Subgroups — C₂×#22558513 #30821538

Order 17152.

994 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11761611	C₂	1	2	yes	maximal
2	C₂×C₄×#11761611	C₂	1	1	yes	maximal, characteristic
3	#22558513	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863118	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11761611	C₄	1	4	yes	—
8	C₂²×#11761611	C₄	1	2	yes	—
9	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
10	C₄×#11761611	C₄	1	4	yes	—
11	C₄×#11761611	C₄	1	8	yes	—
12	C₄×#11761611	C₂²	1	8	yes	—
13	C₄×#11761611	C₂²	1	4	yes	—
14	C₂×#14523953	C₄	1	4	yes	—
15	C₂×#14523953	C₄	1	2	yes	—
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	#16863118	C₂²	1	8	yes	complemented
19	C₈×C₆₇⋊C₈	C₂²	1	4	yes	complemented
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	#322×C₆₇	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₆₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×#11761611	—	2	16	no	—
26	C₂×#11761611	—	2	8	no	—
27	C₂×#11761611	C₂×C₄	1	16	yes	—
28	C₂×#11761611	C₂×C₄	1	8	yes	—
29	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
33	C₄²×D₆₇	C₂×C₄	1	4	yes	—
34	C₄²×D₆₇	C₂×C₄	1	2	yes	—
35	C₄²×D₆₇	C₂³	1	2	yes	—
36	#14523953	C₂×C₄	1	16	yes	—
37	#14523953	C₂×C₄	1	8	yes	—
38	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
39	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
40	C₄×C₆₇⋊C₈	C₂³	1	4	yes	—
41	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
46	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₆₇	C₂³	1	2	yes	—
48	C₄×C₈×C₆₇	C₂³	1	4	yes	—
49	C₂³×D₆₇	C₄²	1	1	yes	characteristic
50	C₂²×Dic₆₇	C₄²	1	1	yes	characteristic
51	C₂²×Dic₆₇	C₄²	1	2	yes	—
52	C₂×C₄×D₆₇	—	2	8	no	—
53	C₂×C₄×D₆₇	—	2	4	no	—
54	C₂×C₄×D₆₇	C₄²	1	2	yes	—
55	C₂×C₄×D₆₇	C₄²	1	4	yes	—
56	C₂×C₄×D₆₇	C₂²×C₄	1	4	yes	—
57	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
58	#11761611	—	2	16	no	—
59	#11761611	—	2	32	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₂×C₆₇⋊C₈	M₁₆	1	16	yes	complemented
62	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
63	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
64	C₄×Dic₆₇	C₄²	1	2	yes	—
65	C₄×Dic₆₇	C₄²	1	4	yes	—
66	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
67	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	—	2	8	no	—
71	C₂×C₈×C₆₇	M₁₆	1	8	yes	complemented
72	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
73	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
74	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
75	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
78	C₂²×D₆₇	—	2	2	no	—
79	C₂²×D₆₇	—	2	4	no	—
80	C₂²×D₆₇	C₂×C₄²	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×M₁₆	1	8	yes	—
