Subgroups — #30744434

Order 17152.

372 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560352	C₂	1	1	yes	maximal, characteristic
2	#22560062	C₂	1	1	yes	maximal, characteristic
3	#22558035	C₂	1	2	yes	maximal
4	#22558032	C₂	1	2	yes	maximal
5	C₆₇×#2863	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
7	C₂×#14523954	—	2	1	no	—
8	#16863151	—	2	1	no	—
9	#16863151	—	2	1	no	—
10	#16863151	C₂²	1	2	yes	—
11	#16863151	C₂²	1	2	yes	—
12	#391×C₆₇	C₄	1	1	yes	characteristic
13	#391×C₆₇	C₄	1	1	yes	characteristic
14	#391×C₆₇	C₂²	1	1	yes	characteristic
15	#389×C₆₇	C₄	1	1	yes	characteristic
16	#389×C₆₇	C₄	1	1	yes	characteristic
17	#389×C₆₇	C₂²	1	1	yes	characteristic
18	#387×C₆₇	—	2	1	no	—
19	#387×C₆₇	—	2	1	no	—
20	#387×C₆₇	—	2	1	no	—
21	#383×C₆₇	—	2	1	no	—
22	C₄³×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
26	#14523954	—	4	2	no	—
27	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	#95×C₆₇	—	2	1	no	—
42	#95×C₆₇	—	2	1	no	—
43	#95×C₆₇	—	2	1	no	—
44	#95×C₆₇	—	2	1	no	—
45	#95×C₆₇	—	2	1	no	—
46	#95×C₆₇	—	2	1	no	—
47	#95×C₆₇	—	2	1	no	—
48	#95×C₆₇	—	2	1	no	—
49	#95×C₆₇	—	2	1	no	—
50	C₂×C₆₇⋊C₈	—	2	4	no	—
51	C₂×C₆₇⋊C₈	—	4	2	no	—
52	C₂×C₆₇⋊C₈	—	4	2	no	—
53	C₂×C₆₇⋊C₈	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₂×Q₈×C₆₇	—	2	2	no	—
57	C₂×Q₈×C₆₇	—	4	1	no	—
58	C₂×Q₈×C₆₇	—	4	1	no	—
59	C₂²×C₄×C₆₇	—	2	1	no	—
60	C₂²×C₄×C₆₇	—	2	1	no	—
61	C₂²×C₄×C₆₇	—	2	1	no	—
62	C₂²×C₄×C₆₇	—	2	1	no	—
63	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₄⋊C₄×C₆₇	—	4	1	no	—
72	C₄⋊C₄×C₆₇	—	4	1	no	—
73	C₄⋊C₄×C₆₇	—	4	2	no	—
74	C₄⋊C₄×C₆₇	—	4	2	no	—
75	C₄⋊C₄×C₆₇	—	4	1	no	—
76	C₄⋊C₄×C₆₇	—	4	1	no	—
77	C₄²×C₆₇	—	2	2	no	—
78	C₄²×C₆₇	—	2	2	no	—
79	C₄²×C₆₇	—	2	1	no	—
80	C₄²×C₆₇	—	2	1	no	—
81	C₄²×C₆₇	—	2	2	no	—
82	C₄²×C₆₇	—	4	1	no	—
83	C₄²×C₆₇	—	4	1	no	—
84	C₄²×C₆₇	—	4	1	no	—
85	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₆₇	—	8	2	no	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	1	no	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	1	no	—
96	C₂×C₄×C₆₇	—	2	1	no	—
97	C₂×C₄×C₆₇	—	2	1	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	1	no	—
102	C₂×C₄×C₆₇	—	2	1	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	4	1	no	—
105	C₂×C₄×C₆₇	—	4	1	no	—
106	C₂×C₄×C₆₇	—	4	1	no	—
107	C₂×C₄×C₆₇	—	4	1	no	—
108	C₂×C₄×C₆₇	—	4	1	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	—	4	1	no	—
114	C₂×C₄×C₆₇	—	4	1	no	—
115	C₂×C₄×C₆₇	—	4	1	no	—
116	C₂×C₄×C₆₇	#104	1	2	yes	—
117	C₂×C₄×C₆₇	#131	1	2	yes	—
118	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#137	1	2	yes	—
121	C₆₇⋊C₈	—	4	4	no	—
122	C₆₇⋊C₈	—	8	2	no	—
123	C₂²×C₆₇	#408	1	1	yes	characteristic
124	C₂²×C₆₇	#410	1	1	yes	characteristic
125	C₂²×C₆₇	#413	1	1	yes	characteristic
126	C₂²×C₆₇	#417	1	1	yes	characteristic
127	C₂²×C₆₇	#419	1	1	yes	characteristic
128	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
129	C₂²×C₆₇	#421	1	1	yes	characteristic
130	C₄×C₆₇	—	2	2	no	—
131	C₄×C₆₇	—	2	1	no	—
132	C₄×C₆₇	—	2	1	no	—
133	C₄×C₆₇	—	2	2	no	—
134	C₄×C₆₇	—	2	2	no	—
135	C₄×C₆₇	—	4	1	no	—
136	C₄×C₆₇	—	4	1	no	—
137	C₄×C₆₇	—	4	1	no	—
138	C₄×C₆₇	—	8	1	no	—
139	C₄×C₆₇	—	8	1	no	—
140	C₄×C₆₇	—	8	1	no	—
141	C₄×C₆₇	—	8	1	no	—
142	#7713	—	67	1	no	maximal, Sylow
143	C₂×C₆₇	#1473	1	1	yes	characteristic
144	C₂×C₆₇	#1485	1	1	yes	characteristic
145	C₂×C₆₇	#1495	1	1	yes	characteristic
146	C₂×C₆₇	#1509	1	1	yes	characteristic
147	C₂×C₆₇	#1515	1	1	yes	characteristic
148	C₂×C₆₇	#1532	1	1	yes	characteristic
