Subgroups — C₄×#16863512 #30854385

Order 17152.

1046 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863512	C₂	1	2	yes	maximal
2	C₄×#14524065	C₂	1	1	yes	maximal, characteristic
3	C₄×#14524051	C₂	1	2	yes	maximal, complemented
4	#22560091	C₂	1	2	yes	maximal, complemented
5	#22560045	C₂	1	4	yes	maximal, complemented
6	C₄×#14523969	C₂	1	2	yes	maximal, complemented
7	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#108×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₂²	1	1	yes	characteristic
10	C₂×#14524051	C₂²	1	4	yes	—
11	C₄×#11761628	C₂²	1	2	yes	—
12	#16863512	C₄	1	16	yes	direct product factor
13	C₄×Dic₂₆₈	C₄	1	2	yes	complemented
14	C₄×#11761617	C₄	1	4	yes	complemented
15	C₄×D₅₃₆	C₄	1	2	yes	complemented
16	#16863434	C₂²	1	1	yes	characteristic
17	C₂×#14523969	C₂²	1	4	yes	—
18	#16863393	C₂²	1	2	yes	—
19	#16863354	C₂²	1	4	yes	—
20	#16863353	—	2	4	no	—
21	C₄×#11761611	—	2	2	no	—
22	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
24	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
25	C₄×C₈×D₆₇	—	2	2	no	—
26	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
27	#16863157	C₂²	1	4	yes	—
28	C₂×#108×C₆₇	C₂²	1	2	yes	—
29	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#341×C₆₇	C₂²	1	2	yes	—
32	C₂×#11761628	C₂³	1	2	yes	—
33	#14524065	C₂×C₄	1	8	yes	—
34	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
35	C₂×#11761617	C₂×C₄	1	2	yes	—
36	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
37	#14524051	—	2	8	no	—
38	#14524051	C₂×C₄	1	16	yes	complemented
39	C₂×#11761611	—	2	4	no	—
40	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
41	C₂×C₈×D₆₇	—	2	4	no	—
42	C₄×Dic₁₃₄	C₂×C₄	1	4	yes	—
43	C₄×D₂₆₈	C₂×C₄	1	4	yes	—
44	C₄×C₂²⋊D₆₇	—	2	4	no	—
45	#14524002	C₂³	1	2	yes	—
46	#14524000	—	2	2	no	—
47	C₂×C₄×Dic₆₇	—	2	2	no	—
48	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
49	C₄²×D₆₇	—	2	2	no	—
50	#14523977	C₂×C₄	1	2	yes	—
51	#14523975	C₂×C₄	1	2	yes	—
52	#14523969	C₂×C₄	1	16	yes	complemented
53	#14523968	C₂×C₄	1	4	yes	—
54	#14523965	C₂×C₄	1	4	yes	—
55	#14523953	—	2	4	no	—
56	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
57	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
58	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
59	C₈×Dic₆₇	—	2	4	no	—
60	C₈×Dic₆₇	—	2	4	no	—
61	C₈×Dic₆₇	C₂×C₄	1	8	yes	complemented
62	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	#108×C₆₇	C₂×C₄	1	8	yes	complemented
67	#97×C₆₇	—	2	2	no	—
68	C₄×C₈×C₆₇	—	2	2	no	—
69	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
71	#11761628	—	2	4	no	—
72	#11761628	C₂²×C₄	1	8	yes	—
73	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
74	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
75	C₂×C₂²⋊D₆₇	—	2	2	no	—
76	C₂²×Dic₆₇	—	2	2	no	—
77	C₂×C₄×D₆₇	—	2	2	no	—
78	C₂×C₄×D₆₇	—	2	4	no	—
79	Dic₂₆₈	C₄²	1	4	yes	complemented
80	#11761617	C₄²	1	8	yes	complemented
81	D₅₃₆	C₄²	1	4	yes	complemented
82	#11761611	—	2	16	no	—
83	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
84	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
85	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
86	C₂×C₆₇⋊C₈	#41	1	4	yes	—
87	C₈×D₆₇	—	2	16	no	—
88	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
89	Dic₆₇⋊C₄	—	2	4	no	—
90	C₂²⋊Dic₆₇	—	2	2	no	—
91	(C₂×D₆₇)⋊C₄	—	2	4	no	—
92	C₄×Dic₆₇	—	2	8	no	—
93	C₄×Dic₆₇	—	2	4	no	—
94	C₄×Dic₆₇	—	2	2	no	—
95	C₄×Dic₆₇	—	2	2	no	—
96	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