84	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
85	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
86	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
87	C₂×Dic₆₇	—	2	2	no	—
88	C₂×Dic₆₇	—	2	4	no	—
89	C₂×Dic₆₇	C₂×C₄²	1	2	yes	—
90	C₂×Dic₆₇	C₂×C₄²	1	4	yes	—
91	C₄×D₆₇	—	2	8	no	—
92	C₄×D₆₇	—	2	4	no	—
93	C₄×D₆₇	—	2	4	no	—
94	C₄×D₆₇	—	2	8	no	—
95	C₈×C₆₇	—	2	8	no	—
96	C₈×C₆₇	C₂×M₁₆	1	8	yes	complemented
97	C₆₇⋊C₈	—	2	4	no	—
98	C₆₇⋊C₈	C₂×M₁₆	1	16	yes	complemented
99	C₂²×C₆₇	C₄×M₁₆	1	2	yes	—
100	C₂²×C₆₇	C₄×M₁₆	1	4	yes	—
101	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
102	C₂×D₆₇	—	2	2	no	—
103	C₂×D₆₇	—	2	4	no	—
104	C₂×D₆₇	—	4	4	no	—
105	C₄×C₆₇	C₄×M₁₆	1	2	yes	—
106	C₄×C₆₇	C₄×M₁₆	1	4	yes	—
107	C₄×C₆₇	C₂²×M₁₆	1	4	yes	—
108	C₄×C₆₇	C₂²×M₁₆	1	2	yes	—
109	Dic₆₇	—	2	2	no	—
110	Dic₆₇	—	2	4	no	—
111	C₂×#1455	—	67	1	no	maximal, Sylow
112	C₂×C₆₇	#1455	1	4	yes	complemented
113	C₂×C₆₇	C₂×C₄×M₁₆	1	1	yes	characteristic
114	C₂×C₆₇	C₂×C₄×M₁₆	1	2	yes	—
115	D₆₇	—	4	2	no	—
116	C₂×C₄×M₁₆	—	67	2	no	—
117	C₂×C₄×M₁₆	—	67	1	no	—
118	#1455	—	67	8	no	—
119	C₂×#322	—	67	2	no	—
120	C₂×#322	D₆₇	1	1	yes	characteristic, complemented
121	C₂×C₈²	—	67	1	no	—
122	C₆₇	C₂×#1455	1	1	yes	characteristic, complemented, Sylow
123	C₂²×M₁₆	—	67	4	no	—
124	C₂²×M₁₆	—	67	2	no	—
125	C₂²×C₄²	—	67	1	no	—
126	C₄×M₁₆	—	67	8	no	—
127	C₄×M₁₆	—	67	4	no	—
128	C₄×M₁₆	—	67	8	no	—
129	C₄×M₁₆	—	67	4	no	—
130	C₂×#97	—	67	4	no	—
131	C₂×#97	—	67	2	no	—
132	C₂×C₄×C₈	—	67	2	no	—
133	C₂×C₄×C₈	—	67	1	no	—
134	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
136	#322	—	67	8	no	—
137	#322	C₂×D₆₇	1	4	yes	complemented
138	C₈²	—	67	4	no	—
139	C₂³×C₄	—	67	2	no	—
140	C₂³×C₄	—	67	1	no	—
141	C₂×M₁₆	—	67	16	no	—
142	C₂×M₁₆	—	67	8	no	—
143	C₂×M₁₆	—	134	16	no	—
144	C₂×M₁₆	—	134	8	no	—
145	C₂²×C₈	—	67	4	no	—
146	C₂²×C₈	—	67	2	no	—
147	C₂²×C₈	C₄×D₆₇	1	4	yes	—
148	C₂²×C₈	C₄×D₆₇	1	2	yes	—
149	C₂×C₄²	—	67	2	no	—
150	C₂×C₄²	—	67	1	no	—
151	C₂×C₄²	—	67	4	no	—
152	C₂×C₄²	—	67	2	no	—
153	C₂×C₄²	—	67	2	no	—
154	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
155	#97	—	67	16	no	—
156	#97	—	67	8	no	—
157	C₄×C₈	—	67	4	no	—
158	C₄×C₈	—	67	2	no	—
159	C₄×C₈	—	67	4	no	—
160	C₄×C₈	—	67	2	no	—
161	C₄×C₈	C₄×D₆₇	1	4	yes	—
162	C₄×C₈	C₄×D₆₇	1	2	yes	—
163	C₄×C₈	C₂²×D₆₇	1	2	yes	—
164	C₄×C₈	C₂²×D₆₇	1	4	yes	—
165	C₂⁴	—	67	1	no	—
166	C₂²×C₄	—	67	1	no	—
167	C₂²×C₄	—	67	2	no	—
168	C₂²×C₄	—	67	2	no	—
169	C₂²×C₄	—	67	4	no	—
170	C₂²×C₄	—	67	4	no	—
171	C₂²×C₄	—	67	2	no	—
172	C₂²×C₄	—	134	8	no	—
173	C₂²×C₄	—	134	4	no	—
174	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
175	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
