Subgroups — #30869365

Order 17152.

694 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569150	C₂	1	1	yes	maximal, characteristic, complemented
2	#22568927	C₂	1	1	yes	maximal, characteristic
3	#22561187	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561038	C₂	1	1	yes	maximal, characteristic
5	#22560859	C₂	1	4	yes	maximal, complemented
6	#22560486	C₂	1	4	yes	maximal, complemented
7	#22560384	C₂	1	1	yes	maximal, characteristic, complemented
8	#22559957	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3135	C₂	1	1	yes	maximal, characteristic
10	C₂×#14524050	C₂²	1	2	yes	—
11	#16863527	—	2	4	no	—
12	#16863468	—	2	4	no	—
13	C₂×#14523979	C₂²	1	4	yes	—
14	#16863423	C₂²	1	2	yes	—
15	#16863420	C₂²	1	1	yes	characteristic
16	#16863420	C₂²	1	1	yes	characteristic
17	C₂×#14523970	C₂²	1	4	yes	—
18	#16863363	C₂²	1	2	yes	—
19	#16863352	C₂²	1	2	yes	—
20	C₂×#14523954	C₂²	1	1	yes	characteristic
21	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
22	#16863155	C₂²	1	4	yes	—
23	#16863150	C₂²	1	2	yes	—
24	#16863150	C₂²	1	2	yes	—
25	#533×C₆₇	C₂²	1	1	yes	characteristic
26	#532×C₆₇	C₂²	1	1	yes	characteristic
27	#491×C₆₇	C₄	1	2	yes	complemented
28	#490×C₆₇	C₄	1	2	yes	complemented
29	#489×C₆₇	C₄	1	4	yes	complemented
30	#416×C₆₇	C₂²	1	4	yes	—
31	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
32	#14524050	—	4	4	no	—
33	C₂×#11761612	—	2	2	no	—
34	C₂×#11761612	C₂³	1	2	yes	—
35	C₂×#11761612	C₂³	1	2	yes	—
36	C₂²×C₆₇⋊C₈	—	2	2	no	—
37	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
39	C₂×C₄×Dic₆₇	—	2	2	no	—
40	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
41	#14523979	—	4	4	no	—
42	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
43	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
44	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
45	#14523970	—	4	4	no	—
46	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
47	#14523954	—	2	2	no	—
48	#14523954	D₄	1	4	yes	—
49	C₄×C₆₇⋊C₈	—	2	2	no	—
50	C₂×#41×C₆₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
52	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
54	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
55	#125×C₆₇	C₂×C₄	1	2	yes	—
56	#124×C₆₇	C₂×C₄	1	2	yes	—
57	C₂²⋊Q₈×C₆₇	—	2	2	no	—
58	#121×C₆₇	—	2	2	no	—
59	C₄×Q₈×C₆₇	—	2	2	no	—
60	C₄×D₄×C₆₇	—	2	2	no	—
61	#117×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	#103×C₆₇	C₂×C₄	1	8	yes	—
64	#102×C₆₇	C₂×C₄	1	8	yes	—
65	#97×C₆₇	C₂×C₄	1	2	yes	—
66	#97×C₆₇	C₂×C₄	1	2	yes	—
67	C₂²×Dic₆₇	—	2	2	no	—
68	#11761612	—	2	4	no	—
69	#11761612	—	2	4	no	—
70	#11761612	—	4	4	no	—
71	C₂×C₆₇⋊C₈	—	2	2	no	—
72	C₂×C₆₇⋊C₈	—	2	2	no	—
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	—	2	2	no	—
75	C₂×C₆₇⋊C₈	—	4	4	no	—
76	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
77	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
78	C₄×Dic₆₇	—	2	2	no	—
79	C₄×Dic₆₇	—	2	2	no	—
80	C₄×Dic₆₇	—	4	4	no	—
81	#41×C₆₇	—	2	4	no	—
82	#41×C₆₇	—	2	4	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	#41	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
