Subgroups — #30827518

Order 17152.

626 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569299	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#22559105	C₂	1	2	yes	maximal, complemented
4	#22559079	C₂	1	4	yes	maximal, complemented
5	#22559015	C₂	1	2	yes	maximal, complemented
6	#22558655	C₂	1	2	yes	maximal
7	C₈×#11761612	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1568	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₆₈	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₃₄	C₂²	1	2	yes	—
11	#16863524	—	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	#16863450	—	2	2	no	—
21	#16863449	C₂²	1	1	yes	characteristic
22	#16863446	—	2	2	no	—
23	C₂×#14523977	C₄	1	1	yes	characteristic
24	C₂×#14523968	C₄	1	2	yes	—
25	C₄×#11761612	C₂²	1	1	yes	characteristic
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	#432×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#335×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	—	2	1	no	—
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)	D₄	1	2	yes	—
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)	D₄	1	1	yes	characteristic
60	C₂²⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
61	#14523954	—	2	1	no	—
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	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	C₂³	1	4	yes	—
70	C₂²⋊C₈×C₆₇	—	2	2	no	—
71	#97×C₆₇	—	2	2	no	—
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	4	no	—
87	C₂×C₆₇⋊C₈	—	2	2	no	—
88	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
89	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
90	C₄⋊Dic₆₇	—	2	2	no	—
91	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
92	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
93	Dic₆₇⋊C₄	—	2	2	no	—
94	Dic₆₇⋊C₄	—	2	2	no	—
95	Dic₆₇⋊C₄	—	2	2	no	—
96	Dic₆₇⋊C₄	—	2	2	no	—
97	C₄×Dic₆₇	—	2	2	no	—
98	C₄×Dic₆₇	—	2	2	no	—
99	C₄×Dic₆₇	—	2	2	no	—
100	C₄×Dic₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	4	no	—
106	C₂×C₈×C₆₇	—	4	1	no	—
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	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₆₇	#41	1	1	yes	characteristic
114	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
115	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
126	C₂×Dic₆₇	—	2	2	no	—
127	C₂×Dic₆₇	—	2	2	no	—
128	C₂×Dic₆₇	—	2	2	no	—
129	C₂×Dic₆₇	—	2	2	no	—
130	C₂×Dic₆₇	—	4	4	no	—
131	C₂×Dic₆₇	—	4	4	no	—
132	Dic₁₃₄	—	2	2	no	—
133	Dic₁₃₄	—	2	2	no	—
134	Dic₁₃₄	—	2	2	no	—
135	Dic₁₃₄	—	4	4	no	—
136	Dic₁₃₄	C₂×M₁₆	1	4	yes	—
137	C₈×C₆₇	—	2	1	no	—
138	C₈×C₆₇	—	2	1	no	—
139	C₈×C₆₇	—	2	1	no	—
140	C₈×C₆₇	—	4	4	no	—
141	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
142	C₆₇⋊C₈	—	2	4	no	—
143	C₆₇⋊C₈	—	2	2	no	—
144	C₆₇⋊C₈	—	2	2	no	—
145	C₂²×C₆₇	—	2	1	no	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	#443	1	2	yes	—
148	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
149	C₄×C₆₇	—	2	1	no	—
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	2	1	no	—
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
154	C₄×C₆₇	#568	1	1	yes	characteristic
155	C₄×C₆₇	#578	1	2	yes	—
156	Dic₆₇	—	2	4	no	—
157	Dic₆₇	—	4	2	no	—
158	Dic₆₇	—	4	2	no	—
159	Dic₆₇	—	4	2	no	—
160	#18058	—	67	1	no	maximal, Sylow
161	C₂×C₆₇	—	2	2	no	—
162	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
163	C₂×C₆₇	#2950	1	1	yes	characteristic
164	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
165	#2984	—	67	2	no	—
166	C₂×C₄×Q₁₆	—	67	1	no	—
167	#1591	—	67	2	no	—
168	#1587	—	67	4	no	—
169	C₈×Q₁₆	—	67	2	no	—
170	#1568	D₆₇	1	1	yes	characteristic
171	#1491	—	67	2	no	—
172	C₈×M₁₆	—	67	1	no	—
173	C₆₇	#18058	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₁₆	—	67	1	no	—
175	C₂×C₄×Q₈	—	67	2	no	—
176	#446	—	67	4	no	—
177	#446	—	134	4	no	—
178	C₈×Q₈	—	67	4	no	—
179	C₄×Q₁₆	—	67	1	no	—
180	C₄×Q₁₆	—	67	2	no	—
181	C₄×Q₁₆	—	67	1	no	—
182	C₄×Q₁₆	—	67	1	no	—
183	C₄×Q₁₆	—	67	2	no	—
184	C₄×Q₁₆	—	67	1	no	—
185	#432	—	134	2	no	—
186	#432	C₂×D₆₇	1	2	yes	—
187	#431	—	67	1	no	—
188	#431	—	134	2	no	—
189	C₂×#107	—	67	1	no	—
190	C₂×#103	—	67	2	no	—
