Subgroups — #30744476

Order 17152.

404 conjugacy classes of subgroups.

The subgroup lattice has 266 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	#22560326	C₂	1	1	yes	maximal, characteristic
3	#22558035	C₂	1	4	yes	maximal
4	C₆₇×#2862	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
6	#16863151	—	2	1	no	—
7	#16863151	—	2	1	no	—
8	#16863151	—	2	1	no	—
9	#16863151	C₂²	1	2	yes	—
10	#16863151	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
13	#391×C₆₇	—	2	1	no	—
14	#391×C₆₇	—	2	1	no	—
15	#391×C₆₇	C₄	1	2	yes	—
16	#391×C₆₇	C₂²	1	2	yes	—
17	#382×C₆₇	—	2	1	no	—
18	#382×C₆₇	—	2	1	no	—
19	C₄³×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₂²×C₆₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
23	C₂²×Q₈×C₆₇	—	2	1	no	—
24	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
25	C₄⋊Q₈×C₆₇	—	4	2	no	—
26	C₄⋊Q₈×C₆₇	—	4	2	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	2	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	#95×C₆₇	—	2	2	no	—
39	#95×C₆₇	—	2	2	no	—
40	#95×C₆₇	—	2	2	no	—
41	#95×C₆₇	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	4	2	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₂×C₆₇⋊C₈	—	4	2	no	—
46	C₂×C₆₇⋊C₈	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₂×Q₈×C₆₇	—	2	4	no	—
49	C₂×Q₈×C₆₇	—	4	2	no	—
50	C₂×Q₈×C₆₇	—	4	2	no	—
51	C₂×Q₈×C₆₇	—	4	2	no	—
52	C₂×Q₈×C₆₇	—	4	2	no	—
53	C₂×Q₈×C₆₇	—	4	2	no	—
54	C₂²×C₄×C₆₇	—	2	2	no	—
55	C₂²×C₄×C₆₇	—	2	2	no	—
56	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₆₇	—	4	2	no	—
64	C₄⋊C₄×C₆₇	—	4	2	no	—
65	C₄⋊C₄×C₆₇	—	4	2	no	—
66	C₄⋊C₄×C₆₇	—	4	2	no	—
67	C₄²×C₆₇	—	2	2	no	—
68	C₄²×C₆₇	—	2	2	no	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	4	1	no	—
71	C₄²×C₆₇	—	4	1	no	—
72	C₄²×C₆₇	—	4	1	no	—
73	C₄²×C₆₇	SD₁₆	1	2	yes	—
74	C₄²×C₆₇	Q₁₆	1	2	yes	—
75	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	Q₈×C₆₇	—	8	4	no	—
77	Q₈×C₆₇	—	8	4	no	—
78	C₂×C₄×C₆₇	—	2	1	no	—
79	C₂×C₄×C₆₇	—	2	1	no	—
80	C₂×C₄×C₆₇	—	2	1	no	—
81	C₂×C₄×C₆₇	—	2	1	no	—
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	—	4	2	no	—
94	C₂×C₄×C₆₇	—	4	2	no	—
95	C₂×C₄×C₆₇	—	4	2	no	—
96	C₂×C₄×C₆₇	—	4	2	no	—
97	C₂×C₄×C₆₇	—	4	2	no	—
98	C₂×C₄×C₆₇	—	4	2	no	—
99	C₂×C₄×C₆₇	#100	1	2	yes	—
100	C₂×C₄×C₆₇	#103	1	2	yes	—
101	C₂×C₄×C₆₇	#103	1	2	yes	—
102	C₂×C₄×C₆₇	#131	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#137	1	2	yes	—
106	C₆₇⋊C₈	—	8	2	no	—
107	C₆₇⋊C₈	—	8	2	no	—
108	C₂²×C₆₇	#408	1	1	yes	characteristic
109	C₂²×C₆₇	#410	1	1	yes	characteristic
110	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
111	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
112	C₂²×C₆₇	#419	1	1	yes	characteristic
113	C₂²×C₆₇	#421	1	1	yes	characteristic
114	C₂²×C₆₇	#421	1	1	yes	characteristic
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	—	2	2	no	—
117	C₄×C₆₇	—	2	2	no	—
118	C₄×C₆₇	—	2	2	no	—
119	C₄×C₆₇	—	2	1	no	—
120	C₄×C₆₇	—	2	1	no	—
121	C₄×C₆₇	—	2	2	no	—
122	C₄×C₆₇	—	4	1	no	—
123	C₄×C₆₇	—	8	2	no	—
124	C₄×C₆₇	—	8	2	no	—
125	#7722	—	67	1	no	maximal, Sylow
126	C₂×C₆₇	#1469	1	1	yes	characteristic
127	C₂×C₆₇	#1481	1	1	yes	characteristic
128	C₂×C₆₇	#1497	1	1	yes	characteristic
