Subgroups — C₂×#22561998 #31008866

Order 17152.

718 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	C₂²×#14523977	C₂	1	1	yes	maximal, characteristic
3	#22561998	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863223	C₂	1	2	yes	maximal, complemented
5	C₂×#16863221	C₂	1	2	yes	maximal, complemented
6	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₆₇	—	2	1	no	—
8	C₂×#14523983	C₄	1	8	yes	complemented
9	C₂×#14523983	C₂²	1	4	yes	—
10	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
11	C₂×#14523977	—	2	1	no	—
12	C₂×#14523977	—	2	2	no	—
13	C₂×#14523977	C₂²	1	2	yes	—
14	C₂×#14523977	C₂²	1	4	yes	—
15	#16863223	C₂²	1	8	yes	complemented
16	#16863221	C₂²	1	8	yes	complemented
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#117×C₆₇	—	2	1	no	—
19	#427×C₆₇	—	2	2	no	—
20	#427×C₆₇	C₂²	1	4	yes	—
21	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#106×C₆₇	—	2	1	no	—
24	C₂³×Dic₆₇	—	4	1	no	—
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	—	2	4	no	—
27	C₂×C₄⋊Dic₆₇	—	4	1	no	—
28	C₂×C₄⋊Dic₆₇	—	4	2	no	—
29	#14523983	C₂×C₄	1	16	yes	complemented
30	C₂×C₆₇⋊C₁₆	C₂×C₄	1	8	yes	complemented
31	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
32	#14523977	—	2	4	no	—
33	#14523977	—	2	2	no	—
34	#14523977	C₂³	1	4	yes	—
35	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	D₄	1	2	yes	—
39	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	#117×C₆₇	—	2	4	no	—
43	#117×C₆₇	—	4	2	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
46	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
47	C₂×C₄²×C₆₇	—	4	1	no	—
48	#107×C₆₇	—	2	1	no	—
49	#107×C₆₇	—	2	1	no	—
50	#107×C₆₇	C₂³	1	2	yes	—
51	#107×C₆₇	C₂³	1	2	yes	—
52	#106×C₆₇	—	2	4	no	—
53	C₂²×Dic₆₇	—	4	4	no	—
54	C₂²×Dic₆₇	—	4	2	no	—
55	C₂²×Dic₆₇	—	8	4	no	—
56	C₄⋊Dic₆₇	—	2	4	no	—
57	C₄⋊Dic₆₇	—	4	4	no	—
58	C₄⋊Dic₆₇	—	4	2	no	—
59	C₆₇⋊C₁₆	C₂²×C₄	1	8	yes	complemented
60	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	—	4	1	no	—
62	C₂²×C₄×C₆₇	—	4	1	no	—
63	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	D₈	1	2	yes	—
66	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
70	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
71	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
77	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₄⋊C₄×C₆₇	—	2	2	no	—
80	C₄⋊C₄×C₆₇	—	4	1	no	—
81	C₄⋊C₄×C₆₇	—	4	1	no	—
82	C₂²⋊C₄×C₆₇	—	4	4	no	—
83	C₄²×C₆₇	—	4	4	no	—
84	C₂³×C₆₇	—	2	2	no	—
85	C₂³×C₆₇	—	2	2	no	—
86	C₂³×C₆₇	#107	1	4	yes	—
87	C₂³×C₆₇	C₂×D₈	1	2	yes	—
88	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₆₇	—	4	2	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	—	8	2	no	—
92	C₂×C₄×C₆₇	—	8	2	no	—
93	C₂×C₄×C₆₇	#107	1	8	yes	—
94	C₂×C₄×C₆₇	#107	1	4	yes	—
95	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
100	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
101	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×Dic₆₇	—	4	4	no	—
103	C₂×Dic₆₇	—	8	8	no	—
104	C₂×Dic₆₇	—	8	4	no	—
105	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
107	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂²×C₆₇	—	2	2	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	2	8	no	—
112	C₂²×C₆₇	C₂×#107	1	4	yes	—
113	C₂²×C₆₇	C₂×#107	1	2	yes	—
114	C₂²×C₆₇	#509	1	2	yes	—
115	C₂²×C₆₇	#510	1	2	yes	—
116	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
117	C₄×C₆₇	—	8	2	no	—
118	C₄×C₆₇	—	8	2	no	—
119	C₄×C₆₇	C₂×#107	1	4	yes	—
120	C₄×C₆₇	C₂×#107	1	2	yes	—
121	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
123	Dic₆₇	—	8	4	no	—
124	C₂×#2159	—	67	1	no	maximal, Sylow
125	C₂×C₆₇	—	2	4	no	—
126	C₂×C₆₇	#2159	1	4	yes	complemented
127	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
128	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
129	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
130	C₂²×M₃₂	—	67	1	no	—
131	C₂×#427	D₆₇	1	1	yes	characteristic
132	C₂²×#107	—	67	1	no	—
133	#2159	—	67	8	no	—
134	C₂×#367	—	67	2	no	—
135	C₂×#366	—	67	2	no	—
136	C₆₇	C₂×#2159	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
138	C₂×#117	—	2	1	no	—
139	C₂²×C₄⋊C₄	—	134	1	no	—
140	C₂×M₃₂	—	67	8	no	—
141	C₂×M₃₂	—	67	4	no	—
142	C₂²×C₁₆	—	67	2	no	—
143	#427	—	2	2	no	—
144	#427	C₂×D₆₇	1	4	yes	—
145	C₂×#107	—	67	2	no	—