97	C₂²×C₄×C₆₇	#41	1	2	yes	—
98	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
99	M₁₆×C₆₇	—	2	4	no	—
100	M₁₆×C₆₇	C₂²×C₄	1	8	yes	—
101	C₂×C₈×C₆₇	—	2	4	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
105	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
108	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
111	C₂²×D₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
116	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
119	C₂²⋊D₆₇	—	2	8	no	—
120	C₂×Dic₆₇	—	2	8	no	—
121	C₂×Dic₆₇	—	2	2	no	—
122	C₂×Dic₆₇	—	2	4	no	—
123	C₂×Dic₆₇	—	2	4	no	—
124	D₂₆₈	—	2	2	no	—
125	D₂₆₈	C₂×C₄²	1	4	yes	—
126	C₄×D₆₇	—	2	8	no	—
127	C₄×D₆₇	—	4	8	no	—
128	Dic₁₃₄	—	2	2	no	—
129	Dic₁₃₄	C₂×C₄²	1	4	yes	—
130	C₈×C₆₇	—	2	8	no	—
131	C₈×C₆₇	—	2	2	no	—
132	C₈×C₆₇	C₂×C₄²	1	2	yes	—
133	C₈×C₆₇	C₂×C₄²	1	2	yes	—
134	C₆₇⋊C₈	C₄×D₄	1	4	yes	—
135	C₆₇⋊C₈	C₄×D₄	1	4	yes	—
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₆₇	C₄×#41	1	2	yes	—
139	C₂×D₆₇	—	2	4	no	—
140	C₂×D₆₇	—	4	4	no	—
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	#443	1	4	yes	complemented
143	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
144	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
145	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
146	Dic₆₇	—	2	8	no	—
147	Dic₆₇	—	2	4	no	—
148	Dic₆₇	—	4	4	no	—
149	C₄×#443	—	67	1	no	maximal, Sylow
150	C₂×C₆₇	—	2	2	no	—
151	C₂×C₆₇	C₄²×D₄	1	1	yes	characteristic
152	C₂×C₆₇	C₂×#443	1	2	yes	—
153	D₆₇	—	4	4	no	—
154	C₂×#443	—	67	2	no	—
155	C₄×#135	—	67	1	no	—
156	C₄×#131	—	67	2	no	—
157	C₄×#108	D₆₇	1	1	yes	characteristic, complemented
158	#1772	—	67	2	no	—
159	#1764	—	67	4	no	—
160	C₄×#104	—	67	2	no	—
161	C₄²×C₈	—	67	1	no	—
162	C₆₇	C₄×#443	1	1	yes	characteristic, complemented, Sylow
163	C₂×#135	—	67	1	no	—
164	C₂×#131	—	67	4	no	—
165	C₄×#41	—	67	2	no	—
166	#443	—	67	16	no	—
167	C₄×Q₁₆	—	67	2	no	—
168	C₄×SD₁₆	—	67	4	no	—
169	C₄×D₈	—	67	2	no	—
170	C₂×#108	C₂×D₆₇	1	2	yes	—
171	#427	—	67	1	no	—
172	C₂×#104	—	67	4	no	—
173	#416	—	67	2	no	—
174	#405	—	67	4	no	—
175	#405	—	134	4	no	—
176	C₄×M₁₆	—	134	2	no	—
177	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
178	C₂×C₄×C₈	—	67	2	no	—
179	C₂×C₄×C₈	—	67	1	no	—
180	C₂×C₄×C₈	—	67	2	no	—
181	C₂×C₄×C₈	—	134	2	no	—
182	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
183	C₄³	—	67	1	no	—
184	#341	C₂×D₆₇	1	2	yes	—
185	#339	—	67	4	no	—
186	C₂×#41	—	67	2	no	—
187	#135	—	67	8	no	—
188	C₂×Q₁₆	—	67	1	no	—
189	C₂×SD₁₆	—	67	2	no	—
190	C₂×D₈	—	67	1	no	—
191	#131	—	67	16	no	—
192	#131	—	134	8	no	—
193	C₂×M₁₆	—	134	4	no	—
194	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
195	C₂²×C₈	—	67	2	no	—
196	C₂²×C₈	—	134	4	no	—
197	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
199	C₄×Q₈	—	67	4	no	—
200	C₄×D₄	—	67	4	no	—
201	C₄×D₄	—	134	4	no	—
202	#117	—	67	2	no	—
203	#117	—	134	2	no	—
204	C₂×C₄²	—	67	2	no	—
205	C₂×C₄²	—	134	2	no	—
206	C₂×C₄²	—	134	2	no	—
207	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
208	#108	C₄×D₆₇	1	8	yes	complemented
209	#107	—	67	2	no	—
210	#106	—	67	2	no	—
211	#104	—	67	16	no	—
212	#103	—	67	4	no	—
213	#102	—	67	4	no	—
214	#97	—	2	2	no	—
215	#97	—	134	4	no	—
216	C₄×C₈	—	2	2	no	—
217	C₄×C₈	—	67	8	no	—
218	C₄×C₈	—	67	2	no	—
219	C₄×C₈	—	67	1	no	—
220	C₄×C₈	—	67	1	no	—
221	C₄×C₈	—	134	4	no	—
222	C₄×C₈	—	134	4	no	—
223	C₄×C₈	C₄×D₆₇	1	2	yes	—
224	C₄×C₈	C₄×D₆₇	1	2	yes	—
225	#41	—	67	8	no	—
