Subgroups — #30761530

Order 17152.

658 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	C₂×#16863352	C₂	1	1	yes	maximal, characteristic
2	#22559942	C₂	1	1	yes	maximal, characteristic
3	#22559925	C₂	1	2	yes	maximal
4	C₂×#16863152	C₂	1	2	yes	maximal
5	C₆₇×#1740	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	1	yes	characteristic
7	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#14524002	C₂²	1	2	yes	—
9	#16863352	—	2	4	no	—
10	C₂×#14523954	C₄	1	2	yes	complemented
11	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	complemented
12	#16863163	—	2	2	no	—
13	#16863158	—	2	4	no	—
14	#16863152	C₄	1	8	yes	complemented
15	C₂×#117×C₆₇	C₂²	1	2	yes	—
16	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
17	#342×C₆₇	—	2	2	no	—
18	C₂×#11761612	—	2	2	no	—
19	C₂×#11761612	—	2	1	no	—
20	C₂×#11761612	—	2	1	no	—
21	C₂×#11761612	C₂×C₄	1	4	yes	—
22	C₂²×C₆₇⋊C₈	—	2	1	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₂²×C₆₇⋊C₈	—	2	1	no	—
25	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₆₇⋊C₈	D₄	1	4	yes	—
29	#14524002	—	2	2	no	—
30	#14524002	—	2	2	no	—
31	#14524002	—	2	2	no	—
32	#14524002	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊Dic₆₇	—	2	2	no	—
34	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
35	C₂×C₄×Dic₆₇	—	2	2	no	—
36	#14523954	—	2	4	no	—
37	C₄×C₆₇⋊C₈	—	2	4	no	—
38	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	#117×C₆₇	—	2	2	no	—
40	#117×C₆₇	—	2	2	no	—
41	#117×C₆₇	—	2	2	no	—
42	#117×C₆₇	—	2	2	no	—
43	#117×C₆₇	—	2	2	no	—
44	#117×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₇	Q₈	1	2	yes	—
47	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
49	C₂×C₄²×C₆₇	—	2	2	no	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
51	C₂²×Dic₆₇	—	4	2	no	—
52	#11761612	—	2	4	no	—
53	#11761612	—	2	4	no	—
54	C₂×C₆₇⋊C₈	—	2	2	no	—
55	C₂×C₆₇⋊C₈	—	2	1	no	—
56	C₂×C₆₇⋊C₈	—	2	1	no	—
57	C₂×C₆₇⋊C₈	—	2	1	no	—
58	C₂×C₆₇⋊C₈	—	2	1	no	—
59	C₂×C₆₇⋊C₈	—	2	2	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₂×C₆₇⋊C₈	—	2	4	no	—
62	C₂×C₆₇⋊C₈	C₄²	1	4	yes	—
63	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
64	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
65	C₄⋊Dic₆₇	—	2	4	no	—
66	C₄⋊Dic₆₇	—	4	2	no	—
67	C₂²⋊Dic₆₇	—	2	4	no	—
68	C₂²⋊Dic₆₇	—	4	2	no	—
69	C₄×Dic₆₇	—	2	4	no	—
70	C₄×Dic₆₇	—	4	2	no	—
71	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	—	4	2	no	—
73	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	2	yes	—
79	C₂²×C₄×C₆₇	#41	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	1	yes	characteristic
83	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
84	C₄⋊C₄×C₆₇	—	2	4	no	—
85	C₄⋊C₄×C₆₇	—	2	4	no	—
86	C₄⋊C₄×C₆₇	—	4	2	no	—
87	C₂²⋊C₄×C₆₇	—	2	4	no	—
88	C₂²⋊C₄×C₆₇	—	2	4	no	—
89	C₂²⋊C₄×C₆₇	—	4	2	no	—
90	C₄²×C₆₇	—	2	2	no	—
91	C₄²×C₆₇	—	2	2	no	—
92	C₄²×C₆₇	—	2	4	no	—
93	C₄²×C₆₇	—	4	2	no	—
94	C₂³×C₆₇	—	4	2	no	—
95	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₆₇	C₄×D₄	1	2	yes	—
97	C₂³×C₆₇	#121	1	2	yes	—
98	C₂³×C₆₇	#124	1	1	yes	characteristic
