Subgroups — #30827559

Order 17152.

618 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569296	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#22559119	C₂	1	2	yes	maximal, complemented
4	#22559061	C₂	1	4	yes	maximal, complemented
5	#22559029	C₂	1	2	yes	maximal, complemented
6	#22558972	C₂	1	1	yes	maximal, characteristic
7	#22558657	C₂	1	2	yes	maximal
8	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₆₈	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₃₄	C₂²	1	2	yes	—
11	#16863526	C₂²	1	4	yes	—
12	#16863525	—	2	4	no	—
13	C₈×Dic₁₃₄	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	#16863453	C₂²	1	2	yes	—
21	#16863450	—	2	2	no	—
22	#16863446	—	2	2	no	—
23	C₂×#14523977	C₄	1	1	yes	characteristic
24	C₂×#14523968	C₄	1	2	yes	—
25	#16863149	C₂²	1	2	yes	—
26	#16863147	C₂²	1	2	yes	—
27	#16863128	C₂²	1	4	yes	—
28	#16863128	C₂²	1	4	yes	—
29	#431×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#322×C₆₇	C₂²	1	2	yes	—
33	#322×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	C₄×Dic₁₃₄	—	2	2	no	—
40	C₄×Dic₁₃₄	—	2	2	no	—
41	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₃₄	C₂³	1	2	yes	—
44	C₄×Dic₁₃₄	C₂³	1	2	yes	—
45	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
46	C₂×Dic₆₇⋊C₄	—	2	2	no	—
47	C₂×C₄×Dic₆₇	—	2	2	no	—
48	#14523977	C₂×C₄	1	2	yes	—
49	#14523977	C₂×C₄	1	1	yes	characteristic
50	#14523977	C₂×C₄	1	1	yes	characteristic
51	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
52	Dic₆₇⋊C₈	—	2	4	no	—
53	Dic₆₇⋊C₈	—	2	4	no	—
54	#14523968	C₂×C₄	1	2	yes	—
55	#14523968	C₂×C₄	1	2	yes	—
56	#14523968	C₂×C₄	1	2	yes	—
57	#14523968	C₂×C₄	1	2	yes	—
58	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
59	#14523954	—	2	2	no	—
60	#14523953	—	2	4	no	—
61	C₈×Dic₆₇	—	2	4	no	—
62	C₂×M₁₆×C₆₇	—	2	1	no	—
63	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
67	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
68	#97×C₆₇	—	2	1	no	—
69	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₆₇	C₂³	1	2	yes	—
72	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	2	yes	—
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	C₂×C₆₇⋊C₈	—	2	4	no	—
86	C₄⋊Dic₆₇	—	2	2	no	—
87	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
88	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
89	Dic₆₇⋊C₄	—	2	2	no	—
90	Dic₆₇⋊C₄	—	2	2	no	—
91	Dic₆₇⋊C₄	—	2	2	no	—
92	Dic₆₇⋊C₄	—	2	2	no	—
93	C₄×Dic₆₇	—	2	2	no	—
94	C₄×Dic₆₇	—	2	2	no	—
95	C₄×Dic₆₇	—	2	2	no	—
96	C₄×Dic₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
100	M₁₆×C₆₇	—	4	2	no	—
101	C₂×C₈×C₆₇	—	2	2	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	4	1	no	—
104	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
109	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₇	#41	1	1	yes	characteristic
112	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	2	2	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	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₆₇	—	2	2	no	—
