₄	—	67	2	no	—
220	C₂×C₄⋊C₄	—	134	2	no	—
221	C₂×C₄²	—	134	2	no	—
222	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
223	#107	—	67	2	no	—
224	#107	—	67	1	no	—
225	#107	—	67	1	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	—	134	4	no	—
228	C₄⋊C₈	—	134	4	no	—
229	C₄⋊C₈	—	134	1	no	—
230	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
231	#103	—	67	2	no	—
232	#103	—	67	2	no	—
233	#103	—	67	2	no	—
234	#103	—	67	2	no	—
235	C₂²⋊C₈	—	134	1	no	—
236	#97	—	67	2	no	—
237	#97	—	134	4	no	—
238	C₄×C₈	—	67	2	no	—
239	C₄×C₈	—	67	2	no	—
240	C₄×C₈	—	134	4	no	—
241	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
242	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
245	C₂×Q₈	—	268	2	no	—
246	C₂×Q₈	—	67	2	no	—
247	C₂×Q₈	—	67	2	no	—
248	C₂×Q₈	—	134	4	no	—
249	C₂×Q₈	—	134	2	no	—
250	C₂²×C₄	—	134	2	no	—
251	C₂²×C₄	—	134	2	no	—
252	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
253	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
254	C₂²×C₄	#11761629	1	1	yes	characteristic
255	Q₁₆	—	67	4	no	—
256	Q₁₆	—	134	4	no	—
257	Q₁₆	—	134	2	no	—
258	M₁₆	—	4	4	no	—
259	M₁₆	—	67	4	no	—
260	M₁₆	—	134	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	67	2	no	—
265	C₂×C₈	—	67	2	no	—
266	C₂×C₈	—	134	2	no	—
267	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
271	C₄⋊C₄	—	67	2	no	—
272	C₄⋊C₄	—	67	2	no	—
273	C₄⋊C₄	—	134	2	no	—
274	C₄⋊C₄	—	134	2	no	—
275	C₄⋊C₄	—	134	2	no	—
276	C₄⋊C₄	—	134	2	no	—
277	C₄⋊C₄	—	134	2	no	—
278	C₄²	—	134	2	no	—
279	C₄²	—	134	2	no	—
280	C₄²	—	134	2	no	—
281	C₄²	—	134	2	no	—
282	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
283	C₄²	D₄×D₆₇	1	1	yes	characteristic
284	C₄²	#11761629	1	1	yes	characteristic
285	C₄²	C₂³×D₆₇	1	1	yes	characteristic
286	C₂³	#14524005	1	1	yes	characteristic
287	Q₈	—	268	4	no	—
288	Q₈	—	67	4	no	—
289	Q₈	—	134	2	no	—
290	Q₈	—	134	2	no	—
291	Q₈	—	134	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	268	4	no	—
296	C₂×C₄	—	536	2	no	—
297	C₂×C₄	—	134	2	no	—
298	C₂×C₄	—	134	2	no	—
299	C₂×C₄	—	134	2	no	—
300	C₂×C₄	—	134	2	no	—
301	C₂×C₄	#14524005	1	1	yes	characteristic
302	C₂×C₄	#14524005	1	1	yes	characteristic
303	C₂×C₄	#14524005	1	1	yes	characteristic
304	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
305	C₂×C₄	#14524062	1	2	yes	—
306	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
308	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	4	no	—
313	C₈	—	268	2	no	—
314	C₈	—	67	4	no	—
315	C₈	—	134	2	no	—
316	C₈	M₁₆×D₆₇	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	#16863490	1	2	yes	—
320	C₂²	C₂×#14524005	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	268	2	no	—
325	C₄	—	268	2	no	—
326	C₄	—	268	2	no	—
327	C₄	—	134	4	no	—
328	C₄	#16863490	1	2	yes	—
329	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
330	C₄	#16864162	1	1	yes	characteristic
331	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
332	C₄	C₂×#14524062	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	#22568944	1	1	yes	characteristic
335	C₂	C₂×#16863490	1	1	yes	characteristic
336	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×#1567	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1567	D₆₇
hypocenter	C₆₇	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #22559105	C₂	semidirect
4. #22559058	C₂	semidirect
5. #22559015	C₂	semidirect
85. Dic₂₆₈	M₁₆	semidirect
87. #11761612	Q₁₆	semidirect
147. C₆₇⋊C₈	C₂×Q₁₆	semidirect
177. C₆₇	M₁₆×Q₁₆	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
#22559105	2
#22559058	2
#22559015	2
#22558637	2
C₈×#11761612	2
C₆₇×#1567	2
M₁₆×Q₁₆	67