129	C₂×C₆₇	#1515	1	1	yes	characteristic
130	C₂×C₆₇	#1518	1	1	yes	characteristic
131	C₂×C₆₇	#1533	1	1	yes	characteristic
132	C₂×C₆₇	#1548	1	1	yes	characteristic
133	#2862	D₆₇	1	1	yes	characteristic
134	#1839	—	67	1	no	—
135	#1831	—	67	1	no	—
136	#1272	—	67	4	no	—
137	C₆₇	#7722	1	1	yes	characteristic, complemented, Sylow
138	C₂×C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
139	C₂×C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
140	C₂×C₄⋊C₈	—	134	1	no	—
141	#391	—	2	1	no	—
142	#391	—	2	1	no	—
143	#391	Dic₆₇	1	2	yes	—
144	#391	C₂×D₆₇	1	2	yes	—
145	#382	—	2	1	no	—
146	#382	—	2	1	no	—
147	C₄³	C₂×D₆₇	1	1	yes	characteristic
148	#336	—	67	2	no	—
149	#336	—	67	2	no	—
150	#336	—	134	1	no	—
151	#336	—	134	1	no	—
152	#336	—	134	1	no	—
153	C₂²×Q₈	—	2	1	no	—
154	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
155	C₂²×C₈	—	134	2	no	—
156	C₂²×C₈	—	134	2	no	—
157	C₄⋊Q₈	—	4	2	no	—
158	C₄⋊Q₈	—	4	2	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
163	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
166	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
170	C₄⋊C₈	—	268	2	no	—
171	#95	—	2	2	no	—
172	#95	—	2	2	no	—
173	#95	—	2	2	no	—
174	#95	—	2	2	no	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
186	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
187	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
190	C₂×C₈	—	268	2	no	—
191	C₂×C₈	—	268	2	no	—
192	C₂×C₈	—	268	2	no	—
193	C₂×C₈	—	268	2	no	—
194	C₂×C₈	—	268	2	no	—
195	C₂×C₈	—	268	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	4	1	no	—
204	C₄²	—	4	1	no	—
205	C₄²	—	4	1	no	—
206	C₄²	#11761615	1	2	yes	—
207	C₄²	#11761618	1	2	yes	—
208	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
209	Q₈	—	8	4	no	—
210	Q₈	—	8	4	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	#14523960	1	2	yes	—
233	C₂×C₄	#14523967	1	2	yes	—
234	C₂×C₄	#14523967	1	2	yes	—
235	C₂×C₄	#14524050	1	2	yes	—
236	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
237	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
238	C₂×C₄	#14524074	1	2	yes	—
239	C₈	—	536	2	no	—
240	C₈	—	536	2	no	—
241	C₂²	#16863363	1	1	yes	characteristic
242	C₂²	#16863370	1	1	yes	characteristic
243	C₂²	C₂×#14523960	1	1	yes	characteristic
244	C₂²	C₂×#14523967	1	1	yes	characteristic
245	C₂²	#16863405	1	1	yes	characteristic
246	C₂²	#16863413	1	1	yes	characteristic
247	C₂²	#16863413	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	2	no	—
258	C₂	#22558561	1	1	yes	characteristic
259	C₂	#22558612	1	1	yes	characteristic
260	C₂	#22558682	1	1	yes	characteristic
261	C₂	#22558760	1	1	yes	characteristic
262	C₂	#22558773	1	1	yes	characteristic
263	C₂	#22558833	1	1	yes	characteristic
264	C₂	#22558890	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×#2862	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2862	D₆₇
hypocenter	C₆₇	#7722
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
137. C₆₇	#7722	semidirect

Maximal subgroups

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

subgroup	index
#22560352	2
#22560326	2
#22558035	2
C₆₇×#2862	2
#7722	67