Subgroups — #349×Dic₆₇ #30770873

Order 17152.

522 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	C₂×M₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863178	C₂	1	1	yes	maximal, characteristic
3	M₃₂×Dic₆₇	C₂	1	2	yes	maximal, complemented
4	#22558292	C₂	1	2	yes	maximal, complemented
5	C₂×#349×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₆₇	C₄	1	1	yes	characteristic
8	C₂×#14523983	C₂²	1	2	yes	—
9	M₁₆×Dic₆₇	—	2	2	no	—
10	M₁₆×Dic₆₇	—	2	2	no	—
11	C₂×#14523953	C₄	1	1	yes	characteristic
12	C₂×#14523953	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
15	#16863178	C₄	1	4	yes	complemented
16	#16863171	—	2	2	no	—
17	C₁₆×Dic₆₇	—	2	2	no	—
18	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
20	#349×C₆₇	C₄	1	4	yes	complemented
21	C₂³×Dic₆₇	C₈	1	2	yes	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	C₈	1	4	yes	—
25	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₆₇	—	2	1	no	—
28	C₂×C₄×Dic₆₇	—	2	1	no	—
29	C₂×C₄×Dic₆₇	—	2	2	no	—
30	C₂×C₄×Dic₆₇	C₈	1	2	yes	—
31	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
33	#14523983	—	2	2	no	—
34	#14523983	C₂×C₄	1	4	yes	complemented
35	C₂×C₆₇⋊C₁₆	—	2	2	no	—
36	#14523953	—	2	2	no	—
37	#14523953	—	2	2	no	—
38	C₈×Dic₆₇	—	2	1	no	—
39	C₈×Dic₆₇	—	2	1	no	—
40	C₈×Dic₆₇	D₄	1	2	yes	—
41	C₈×Dic₆₇	D₄	1	2	yes	—
42	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	M₃₂×C₆₇	—	2	2	no	—
50	M₃₂×C₆₇	C₂×C₄	1	4	yes	complemented
51	C₂×C₁₆×C₆₇	—	2	2	no	—
52	C₂²×Dic₆₇	—	2	2	no	—
53	C₂²×Dic₆₇	—	2	2	no	—
54	C₂²×Dic₆₇	—	4	4	no	—
55	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
56	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
57	#11761612	—	2	8	no	—
58	C₂×C₆₇⋊C₈	—	2	1	no	—
59	C₂×C₆₇⋊C₈	—	2	1	no	—
60	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
61	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
62	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
63	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
64	C₄×Dic₆₇	—	2	1	no	—
65	C₄×Dic₆₇	—	2	1	no	—
66	C₄×Dic₆₇	—	2	2	no	—
67	C₄×Dic₆₇	—	2	2	no	—
68	C₄×Dic₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄×Dic₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄×Dic₆₇	M₁₆	1	2	yes	—
71	C₆₇⋊C₁₆	—	2	4	no	—
72	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	—	2	1	no	—
75	C₂²×C₄×C₆₇	—	2	1	no	—
76	C₂²×C₄×C₆₇	—	2	1	no	—
77	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	M₁₆×C₆₇	—	2	4	no	—
82	M₁₆×C₆₇	—	2	4	no	—
83	C₂×C₈×C₆₇	—	2	1	no	—
84	C₂×C₈×C₆₇	—	2	1	no	—
85	C₂×C₈×C₆₇	C₄²	1	2	yes	—
86	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₆₇	#41	1	2	yes	—
90	C₁₆×C₆₇	—	2	4	no	—
91	C₂³×C₆₇	—	2	1	no	—
92	C₂³×C₆₇	—	2	1	no	—
93	C₂³×C₆₇	—	4	2	no	—
94	C₂³×C₆₇	C₄×C₈	1	2	yes	—
95	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	—	2	1	no	—
97	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
109	C₂×Dic₆₇	—	2	4	no	—
110	C₂×Dic₆₇	—	2	2	no	—
111	C₂×Dic₆₇	—	4	4	no	—
112	C₂×Dic₆₇	—	4	4	no	—
113	C₂×Dic₆₇	—	4	2	no	—
114	C₂×Dic₆₇	C₂²⋊C₈	1	2	yes	—
115	C₂×Dic₆₇	C₂²⋊C₈	1	2	yes	—
116	C₈×C₆₇	—	2	2	no	—
117	C₈×C₆₇	C₄×D₄	1	2	yes	—
118	C₈×C₆₇	C₄×D₄	1	2	yes	—
119	C₆₇⋊C₈	—	2	2	no	—
120	C₆₇⋊C₈	C₂²⋊C₈	1	4	yes	—
121	C₂²×C₆₇	—	2	2	no	—
122	C₂²×C₆₇	—	2	2	no	—
123	C₂²×C₆₇	—	4	2	no	—
124	C₂²×C₆₇	—	4	2	no	—
125	C₂²×C₆₇	—	4	2	no	—
126	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
127	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₂²×C₆₇	#431	1	1	yes	characteristic
129	C₄×C₆₇	—	2	2	no	—
130	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
131	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
132	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₄×C₆₇	#431	1	1	yes	characteristic
134	Dic₆₇	—	2	2	no	—
135	Dic₆₇	—	4	2	no	—
136	Dic₆₇	#349	1	4	yes	direct product factor
137	C₄×#349	—	67	1	no	maximal, Sylow
138	C₂×C₆₇	—	2	1	no	—
139	C₂×C₆₇	—	2	1	no	—
140	C₂×C₆₇	—	4	2	no	—
141	C₂×C₆₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
142	C₂×C₆₇	C₂×#349	1	2	yes	—
143	C₂×C₄×M₁₆	—	67	1	no	—
144	C₂×#349	—	67	1	no	—
145	C₂×#349	D₆₇	1	1	yes	characteristic
146	C₄×M₃₂	—	67	2	no	—
147	#1377	—	67	2	no	—
148	C₆₇	C₄×#349	1	1	yes	characteristic, complemented, Sylow
149	C₂²×M₁₆	—	67	1	no	—
150	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
151	C₂²×C₄²	—	67	1	no	—
152	C₂×M₃₂	—	67	2	no	—
