Subgroups — #30761544

Order 17152.

642 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863167	C₂	1	2	yes	maximal
2	C₂×#16863161	C₂	1	1	yes	maximal, characteristic
3	#22559926	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863152	C₂	1	2	yes	maximal
5	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂×#14524002	C₂²	1	2	yes	—
8	#16863167	—	2	4	no	—
9	#16863161	—	2	2	no	—
10	#16863159	—	2	2	no	—
11	#16863152	C₄	1	8	yes	complemented
12	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
15	#405×C₆₇	—	2	4	no	—
16	C₂×#97×C₆₇	C₄	1	2	yes	complemented
17	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	complemented
18	C₂×#11761612	—	2	2	no	—
19	C₂×#11761612	—	2	2	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	C₂×C₄	1	4	yes	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	#14524002	—	2	2	no	—
26	#14524002	—	2	2	no	—
27	#14524002	—	2	2	no	—
28	#14524002	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
31	C₂×C₄×Dic₆₇	—	2	2	no	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₆₇	—	2	1	no	—
34	C₂×M₁₆×C₆₇	—	2	1	no	—
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
37	C₂²×C₈×C₆₇	—	2	2	no	—
38	C₂²×C₈×C₆₇	—	2	2	no	—
39	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	D₄	1	2	yes	—
43	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
44	#117×C₆₇	—	2	2	no	—
45	#117×C₆₇	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
47	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
49	#97×C₆₇	—	2	4	no	—
50	C₄×C₈×C₆₇	—	2	4	no	—
51	C₂²×Dic₆₇	—	4	2	no	—
52	#11761612	—	2	4	no	—
53	#11761612	—	2	4	no	—
54	#11761612	—	4	2	no	—
55	#11761612	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	2	4	no	—
57	C₂×C₆₇⋊C₈	—	4	2	no	—
58	C₂×C₆₇⋊C₈	—	4	2	no	—
59	C₄⋊Dic₆₇	—	2	4	no	—
60	C₄⋊Dic₆₇	—	4	2	no	—
61	C₂²⋊Dic₆₇	—	2	4	no	—
62	C₂²⋊Dic₆₇	—	4	2	no	—
63	C₄×Dic₆₇	—	2	4	no	—
64	C₄×Dic₆₇	—	4	2	no	—
65	C₂⁴×C₆₇	#41	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	#41	1	2	yes	—
73	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	#41	1	2	yes	—
76	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
77	M₁₆×C₆₇	—	2	4	no	—
78	M₁₆×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	—	2	1	no	—
80	C₂×C₈×C₆₇	—	2	1	no	—
81	C₂×C₈×C₆₇	—	2	1	no	—
82	C₂×C₈×C₆₇	—	2	1	no	—
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₂×C₈×C₆₇	—	2	4	no	—
87	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
88	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₂²⋊C₄×C₆₇	—	2	4	no	—
92	C₄²×C₆₇	—	2	2	no	—
93	C₄²×C₆₇	C₄²	1	4	yes	—
94	C₂³×C₆₇	—	4	2	no	—
95	C₂³×C₆₇	#117	1	1	yes	characteristic
96	C₂³×C₆₇	C₄×D₄	1	2	yes	—
97	C₂³×C₆₇	#124	1	1	yes	characteristic
98	C₂³×C₆₇	#125	1	1	yes	characteristic
99	C₂³×C₆₇	#126	1	2	yes	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#117	1	2	yes	—
106	C₂×C₄×C₆₇	#117	1	2	yes	—
107	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₆₇	#121	1	2	yes	—
114	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₆₇	#123	1	2	yes	—
116	C₂×C₄×C₆₇	#123	1	2	yes	—
117	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
119	C₂×C₄×C₆₇	#126	1	2	yes	—
120	C₂×Dic₆₇	—	4	4	no	—
121	C₂×Dic₆₇	—	8	4	no	—
122	C₈×C₆₇	—	2	2	no	—
123	C₈×C₆₇	—	2	2	no	—
124	C₈×C₆₇	—	2	4	no	—
125	C₆₇⋊C₈	—	4	4	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	C₄×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₆₇	#382	1	1	yes	characteristic
133	C₂²×C₆₇	#383	1	1	yes	characteristic
134	C₂²×C₆₇	#387	1	2	yes	—
135	C₂²×C₆₇	#388	1	2	yes	—
136	C₄×C₆₇	—	4	4	no	—
137	C₄×C₆₇	#376	1	1	yes	characteristic
138	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₆₇	#380	1	1	yes	characteristic
140	C₄×C₆₇	#381	1	1	yes	characteristic
141	C₄×C₆₇	#387	1	2	yes	—
142	C₄×C₆₇	#388	1	2	yes	—
143	Dic₆₇	—	8	4	no	—
144	#10817	—	67	1	no	maximal, Sylow
145	C₂×C₆₇	—	2	2	no	—
146	C₂×C₆₇	—	2	1	no	—
147	C₂×C₆₇	—	2	1	no	—
148	C₂×C₆₇	—	2	2	no	—
149	C₂×C₆₇	#1434	1	1	yes	characteristic
150	C₂×C₆₇	#1833	1	2	yes	—
151	C₂×#405	D₆₇	1	1	yes	characteristic
152	C₂×#344	—	67	2	no	—
153	C₂×#341	—	67	1	no	—
154	#1735	—	67	1	no	—
155	C₂×H(3,ℤ/4)	—	67	2	no	—
156	C₆₇	#10817	1	1	yes	characteristic, complemented, Sylow
157	C₂²×M₁₆	—	67	2	no	—
158	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
159	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
160	C₂×#117	—	67	2	no	—
161	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
162	#405	—	2	4	no	—