191	C₄×M₁₆	—	67	1	no	—
192	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
193	#335	C₂×D₆₇	1	2	yes	—
194	#334	C₂×D₆₇	1	2	yes	—
195	Q₈⋊C₈	—	67	4	no	—
196	Q₈⋊C₈	—	67	4	no	—
197	#322	—	67	2	no	—
198	C₈²	—	67	2	no	—
199	C₂²×Q₈	—	67	2	no	—
200	C₂×Q₁₆	—	67	1	no	—
201	C₂×Q₁₆	—	67	2	no	—
202	C₂×Q₁₆	—	67	1	no	—
203	C₂×Q₁₆	—	134	4	no	—
204	C₂×M₁₆	—	134	1	no	—
205	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
206	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
207	C₄×Q₈	—	67	2	no	—
208	C₄×Q₈	—	67	2	no	—
209	C₄×Q₈	—	67	2	no	—
210	C₄×Q₈	—	67	2	no	—
211	C₄×Q₈	—	134	2	no	—
212	C₄×Q₈	—	134	2	no	—
213	C₂×C₄⋊C₄	—	67	2	no	—
214	C₂×C₄⋊C₄	—	134	2	no	—
215	C₂×C₄²	—	134	2	no	—
216	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
217	#107	—	67	2	no	—
218	#107	—	67	1	no	—
219	#107	—	67	1	no	—
220	C₄⋊C₈	—	67	2	no	—
221	C₄⋊C₈	—	134	4	no	—
222	C₄⋊C₈	—	134	4	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	—	2	2	no	—
229	C₂²⋊C₈	—	67	1	no	—
230	C₂²⋊C₈	—	67	1	no	—
231	#97	—	2	2	no	—
232	#97	—	134	4	no	—
233	#97	—	134	1	no	—
234	C₄×C₈	—	67	2	no	—
235	C₄×C₈	—	67	2	no	—
236	C₄×C₈	—	134	4	no	—
237	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
238	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
241	C₂×Q₈	—	268	2	no	—
242	C₂×Q₈	—	67	2	no	—
243	C₂×Q₈	—	67	2	no	—
244	C₂×Q₈	—	134	4	no	—
245	C₂×Q₈	—	134	2	no	—
246	C₂²×C₄	—	134	2	no	—
247	C₂²×C₄	—	134	2	no	—
248	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
249	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
250	C₂²×C₄	#11761629	1	1	yes	characteristic
251	Q₁₆	—	67	4	no	—
252	Q₁₆	—	134	4	no	—
253	Q₁₆	—	134	2	no	—
254	M₁₆	—	134	4	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	67	2	no	—
259	C₂×C₈	—	67	2	no	—
260	C₂×C₈	—	134	2	no	—
261	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
262	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
263	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
264	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
265	C₄⋊C₄	—	67	2	no	—
266	C₄⋊C₄	—	67	2	no	—
267	C₄⋊C₄	—	134	2	no	—
268	C₄⋊C₄	—	134	2	no	—
269	C₄⋊C₄	—	134	2	no	—
270	C₄⋊C₄	—	134	2	no	—
271	C₄⋊C₄	—	134	2	no	—
272	C₄²	—	134	2	no	—
273	C₄²	—	134	2	no	—
274	C₄²	—	134	2	no	—
275	C₄²	—	134	2	no	—
276	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
277	C₄²	D₄×D₆₇	1	1	yes	characteristic
278	C₄²	#11761629	1	1	yes	characteristic
279	C₄²	C₂³×D₆₇	1	1	yes	characteristic
280	C₂³	#14524005	1	1	yes	characteristic
281	Q₈	—	268	4	no	—
282	Q₈	—	67	4	no	—
283	Q₈	—	134	2	no	—
284	Q₈	—	134	2	no	—
285	Q₈	—	134	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	268	4	no	—
291	C₂×C₄	—	268	4	no	—
292	C₂×C₄	—	134	2	no	—
293	C₂×C₄	—	134	2	no	—
294	C₂×C₄	—	134	2	no	—
295	C₂×C₄	—	134	2	no	—
296	C₂×C₄	#14524005	1	1	yes	characteristic
297	C₂×C₄	#14524005	1	1	yes	characteristic
298	C₂×C₄	#14524005	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
301	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	4	4	no	—
306	C₈	—	134	4	no	—
307	C₈	—	134	2	no	—
308	C₈	—	134	2	no	—
309	C₈	M₁₆×D₆₇	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	#16863512	1	2	yes	—
313	C₂²	C₂×#14524005	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	268	2	no	—
319	C₄	—	268	2	no	—
320	C₄	—	268	2	no	—
321	C₄	—	134	4	no	—
322	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
323	C₄	#16864162	1	1	yes	characteristic
324	C₄	#16864210	1	2	yes	—
325	C₂	—	2	2	no	—
326	C₂	#22568944	1	1	yes	characteristic
327	C₂	#22569076	1	1	yes	characteristic
328	C₂	C₂×#16863512	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×#1568	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1568	D₆₇
hypocenter	C₆₇	#18058
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #22559105	C₂	semidirect
4. #22559079	C₂	semidirect
5. #22559015	C₂	semidirect
85. Dic₂₆₈	M₁₆	semidirect
173. C₆₇	#18058	semidirect

Maximal subgroups

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

subgroup	index
#22569299	2
C₂×C₄×Dic₂₆₈	2
#22559105	2
#22559079	2
#22559015	2
#22558655	2
C₈×#11761612	2
C₆₇×#1568	2
#18058	67