125	C₂×Dic₆₇	—	2	2	no	—
126	C₂×Dic₆₇	—	2	2	no	—
127	C₂×Dic₆₇	—	2	2	no	—
128	C₂×Dic₆₇	—	4	4	no	—
129	C₂×Dic₆₇	—	4	4	no	—
130	Dic₁₃₄	—	2	2	no	—
131	Dic₁₃₄	—	2	2	no	—
132	Dic₁₃₄	—	2	2	no	—
133	Dic₁₃₄	—	4	4	no	—
134	Dic₁₃₄	C₂×M₁₆	1	4	yes	—
135	C₈×C₆₇	—	2	1	no	—
136	C₈×C₆₇	—	2	1	no	—
137	C₈×C₆₇	—	2	1	no	—
138	C₈×C₆₇	—	2	4	no	—
139	C₈×C₆₇	—	4	2	no	—
140	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
141	C₆₇⋊C₈	—	4	4	no	—
142	C₂²×C₆₇	—	2	1	no	—
143	C₂²×C₆₇	—	2	1	no	—
144	C₂²×C₆₇	#444	1	2	yes	—
145	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
146	C₄×C₆₇	—	2	1	no	—
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	—	2	1	no	—
150	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
151	C₄×C₆₇	#568	1	1	yes	characteristic
152	C₄×C₆₇	#578	1	2	yes	—
153	Dic₆₇	—	2	4	no	—
154	Dic₆₇	—	4	2	no	—
155	Dic₆₇	—	4	2	no	—
156	Dic₆₇	—	4	2	no	—
157	#18062	—	67	1	no	maximal, Sylow
158	C₂×C₆₇	—	2	2	no	—
159	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
160	C₂×C₆₇	#2950	1	1	yes	characteristic
161	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
162	#2984	—	67	2	no	—
163	C₂×C₄×Q₁₆	—	67	1	no	—
164	#1593	—	67	2	no	—
165	#1584	—	67	4	no	—
166	#1579	—	67	2	no	—
167	#1568	—	67	1	no	—
168	#1491	—	67	2	no	—
169	#1451	D₆₇	1	1	yes	characteristic
170	C₆₇	#18062	1	1	yes	characteristic, complemented, Sylow
171	C₂²×Q₁₆	—	67	1	no	—
172	C₂×C₄×Q₈	—	67	2	no	—
173	#446	—	67	4	no	—
174	#446	—	134	4	no	—
175	C₈×Q₈	—	67	4	no	—
176	C₄×Q₁₆	—	67	1	no	—
177	C₄×Q₁₆	—	67	2	no	—
178	C₄×Q₁₆	—	67	1	no	—
179	C₄×Q₁₆	—	67	1	no	—
180	C₄×Q₁₆	—	67	2	no	—
181	C₄×Q₁₆	—	67	1	no	—
182	#432	—	67	2	no	—
183	#432	—	134	2	no	—
184	#431	—	134	2	no	—
185	#431	C₂×D₆₇	1	1	yes	characteristic
186	C₂×#107	—	67	1	no	—
187	C₂×#103	—	67	2	no	—
188	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
190	#335	—	67	2	no	—
191	#334	—	67	2	no	—
192	Q₈⋊C₈	—	67	4	no	—
193	Q₈⋊C₈	—	67	4	no	—
194	#322	C₂×D₆₇	1	2	yes	—
195	#322	C₂×D₆₇	1	2	yes	—
196	C₂²×Q₈	—	67	2	no	—
197	C₂×Q₁₆	—	67	1	no	—
198	C₂×Q₁₆	—	67	2	no	—
199	C₂×Q₁₆	—	67	1	no	—
200	C₂×Q₁₆	—	134	4	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
203	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
204	C₄×Q₈	—	67	2	no	—
205	C₄×Q₈	—	67	2	no	—
206	C₄×Q₈	—	67	2	no	—
207	C₄×Q₈	—	67	2	no	—
208	C₄×Q₈	—	134	2	no	—
209	C₄×Q₈	—	134	2	no	—
210	C₂×C₄⋊C₄	—	67	2	no	—
211	C₂×C₄⋊C₄	—	134	2	no	—
212	C₂×C₄²	—	134	2	no	—
213	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
214	#107	—	67	2	no	—
215	#107	—	67	1	no	—
216	#107	—	67	1	no	—
217	C₄⋊C₈	—	67	4	no	—
218	C₄⋊C₈	—	134	4	no	—
219	C₄⋊C₈	—	134	4	no	—
220	C₄⋊C₈	D₂₆₈	1	2	yes	—
221	#103	—	67	2	no	—
222	#103	—	67	2	no	—
223	#103	—	67	2	no	—
224	#103	—	67	2	no	—
225	C₂²⋊C₈	—	134	2	no	—
226	C₂²⋊C₈	D₂₆₈	1	2	yes	—
227	#97	—	2	1	no	—
228	#97	—	134	4	no	—
229	#97	—	134	2	no	—
230	C₄×C₈	—	134	4	no	—
231	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
