Subgroups — #30770949

Order 17152.

420 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863173	C₂	1	2	yes	maximal, complemented
3	#22559985	C₂	1	1	yes	maximal, characteristic
4	#22558288	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2112	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	—	2	4	no	—
7	C₂×#14523983	C₄	1	2	yes	complemented
8	C₂×#14523983	C₂²	1	1	yes	characteristic
9	C₂×#14523983	C₂²	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₁₆	C₄	1	2	yes	complemented
11	C₂×C₂²⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
12	C₂×#14523953	C₂²	1	2	yes	—
13	C₄×C₂²⋊Dic₆₇	C₄	1	1	yes	characteristic
14	#16863173	—	2	4	no	—
15	#16863150	C₄	1	2	yes	—
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
18	C₂²⋊C₁₆×C₆₇	C₄	1	2	yes	complemented
19	C₂²⋊C₁₆×C₆₇	C₄	1	2	yes	complemented
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
22	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
23	#14523983	—	2	4	no	—
24	#14523983	—	2	2	no	—
25	#14523983	D₄	1	4	yes	—
26	C₂×C₆₇⋊C₁₆	—	2	4	no	—
27	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
29	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
30	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
31	#14523953	—	2	2	no	—
32	#14523953	—	2	2	no	—
33	C₆₇⋊#95	—	2	1	no	—
34	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	—	2	2	no	—
36	C₂²×C₈×C₆₇	—	2	2	no	—
37	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	M₃₂×C₆₇	—	2	2	no	—
44	M₃₂×C₆₇	—	2	2	no	—
45	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
46	C₂²×Dic₆₇	—	2	2	no	—
47	C₂×C₆₇⋊C₈	—	2	2	no	—
48	C₂×C₆₇⋊C₈	—	2	2	no	—
49	C₂×C₆₇⋊C₈	—	2	2	no	—
50	C₂²⋊Dic₆₇	—	4	2	no	—
51	C₄×Dic₆₇	—	2	2	no	—
52	C₄×Dic₆₇	—	2	2	no	—
53	C₆₇⋊C₁₆	—	2	2	no	—
54	C₆₇⋊C₁₆	C₂²⋊C₄	1	4	yes	complemented
55	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	—	2	2	no	—
57	C₂²×C₄×C₆₇	—	2	2	no	—
58	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	—	2	1	no	—
66	C₂×C₈×C₆₇	—	2	1	no	—
67	C₂×C₈×C₆₇	—	2	1	no	—
68	C₂×C₈×C₆₇	—	2	1	no	—
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
73	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₆₇	#41	1	2	yes	—
77	C₁₆×C₆₇	—	2	4	no	—
78	C₂³×C₆₇	—	2	2	no	—
79	C₂³×C₆₇	—	2	2	no	—
80	C₂³×C₆₇	—	4	1	no	—
81	C₂³×C₆₇	#97	1	1	yes	characteristic
82	C₂³×C₆₇	#97	1	1	yes	characteristic
83	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₆₇	—	2	2	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	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#131	1	2	yes	—
97	C₂×Dic₆₇	—	4	2	no	—
98	C₂×Dic₆₇	—	4	2	no	—
99	C₂×Dic₆₇	—	4	2	no	—
100	C₈×C₆₇	—	2	1	no	—
101	C₈×C₆₇	—	2	1	no	—
102	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₈×C₆₇	#117	1	1	yes	characteristic
104	C₈×C₆₇	C₄×D₄	1	2	yes	—
105	C₆₇⋊C₈	—	4	2	no	—
106	C₂²×C₆₇	—	2	1	no	—
107	C₂²×C₆₇	—	2	1	no	—
108	C₂²×C₆₇	—	2	1	no	—
109	C₂²×C₆₇	—	2	1	no	—
110	C₂²×C₆₇	—	2	1	no	—
111	C₂²×C₆₇	—	2	1	no	—
112	C₂²×C₆₇	—	2	1	no	—
113	C₂²×C₆₇	—	4	4	no	—
114	C₂²×C₆₇	#347	1	2	yes	complemented
115	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
116	C₂²×C₆₇	#435	1	2	yes	—
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	—	2	1	no	—
119	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₆₇	#405	1	1	yes	characteristic
121	C₄×C₆₇	#435	1	2	yes	—
122	Dic₆₇	—	8	2	no	—
123	#12616	—	67	1	no	maximal, Sylow
124	C₂×C₆₇	—	2	2	no	—
125	C₂×C₆₇	—	2	2	no	—
126	C₂×C₆₇	—	4	1	no	—
127	C₂×C₆₇	#1752	1	1	yes	characteristic
128	C₂×C₆₇	C₂×#347	1	1	yes	characteristic
129	C₂×C₆₇	#2110	1	1	yes	characteristic
130	C₂²×M₃₂	—	67	1	no	—
131	#2112	D₆₇	1	1	yes	characteristic
132	C₂×#347	—	67	2	no	—
133	#1752	—	67	1	no	—
134	#1375	—	67	2	no	—
135	C₆₇	#12616	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
137	C₂×M₃₂	—	67	1	no	—
138	C₂×M₃₂	—	67	2	no	—
139	C₂×M₃₂	—	67	1	no	—
140	C₂×M₃₂	—	134	4	no	—
141	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