163	C₂×#97	Dic₆₇	1	2	yes	complemented
164	C₂×C₄×C₈	Dic₆₇	1	2	yes	complemented
165	#344	—	134	4	no	—
166	#341	—	134	2	no	—
167	#340	—	134	2	no	—
168	H(3,ℤ/4)	—	67	8	no	—
169	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	67	4	no	—
174	C₂×M₁₆	—	134	2	no	—
175	C₂×M₁₆	—	134	2	no	—
176	C₂×M₁₆	—	134	2	no	—
177	C₂×M₁₆	—	134	2	no	—
178	C₂×M₁₆	—	134	2	no	—
179	C₂×M₁₆	C₂²⋊D₆₇	1	4	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	134	2	no	—
183	C₂²×C₈	Dic₁₃₄	1	2	yes	—
184	C₂²×C₈	D₂₆₈	1	2	yes	—
185	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
186	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
187	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
188	#117	—	2	2	no	—
189	#117	—	67	4	no	—
190	#117	—	134	2	no	—
191	#117	—	134	2	no	—
192	#117	—	134	2	no	—
193	#117	C₄×D₆₇	1	4	yes	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	134	2	no	—
196	C₂×C₂²⋊C₄	—	2	1	no	—
197	C₂×C₂²⋊C₄	—	134	2	no	—
198	C₂×C₄²	—	134	2	no	—
199	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
200	#97	—	2	4	no	—
201	C₄×C₈	—	2	4	no	—
202	C₂⁴	#11761628	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	268	2	no	—
205	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
206	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
208	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
209	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
210	C₂²×C₄	#11761628	1	2	yes	—
211	C₂²×C₄	#11761628	1	1	yes	characteristic
212	C₂²×C₄	#11761628	1	1	yes	characteristic
213	C₂²×C₄	#11761628	1	2	yes	—
214	C₂²×C₄	#11761628	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	268	2	no	—
218	M₁₆	—	268	2	no	—
219	M₁₆	—	134	4	no	—
220	M₁₆	—	134	4	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	268	2	no	—
230	C₂×C₈	—	268	2	no	—
231	C₂×C₈	—	134	4	no	—
232	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
233	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
234	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	268	2	no	—
237	C₄⋊C₄	—	134	4	no	—
238	C₂²⋊C₄	—	2	4	no	—
239	C₂²⋊C₄	—	268	2	no	—
240	C₂²⋊C₄	—	134	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	268	2	no	—
243	C₄²	—	134	4	no	—
244	C₄²	C₄×Dic₆₇	1	4	yes	—
245	C₂³	—	4	2	no	—
246	C₂³	#14524002	1	1	yes	characteristic
247	C₂³	C₄×C₂²⋊D₆₇	1	2	yes	—
248	C₂³	#14524031	1	1	yes	characteristic
249	C₂³	#14524035	1	1	yes	characteristic
250	C₂³	#14524037	1	2	yes	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	268	4	no	—
254	C₂×C₄	—	536	4	no	—
255	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
258	C₂×C₄	#14524000	1	2	yes	—
259	C₂×C₄	#14524000	1	2	yes	—
260	C₂×C₄	#14524002	1	1	yes	characteristic
261	C₂×C₄	#14524002	1	1	yes	characteristic
262	C₂×C₄	#14524002	1	1	yes	characteristic
263	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
264	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
265	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
266	C₂×C₄	#14524017	1	2	yes	—
267	C₂×C₄	#14524022	1	2	yes	—
268	C₂×C₄	#14524026	1	2	yes	—
269	C₂×C₄	#14524026	1	2	yes	—
270	C₂×C₄	#14524031	1	1	yes	characteristic
271	C₂×C₄	#14524035	1	1	yes	characteristic
272	C₂×C₄	#14524037	1	2	yes	—
273	C₈	—	2	2	no	—
274	C₈	—	2	2	no	—
275	C₈	—	2	4	no	—
276	C₈	—	268	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
283	C₂²	#16863262	1	1	yes	characteristic
284	C₂²	#16863264	1	1	yes	characteristic
285	C₂²	#16863280	1	2	yes	—
286	C₂²	#16863287	1	2	yes	—
287	C₄	—	4	4	no	—
288	C₄	—	536	4	no	—
289	C₄	#16863242	1	1	yes	characteristic
290	C₄	C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
291	C₄	#16863256	1	1	yes	characteristic
292	C₄	#16863259	1	1	yes	characteristic
293	C₄	#16863280	1	2	yes	—
294	C₄	#16863287	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	#22558436	1	1	yes	characteristic
300	C₂	#22560333	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524002
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#405×C₆₇	C₂
socle	C₂²×C₆₇	#382
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#405	D₆₇
hypocenter	C₆₇	#10817
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
11. #16863152	C₄	semidirect
16. C₂×#97×C₆₇	C₄	semidirect
17. C₂×C₄×C₈×C₆₇	C₄	semidirect
156. C₆₇	#10817	semidirect
163. C₂×#97	Dic₆₇	semidirect
164. C₂×C₄×C₈	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863167	2
C₂×#16863161	2
#22559926	2
C₂×#16863152	2
C₂×#405×C₆₇	2
#10817	67