149	C₂×C₆₇	#1559	1	1	yes	characteristic
150	#2863	D₆₇	1	1	yes	characteristic
151	#1839	—	67	1	no	—
152	#1767	—	67	1	no	—
153	#1272	—	67	2	no	—
154	#1271	—	67	2	no	—
155	C₆₇	#7713	1	1	yes	characteristic, complemented, Sylow
156	C₂×C₄⋊C₈	—	134	1	no	—
157	C₂×#97	—	134	1	no	—
158	#391	Dic₆₇	1	1	yes	characteristic
159	#391	Dic₆₇	1	1	yes	characteristic
160	#391	C₂×D₆₇	1	1	yes	characteristic
161	#389	Dic₆₇	1	1	yes	characteristic
162	#389	Dic₆₇	1	1	yes	characteristic
163	#389	C₂×D₆₇	1	1	yes	characteristic
164	#387	—	2	1	no	—
165	#387	—	2	1	no	—
166	#387	—	2	1	no	—
167	#383	—	2	1	no	—
168	C₄³	C₂×D₆₇	1	1	yes	characteristic
169	#336	—	67	2	no	—
170	#336	—	67	2	no	—
171	#336	—	134	1	no	—
172	#336	—	134	1	no	—
173	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
174	C₂²×C₈	—	134	2	no	—
175	C₂²×C₈	—	134	2	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
180	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
182	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
183	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
186	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
188	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
189	C₄⋊C₈	—	134	4	no	—
190	#97	—	268	2	no	—
191	#95	—	2	1	no	—
192	#95	—	2	1	no	—
193	#95	—	2	1	no	—
194	#95	—	2	1	no	—
195	#95	—	2	1	no	—
196	#95	—	2	1	no	—
197	#95	—	2	1	no	—
198	#95	—	2	1	no	—
199	#95	—	2	1	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	4	1	no	—
202	C₂×Q₈	—	4	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
211	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
214	C₂×C₈	—	268	2	no	—
215	C₂×C₈	—	268	2	no	—
216	C₂×C₈	—	268	2	no	—
217	C₂×C₈	—	268	2	no	—
218	C₂×C₈	—	268	2	no	—
219	C₂×C₈	—	134	4	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	4	1	no	—
222	C₄⋊C₄	—	4	1	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	1	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	4	1	no	—
234	C₄²	—	4	1	no	—
235	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
236	Q₈	—	8	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	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	#14523970	1	2	yes	—
267	C₂×C₄	#14524050	1	2	yes	—
268	C₂×C₄	#14524068	1	1	yes	characteristic
269	C₂×C₄	#14524073	1	1	yes	characteristic
270	C₂×C₄	#14524074	1	2	yes	—
271	C₈	—	268	4	no	—
272	C₈	—	536	2	no	—
273	C₂²	#16863363	1	1	yes	characteristic
274	C₂²	#16863370	1	1	yes	characteristic
275	C₂²	#16863380	1	1	yes	characteristic
276	C₂²	#16863397	1	1	yes	characteristic
277	C₂²	#16863405	1	1	yes	characteristic
278	C₂²	C₂×#14523970	1	1	yes	characteristic
279	C₂²	#16863413	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	1	no	—
288	C₄	—	8	1	no	—
289	C₄	—	8	1	no	—
290	C₄	—	8	1	no	—
291	C₄	—	8	1	no	—
292	C₂	#22558578	1	1	yes	characteristic
293	C₂	#22558630	1	1	yes	characteristic
294	C₂	#22558674	1	1	yes	characteristic
295	C₂	#22558734	1	1	yes	characteristic
296	C₂	#22558760	1	1	yes	characteristic
297	C₂	#22558829	1	1	yes	characteristic
298	C₂	#22558939	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2863	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2863	D₆₇
hypocenter	C₆₇	#7713
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2863

Complements

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

N	H ≅ G/N	type
155. C₆₇	#7713	semidirect

Maximal subgroups

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

subgroup	index
#22560352	2
#22560062	2
#22558035	2
#22558032	2
C₆₇×#2863	2
#7713	67