Subgroups — C₂×#22558187 #30765629

Order 17152.

382 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#22558187	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
6	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
7	C₈×Dic₁₃₄	—	2	1	no	—
8	C₈×Dic₁₃₄	—	2	1	no	—
9	C₈×Dic₁₃₄	C₄	1	2	yes	—
10	C₈×Dic₁₃₄	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
12	C₂×Dic₆₇⋊C₈	—	2	1	no	—
13	C₂×C₈×Dic₆₇	—	2	1	no	—
14	C₄×C₆₇⋊C₁₆	—	2	1	no	—
15	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₆₇	—	2	1	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₁₆×C₆₇	—	2	1	no	—
19	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
20	C₂²×Dic₁₃₄	C₈	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	—	2	1	no	—
22	C₄×Dic₁₃₄	—	2	1	no	—
23	C₄×Dic₁₃₄	—	2	1	no	—
24	C₄×Dic₁₃₄	C₈	1	2	yes	—
25	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
27	C₂×Dic₆₇⋊C₄	—	2	1	no	—
28	C₂×C₄×Dic₆₇	—	2	1	no	—
29	C₂×C₆₇⋊C₁₆	—	2	4	no	—
30	C₂×C₆₇⋊C₁₆	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
32	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
33	Dic₆₇⋊C₈	—	2	2	no	—
34	Dic₆₇⋊C₈	—	2	2	no	—
35	C₈×Dic₆₇	—	2	2	no	—
36	C₈×Dic₆₇	—	2	2	no	—
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₆₇	—	2	2	no	—
41	C₂×C₁₆×C₆₇	—	4	2	no	—
42	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₃₄	—	2	1	no	—
47	C₂×Dic₁₃₄	—	2	2	no	—
48	C₂×Dic₁₃₄	C₁₆	1	4	yes	complemented
49	C₂×Dic₁₃₄	C₂×C₈	1	2	yes	—
50	C₂²×Dic₆₇	—	2	1	no	—
51	C₂²×Dic₆₇	—	2	1	no	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	—	4	2	no	—
54	C₄⋊Dic₆₇	—	2	1	no	—
55	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
56	Dic₆₇⋊C₄	—	2	2	no	—
57	Dic₆₇⋊C₄	—	2	2	no	—
58	C₄×Dic₆₇	—	2	2	no	—
59	C₄×Dic₆₇	—	2	2	no	—
60	C₆₇⋊C₁₆	—	2	4	no	—
61	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
68	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
69	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
74	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₁₆×C₆₇	—	4	2	no	—
76	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#103	1	2	yes	—
82	C₂×C₄×C₆₇	#104	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
84	C₂×C₄×C₆₇	M₃₂	1	2	yes	—
85	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
88	C₂×Dic₆₇	—	2	4	no	—
89	C₂×Dic₆₇	—	2	2	no	—
90	C₂×Dic₆₇	—	2	2	no	—
91	C₂×Dic₆₇	—	4	2	no	—
92	Dic₁₃₄	—	2	4	no	—
93	Dic₁₃₄	—	2	2	no	—
94	Dic₁₃₄	C₂×C₁₆	1	4	yes	complemented
95	C₈×C₆₇	—	2	1	no	—
96	C₈×C₆₇	—	2	1	no	—
97	C₈×C₆₇	#103	1	2	yes	—
98	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
100	C₆₇⋊C₈	—	4	2	no	—
101	C₂²×C₆₇	Q₈⋊C₈	1	2	yes	—
102	C₂²×C₆₇	C₂²⋊C₁₆	1	2	yes	—
103	C₂²×C₆₇	#350	1	2	yes	—
104	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
105	C₄×C₆₇	—	2	1	no	—
106	C₄×C₆₇	—	2	1	no	—
107	C₄×C₆₇	Q₈⋊C₈	1	2	yes	—
108	C₄×C₆₇	C₂²⋊C₁₆	1	2	yes	—
109	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
110	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
111	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
112	C₄×C₆₇	C₂×M₃₂	1	1	yes	characteristic
113	Dic₆₇	—	2	4	no	—
114	Dic₆₇	—	4	2	no	—
115	C₂×Q₈⋊C₁₆	—	67	1	no	maximal, Sylow
116	C₂×C₆₇	Q₈⋊C₁₆	1	4	yes	complemented
117	C₂×C₆₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
118	C₂×C₆₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
119	C₂×C₆₇	C₂×#350	1	1	yes	characteristic
120	C₂×C₈×Q₈	—	67	1	no	—
121	C₂×C₄⋊C₁₆	D₆₇	1	1	yes	characteristic
122	C₂×C₄×C₁₆	—	67	1	no	—
123	Q₈⋊C₁₆	—	67	4	no	—
124	C₆₇	C₂×Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
125	C₂×C₄×Q₈	—	67	1	no	—
126	C₂²×C₁₆	—	2	1	no	—
127	C₂²×C₁₆	—	134	1	no	—
128	C₈×Q₈	—	67	2	no	—
129	C₈×Q₈	—	67	2	no	—
130	C₈×Q₈	—	134	1	no	—
131	C₈×Q₈	—	134	1	no	—
132	C₂×C₄⋊C₈	—	67	1	no	—
133	C₂×C₄⋊C₈	—	134	1	no	—
134	C₂×C₄×C₈	—	134	1	no	—
135	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
136	C₄⋊C₁₆	—	2	1	no	—
137	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
138	C₄×C₁₆	—	67	2	no	—
139	C₄×C₁₆	—	134	1	no	—
