Subgroups — #30748738

Order 17152.

486 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560247	C₂	1	2	yes	maximal, complemented
2	#22558547	C₂	1	2	yes	maximal
3	#22558086	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523979	—	2	2	no	—
6	#16863360	C₂²	1	4	yes	—
7	#16863161	—	2	2	no	—
8	#16863122	—	2	2	no	—
9	#16863120	—	2	2	no	—
10	C₂³×C₈×C₆₇	C₄	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₈×D₄×C₆₇	—	2	2	no	—
14	C₈×D₄×C₆₇	—	2	2	no	—
15	C₂×C₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
16	C₂×C₂²⋊C₈×C₆₇	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
19	C₂×#11761612	—	2	4	no	—
20	#14523979	—	4	4	no	—
21	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
22	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
23	C₂²×D₄×C₆₇	Q₈	1	1	yes	characteristic
24	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	—	2	1	no	—
27	C₂²×C₈×C₆₇	—	2	1	no	—
28	C₂²×C₈×C₆₇	—	4	2	no	—
29	C₂²×C₈×C₆₇	—	4	2	no	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	D₄	1	2	yes	—
35	C₂²×C₈×C₆₇	D₄	1	2	yes	—
36	C₄×D₄×C₆₇	—	2	2	no	—
37	C₄×D₄×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₆₇	Q₈	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₆₇	—	2	2	no	—
43	C₂²⋊C₈×C₆₇	—	2	2	no	—
44	C₂²⋊C₈×C₆₇	—	2	2	no	—
45	C₄×C₈×C₆₇	—	2	1	no	—
46	C₄×C₈×C₆₇	—	2	1	no	—
47	#11761612	—	4	4	no	—
48	#11761612	—	4	4	no	—
49	C₂×C₆₇⋊C₈	—	4	4	no	—
50	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×D₄×C₆₇	—	2	2	no	—
53	C₂×D₄×C₆₇	—	2	2	no	—
54	C₂×D₄×C₆₇	—	2	2	no	—
55	C₂²×C₄×C₆₇	—	2	1	no	—
56	C₂²×C₄×C₆₇	—	2	1	no	—
57	C₂²×C₄×C₆₇	—	4	1	no	—
58	C₂²×C₄×C₆₇	—	4	1	no	—
59	C₂²×C₄×C₆₇	—	4	1	no	—
60	C₂²×C₄×C₆₇	—	4	1	no	—
61	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	#41	1	2	yes	—
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	2	4	no	—
70	C₂×C₈×C₆₇	—	2	1	no	—
71	C₂×C₈×C₆₇	—	2	1	no	—
72	C₂×C₈×C₆₇	—	4	1	no	—
73	C₂×C₈×C₆₇	—	4	2	no	—
74	C₂×C₈×C₆₇	—	4	2	no	—
75	C₂×C₈×C₆₇	—	4	2	no	—
76	C₂×C₈×C₆₇	—	4	1	no	—
77	C₂×C₈×C₆₇	—	4	1	no	—
78	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₆₇	—	2	2	no	—
82	C₂²⋊C₄×C₆₇	—	2	2	no	—
83	C₂²⋊C₄×C₆₇	—	2	2	no	—
84	C₄²×C₆₇	—	2	1	no	—
85	C₄²×C₆₇	—	2	1	no	—
86	C₂³×C₆₇	—	2	1	no	—
87	C₂³×C₆₇	—	2	1	no	—
88	C₂³×C₆₇	—	2	1	no	—
89	C₂³×C₆₇	—	2	1	no	—
90	C₂³×C₆₇	—	4	1	no	—
91	C₂³×C₆₇	—	4	1	no	—
92	C₂³×C₆₇	—	4	1	no	—
93	C₂³×C₆₇	—	4	1	no	—
94	C₂³×C₆₇	#108	1	2	yes	complemented
95	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
97	D₄×C₆₇	—	2	8	no	—
98	C₂×C₄×C₆₇	—	2	1	no	—
99	C₂×C₄×C₆₇	—	2	1	no	—
100	C₂×C₄×C₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	1	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	4	1	no	—
107	C₂×C₄×C₆₇	—	4	2	no	—
108	C₂×C₄×C₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	—	4	1	no	—
110	C₂×C₄×C₆₇	—	4	1	no	—
111	C₂×C₄×C₆₇	—	4	1	no	—
112	C₂×C₄×C₆₇	—	4	1	no	—
113	C₂×C₄×C₆₇	#108	1	2	yes	complemented
114	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
116	C₂×C₄×C₆₇	#121	1	2	yes	—
117	C₈×C₆₇	—	2	1	no	—
118	C₈×C₆₇	—	2	2	no	—
119	C₈×C₆₇	—	4	1	no	—
120	C₈×C₆₇	—	4	1	no	—
121	C₈×C₆₇	#99	1	2	yes	—
122	C₆₇⋊C₈	—	8	4	no	—
123	C₂²×C₆₇	—	2	1	no	—
124	C₂²×C₆₇	—	2	1	no	—
125	C₂²×C₆₇	—	2	2	no	—
126	C₂²×C₆₇	—	2	2	no	—
127	C₂²×C₆₇	—	2	2	no	—
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	—	4	2	no	—
130	C₂²×C₆₇	—	4	2	no	—
131	C₂²×C₆₇	—	4	2	no	—
132	C₂²×C₆₇	—	4	2	no	—
133	C₂²×C₆₇	—	4	1	no	—
134	C₂²×C₆₇	—	4	1	no	—
135	C₂²×C₆₇	#380	1	1	yes	characteristic
136	C₂²×C₆₇	C₂×#108	1	1	yes	characteristic
137	C₂²×C₆₇	C₂×#108	1	1	yes	characteristic
138	C₄×C₆₇	—	2	2	no	—
139	C₄×C₆₇	—	2	1	no	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	4	1	no	—
142	C₄×C₆₇	—	4	1	no	—
143	C₄×C₆₇	C₂×#99	1	1	yes	characteristic
144	C₄×C₆₇	#410	1	1	yes	characteristic
145	#8611	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	—	2	1	no	—
147	C₂×C₆₇	—	2	1	no	—
148	C₂×C₆₇	—	4	2	no	—
149	C₂×C₆₇	—	4	2	no	—
150	C₂×C₆₇	#1799	1	1	yes	characteristic
151	C₂×C₆₇	#1810	1	1	yes	characteristic
152	C₂×C₆₇	#1813	1	1	yes	characteristic
153	C₂×C₈×D₄	D₆₇	1	1	yes	characteristic
154	#1810	—	67	2	no	—
