Subgroups — #30744178

Order 17152.

370 conjugacy classes of subgroups.

The subgroup lattice has 286 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	#22559955	C₂	1	1	yes	maximal, characteristic
3	#22558032	C₂	1	2	yes	maximal
4	#22558032	C₂	1	2	yes	maximal
5	C₆₇×#2864	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
7	#16863151	—	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	#389×C₆₇	C₄	1	1	yes	characteristic
13	#389×C₆₇	C₄	1	1	yes	characteristic
14	#389×C₆₇	C₂²	1	1	yes	characteristic
15	#389×C₆₇	C₂²	1	1	yes	characteristic
16	#387×C₆₇	—	2	1	no	—
17	#387×C₆₇	—	2	1	no	—
18	#387×C₆₇	—	2	1	no	—
19	#387×C₆₇	—	2	1	no	—
20	#382×C₆₇	C₄	1	1	yes	characteristic
21	#382×C₆₇	C₄	1	1	yes	characteristic
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	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
28	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
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	C₂×C₆₇⋊C₈	—	2	4	no	—
50	C₂×C₆₇⋊C₈	—	4	2	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	C₂²×C₄×C₆₇	—	2	1	no	—
56	C₂²×C₄×C₆₇	—	2	1	no	—
57	C₂²×C₄×C₆₇	—	2	1	no	—
58	C₂²×C₄×C₆₇	—	2	1	no	—
59	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₆₇	—	2	2	no	—
67	C₄⋊C₄×C₆₇	—	2	2	no	—
68	C₄⋊C₄×C₆₇	—	4	1	no	—
69	C₄⋊C₄×C₆₇	—	4	1	no	—
70	C₄⋊C₄×C₆₇	—	4	2	no	—
71	C₄⋊C₄×C₆₇	—	4	2	no	—
72	C₄⋊C₄×C₆₇	—	4	2	no	—
73	C₄⋊C₄×C₆₇	—	4	2	no	—
74	C₄²×C₆₇	—	2	2	no	—
75	C₄²×C₆₇	—	2	2	no	—
76	C₄²×C₆₇	—	2	2	no	—
77	C₄²×C₆₇	—	2	2	no	—
78	C₄²×C₆₇	—	2	2	no	—
79	C₄²×C₆₇	—	4	1	no	—
80	C₄²×C₆₇	—	4	1	no	—
81	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	1	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	—	4	1	no	—
99	C₂×C₄×C₆₇	—	4	1	no	—
100	C₂×C₄×C₆₇	—	4	1	no	—
101	C₂×C₄×C₆₇	—	4	1	no	—
102	C₂×C₄×C₆₇	—	4	1	no	—
103	C₂×C₄×C₆₇	—	4	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	#101	1	2	yes	—
111	C₂×C₄×C₆₇	#104	1	2	yes	—
112	C₂×C₄×C₆₇	#131	1	2	yes	—
113	C₂×C₄×C₆₇	#135	1	2	yes	—
114	C₂×C₄×C₆₇	#135	1	2	yes	—
115	C₆₇⋊C₈	—	4	4	no	—
116	C₆₇⋊C₈	—	8	2	no	—
117	C₂²×C₆₇	#408	1	1	yes	characteristic
118	C₂²×C₆₇	#410	1	1	yes	characteristic
119	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
120	C₂²×C₆₇	#416	1	1	yes	characteristic
121	C₂²×C₆₇	#416	1	1	yes	characteristic
122	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
123	C₂²×C₆₇	#421	1	1	yes	characteristic
124	C₄×C₆₇	—	2	2	no	—
125	C₄×C₆₇	—	2	2	no	—
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	2	2	no	—
128	C₄×C₆₇	—	2	2	no	—
129	C₄×C₆₇	—	4	1	no	—
130	C₄×C₆₇	—	4	1	no	—
131	C₄×C₆₇	—	8	1	no	—
132	C₄×C₆₇	—	8	1	no	—
133	C₄×C₆₇	—	8	1	no	—
134	C₄×C₆₇	—	8	1	no	—
135	#7657	—	67	1	no	maximal, Sylow
136	C₂×C₆₇	#1470	1	1	yes	characteristic
137	C₂×C₆₇	#1482	1	1	yes	characteristic
138	C₂×C₆₇	#1494	1	1	yes	characteristic
139	C₂×C₆₇	#1500	1	1	yes	characteristic
140	C₂×C₆₇	#1521	1	1	yes	characteristic
141	C₂×C₆₇	#1531	1	1	yes	characteristic
