Subgroups — C₂×#22561814 #31008489

Order 17152.

738 conjugacy classes of subgroups.

The subgroup lattice has 238 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	2	yes	maximal
2	#22561814	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863174	C₂	1	4	yes	maximal, complemented
4	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₄	1	4	yes	complemented
6	C₂×#14523983	C₄	1	8	yes	complemented
7	C₂×#14523983	C₂²	1	4	yes	—
8	C₂×#14523983	C₂²	1	8	yes	—
9	C₂²×C₆₇⋊C₁₆	C₄	1	4	yes	complemented
10	#16863174	C₂²	1	16	yes	complemented
11	C₂²×M₁₆×C₆₇	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
14	#405×C₆₇	C₄	1	4	yes	—
15	#405×C₆₇	C₂²	1	4	yes	—
16	C₂×#97×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
18	#14523983	C₂×C₄	1	8	yes	—
19	#14523983	C₂×C₄	1	16	yes	complemented
20	#14523983	C₂³	1	8	yes	—
21	C₂×C₆₇⋊C₁₆	C₂×C₄	1	8	yes	—
22	C₂×C₆₇⋊C₁₆	C₂×C₄	1	16	yes	complemented
23	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆×C₆₇	—	2	2	no	—
25	C₂×M₁₆×C₆₇	—	2	2	no	—
26	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
35	#117×C₆₇	—	2	2	no	—
36	#117×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
40	#97×C₆₇	C₂×C₄	1	4	yes	—
41	#97×C₆₇	C₂³	1	4	yes	—
42	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
43	C₆₇⋊C₁₆	C₂²×C₄	1	16	yes	—
44	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	—	2	2	no	—
46	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
47	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
48	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
50	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
51	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
54	M₁₆×C₆₇	—	2	8	no	—
55	C₂×C₈×C₆₇	—	2	2	no	—
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	C₄²	1	4	yes	—
58	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₄²	1	2	yes	—
60	C₂×C₈×C₆₇	C₄²	1	4	yes	—
61	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₆₇	—	2	4	no	—
71	C₂²⋊C₄×C₆₇	—	2	4	no	—
72	C₄²×C₆₇	—	2	2	no	—
73	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
74	C₂³×C₆₇	—	4	2	no	—
75	C₂³×C₆₇	#97	1	2	yes	—
76	C₂³×C₆₇	#97	1	2	yes	—
77	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	—	4	4	no	—
81	C₂×C₄×C₆₇	#97	1	2	yes	—
82	C₂×C₄×C₆₇	#97	1	4	yes	—
83	C₂×C₄×C₆₇	#97	1	4	yes	—
84	C₂×C₄×C₆₇	#97	1	2	yes	—
85	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
86	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
91	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
94	C₈×C₆₇	—	2	4	no	—
95	C₈×C₆₇	C₂×C₄²	1	2	yes	—
96	C₈×C₆₇	C₂×C₄²	1	2	yes	—
97	C₈×C₆₇	C₂×C₄²	1	2	yes	—
98	C₈×C₆₇	C₂³×C₄	1	2	yes	—
99	C₂²×C₆₇	—	2	2	no	—
100	C₂²×C₆₇	—	2	4	no	—
101	C₂²×C₆₇	—	4	2	no	—
102	C₂²×C₆₇	—	4	2	no	—
103	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
104	C₂²×C₆₇	C₂×#97	1	4	yes	—
105	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
106	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
107	C₄×C₆₇	—	4	4	no	—
108	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
109	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
110	C₄×C₆₇	C₂×#97	1	4	yes	—
111	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
112	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
113	C₂×#2110	—	67	1	no	maximal, Sylow
114	C₂×C₆₇	—	2	1	no	—
115	C₂×C₆₇	—	2	1	no	—
116	C₂×C₆₇	—	2	4	no	—
117	C₂×C₆₇	#2110	1	2	yes	complemented
118	C₂×C₆₇	C₂²×#97	1	1	yes	characteristic
119	C₂²×M₃₂	—	67	2	no	—
120	C₂×#405	D₆₇	1	1	yes	characteristic
121	#2110	—	67	8	no	—
122	C₂×#347	—	67	4	no	—
123	C₆₇	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
124	C₂²×M₁₆	Dic₆₇	1	1	yes	characteristic
125	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
126	C₂×#117	Dic₆₇	1	1	yes	characteristic
127	C₂×M₃₂	—	67	4	no	—
128	C₂×M₃₂	—	67	8	no	—
129	C₂×M₃₂	—	67	4	no	—
130	C₂×M₃₂	—	67	8	no	—
131	C₂²×C₁₆	—	67	4	no	—
132	#405	Dic₆₇	1	4	yes	—
133	#405	C₂×D₆₇	1	4	yes	—
134	C₂×#97	C₂×D₆₇	1	2	yes	—
135	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
136	#347	—	67	16	no	—
137	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
138	C₂×M₁₆	—	2	2	no	—
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
141	C₂²×C₈	C₄×D₆₇	1	2	yes	—
142	C₂²×C₈	C₄×D₆₇	1	4	yes	—
