Subgroups — #30761528

Order 17152.

658 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863351	C₂	1	1	yes	maximal, characteristic
2	#22559941	C₂	1	2	yes	maximal
3	#22559926	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863152	C₂	1	2	yes	maximal
5	C₆₇×#1735	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	1	yes	characteristic
7	C₂×#14524002	C₂²	1	2	yes	—
8	C₂×#14524002	C₂²	1	1	yes	characteristic
9	#16863351	—	2	2	no	—
10	#16863351	—	2	2	no	—
11	C₂×#14523953	C₄	1	2	yes	complemented
12	C₂×C₈×Dic₆₇	C₄	1	2	yes	complemented
13	#16863162	—	2	4	no	—
14	#16863159	—	2	2	no	—
15	#16863152	C₄	1	8	yes	complemented
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#117×C₆₇	C₂²	1	2	yes	—
18	#340×C₆₇	—	2	2	no	—
19	C₂×#11761612	—	2	1	no	—
20	C₂×#11761612	—	2	1	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	C₂×C₄	1	4	yes	—
23	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
24	#14524002	—	2	1	no	—
25	#14524002	—	2	1	no	—
26	#14524002	—	2	1	no	—
27	#14524002	—	2	1	no	—
28	#14524002	—	2	2	no	—
29	#14524002	—	2	2	no	—
30	#14524002	—	2	2	no	—
31	#14524002	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊Dic₆₇	—	2	2	no	—
33	C₂×C₄⋊Dic₆₇	Q₈	1	2	yes	—
34	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
35	C₂×C₂²⋊Dic₆₇	D₄	1	2	yes	—
36	C₂×C₄×Dic₆₇	—	2	2	no	—
37	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
38	#14523953	—	2	4	no	—
39	C₈×Dic₆₇	—	2	4	no	—
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	—	2	2	no	—
42	C₂²×C₈×C₆₇	—	2	2	no	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	D₄	1	2	yes	—
46	C₂²×C₈×C₆₇	D₄	1	2	yes	—
47	#117×C₆₇	—	2	2	no	—
48	#117×C₆₇	—	2	2	no	—
49	#117×C₆₇	—	2	2	no	—
50	#117×C₆₇	C₂×C₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₄²×C₆₇	—	2	2	no	—
54	C₂²×Dic₆₇	—	4	2	no	—
55	C₂²×Dic₆₇	C₄⋊C₄	1	4	yes	—
56	#11761612	—	2	4	no	—
57	#11761612	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	2	2	no	—
59	C₂×C₆₇⋊C₈	—	2	2	no	—
60	C₂×C₆₇⋊C₈	C₄²	1	4	yes	—
61	C₄⋊Dic₆₇	—	2	2	no	—
62	C₄⋊Dic₆₇	—	2	2	no	—
63	C₄⋊Dic₆₇	—	2	4	no	—
64	C₄⋊Dic₆₇	—	4	2	no	—
65	C₂²⋊Dic₆₇	—	2	2	no	—
66	C₂²⋊Dic₆₇	—	2	2	no	—
67	C₂²⋊Dic₆₇	—	2	4	no	—
68	C₂²⋊Dic₆₇	—	4	2	no	—
69	C₄×Dic₆₇	—	2	2	no	—
70	C₄×Dic₆₇	—	2	2	no	—
71	C₄×Dic₆₇	—	2	4	no	—
72	C₄×Dic₆₇	—	4	2	no	—
73	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	—	4	2	no	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	#41	1	2	yes	—
82	C₂²×C₄×C₆₇	#41	1	2	yes	—
83	C₂²×C₄×C₆₇	#41	1	2	yes	—
84	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
85	C₂×C₈×C₆₇	—	2	1	no	—
86	C₂×C₈×C₆₇	—	2	1	no	—
87	C₂×C₈×C₆₇	—	2	1	no	—
88	C₂×C₈×C₆₇	—	2	1	no	—
89	C₂×C₈×C₆₇	—	2	4	no	—
90	C₂×C₈×C₆₇	—	2	4	no	—
91	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
92	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
93	C₄⋊C₄×C₆₇	—	2	4	no	—
94	C₄⋊C₄×C₆₇	—	4	2	no	—
95	C₂²⋊C₄×C₆₇	—	2	4	no	—
96	C₂²⋊C₄×C₆₇	—	4	2	no	—
97	C₄²×C₆₇	—	2	4	no	—
98	C₄²×C₆₇	—	4	2	no	—
99	C₂³×C₆₇	—	4	2	no	—
100	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₆₇	C₄×D₄	1	2	yes	—
102	C₂³×C₆₇	#121	1	1	yes	characteristic
103	C₂³×C₆₇	#121	1	1	yes	characteristic
104	C₂³×C₆₇	#124	1	2	yes	—
