Subgroups — C₂×#22559015 #30825201

Order 17152.

734 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863520	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#22559015	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863126	C₂	1	2	yes	maximal
5	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₆₈	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₃₄	C₂²	1	2	yes	—
9	#16863520	—	2	4	no	—
10	#16863520	C₂²	1	8	yes	—
11	C₄×Dic₂₆₈	C₄	1	4	yes	—
12	C₄×Dic₂₆₈	C₂²	1	4	yes	—
13	C₂×#14523977	C₄	1	1	yes	characteristic
14	C₂×Dic₆₇⋊C₈	—	2	2	no	—
15	C₂×#14523968	C₄	1	2	yes	—
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₆₇	—	2	2	no	—
19	#16863126	C₂²	1	8	yes	—
20	C₈×C₆₇⋊C₈	C₂²	1	4	yes	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#335×C₆₇	C₂²	1	4	yes	—
24	C₂²×Dic₁₃₄	C₂×C₄	1	2	yes	—
25	C₂×Dic₂₆₈	C₈	1	8	yes	complemented
26	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
27	C₂²×C₆₇⋊C₈	—	2	1	no	—
28	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
29	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
30	C₄×Dic₁₃₄	—	2	2	no	—
31	C₄×Dic₁₃₄	—	2	2	no	—
32	C₄×Dic₁₃₄	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₃₄	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
35	C₂×Dic₆₇⋊C₄	—	2	2	no	—
36	C₂×C₄×Dic₆₇	—	2	2	no	—
37	#14523977	C₂×C₄	1	4	yes	—
38	Dic₆₇⋊C₈	—	2	8	no	—
39	#14523968	C₂×C₄	1	8	yes	—
40	C₄×C₆₇⋊C₈	—	2	1	no	—
41	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
42	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
43	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
44	C₈×Dic₆₇	—	2	8	no	—
45	C₂²×C₈×C₆₇	—	2	2	no	—
46	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₆₇	—	2	2	no	—
50	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
51	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₆₇	C₂³	1	2	yes	—
53	C₂×Dic₁₃₄	—	2	2	no	—
54	C₂×Dic₁₃₄	—	2	4	no	—
55	C₂×Dic₁₃₄	C₂×C₈	1	8	yes	—
56	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
57	C₂²×Dic₆₇	—	2	2	no	—
58	C₂²×Dic₆₇	—	2	2	no	—
59	Dic₂₆₈	C₂×C₈	1	16	yes	complemented
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₂×C₆₇⋊C₈	—	2	2	no	—
62	C₂×C₆₇⋊C₈	—	2	2	no	—
63	C₂×C₆₇⋊C₈	Q₁₆	1	4	yes	complemented
64	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
65	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
66	C₄⋊Dic₆₇	—	2	2	no	—
67	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
68	Dic₆₇⋊C₄	—	2	4	no	—
69	Dic₆₇⋊C₄	—	2	4	no	—
70	C₄×Dic₆₇	—	2	4	no	—
71	C₄×Dic₆₇	—	2	4	no	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
75	C₂×C₈×C₆₇	—	2	2	no	—
76	C₂×C₈×C₆₇	—	2	4	no	—
77	C₂×C₈×C₆₇	—	4	4	no	—
78	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
79	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
81	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₆₇	#41	1	1	yes	characteristic
84	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
85	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₆₇	#131	1	2	yes	—
92	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄×C₆₇	#135	1	2	yes	—
94	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
97	C₂×Dic₆₇	—	2	4	no	—
98	C₂×Dic₆₇	—	2	8	no	—
99	C₂×Dic₆₇	—	2	4	no	—
100	C₂×Dic₆₇	—	4	4	no	—
101	Dic₁₃₄	—	2	4	no	—
102	Dic₁₃₄	—	2	8	no	—
103	Dic₁₃₄	C₂²×C₈	1	8	yes	—
104	C₈×C₆₇	—	2	2	no	—
105	C₈×C₆₇	—	4	4	no	—
106	C₈×C₆₇	C₂²×C₈	1	4	yes	—
107	C₆₇⋊C₈	—	2	4	no	—
108	C₆₇⋊C₈	—	2	2	no	—
109	C₆₇⋊C₈	C₂×Q₁₆	1	4	yes	complemented
110	C₂²×C₆₇	C₈×D₄	1	2	yes	—
111	C₂²×C₆₇	C₄×Q₁₆	1	2	yes	—
112	C₂²×C₆₇	#443	1	2	yes	—
113	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
114	C₄×C₆₇	—	2	1	no	—
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	C₈×D₄	1	2	yes	—
117	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
118	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
119	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
120	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
121	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
122	Dic₆₇	—	2	8	no	—
123	Dic₆₇	—	4	4	no	—
124	C₂×C₈×Q₁₆	—	67	1	no	maximal, Sylow
125	C₂×C₆₇	C₈×Q₁₆	1	4	yes	complemented
126	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
127	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
128	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
129	C₂×C₈×Q₈	—	67	2	no	—
130	C₂×C₄×Q₁₆	—	67	1	no	—
131	C₈×Q₁₆	—	67	8	no	—
132	C₂×#335	D₆₇	1	1	yes	characteristic
133	C₂×Q₈⋊C₈	—	67	2	no	—
134	C₂×C₈²	—	67	1	no	—