142	C₂×C₆₇	#1544	1	1	yes	characteristic
143	#2864	D₆₇	1	1	yes	characteristic
144	#1839	—	67	1	no	—
145	#1744	—	67	1	no	—
146	#1271	—	67	2	no	—
147	#1271	—	67	2	no	—
148	C₆₇	#7657	1	1	yes	characteristic, complemented, Sylow
149	C₂×C₄×C₈	—	134	1	no	—
150	#389	Dic₆₇	1	1	yes	characteristic
151	#389	Dic₆₇	1	1	yes	characteristic
152	#389	C₂×D₆₇	1	1	yes	characteristic
153	#389	C₂×D₆₇	1	1	yes	characteristic
154	#387	—	2	1	no	—
155	#387	—	2	1	no	—
156	#387	—	2	1	no	—
157	#387	—	2	1	no	—
158	#382	Dic₆₇	1	1	yes	characteristic
159	#382	Dic₆₇	1	1	yes	characteristic
160	C₄³	C₂×D₆₇	1	1	yes	characteristic
161	#336	—	67	2	no	—
162	#336	—	67	2	no	—
163	#336	—	134	1	no	—
164	#336	—	134	1	no	—
165	#336	—	134	1	no	—
166	C₂²×C₈	—	134	2	no	—
167	C₂²×C₈	—	134	2	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
176	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
177	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
182	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
183	C₄×C₈	—	134	4	no	—
184	#95	—	2	1	no	—
185	#95	—	2	1	no	—
186	#95	—	2	1	no	—
187	#95	—	2	1	no	—
188	#95	—	2	1	no	—
189	#95	—	2	1	no	—
190	#95	—	2	1	no	—
191	#95	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
203	C₂×C₈	—	268	2	no	—
204	C₂×C₈	—	268	2	no	—
205	C₂×C₈	—	268	2	no	—
206	C₂×C₈	—	268	2	no	—
207	C₂×C₈	—	268	2	no	—
208	C₂×C₈	—	134	4	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	1	no	—
224	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	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	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	#14523962	1	2	yes	—
254	C₂×C₄	#14523970	1	2	yes	—
255	C₂×C₄	#14524050	1	2	yes	—
256	C₂×C₄	#14524063	1	2	yes	—
257	C₂×C₄	#14524063	1	2	yes	—
258	C₈	—	268	4	no	—
259	C₈	—	536	2	no	—
260	C₂²	#16863363	1	1	yes	characteristic
261	C₂²	#16863370	1	1	yes	characteristic
262	C₂²	C₂×#14523962	1	1	yes	characteristic
263	C₂²	#16863392	1	1	yes	characteristic
264	C₂²	#16863392	1	1	yes	characteristic
265	C₂²	C₂×#14523970	1	1	yes	characteristic
266	C₂²	#16863413	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	8	1	no	—
275	C₄	—	8	1	no	—
276	C₄	—	8	1	no	—
277	C₄	—	8	1	no	—
278	C₂	#22558565	1	1	yes	characteristic
279	C₂	#22558617	1	1	yes	characteristic
280	C₂	#22558669	1	1	yes	characteristic
281	C₂	#22558695	1	1	yes	characteristic
282	C₂	#22558786	1	1	yes	characteristic
283	C₂	#22558824	1	1	yes	characteristic
284	C₂	#22558874	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×#2864	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2864	D₆₇
hypocenter	C₆₇	#7657
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2864

Complements

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

N	H ≅ G/N	type
148. C₆₇	#7657	semidirect

Maximal subgroups

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

subgroup	index
#22560352	2
#22559955	2
#22558032	2
C₆₇×#2864	2
#7657	67