105	C₂×C₄×C₆₇	—	4	4	no	—
106	C₂×C₄×C₆₇	—	8	4	no	—
107	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#117	1	2	yes	—
111	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#117	1	2	yes	—
114	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#123	1	2	yes	—
121	C₂×C₄×C₆₇	#123	1	2	yes	—
122	C₂×C₄×C₆₇	#125	1	2	yes	—
123	C₂×C₄×C₆₇	#126	1	2	yes	—
124	C₂×C₄×C₆₇	#126	1	2	yes	—
125	C₂×Dic₆₇	—	2	4	no	—
126	C₂×Dic₆₇	—	4	4	no	—
127	C₂×Dic₆₇	—	4	4	no	—
128	C₂×Dic₆₇	—	8	4	no	—
129	C₈×C₆₇	—	2	2	no	—
130	C₈×C₆₇	—	2	2	no	—
131	C₆₇⋊C₈	—	2	4	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	—	4	2	no	—
136	C₂²×C₆₇	—	4	2	no	—
137	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₆₇	#380	1	1	yes	characteristic
139	C₂²×C₆₇	#382	1	1	yes	characteristic
140	C₂²×C₆₇	#388	1	2	yes	—
141	C₂²×C₆₇	#388	1	2	yes	—
142	C₄×C₆₇	—	8	4	no	—
143	C₄×C₆₇	#376	1	1	yes	characteristic
144	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₄×C₆₇	#381	1	1	yes	characteristic
146	C₄×C₆₇	#383	1	1	yes	characteristic
147	C₄×C₆₇	#387	1	2	yes	—
148	C₄×C₆₇	#387	1	2	yes	—
149	Dic₆₇	—	4	4	no	—
150	Dic₆₇	—	8	4	no	—
151	#10814	—	67	1	no	maximal, Sylow
152	C₂×C₆₇	—	2	2	no	—
153	C₂×C₆₇	—	2	2	no	—
154	C₂×C₆₇	—	2	1	no	—
155	C₂×C₆₇	—	2	1	no	—
156	C₂×C₆₇	#1434	1	1	yes	characteristic
157	C₂×C₆₇	#1812	1	2	yes	—
158	C₂×#405	—	67	1	no	—
159	#1740	—	67	2	no	—
160	#1735	—	67	1	no	—
161	#1735	D₆₇	1	1	yes	characteristic
162	C₂×H(3,ℤ/4)	—	67	2	no	—
163	C₆₇	#10814	1	1	yes	characteristic, complemented, Sylow
164	C₂²×M₁₆	—	67	1	no	—
165	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
166	C₂×#117	—	67	1	no	—
167	C₂×#117	—	67	2	no	—
168	C₂×#117	C₂×D₆₇	1	2	yes	—
169	#405	—	134	2	no	—
170	#405	—	134	2	no	—
171	C₂×#97	—	67	2	no	—
172	C₂×C₄×C₈	—	67	2	no	—
173	#342	—	134	4	no	—
174	#340	—	2	2	no	—
175	#340	—	134	2	no	—
176	H(3,ℤ/4)	—	67	8	no	—
177	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
178	C₂×M₁₆	—	67	4	no	—
179	C₂×M₁₆	—	134	1	no	—
180	C₂×M₁₆	—	134	1	no	—
181	C₂×M₁₆	—	134	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	67	2	no	—
185	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
187	C₂²×C₈	D₂₆₈	1	2	yes	—
188	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	67	4	no	—
193	#117	—	134	2	no	—
194	#117	—	134	2	no	—
195	#117	—	134	2	no	—
196	#117	—	134	1	no	—
197	#117	—	134	1	no	—
198	#117	—	134	1	no	—
199	#117	—	134	1	no	—
200	#117	C₄×D₆₇	1	4	yes	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	67	2	no	—
203	C₂×C₄⋊C₄	—	134	2	no	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₂²⋊C₄	—	67	2	no	—
206	C₂×C₂²⋊C₄	—	134	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	67	2	no	—
209	C₂×C₄²	—	134	2	no	—
210	#97	—	134	4	no	—
211	C₄×C₈	—	134	4	no	—
212	C₂⁴	D₄×D₆₇	1	1	yes	characteristic
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	268	2	no	—
215	C₂²×C₄	—	67	4	no	—
216	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
218	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
220	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
221	C₂²×C₄	#11761627	1	2	yes	—
222	C₂²×C₄	#11761627	1	1	yes	characteristic
223	C₂²×C₄	#11761627	1	2	yes	—