226	#41	—	134	4	no	—
227	C₂×Q₈	—	67	2	no	—
228	C₂×D₄	—	67	2	no	—
229	C₂×D₄	—	134	2	no	—
230	C₂²×C₄	—	134	4	no	—
231	C₂²×C₄	—	134	2	no	—
232	C₂²×C₄	—	134	2	no	—
233	C₂²×C₄	#11761629	1	2	yes	—
234	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
235	Q₁₆	—	67	4	no	—
236	SD₁₆	—	67	8	no	—
237	D₈	—	67	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	134	16	no	—
240	M₁₆	C₂×C₄×D₆₇	1	8	yes	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	67	4	no	—
244	C₂×C₈	—	67	2	no	—
245	C₂×C₈	—	67	2	no	—
246	C₂×C₈	—	67	4	no	—
247	C₂×C₈	—	134	16	no	—
248	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
249	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
250	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
251	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
252	C₄⋊C₄	—	67	4	no	—
253	C₄⋊C₄	—	134	4	no	—
254	C₂²⋊C₄	—	134	2	no	—
255	C₂²⋊C₄	—	134	4	no	—
256	C₄²	—	67	4	no	—
257	C₄²	—	134	8	no	—
258	C₄²	—	134	4	no	—
259	C₄²	—	134	2	no	—
260	C₄²	—	134	2	no	—
261	C₄²	C₂×C₄×D₆₇	1	2	yes	—
262	C₄²	D₄×D₆₇	1	1	yes	characteristic
263	C₄²	D₄×D₆₇	1	1	yes	characteristic
264	C₂³	—	134	2	no	—
265	C₂³	C₂×#11761629	1	1	yes	characteristic
266	Q₈	—	67	4	no	—
267	Q₈	—	134	2	no	—
268	D₄	—	67	4	no	—
269	D₄	—	134	8	no	—
270	D₄	—	134	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	268	8	no	—
273	C₂×C₄	—	134	4	no	—
274	C₂×C₄	—	134	8	no	—
275	C₂×C₄	—	134	2	no	—
276	C₂×C₄	—	134	4	no	—
277	C₂×C₄	—	134	8	no	—
278	C₂×C₄	#14524005	1	2	yes	—
279	C₂×C₄	#14524005	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
281	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
282	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
283	C₂×C₄	#41×D₆₇	1	2	yes	—
284	C₈	—	2	8	no	—
285	C₈	—	2	2	no	—
286	C₈	—	67	4	no	—
287	C₈	—	67	4	no	—
288	C₈	C₄²×D₆₇	1	2	yes	—
289	C₈	C₄²×D₆₇	1	2	yes	—
290	C₂²	—	2	2	no	—
291	C₂²	—	268	4	no	—
292	C₂²	—	134	4	no	—
293	C₂²	C₂×#14524005	1	1	yes	characteristic
294	C₂²	C₄×#11761629	1	2	yes	—
295	C₄	—	2	2	no	—
296	C₄	—	268	4	no	—
297	C₄	—	134	8	no	—
298	C₄	—	134	4	no	—
299	C₄	#16863512	1	4	yes	direct product factor
300	C₄	C₂×C₄²×D₆₇	1	1	yes	characteristic
301	C₄	C₄×D₄×D₆₇	1	2	yes	—
302	C₄	#16863864	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	268	4	no	—
305	C₂	C₄×#14524005	1	1	yes	characteristic
306	C₂	C₂×#16863512	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#108×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#108	D₆₇
hypocenter	C₆₇	C₄×#443
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
3. C₄×#14524051	C₂	semidirect
4. #22560091	C₂	semidirect
5. #22560045	C₂	semidirect
6. C₄×#14523969	C₂	semidirect
7. C₄×C₈×Dic₆₇	C₂	semidirect
8. C₄×#108×C₆₇	C₂	semidirect
12. #16863512	C₄	direct (N × H)
13. C₄×Dic₂₆₈	C₄	semidirect
14. C₄×#11761617	C₄	semidirect
15. C₄×D₅₃₆	C₄	semidirect
38. #14524051	C₂×C₄	semidirect
52. #14523969	C₂×C₄	semidirect
61. C₈×Dic₆₇	C₂×C₄	semidirect
66. #108×C₆₇	C₂×C₄	semidirect
79. Dic₂₆₈	C₄²	semidirect
80. #11761617	C₄²	semidirect
81. D₅₃₆	C₄²	semidirect
142. C₄×C₆₇	#443	semidirect
157. C₄×#108	D₆₇	semidirect
162. C₆₇	C₄×#443	semidirect
208. #108	C₄×D₆₇	semidirect
299. C₄	#16863512	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16863512	2
C₄×#14524065	2
C₄×#14524051	2
#22560091	2
#22560045	2
C₄×#14523969	2
C₄×C₈×Dic₆₇	2
C₄×#108×C₆₇	2
C₄×#443	67

Subgroups — C4×#16863512 #30854385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863512 #30854385