99	C₂³×C₆₇	#124	1	1	yes	characteristic
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	—	4	4	no	—
103	C₂×C₄×C₆₇	—	8	4	no	—
104	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	#117	1	2	yes	—
111	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₆₇	#123	1	2	yes	—
117	C₂×C₄×C₆₇	#123	1	2	yes	—
118	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
119	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#126	1	2	yes	—
121	C₂×C₄×C₆₇	#126	1	2	yes	—
122	C₂×Dic₆₇	—	4	4	no	—
123	C₂×Dic₆₇	—	8	4	no	—
124	C₆₇⋊C₈	—	2	4	no	—
125	C₆₇⋊C₈	—	2	2	no	—
126	C₆₇⋊C₈	—	2	2	no	—
127	C₂²×C₆₇	—	2	4	no	—
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	—	4	2	no	—
130	C₂²×C₆₇	—	4	1	no	—
131	C₂²×C₆₇	—	4	1	no	—
132	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₆₇	#380	1	1	yes	characteristic
134	C₂²×C₆₇	#382	1	1	yes	characteristic
135	C₂²×C₆₇	#388	1	2	yes	—
136	C₂²×C₆₇	#388	1	2	yes	—
137	C₄×C₆₇	—	4	4	no	—
138	C₄×C₆₇	—	8	4	no	—
139	C₄×C₆₇	#376	1	1	yes	characteristic
140	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
141	C₄×C₆₇	#381	1	1	yes	characteristic
142	C₄×C₆₇	#383	1	1	yes	characteristic
143	C₄×C₆₇	#387	1	2	yes	—
144	C₄×C₆₇	#387	1	2	yes	—
145	Dic₆₇	—	8	4	no	—
146	#10814	—	67	1	no	maximal, Sylow
147	C₂×C₆₇	—	2	2	no	—
148	C₂×C₆₇	—	2	2	no	—
149	C₂×C₆₇	—	2	1	no	—
150	C₂×C₆₇	—	2	1	no	—
151	C₂×C₆₇	#1434	1	1	yes	characteristic
152	C₂×C₆₇	#1812	1	2	yes	—
153	C₂×#405	—	67	1	no	—
154	#1740	—	67	1	no	—
155	#1740	D₆₇	1	1	yes	characteristic
156	#1735	—	67	2	no	—
157	C₂×H(3,ℤ/4)	—	67	2	no	—
158	C₆₇	#10814	1	1	yes	characteristic, complemented, Sylow
159	C₂²×M₁₆	—	67	1	no	—
160	C₂³×C₈	—	67	1	no	—
161	C₂×#117	—	67	2	no	—
162	C₂×#117	C₂×D₆₇	1	2	yes	—
163	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
164	#405	—	134	4	no	—
165	C₂×#97	—	67	2	no	—
166	C₂×C₄×C₈	—	67	2	no	—
167	#342	—	2	2	no	—
168	#342	—	134	2	no	—
169	#340	—	134	4	no	—
170	H(3,ℤ/4)	—	67	8	no	—
171	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
172	C₂×M₁₆	—	67	4	no	—
173	C₂×M₁₆	—	134	2	no	—
174	C₂×M₁₆	—	134	1	no	—
175	C₂×M₁₆	—	134	1	no	—
176	C₂²×C₈	—	67	1	no	—
177	C₂²×C₈	—	67	1	no	—
178	C₂²×C₈	—	67	2	no	—
179	C₂²×C₈	—	67	4	no	—
180	C₂²×C₈	—	134	1	no	—
181	C₂²×C₈	—	134	2	no	—
182	C₂²×C₈	—	134	1	no	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	—	67	4	no	—
189	#117	—	134	2	no	—
190	#117	—	134	2	no	—
191	#117	—	134	2	no	—
192	#117	C₄×D₆₇	1	4	yes	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	134	2	no	—
195	C₂×C₄⋊C₄	Dic₁₃₄	1	2	yes	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₂²⋊C₄	—	134	2	no	—
198	C₂×C₂²⋊C₄	C₂²⋊D₆₇	1	2	yes	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	134	2	no	—
201	C₂×C₄²	C₄×D₆₇	1	2	yes	—
202	#97	—	134	4	no	—
203	C₄×C₈	—	134	4	no	—
204	C₂⁴	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	268	2	no	—
207	C₂²×C₄	Dic₆₇⋊C₄	1	4	yes	—
208	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
210	C₂²×C₄	D₄×D₆₇	1	2	yes	—
211	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
212	C₂²×C₄	#11761627	1	1	yes	characteristic