142	C₂²×C₁₆	—	67	2	no	—
143	C₂×C₂²⋊C₈	—	67	1	no	—
144	C₂×#97	—	67	2	no	—
145	C₄×C₂²⋊C₄	—	67	1	no	—
146	C₂²⋊C₁₆	Dic₆₇	1	2	yes	complemented
147	C₂²⋊C₁₆	Dic₆₇	1	2	yes	complemented
148	#347	—	134	4	no	—
149	#336	—	67	2	no	—
150	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	67	2	no	—
154	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
155	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
156	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
158	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	—	134	1	no	—
161	C₂×C₄²	—	67	2	no	—
162	M₃₂	—	2	2	no	—
163	M₃₂	—	2	2	no	—
164	M₃₂	—	67	4	no	—
165	M₃₂	—	134	4	no	—
166	M₃₂	—	134	2	no	—
167	C₂×C₁₆	—	67	2	no	—
168	C₂×C₁₆	—	67	2	no	—
169	C₂×C₁₆	—	134	4	no	—
170	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
171	C₂²⋊C₈	—	134	1	no	—
172	C₂²⋊C₈	—	134	1	no	—
173	#97	—	134	2	no	—
174	#97	—	134	2	no	—
175	#95	—	134	1	no	—
176	C₂⁴	#11761611	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	134	2	no	—
180	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
182	C₂²×C₄	#11761611	1	1	yes	characteristic
183	C₂²×C₄	#11761611	1	1	yes	characteristic
184	C₂²×C₄	#11761611	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	134	2	no	—
194	C₂×C₈	—	134	2	no	—
195	C₂×C₈	—	134	2	no	—
196	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
197	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
198	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
199	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
200	C₂×C₈	D₄×D₆₇	1	2	yes	—
201	C₂×C₈	#11761627	1	2	yes	—
202	C₂²⋊C₄	—	268	2	no	—
203	C₄²	—	134	2	no	—
204	C₄²	—	134	2	no	—
205	C₁₆	—	2	4	no	—
206	C₁₆	—	67	4	no	—
207	C₁₆	—	134	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	1	no	—
211	C₂³	#14523953	1	1	yes	characteristic
212	C₂³	#14523953	1	1	yes	characteristic
213	C₂³	C₂×#11761611	1	1	yes	characteristic
214	C₂×C₄	—	2	2	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	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	268	2	no	—
222	C₂×C₄	—	268	2	no	—
223	C₂×C₄	—	268	2	no	—
224	C₂×C₄	#14523953	1	1	yes	characteristic
225	C₂×C₄	#14523953	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
227	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
228	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
229	C₂×C₄	#14524051	1	2	yes	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	268	2	no	—
233	C₈	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
234	C₈	#14523999	1	1	yes	characteristic
235	C₈	D₄×Dic₆₇	1	2	yes	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	#16863173	1	2	yes	complemented
245	C₂²	C₂×#14523953	1	1	yes	characteristic
246	C₂²	#16863464	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	536	2	no	—
250	C₄	C₂²⋊C₄×Dic₆₇	1	1	yes	characteristic
251	C₄	#16863354	1	1	yes	characteristic
252	C₄	#16863464	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	4	1	no	—
256	C₂	#22559988	1	1	yes	characteristic
257	C₂	C₂×#16863173	1	1	yes	characteristic
258	C₂	#22561813	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2112	C₂
socle	C₂²×C₆₇	C₂×#97
prime radical	C₂⁴×C₆₇	M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2112	D₆₇
hypocenter	C₆₇	#12616
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #2112

Complements

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

N	H ≅ G/N	type
2. C₂×#16863173	C₂	semidirect
4. #22558288	C₂	semidirect
7. C₂×#14523983	C₄	semidirect
10. C₂²×C₆₇⋊C₁₆	C₄	semidirect
18. C₂²⋊C₁₆×C₆₇	C₄	semidirect
19. C₂²⋊C₁₆×C₆₇	C₄	semidirect
54. C₆₇⋊C₁₆	C₂²⋊C₄	semidirect
114. C₂²×C₆₇	#347	semidirect
135. C₆₇	#12616	semidirect
146. C₂²⋊C₁₆	Dic₆₇	semidirect
147. C₂²⋊C₁₆	Dic₆₇	semidirect
244. C₂²	#16863173	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523983	2
C₂×#16863173	2
#22559985	2
#22558288	2
C₆₇×#2112	2
#12616	67