94	C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₄⋊C₄×C₆₇	—	2	4	no	—
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
98	C₂²⋊C₄×C₆₇	—	2	2	no	—
99	C₂²⋊C₄×C₆₇	—	2	4	no	—
100	C₄²×C₆₇	—	2	2	no	—
101	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₆₇	C₂×D₄	1	2	yes	—
104	C₂³×C₆₇	—	2	2	no	—
105	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
106	Q₈×C₆₇	—	2	4	no	—
107	Q₈×C₆₇	—	8	2	no	—
108	D₄×C₆₇	—	2	4	no	—
109	D₄×C₆₇	—	4	4	no	—
110	D₄×C₆₇	—	8	2	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	—	4	2	no	—
115	C₂×C₄×C₆₇	—	4	2	no	—
116	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
121	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
122	C₂×Dic₆₇	—	2	4	no	—
123	C₂×Dic₆₇	—	4	2	no	—
124	C₂×Dic₆₇	—	4	2	no	—
125	C₈×C₆₇	—	2	4	no	—
126	C₆₇⋊C₈	—	2	2	no	—
127	C₆₇⋊C₈	—	2	2	no	—
128	C₆₇⋊C₈	—	4	2	no	—
129	C₆₇⋊C₈	—	4	2	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	4	2	no	—
132	C₂²×C₆₇	—	4	4	no	—
133	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₆₇	#542	1	2	yes	—
135	C₄×C₆₇	—	2	2	no	—
136	C₄×C₆₇	—	2	2	no	—
137	C₄×C₆₇	—	4	2	no	—
138	C₄×C₆₇	—	4	4	no	—
139	C₄×C₆₇	#520	1	2	yes	—
140	C₄×C₆₇	#535	1	1	yes	characteristic
141	C₄×C₆₇	#536	1	1	yes	characteristic
142	Dic₆₇	—	4	4	no	—
143	#22237	—	67	1	no	maximal, Sylow
144	C₂×C₆₇	—	2	2	no	—
145	C₂×C₆₇	—	8	2	no	—
146	C₂×C₆₇	#2326	1	1	yes	characteristic
147	C₂×C₆₇	#2956	1	2	yes	—
148	#3135	D₆₇	1	1	yes	characteristic
149	#2961	—	67	1	no	—
150	#2932	—	67	1	no	—
151	#1994	—	67	1	no	—
152	#1962	—	67	1	no	—
153	#1929	—	67	4	no	—
154	#1866	—	67	4	no	—
155	#1849	—	67	1	no	—
156	#1745	—	67	1	no	—
157	C₆₇	#22237	1	1	yes	characteristic, complemented, Sylow
158	C₂×#131	—	67	2	no	—
159	#533	C₂×D₆₇	1	1	yes	characteristic
160	#532	C₂×D₆₇	1	1	yes	characteristic
161	#491	Dic₆₇	1	2	yes	complemented
162	#490	Dic₆₇	1	2	yes	complemented
163	#489	Dic₆₇	1	4	yes	complemented
164	#446	—	134	4	no	—
165	#435	—	134	4	no	—
166	C₂×#108	—	67	4	no	—
167	#424	—	67	2	no	—
168	#423	—	67	1	no	—
169	#423	—	67	1	no	—
170	C₂×#104	—	67	4	no	—
171	#416	C₂×D₆₇	1	4	yes	—
172	#408	—	67	2	no	—
173	#405	—	67	2	no	—
174	C₂×#97	—	67	1	no	—
175	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
176	C₄³	—	67	1	no	—
177	#339	—	67	4	no	—
178	#336	—	67	2	no	—
179	#336	—	67	2	no	—
180	C₂×#41	C₂²×D₆₇	1	2	yes	—
181	#131	—	268	4	no	—
182	C₂×M₁₆	—	67	2	no	—
183	C₂×M₁₆	—	67	2	no	—
184	C₂×M₁₆	—	134	2	no	—
185	C₂²×C₈	—	67	1	no	—
186	C₂²×C₈	—	67	1	no	—
187	C₂²×C₈	—	134	2	no	—
188	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
189	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic
190	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
191	#127	C₂×Dic₆₇	1	1	yes	characteristic
192	#125	C₂×Dic₆₇	1	2	yes	—
193	#124	C₂×Dic₆₇	1	2	yes	—
194	C₂²⋊Q₈	—	2	2	no	—
195	#121	—	2	2	no	—
196	C₄×Q₈	—	2	2	no	—
197	C₄×D₄	—	2	2	no	—
198	#117	C₂²×D₆₇	1	2	yes	—
199	C₂×C₄²	—	67	2	no	—
200	C₂×C₄²	—	134	2	no	—
201	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