176	M₁₆	—	134	32	no	—
177	M₁₆	—	134	16	no	—
178	C₂×C₈	—	2	8	no	—
179	C₂×C₈	—	67	16	no	—
180	C₂×C₈	—	67	8	no	—
181	C₂×C₈	—	67	4	no	—
182	C₂×C₈	—	134	4	no	—
183	C₂×C₈	#11761611	1	8	yes	complemented
184	C₂×C₈	C₂×C₄×D₆₇	1	8	yes	—
185	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
186	C₄²	—	67	2	no	—
187	C₄²	—	67	4	no	—
188	C₄²	—	67	4	no	—
189	C₄²	—	67	2	no	—
190	C₄²	C₂×C₄×D₆₇	1	2	yes	—
191	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
192	C₄²	C₂³×D₆₇	1	1	yes	characteristic
193	C₂³	—	67	2	no	—
194	C₂³	—	134	2	no	—
195	C₂³	—	134	4	no	—
196	C₂³	C₄²×D₆₇	1	1	yes	characteristic
197	C₂×C₄	—	67	2	no	—
198	C₂×C₄	—	67	4	no	—
199	C₂×C₄	—	134	2	no	—
200	C₂×C₄	—	134	4	no	—
201	C₂×C₄	—	134	4	no	—
202	C₂×C₄	—	134	8	no	—
203	C₂×C₄	—	134	8	no	—
204	C₂×C₄	—	134	4	no	—
205	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
206	C₂×C₄	C₄²×D₆₇	1	2	yes	—
207	C₂×C₄	M₁₆×D₆₇	1	8	yes	—
208	C₂×C₄	C₂×#11761611	1	4	yes	—
209	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
210	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
211	C₈	—	2	8	no	—
212	C₈	—	67	16	no	—
213	C₈	—	134	4	no	—
214	C₈	C₂×#11761611	1	8	yes	complemented
215	C₂²	—	268	4	no	—
216	C₂²	—	134	2	no	—
217	C₂²	—	134	4	no	—
218	C₂²	C₄×#11761611	1	2	yes	—
219	C₂²	#16863348	1	4	yes	—
220	C₂²	C₂×C₄²×D₆₇	1	1	yes	characteristic
221	C₄	—	134	2	no	—
222	C₄	—	134	4	no	—
223	C₄	C₄×#11761611	1	2	yes	—
224	C₄	#16863348	1	4	yes	—
225	C₄	C₂×M₁₆×D₆₇	1	4	yes	—
226	C₄	C₂²×#11761611	1	2	yes	—
227	C₂	—	268	2	no	—
228	C₂	#22558513	1	4	yes	direct product factor
229	C₂	C₂×C₄×#11761611	1	1	yes	characteristic
230	C₂	C₂×#16863348	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂²×C₄²
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄²
prime radical	C₂³×D₆₇	C₄²
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	C₂×#1455
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #22558513	C₂	direct (N × H)
4. C₂×#16863118	C₂	semidirect
5. C₂×C₈×C₆₇⋊C₈	C₂	semidirect
6. C₂×#322×C₆₇	C₂	semidirect
18. #16863118	C₂²	semidirect
19. C₈×C₆₇⋊C₈	C₂²	semidirect
22. #322×C₆₇	C₂²	semidirect
61. C₂×C₆₇⋊C₈	M₁₆	semidirect
71. C₂×C₈×C₆₇	M₁₆	semidirect
96. C₈×C₆₇	C₂×M₁₆	semidirect
98. C₆₇⋊C₈	C₂×M₁₆	semidirect
112. C₂×C₆₇	#1455	semidirect
120. C₂×#322	D₆₇	semidirect
122. C₆₇	C₂×#1455	semidirect
137. #322	C₂×D₆₇	semidirect
183. C₂×C₈	#11761611	semidirect
214. C₈	C₂×#11761611	semidirect
228. C₂	#22558513	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×C₄×#11761611	2
#22558513	2
C₂×#16863118	2
C₂×C₈×C₆₇⋊C₈	2
C₂×#322×C₆₇	2
C₂×#1455	67

Subgroups — C2×#22558513 #30821538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558513 #30821538