213	C₂²×C₄	#11761627	1	2	yes	—
214	C₂²×C₄	#11761627	1	1	yes	characteristic
215	C₂²×C₄	#11761628	1	2	yes	—
216	C₂²×C₄	#11761629	1	1	yes	characteristic
217	C₂²×C₄	#11761629	1	1	yes	characteristic
218	M₁₆	—	134	4	no	—
219	M₁₆	—	134	4	no	—
220	C₂×C₈	—	67	4	no	—
221	C₂×C₈	—	67	2	no	—
222	C₂×C₈	—	67	2	no	—
223	C₂×C₈	—	134	2	no	—
224	C₂×C₈	—	134	1	no	—
225	C₂×C₈	—	134	1	no	—
226	C₂×C₈	—	134	1	no	—
227	C₂×C₈	—	134	1	no	—
228	C₂×C₈	—	134	2	no	—
229	C₂×C₈	—	134	4	no	—
230	C₂×C₈	—	134	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	268	2	no	—
235	C₄⋊C₄	—	134	4	no	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	268	2	no	—
240	C₂²⋊C₄	—	134	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	268	2	no	—
246	C₄²	—	134	4	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
249	C₂³	C₄×C₂²⋊D₆₇	1	2	yes	—
250	C₂³	#14524015	1	2	yes	—
251	C₂³	#14524030	1	1	yes	characteristic
252	C₂³	#14524032	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	—	268	4	no	—
258	C₂×C₄	—	536	4	no	—
259	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
260	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
261	C₂×C₄	#14523999	1	2	yes	—
262	C₂×C₄	#14523999	1	1	yes	characteristic
263	C₂×C₄	#14524000	1	1	yes	characteristic
264	C₂×C₄	#14524001	1	1	yes	characteristic
265	C₂×C₄	#14524001	1	2	yes	—
266	C₂×C₄	#14524001	1	1	yes	characteristic
267	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
268	C₂×C₄	#14524005	1	2	yes	—
269	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
270	C₂×C₄	#14524020	1	2	yes	—
271	C₂×C₄	#14524025	1	2	yes	—
272	C₂×C₄	#14524028	1	2	yes	—
273	C₂×C₄	#14524034	1	1	yes	characteristic
274	C₂×C₄	#14524034	1	1	yes	characteristic
275	C₂×C₄	#14524036	1	2	yes	—
276	C₂×C₄	#14524038	1	2	yes	—
277	C₈	—	134	4	no	—
278	C₈	—	134	2	no	—
279	C₈	—	134	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	C₄×Dic₆₇⋊C₄	1	1	yes	characteristic
286	C₂²	#16863254	1	1	yes	characteristic
287	C₂²	#16863261	1	1	yes	characteristic
288	C₂²	#16863286	1	2	yes	—
289	C₂²	#16863290	1	2	yes	—
290	C₄	—	4	4	no	—
291	C₄	—	8	4	no	—
292	C₄	—	536	4	no	—
293	C₄	#16863241	1	1	yes	characteristic
294	C₄	#16863243	1	1	yes	characteristic
295	C₄	#16863258	1	1	yes	characteristic
296	C₄	#16863263	1	1	yes	characteristic
297	C₄	#16863279	1	2	yes	—
298	C₄	#16863283	1	2	yes	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	#22558435	1	1	yes	characteristic
304	C₂	#22560253	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14523999
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₆₇×#1740	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	#1740	D₆₇
hypocenter	C₆₇	#10814
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1740

Complements

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

N	H ≅ G/N	type
10. C₂×#14523954	C₄	semidirect
11. C₂×C₄×C₆₇⋊C₈	C₄	semidirect
14. #16863152	C₄	semidirect
158. 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×#16863352	2
#22559942	2
#22559925	2
C₂×#16863152	2
C₆₇×#1740	2
#10814	67