140	C₂²×Q₈	—	67	1	no	—
141	C₂²×C₈	—	134	1	no	—
142	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
143	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
144	C₄×Q₈	—	67	2	no	—
145	C₄×Q₈	—	67	2	no	—
146	C₄×Q₈	—	134	1	no	—
147	C₄×Q₈	—	134	1	no	—
148	C₂×C₄⋊C₄	—	67	1	no	—
149	C₂×C₄⋊C₄	—	134	1	no	—
150	C₂×C₄²	—	134	1	no	—
151	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	4	2	no	—
154	C₂×C₁₆	—	134	4	no	—
155	C₂×C₁₆	—	134	2	no	—
156	C₄⋊C₈	—	67	2	no	—
157	C₄⋊C₈	—	134	2	no	—
158	C₄⋊C₈	—	134	2	no	—
159	C₄⋊C₈	—	134	1	no	—
160	C₄×C₈	—	134	2	no	—
161	C₄×C₈	—	134	2	no	—
162	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
163	C₄×C₈	D₂₆₈	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
165	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
166	C₂×Q₈	—	67	4	no	—
167	C₂×Q₈	—	67	2	no	—
168	C₂×Q₈	—	134	1	no	—
169	C₂×Q₈	—	134	2	no	—
170	C₂²×C₄	—	134	1	no	—
171	C₂²×C₄	—	134	1	no	—
172	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
173	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
174	C₂²×C₄	#11761611	1	1	yes	characteristic
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	268	2	no	—
177	C₂×C₈	—	134	2	no	—
178	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
179	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
180	C₂×C₈	#11761616	1	2	yes	—
181	C₂×C₈	#11761618	1	2	yes	—
182	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
183	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
184	C₄⋊C₄	—	67	2	no	—
185	C₄⋊C₄	—	134	1	no	—
186	C₄⋊C₄	—	134	2	no	—
187	C₄⋊C₄	—	134	2	no	—
188	C₄²	—	134	2	no	—
189	C₄²	—	134	2	no	—
190	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
191	C₄²	C₈×D₆₇	1	1	yes	characteristic
192	C₄²	#11761611	1	1	yes	characteristic
193	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
194	C₁₆	—	4	2	no	—
195	C₁₆	—	134	4	no	—
196	C₂³	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
197	Q₈	—	67	4	no	—
198	Q₈	—	134	2	no	—
199	Q₈	—	134	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	268	2	no	—
202	C₂×C₄	—	134	2	no	—
203	C₂×C₄	—	134	2	no	—
204	C₂×C₄	—	134	4	no	—
205	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
206	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
207	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
208	C₂×C₄	#14523966	1	2	yes	—
209	C₂×C₄	#14523969	1	2	yes	—
210	C₂×C₄	C₁₆×D₆₇	1	2	yes	—
211	C₂×C₄	#14523982	1	2	yes	—
212	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
213	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
214	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	268	2	no	—
218	C₈	#14523966	1	2	yes	—
219	C₈	C₂×#11761616	1	1	yes	characteristic
220	C₈	C₂×#11761618	1	1	yes	characteristic
221	C₂²	#16863126	1	2	yes	—
222	C₂²	(C₂×D₆₇)⋊C₁₆	1	2	yes	—
223	C₂²	#16863181	1	2	yes	—
224	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	268	2	no	—
228	C₄	—	134	4	no	—
229	C₄	#16863126	1	2	yes	—
230	C₄	(C₂×D₆₇)⋊C₁₆	1	2	yes	—
231	C₄	C₂×#14523966	1	1	yes	characteristic
232	C₄	C₂×#14523969	1	1	yes	characteristic
233	C₄	C₂×C₁₆×D₆₇	1	1	yes	characteristic
234	C₄	C₂×#14523982	1	1	yes	characteristic
235	C₂	#22558187	1	4	yes	direct product factor
236	C₂	C₂×#16863126	1	1	yes	characteristic
237	C₂	C₂×(C₂×D₆₇)⋊C₁₆	1	1	yes	characteristic
238	C₂	C₂×#16863181	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×C₄⋊C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²⋊C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	C₂×Q₈⋊C₁₆
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #22558187	C₂	direct (N × H)
48. C₂×Dic₁₃₄	C₁₆	semidirect
94. Dic₁₃₄	C₂×C₁₆	semidirect
116. C₂×C₆₇	Q₈⋊C₁₆	semidirect
124. C₆₇	C₂×Q₈⋊C₁₆	semidirect
235. C₂	#22558187	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×C₈×Dic₁₃₄	2
C₂×C₄×C₆₇⋊C₁₆	2
#22558187	2
C₂×C₄⋊C₁₆×C₆₇	2
C₂×Q₈⋊C₁₆	67

Subgroups — C2×#22558187 #30765629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558187 #30765629