153	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
154	C₄×M₁₆	—	134	2	no	—
155	C₄×M₁₆	—	134	2	no	—
156	C₂×#97	—	67	1	no	—
157	C₂×#97	—	67	1	no	—
158	C₂×C₄×C₈	—	67	1	no	—
159	C₂×C₄×C₈	—	67	1	no	—
160	#349	—	67	4	no	—
161	#349	Dic₆₇	1	4	yes	direct product factor
162	#346	—	134	2	no	—
163	C₄×C₁₆	—	134	2	no	—
164	C₂³×C₄	—	67	2	no	—
165	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	67	4	no	—
169	C₂×M₁₆	—	134	2	no	—
170	C₂×M₁₆	—	134	2	no	—
171	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
172	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
173	C₂²×C₈	—	67	1	no	—
174	C₂²×C₈	—	67	1	no	—
175	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
177	C₂×C₄²	—	67	1	no	—
178	C₂×C₄²	—	67	2	no	—
179	C₂×C₄²	—	67	1	no	—
180	C₂×C₄²	—	134	1	no	—
181	C₂×C₄²	—	134	1	no	—
182	C₂×C₄²	—	134	2	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	—	67	4	no	—
185	M₃₂	—	134	2	no	—
186	M₃₂	C₂×Dic₆₇	1	4	yes	complemented
187	C₂×C₁₆	—	2	2	no	—
188	C₂×C₁₆	—	134	2	no	—
189	#97	—	134	2	no	—
190	#97	—	134	2	no	—
191	C₄×C₈	—	67	2	no	—
192	C₄×C₈	—	67	2	no	—
193	C₄×C₈	—	134	1	no	—
194	C₄×C₈	—	134	1	no	—
195	C₂⁴	C₈×D₆₇	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	268	4	no	—
201	C₂²×C₄	—	67	2	no	—
202	C₂²×C₄	—	67	2	no	—
203	C₂²×C₄	—	134	2	no	—
204	C₂²×C₄	—	134	2	no	—
205	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
206	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
207	C₂²×C₄	C₈×D₆₇	1	2	yes	—
208	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
209	M₁₆	—	2	4	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	134	8	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	67	2	no	—
215	C₂×C₈	—	67	2	no	—
216	C₂×C₈	—	67	2	no	—
217	C₂×C₈	—	67	2	no	—
218	C₂×C₈	—	134	1	no	—
219	C₂×C₈	—	134	1	no	—
220	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
221	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
222	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
223	C₂×C₈	D₄×D₆₇	1	2	yes	—
224	C₂×C₈	#11761627	1	2	yes	—
225	C₄²	—	67	1	no	—
226	C₄²	—	67	1	no	—
227	C₄²	—	67	2	no	—
228	C₄²	—	134	2	no	—
229	C₄²	—	134	2	no	—
230	C₄²	—	134	1	no	—
231	C₄²	—	134	1	no	—
232	C₁₆	—	2	4	no	—
233	C₁₆	—	134	4	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₈×Dic₆₇	1	2	yes	—
238	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	268	4	no	—
246	C₂×C₄	—	268	4	no	—
247	C₂×C₄	—	268	2	no	—
248	C₂×C₄	—	67	2	no	—
249	C₂×C₄	—	67	2	no	—
250	C₂×C₄	—	134	2	no	—
251	C₂×C₄	—	134	4	no	—
252	C₂×C₄	C₈×Dic₆₇	1	2	yes	—
253	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
254	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
255	C₂×C₄	#14523999	1	1	yes	characteristic
256	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
257	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
258	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	67	4	no	—
261	C₈	—	134	2	no	—
262	C₈	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
263	C₈	D₄×Dic₆₇	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	C₂×C₈×Dic₆₇	1	1	yes	characteristic
270	C₂²	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
271	C₂²	#16863446	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	268	2	no	—
274	C₄	—	67	4	no	—
275	C₄	—	134	2	no	—
276	C₄	C₂²⋊C₄×Dic₆₇	1	1	yes	characteristic
277	C₄	M₁₆×Dic₆₇	1	1	yes	characteristic
278	C₄	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
279	C₄	#16863446	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	4	2	no	—
283	C₂	C₂²⋊C₈×Dic₆₇	1	1	yes	characteristic
284	C₂	#349×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#349×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₆₇	C₂×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#349	D₆₇
hypocenter	C₆₇	C₄×#349
LCS	G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
3. M₃₂×Dic₆₇	C₂	semidirect
4. #22558292	C₂	semidirect
15. #16863178	C₄	semidirect
20. #349×C₆₇	C₄	semidirect
34. #14523983	C₂×C₄	semidirect
50. M₃₂×C₆₇	C₂×C₄	semidirect
136. Dic₆₇	#349	direct (N × H)
148. C₆₇	C₄×#349	semidirect
161. #349	Dic₆₇	direct (N × H)
186. M₃₂	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×Dic₆₇	2
C₂×#16863178	2
M₃₂×Dic₆₇	2
#22558292	2
C₂×#349×C₆₇	2
C₄×#349	67

Subgroups — #349×Dic67 #30770873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #349×Dic₆₇ #30770873