Subgroups — #30854620

Order 17152.

1090 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×#14524002	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₆₇	C₂	1	4	yes	maximal, complemented
4	C₄⋊C₈×Dic₆₇	C₂	1	4	yes	maximal, complemented
5	#22559983	C₂	1	4	yes	maximal
6	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
8	C₂×#14524002	C₄	1	2	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	8	yes	complemented
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₈×Dic₆₇	C₂²	1	8	yes	—
13	C₄×C₄⋊Dic₆₇	C₄	1	2	yes	—
14	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
15	C₄×C₂²⋊Dic₆₇	C₄	1	2	yes	—
16	C₄×C₂²⋊Dic₆₇	C₂²	1	2	yes	—
17	#16863242	C₄	1	2	yes	—
18	C₄²×Dic₆₇	C₂²	1	2	yes	—
19	#16863150	C₂²	1	8	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#431×C₆₇	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₆₇⋊C₈	—	4	4	no	—
26	C₂²×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
27	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
28	#14524002	C₈	1	16	yes	complemented
29	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊Dic₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
34	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	8	yes	—
35	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	8	yes	—
36	C₈×Dic₆₇	—	4	8	no	—
37	C₆₇⋊#95	C₂×C₄	1	4	yes	—
38	C₆₇⋊#95	C₂×C₄	1	4	yes	—
39	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
50	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
51	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
52	C₂²×Dic₆₇	C₂²×C₄	1	8	yes	—
53	C₂×C₆₇⋊C₈	—	2	8	no	—
54	C₂×C₆₇⋊C₈	—	4	8	no	—
55	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
56	C₄⋊Dic₆₇	C₂×C₈	1	16	yes	complemented
57	C₂²⋊Dic₆₇	C₂×C₈	1	16	yes	—
58	C₄×Dic₆₇	—	2	4	no	—
59	C₄×Dic₆₇	—	2	4	no	—
60	C₄×Dic₆₇	—	4	4	no	—
61	C₄×Dic₆₇	C₂×C₈	1	16	yes	complemented
62	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	4	2	no	—
66	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
67	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
68	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₆₇	—	2	4	no	—
78	C₂×C₈×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
80	C₄²×C₆₇	C₄²	1	4	yes	—
81	C₄²×C₆₇	C₄²	1	4	yes	—
82	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
84	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
85	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂³×C₆₇	—	2	2	no	—
87	C₂³×C₆₇	—	2	2	no	—
88	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
89	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
90	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	4	4	no	—
98	C₂×C₄×C₆₇	C₄×C₈	1	16	yes	complemented
99	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
105	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
109	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
110	C₂×Dic₆₇	—	2	8	no	—
111	C₂×Dic₆₇	—	2	8	no	—
112	C₂×Dic₆₇	C₂²×C₈	1	16	yes	—
113	C₈×C₆₇	—	2	8	no	—
114	C₆₇⋊C₈	—	2	8	no	—
115	C₂²×C₆₇	—	2	2	no	—
116	C₂²×C₆₇	—	2	8	no	—
117	C₂²×C₆₇	—	2	2	no	—
118	C₂²×C₆₇	C₂×C₄×C₈	1	4	yes	—
119	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
120	C₂²×C₆₇	#520	1	2	yes	—
121	C₂²×C₆₇	C₂³×C₈	1	2	yes	—
122	C₂²×C₆₇	#568	1	2	yes	—
123	C₄×C₆₇	—	2	2	no	—
124	C₄×C₆₇	—	2	2	no	—
125	C₄×C₆₇	—	2	2	no	—
126	C₄×C₆₇	C₂×C₄×C₈	1	4	yes	—
127	C₄×C₆₇	C₂×C₄×C₈	1	4	yes	—
128	C₄×C₆₇	#518	1	1	yes	characteristic
129	C₄×C₆₇	#519	1	1	yes	characteristic
130	C₄×C₆₇	#520	1	2	yes	—
131	Dic₆₇	—	2	16	no	—
132	#20543	—	67	1	no	maximal, Sylow
133	C₂×C₆₇	—	2	4	no	—
134	C₂×C₆₇	#2274	1	1	yes	characteristic
135	C₂×C₆₇	C₂²×C₄×C₈	1	1	yes	characteristic
136	C₂×C₆₇	#2873	1	1	yes	characteristic
137	C₂×C₆₇	#2988	1	4	yes	—
138	C₂×#431	D₆₇	1	1	yes	characteristic
139	C₂²×C₄⋊C₈	—	67	1	no	—
140	C₄×#117	—	67	1	no	—
141	C₈×C₄⋊C₄	—	67	4	no	—
142	C₄×C₄⋊C₈	—	67	4	no	—
143	C₈×C₂²⋊C₄	—	67	4	no	—
144	C₆₇	#20543	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₈	—	67	2	no	—
146	C₂×#117	—	67	2	no	—
147	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
148	#431	Dic₆₇	1	8	yes	complemented
149	C₂×C₄⋊C₈	—	67	8	no	—
150	C₂×C₄⋊C₈	—	67	2	no	—
151	C₂×C₄⋊C₈	—	67	2	no	—
152	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