143	C₂²×C₈	C₄×D₆₇	1	2	yes	—
144	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
145	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
146	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
149	#117	—	2	2	no	—
150	#117	C₂×Dic₆₇	1	4	yes	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₂²⋊C₄	—	2	1	no	—
153	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
154	M₃₂	—	67	8	no	—
155	M₃₂	—	67	16	no	—
156	M₃₂	—	67	8	no	—
157	C₂×C₁₆	—	67	8	no	—
158	C₂×C₁₆	—	67	16	no	—
159	#97	C₂×Dic₆₇	1	4	yes	—
160	#97	C₂²×D₆₇	1	4	yes	—
161	C₄×C₈	C₂×Dic₆₇	1	8	yes	—
162	C₂⁴	#11761612	1	1	yes	characteristic
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
165	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
166	C₂²×C₄	#11761612	1	1	yes	characteristic
167	C₂²×C₄	#11761612	1	2	yes	—
168	C₂²×C₄	#11761612	1	2	yes	—
169	C₂²×C₄	#11761612	1	2	yes	—
170	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
172	M₁₆	—	2	8	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
176	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
177	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
178	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
179	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
180	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
181	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
182	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
183	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
184	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
185	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
186	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
187	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
188	C₄⋊C₄	—	2	4	no	—
189	C₂²⋊C₄	—	2	4	no	—
190	C₄²	—	2	2	no	—
191	C₄²	C₂²×Dic₆₇	1	4	yes	—
192	C₁₆	—	67	16	no	—
193	C₂³	—	4	2	no	—
194	C₂³	#14523954	1	2	yes	—
195	C₂³	#14523954	1	2	yes	—
196	C₂³	C₂×#11761612	1	1	yes	characteristic
197	C₂³	C₂×#11761612	1	2	yes	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	#14523954	1	2	yes	—
201	C₂×C₄	#14523954	1	4	yes	—
202	C₂×C₄	#14523954	1	4	yes	—
203	C₂×C₄	#14523954	1	2	yes	—
204	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
205	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
207	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
208	C₂×C₄	C₂×#11761612	1	2	yes	—
209	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
210	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
211	C₂×C₄	C₂×#11761612	1	4	yes	—
212	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
213	C₈	—	2	4	no	—
214	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
215	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
216	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
217	C₈	C₂²×C₄×D₆₇	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₂×#14523954	1	1	yes	characteristic
223	C₂²	C₂×#14523954	1	4	yes	—
224	C₂²	C₂×#14523954	1	1	yes	characteristic
225	C₂²	C₂²×#11761612	1	1	yes	characteristic
226	C₄	—	4	4	no	—
227	C₄	C₂×#14523954	1	1	yes	characteristic
228	C₄	C₂×#14523954	1	4	yes	—
229	C₄	C₂×#14523954	1	1	yes	characteristic
230	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
231	C₄	C₂²×#11761612	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	2	4	no	—
235	C₂	#22561814	1	2	yes	direct product factor
236	C₂	C₂²×#14523954	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#405×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#97
prime radical	C₂⁴×C₆₇	M₁₆
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#405	D₆₇
hypocenter	C₆₇	C₂×#2110
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #22561814	C₂	direct (N × H)
3. C₂×#16863174	C₂	semidirect
5. C₂×#14523983	C₄	semidirect
6. C₂×#14523983	C₄	semidirect
9. C₂²×C₆₇⋊C₁₆	C₄	semidirect
10. #16863174	C₂²	semidirect
19. #14523983	C₂×C₄	semidirect
22. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
117. C₂×C₆₇	#2110	semidirect
123. C₆₇	C₂×#2110	semidirect
235. C₂	#22561814	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
#22561814	2
C₂×#16863174	2
C₂×#405×C₆₇	2
C₂×#2110	67

Subgroups — C2×#22561814 #31008489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22561814 #31008489