146	C₂×#107	—	67	4	no	—
147	C₂×#107	—	134	1	no	—
148	C₂×#107	—	134	2	no	—
149	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
150	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
151	C₂×#106	—	2	1	no	—
152	#367	—	67	8	no	—
153	#366	—	67	8	no	—
154	C₂³×C₄	—	268	1	no	—
155	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
156	C₂²×C₈	Dic₁₃₄	1	4	yes	—
157	C₂²×C₈	C₄×D₆₇	1	4	yes	—
158	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
160	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
162	#117	—	2	4	no	—
163	#117	—	4	2	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	268	2	no	—
167	C₂×C₄⋊C₄	—	268	1	no	—
168	C₂×C₄⋊C₄	—	134	4	no	—
169	C₂×C₄⋊C₄	—	134	2	no	—
170	C₂×C₂²⋊C₄	—	4	1	no	—
171	C₂×C₄²	—	4	1	no	—
172	M₃₂	—	67	16	no	—
173	C₂×C₁₆	—	67	8	no	—
174	C₂×C₁₆	—	67	4	no	—
175	#107	—	2	1	no	—
176	#107	—	2	1	no	—
177	#107	—	67	4	no	—
178	#107	—	134	4	no	—
179	#107	—	134	2	no	—
180	#107	C₂²×D₆₇	1	2	yes	—
181	#107	C₂²×D₆₇	1	2	yes	—
182	#106	—	2	4	no	—
183	C₂⁴	#11761613	1	1	yes	characteristic
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	268	4	no	—
187	C₂²×C₄	—	268	2	no	—
188	C₂²×C₄	—	536	4	no	—
189	C₂²×C₄	Dic₆₇⋊C₄	1	4	yes	—
190	C₂²×C₄	#11761613	1	2	yes	—
191	C₂²×C₄	#11761613	1	1	yes	characteristic
192	C₂²×C₄	#11761618	1	4	yes	—
193	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
195	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
196	C₂×C₈	Dic₆₇⋊C₄	1	8	yes	—
197	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
198	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
199	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
200	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
201	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
202	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
203	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₄⋊C₄	—	4	1	no	—
208	C₄⋊C₄	—	268	4	no	—
209	C₄⋊C₄	—	268	2	no	—
210	C₄⋊C₄	—	134	4	no	—
211	C₂²⋊C₄	—	4	4	no	—
212	C₄²	—	4	4	no	—
213	C₁₆	—	67	8	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	#14523976	1	4	yes	—
217	C₂³	C₂×#11761613	1	2	yes	—
218	C₂³	C₂×#11761613	1	1	yes	characteristic
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	268	4	no	—
224	C₂×C₄	—	536	8	no	—
225	C₂×C₄	—	536	4	no	—
226	C₂×C₄	#14523976	1	8	yes	—
227	C₂×C₄	#14523976	1	4	yes	—
228	C₂×C₄	C₂×Dic₆₇⋊C₄	1	4	yes	—
229	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
230	C₂×C₄	C₂×#11761613	1	2	yes	—
231	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
232	C₂×C₄	C₂×#11761618	1	4	yes	—
233	C₂×C₄	C₂×#11761618	1	2	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
235	C₈	C₂×Dic₆₇⋊C₄	1	2	yes	—
236	C₈	C₂×Dic₆₇⋊C₄	1	4	yes	—
237	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
238	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	8	no	—
242	C₂²	C₂×#14523976	1	4	yes	—
243	C₂²	C₂×#14523976	1	2	yes	—
244	C₂²	#16863831	1	2	yes	—
245	C₂²	#16863836	1	2	yes	—
246	C₂²	C₂²×#11761613	1	1	yes	characteristic
247	C₄	—	8	2	no	—
248	C₄	—	8	2	no	—
249	C₄	—	536	4	no	—
250	C₄	C₂×#14523976	1	4	yes	—
251	C₄	C₂×#14523976	1	2	yes	—
252	C₄	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
253	C₄	C₂²×#11761618	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	#22561998	1	4	yes	direct product factor
256	C₂	C₂²×#14523976	1	1	yes	characteristic
257	C₂	C₂×#16863831	1	1	yes	characteristic
258	C₂	C₂×#16863836	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#427×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#427	D₆₇
hypocenter	C₆₇	C₂×#2159
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #22561998	C₂	direct (N × H)
4. C₂×#16863223	C₂	semidirect
5. C₂×#16863221	C₂	semidirect
8. C₂×#14523983	C₄	semidirect
15. #16863223	C₂²	semidirect
16. #16863221	C₂²	semidirect
29. #14523983	C₂×C₄	semidirect
30. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
59. C₆₇⋊C₁₆	C₂²×C₄	semidirect
126. C₂×C₆₇	#2159	semidirect
136. C₆₇	C₂×#2159	semidirect
255. C₂	#22561998	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523983	2
C₂²×#14523977	2
#22561998	2
C₂×#16863223	2
C₂×#16863221	2
C₂×#427×C₆₇	2
C₂×#2159	67

Subgroups — C2×#22561998 #31008866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22561998 #31008866