135	C₆₇	C₂×C₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₁₆	—	67	1	no	—
137	C₂×C₄×Q₈	—	67	2	no	—
138	C₈×Q₈	—	67	8	no	—
139	C₈×Q₈	—	134	4	no	—
140	C₄×Q₁₆	—	67	4	no	—
141	C₄×Q₁₆	—	67	4	no	—
142	C₂×#107	—	67	1	no	—
143	C₂×C₄⋊C₈	—	134	2	no	—
144	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
145	C₂×#103	—	67	2	no	—
146	C₂×C₄×C₈	—	67	1	no	—
147	C₂×C₄×C₈	—	67	1	no	—
148	C₂×C₄×C₈	—	134	2	no	—
149	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
150	#335	C₂×D₆₇	1	4	yes	—
151	Q₈⋊C₈	—	67	8	no	—
152	C₈²	—	67	4	no	—
153	C₂²×Q₈	—	67	2	no	—
154	C₂×Q₁₆	—	67	8	no	—
155	C₂×Q₁₆	—	67	4	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	67	1	no	—
158	C₂²×C₈	—	67	1	no	—
159	C₂²×C₈	—	134	1	no	—
160	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
161	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
162	C₄×Q₈	—	67	4	no	—
163	C₄×Q₈	—	67	4	no	—
164	C₄×Q₈	—	134	2	no	—
165	C₄×Q₈	—	134	2	no	—
166	C₂×C₄⋊C₄	—	67	2	no	—
167	C₂×C₄⋊C₄	—	134	2	no	—
168	C₂×C₄²	—	134	2	no	—
169	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
170	#107	—	67	4	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	134	8	no	—
173	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
174	#103	—	67	8	no	—
175	C₄×C₈	—	67	2	no	—
176	C₄×C₈	—	67	2	no	—
177	C₄×C₈	—	67	2	no	—
178	C₄×C₈	—	134	8	no	—
179	C₄×C₈	—	134	1	no	—
180	C₄×C₈	C₄×D₆₇	1	2	yes	—
181	C₄×C₈	C₂²×D₆₇	1	2	yes	—
182	C₂×Q₈	—	67	8	no	—
183	C₂×Q₈	—	67	4	no	—
184	C₂×Q₈	—	134	2	no	—
185	C₂×Q₈	—	134	4	no	—
186	C₂²×C₄	—	134	2	no	—
187	C₂²×C₄	—	134	2	no	—
188	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
189	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
190	C₂²×C₄	#11761629	1	1	yes	characteristic
191	Q₁₆	—	67	16	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	67	4	no	—
196	C₂×C₈	—	67	2	no	—
197	C₂×C₈	—	67	2	no	—
198	C₂×C₈	—	134	4	no	—
199	C₂×C₈	—	134	2	no	—
200	C₂×C₈	—	134	2	no	—
201	C₂×C₈	C₈×D₆₇	1	4	yes	—
202	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
203	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
204	C₄⋊C₄	—	67	4	no	—
205	C₄⋊C₄	—	134	2	no	—
206	C₄⋊C₄	—	134	4	no	—
207	C₄⋊C₄	—	134	4	no	—
208	C₄²	—	134	4	no	—
209	C₄²	—	134	4	no	—
210	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
211	C₄²	D₄×D₆₇	1	1	yes	characteristic
212	C₄²	#11761629	1	1	yes	characteristic
213	C₄²	C₂³×D₆₇	1	1	yes	characteristic
214	C₂³	#14524005	1	1	yes	characteristic
215	Q₈	—	67	8	no	—
216	Q₈	—	134	4	no	—
217	Q₈	—	134	8	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	268	4	no	—
220	C₂×C₄	—	134	4	no	—
221	C₂×C₄	—	134	4	no	—
222	C₂×C₄	—	134	8	no	—
223	C₂×C₄	#14524005	1	1	yes	characteristic
224	C₂×C₄	#14524005	1	1	yes	characteristic
225	C₂×C₄	#14524005	1	1	yes	characteristic
226	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
227	C₂×C₄	#14524051	1	2	yes	—
228	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
229	C₂×C₄	#14524062	1	2	yes	—
230	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
232	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	4	4	no	—
235	C₈	—	67	4	no	—
236	C₈	—	134	4	no	—
237	C₈	—	134	2	no	—
238	C₈	C₂×C₈×D₆₇	1	4	yes	—
239	C₂²	#16863461	1	2	yes	—
240	C₂²	#16863490	1	2	yes	—
241	C₂²	#16863512	1	2	yes	—
242	C₂²	C₂×#14524005	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	268	4	no	—
246	C₄	—	134	8	no	—
247	C₄	#16863461	1	2	yes	—
248	C₄	#16863490	1	2	yes	—
249	C₄	C₂²×C₈×D₆₇	1	1	yes	characteristic
250	C₄	C₂×#14524051	1	1	yes	characteristic
251	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
252	C₄	C₂×#14524062	1	1	yes	characteristic
253	C₂	#22559015	1	4	yes	direct product factor
254	C₂	C₂×#16863461	1	1	yes	characteristic
255	C₂	C₂×#16863490	1	1	yes	characteristic
256	C₂	C₂×#16863512	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₄×C₆₇	C₂³×C₈
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#335×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#335	D₆₇
hypocenter	C₆₇	C₂×C₈×Q₁₆
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #22559015	C₂	direct (N × H)
25. C₂×Dic₂₆₈	C₈	semidirect
59. Dic₂₆₈	C₂×C₈	semidirect
63. C₂×C₆₇⋊C₈	Q₁₆	semidirect
109. C₆₇⋊C₈	C₂×Q₁₆	semidirect
125. C₂×C₆₇	C₈×Q₁₆	semidirect
135. C₆₇	C₂×C₈×Q₁₆	semidirect
253. C₂	#22559015	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×#16863520	2
C₂×C₄×Dic₂₆₈	2
#22559015	2
C₂×#16863126	2
C₂×C₈×C₆₇⋊C₈	2
C₂×#335×C₆₇	2
C₂×C₈×Q₁₆	67

Subgroups — C2×#22559015 #30825201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559015 #30825201