155	#1465	—	67	2	no	—
156	#1293	—	67	2	no	—
157	C₆₇	#8611	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₈	Dic₆₇	1	1	yes	characteristic
159	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
160	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
161	C₈×D₄	—	2	2	no	—
162	C₈×D₄	—	2	2	no	—
163	C₂×#108	—	134	2	no	—
164	C₂×C₄⋊C₈	Dic₆₇	1	1	yes	characteristic
165	#407	—	67	4	no	—
166	C₂×C₂²⋊C₈	Dic₆₇	1	1	yes	characteristic
167	C₂×C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
168	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
169	#341	—	134	2	no	—
170	#324	—	134	2	no	—
171	#323	—	134	2	no	—
172	C₂²×D₄	Dic₁₃₄	1	1	yes	characteristic
173	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
174	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
175	C₂×M₁₆	—	134	4	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	4	2	no	—
179	C₂²×C₈	—	4	2	no	—
180	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
181	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
182	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
183	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
184	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
185	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
186	C₄×D₄	—	2	2	no	—
187	C₄×D₄	—	2	2	no	—
188	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
189	C₂×C₂²⋊C₄	Dic₁₃₄	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	D₂₆₈	1	1	yes	characteristic
191	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
192	#108	—	268	4	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	2	2	no	—
196	C₂²⋊C₈	—	134	4	no	—
197	C₂²⋊C₈	—	134	4	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₂⁴	C₄⋊Dic₆₇	1	1	yes	characteristic
201	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
212	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
213	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
214	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
216	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
217	C₂²×C₄	#11761628	1	2	yes	—
218	M₁₆	—	268	4	no	—
219	M₁₆	—	268	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	268	4	no	—
231	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
233	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	2	2	no	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	4	1	no	—
244	C₂³	—	4	1	no	—
245	C₂³	—	4	1	no	—
246	C₂³	—	4	1	no	—
247	C₂³	#14523979	1	2	yes	complemented
248	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
249	C₂³	#14524022	1	2	yes	—
250	D₄	—	2	8	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	#14523979	1	2	yes	complemented
267	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
268	C₂×C₄	#14524002	1	1	yes	characteristic
269	C₂×C₄	#14524017	1	2	yes	—
270	C₈	—	2	1	no	—
271	C₈	—	2	2	no	—
272	C₈	—	4	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	536	4	no	—
275	C₈	#14523958	1	2	yes	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	#16863256	1	1	yes	characteristic
289	C₂²	C₂×#14523979	1	1	yes	characteristic
290	C₂²	C₂×#14523979	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	C₂×#14523958	1	1	yes	characteristic
297	C₄	#16863370	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	4	2	no	—
301	C₂	—	4	2	no	—
302	C₂	#22560197	1	1	yes	characteristic
303	C₂	#22560247	1	1	yes	characteristic
304	C₂	#22560259	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863256
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×C₈×D₄×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂²×D₄×C₆₇	Q₈
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×C₈×D₄	D₆₇
hypocenter	C₆₇	#8611
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #22560247	C₂	semidirect
3. #22558086	C₂	semidirect
94. C₂³×C₆₇	#108	semidirect
113. C₂×C₄×C₆₇	#108	semidirect
157. C₆₇	#8611	semidirect
247. C₂³	#14523979	semidirect
266. C₂×C₄	#14523979	semidirect

Maximal subgroups

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

subgroup	index
#22560247	2
#22558547	2
#22558086	2
C₂×C₈×D₄×C₆₇	2
#8611	67