232	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
233	C₄×C₈	C₂²×D₆₇	1	2	yes	—
234	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₆₇	1	2	yes	—
237	C₂×Q₈	—	268	2	no	—
238	C₂×Q₈	—	67	2	no	—
239	C₂×Q₈	—	67	2	no	—
240	C₂×Q₈	—	134	4	no	—
241	C₂×Q₈	—	134	2	no	—
242	C₂²×C₄	—	134	2	no	—
243	C₂²×C₄	—	134	2	no	—
244	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
246	C₂²×C₄	#11761628	1	1	yes	characteristic
247	Q₁₆	—	67	4	no	—
248	Q₁₆	—	134	4	no	—
249	Q₁₆	—	134	2	no	—
250	M₁₆	—	4	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	134	4	no	—
255	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
257	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
258	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
259	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
260	C₄⋊C₄	—	67	2	no	—
261	C₄⋊C₄	—	67	2	no	—
262	C₄⋊C₄	—	134	2	no	—
263	C₄⋊C₄	—	134	2	no	—
264	C₄⋊C₄	—	134	2	no	—
265	C₄⋊C₄	—	134	2	no	—
266	C₄⋊C₄	—	134	2	no	—
267	C₄²	—	134	2	no	—
268	C₄²	—	134	2	no	—
269	C₄²	—	134	2	no	—
270	C₄²	—	134	2	no	—
271	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
272	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
273	C₄²	#11761628	1	1	yes	characteristic
274	C₄²	C₂³×D₆₇	1	1	yes	characteristic
275	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
276	Q₈	—	268	4	no	—
277	Q₈	—	67	4	no	—
278	Q₈	—	134	2	no	—
279	Q₈	—	134	2	no	—
280	Q₈	—	134	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	268	4	no	—
286	C₂×C₄	—	268	4	no	—
287	C₂×C₄	—	134	2	no	—
288	C₂×C₄	—	134	2	no	—
289	C₂×C₄	—	134	2	no	—
290	C₂×C₄	—	134	2	no	—
291	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
292	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
293	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
296	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	4	no	—
301	C₈	—	4	2	no	—
302	C₈	—	268	4	no	—
303	C₈	M₁₆×D₆₇	1	2	yes	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	#16863519	1	2	yes	—
307	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	268	2	no	—
313	C₄	—	268	2	no	—
314	C₄	—	268	2	no	—
315	C₄	—	134	4	no	—
316	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
317	C₄	#16864162	1	1	yes	characteristic
318	C₄	#16864213	1	2	yes	—
319	C₂	—	2	2	no	—
320	C₂	M₁₆×D₂₆₈	1	1	yes	characteristic
321	C₂	#22569078	1	1	yes	characteristic
322	C₂	C₂×#16863519	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1451	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1451	D₆₇
hypocenter	C₆₇	#18062
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #22559119	C₂	semidirect
4. #22559061	C₂	semidirect
5. #22559029	C₂	semidirect
84. Dic₂₆₈	M₁₆	semidirect
170. C₆₇	#18062	semidirect

Maximal subgroups

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

subgroup	index
#22569296	2
C₂×C₄×Dic₂₆₈	2
#22559119	2
#22559061	2
#22559029	2
#22558972	2
#22558657	2
C₆₇×#1451	2
#18062	67