202	#108	—	268	4	no	—
203	C₄⋊C₈	—	134	4	no	—
204	C₄⋊C₈	—	134	2	no	—
205	C₄⋊C₈	—	134	2	no	—
206	#104	—	268	4	no	—
207	#103	C₂×Dic₆₇	1	8	yes	—
208	#102	C₂×Dic₆₇	1	8	yes	—
209	C₂²⋊C₈	—	134	4	no	—
210	#97	—	67	4	no	—
211	#97	—	134	2	no	—
212	#97	C₂×Dic₆₇	1	2	yes	—
213	#97	C₂×Dic₆₇	1	2	yes	—
214	C₄×C₈	—	134	2	no	—
215	#41	—	2	4	no	—
216	#41	—	2	4	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	134	2	no	—
224	C₂²×C₄	#11761627	1	2	yes	—
225	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
226	M₁₆	—	268	4	no	—
227	M₁₆	—	134	4	no	—
228	M₁₆	—	134	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	268	4	no	—
232	C₂×C₈	—	67	2	no	—
233	C₂×C₈	—	67	2	no	—
234	C₂×C₈	—	134	2	no	—
235	C₂×C₈	—	134	2	no	—
236	C₂×C₈	—	134	2	no	—
237	C₂×C₈	—	134	2	no	—
238	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	4	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	268	4	no	—
247	C₄²	—	134	2	no	—
248	C₄²	—	134	2	no	—
249	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
250	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
251	C₄²	D₄×D₆₇	1	2	yes	—
252	C₂³	—	2	2	no	—
253	C₂³	C₂×#11761627	1	1	yes	characteristic
254	Q₈	—	2	4	no	—
255	Q₈	—	8	2	no	—
256	D₄	—	2	4	no	—
257	D₄	—	4	4	no	—
258	D₄	—	8	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	268	2	no	—
265	C₂×C₄	—	268	2	no	—
266	C₂×C₄	—	134	4	no	—
267	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
268	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
269	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
270	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
271	C₂×C₄	#14524089	1	2	yes	—
272	C₂×C₄	#14524090	1	2	yes	—
273	C₈	—	2	4	no	—
274	C₈	—	268	2	no	—
275	C₈	—	268	2	no	—
276	C₈	—	134	2	no	—
277	C₈	—	134	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
282	C₂²	#16864015	1	2	yes	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	2	no	—
286	C₄	—	4	4	no	—
287	C₄	—	268	4	no	—
288	C₄	#16863879	1	2	yes	—
289	C₄	#16863964	1	1	yes	characteristic
290	C₄	#16863966	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	—	8	2	no	—
293	C₂	#22562922	1	1	yes	characteristic
294	C₂	#22569119	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#3135	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#532×C₆₇	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3135	D₆₇
hypocenter	C₆₇	#22237
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3135

Complements

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

N	H ≅ G/N	type
1. #22569150	C₂	semidirect
3. #22561187	C₂	semidirect
5. #22560859	C₂	semidirect
6. #22560486	C₂	semidirect
7. #22560384	C₂	semidirect
8. #22559957	C₂	semidirect
27. #491×C₆₇	C₄	semidirect
28. #490×C₆₇	C₄	semidirect
29. #489×C₆₇	C₄	semidirect
157. C₆₇	#22237	semidirect
161. #491	Dic₆₇	semidirect
162. #490	Dic₆₇	semidirect
163. #489	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569150	2
#22568927	2
#22561187	2
#22561038	2
#22560859	2
#22560486	2
#22560384	2
#22559957	2
C₆₇×#3135	2
#22237	67