224	C₂²×C₄	#11761628	1	2	yes	—
225	C₂²×C₄	#11761629	1	1	yes	characteristic
226	M₁₆	—	134	4	no	—
227	M₁₆	—	134	4	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	67	4	no	—
235	C₂×C₈	—	134	2	no	—
236	C₂×C₈	—	134	2	no	—
237	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
238	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	268	2	no	—
242	C₄⋊C₄	—	134	4	no	—
243	C₄⋊C₄	—	134	2	no	—
244	C₄⋊C₄	—	134	2	no	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	268	2	no	—
248	C₂²⋊C₄	—	134	4	no	—
249	C₂²⋊C₄	—	134	2	no	—
250	C₂²⋊C₄	—	134	2	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	4	2	no	—
253	C₄²	—	268	2	no	—
254	C₄²	—	134	4	no	—
255	C₄²	—	134	2	no	—
256	C₄²	—	134	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
259	C₂³	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
260	C₂³	#14524014	1	1	yes	characteristic
261	C₂³	#14524018	1	1	yes	characteristic
262	C₂³	#14524029	1	2	yes	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	—	268	4	no	—
266	C₂×C₄	—	268	4	no	—
267	C₂×C₄	—	536	4	no	—
268	C₂×C₄	—	134	4	no	—
269	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
270	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
271	C₂×C₄	#14523999	1	1	yes	characteristic
272	C₂×C₄	#14523999	1	2	yes	—
273	C₂×C₄	#14523999	1	1	yes	characteristic
274	C₂×C₄	#14524000	1	1	yes	characteristic
275	C₂×C₄	#14524001	1	2	yes	—
276	C₂×C₄	#14524001	1	1	yes	characteristic
277	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
278	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
279	C₂×C₄	#14524008	1	2	yes	—
280	C₂×C₄	#14524019	1	1	yes	characteristic
281	C₂×C₄	#14524023	1	1	yes	characteristic
282	C₂×C₄	#14524024	1	2	yes	—
283	C₂×C₄	#14524027	1	2	yes	—
284	C₂×C₄	#14524033	1	2	yes	—
285	C₂×C₄	#14524036	1	2	yes	—
286	C₂×C₄	#14524038	1	2	yes	—
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	134	4	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	#16863247	1	1	yes	characteristic
296	C₂²	#16863253	1	1	yes	characteristic
297	C₂²	#16863260	1	1	yes	characteristic
298	C₂²	#16863284	1	2	yes	—
299	C₂²	#16863288	1	2	yes	—
300	C₄	—	8	4	no	—
301	C₄	—	268	4	no	—
302	C₄	—	536	4	no	—
303	C₄	#16863241	1	1	yes	characteristic
304	C₄	C₄×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
305	C₄	#16863257	1	1	yes	characteristic
306	C₄	#16863263	1	1	yes	characteristic
307	C₄	#16863277	1	2	yes	—
308	C₄	#16863281	1	2	yes	—
309	C₂	—	2	2	no	—
310	C₂	—	2	2	no	—
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	#22558433	1	1	yes	characteristic
314	C₂	#22560252	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524001
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₆₇×#1735	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	#1735	D₆₇
hypocenter	C₆₇	#10814
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1735

Complements

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

N	H ≅ G/N	type
11. C₂×#14523953	C₄	semidirect
12. C₂×C₈×Dic₆₇	C₄	semidirect
15. #16863152	C₄	semidirect
163. C₆₇	#10814	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863351	2
#22559941	2
#22559926	2
C₂×#16863152	2
C₆₇×#1735	2
#10814	67