153	C₂×C₂²⋊C₈	C₂×D₆₇	1	2	yes	—
154	C₂×C₄×C₈	—	67	8	no	—
155	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
156	C₄×C₄⋊C₄	—	67	2	no	—
157	C₄×C₄⋊C₄	—	67	2	no	—
158	C₄×C₂²⋊C₄	—	67	2	no	—
159	C₄×C₂²⋊C₄	—	67	2	no	—
160	#376	—	67	2	no	—
161	C₄³	—	67	2	no	—
162	#336	—	67	8	no	—
163	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
164	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
165	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
166	C₂²×C₈	—	268	4	no	—
167	C₂²×C₈	—	67	8	no	—
168	C₂²×C₈	—	67	4	no	—
169	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
170	#117	—	67	16	no	—
171	C₂×C₄⋊C₄	—	67	4	no	—
172	C₂×C₂²⋊C₄	—	67	4	no	—
173	C₂×C₄²	—	67	4	no	—
174	C₂×C₄²	—	67	4	no	—
175	C₂×C₄²	—	67	4	no	—
176	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
177	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₆₇	1	4	yes	—
179	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
180	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
181	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
182	C₄⋊C₈	—	67	8	no	—
183	C₄⋊C₈	—	67	8	no	—
184	C₄⋊C₈	C₂×Dic₆₇	1	8	yes	—
185	C₂²⋊C₈	C₂×Dic₆₇	1	8	yes	—
186	C₄×C₈	—	268	8	no	—
187	C₄×C₈	C₂×Dic₆₇	1	8	yes	—
188	#95	—	67	4	no	—
189	#95	—	67	4	no	—
190	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	67	8	no	—
195	C₂²×C₄	C₄×Dic₆₇	1	4	yes	—
196	C₂²×C₄	C₄×Dic₆₇	1	4	yes	—
197	C₂²×C₄	C₈×D₆₇	1	8	yes	—
198	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
202	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
204	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
205	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	268	8	no	—
209	C₂×C₈	—	67	8	no	—
210	C₂×C₈	—	134	8	no	—
211	C₂×C₈	C₂²×Dic₆₇	1	8	yes	—
212	C₄⋊C₄	—	67	16	no	—
213	C₂²⋊C₄	—	67	16	no	—
214	C₄²	—	268	4	no	—
215	C₄²	—	67	16	no	—
216	C₄²	—	134	4	no	—
217	C₄²	—	134	4	no	—
218	C₄²	C₄×Dic₆₇	1	4	yes	—
219	C₄²	C₄×Dic₆₇	1	4	yes	—
220	C₄²	C₂×C₄×D₆₇	1	2	yes	—
221	C₄²	C₂×C₄×D₆₇	1	2	yes	—
222	C₄²	C₂²×Dic₆₇	1	2	yes	—
223	C₄²	C₂²×Dic₆₇	1	2	yes	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂×C₄×Dic₆₇	1	2	yes	—
227	C₂³	C₂×C₈×D₆₇	1	4	yes	—
228	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	67	16	no	—
237	C₂×C₄	—	134	8	no	—
238	C₂×C₄	—	134	8	no	—
239	C₂×C₄	C₈×Dic₆₇	1	16	yes	complemented
240	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
241	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
242	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
244	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
245	C₂×C₄	C₂×C₈×D₆₇	1	4	yes	—
246	C₂×C₄	C₂×C₈×D₆₇	1	4	yes	—
247	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
248	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
249	C₂×C₄	#14524089	1	2	yes	—
250	C₂×C₄	#14524090	1	2	yes	—
251	C₈	—	2	8	no	—
252	C₈	—	134	8	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	2	2	no	—
256	C₂²	C₂×C₈×Dic₆₇	1	4	yes	—
257	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
258	C₂²	#16863876	1	2	yes	—
259	C₂²	C₂²×C₈×D₆₇	1	2	yes	—
260	C₂²	#16864162	1	2	yes	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	134	16	no	—
265	C₄	C₂×C₈×Dic₆₇	1	4	yes	—
266	C₄	C₂×C₈×Dic₆₇	1	4	yes	—
267	C₄	#16863870	1	1	yes	characteristic
268	C₄	#16863873	1	1	yes	characteristic
269	C₄	#16863879	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	#22562565	1	1	yes	characteristic
272	C₂	C₂²×C₈×Dic₆₇	1	1	yes	characteristic
273	C₂	#22568627	1	1	yes	characteristic
274	C₂	#22569317	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator	C₂×C₆₇	C₂²×C₄×C₈
Fitting	C₂×#431×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#431	D₆₇
hypocenter	C₆₇	#20543
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. C₈×C₄⋊Dic₆₇	C₂	semidirect
4. C₄⋊C₈×Dic₆₇	C₂	semidirect
9. C₂×C₄⋊(C₆₇⋊C₈)	C₄	semidirect
21. #431×C₆₇	C₄	semidirect
28. #14524002	C₈	semidirect
56. C₄⋊Dic₆₇	C₂×C₈	semidirect
61. C₄×Dic₆₇	C₂×C₈	semidirect
98. C₂×C₄×C₆₇	C₄×C₈	semidirect
144. C₆₇	#20543	semidirect
148. #431	Dic₆₇	semidirect
239. C₂×C₄	C₈×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₆₇⋊C₈)	2
C₄×#14524002	2
C₈×C₄⋊Dic₆₇	2
C₄⋊C₈×Dic₆₇	2
#22559983	2
C₂×#431×C₆₇	2
#20543	67