Subgroups — C₂×#22558150 #30764971

Order 17152.

394 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863422	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#22558150	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₂²⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	C₄	1	1	yes	characteristic
6	C₂×#14524002	C₄	1	1	yes	characteristic
7	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
8	#16863422	—	2	1	no	—
9	#16863422	—	2	1	no	—
10	#16863422	C₄	1	2	yes	—
11	#16863422	C₂²	1	2	yes	—
12	C₂×Dic₆₇⋊C₈	—	2	1	no	—
13	C₂×Dic₆₇⋊C₈	—	2	1	no	—
14	C₂²⋊(C₆₇⋊C₁₆)	—	2	1	no	—
15	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₆₇	—	2	1	no	—
18	C₂²⋊C₁₆×C₆₇	—	2	1	no	—
19	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
20	C₂×#11761612	—	2	1	no	—
21	C₂×#11761612	—	4	2	no	—
22	C₂×#11761612	C₂×C₄	1	2	yes	—
23	C₂²×C₆₇⋊C₈	—	2	1	no	—
24	#14524002	—	2	1	no	—
25	#14524002	—	2	1	no	—
26	#14524002	C₈	1	2	yes	—
27	#14524002	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
29	C₂×C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
31	C₂×C₄×Dic₆₇	—	2	1	no	—
32	C₂×C₆₇⋊C₁₆	—	2	2	no	—
33	C₂×C₆₇⋊C₁₆	—	4	2	no	—
34	Dic₆₇⋊C₈	—	2	2	no	—
35	Dic₆₇⋊C₈	—	2	2	no	—
36	Dic₆₇⋊C₈	—	2	2	no	—
37	Dic₆₇⋊C₈	—	2	2	no	—
38	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	—	4	2	no	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₆₇	—	2	2	no	—
47	C₂×C₁₆×C₆₇	—	4	2	no	—
48	C₂²×Dic₆₇	—	2	1	no	—
49	C₂²×Dic₆₇	—	2	1	no	—
50	#11761612	—	4	2	no	—
51	#11761612	—	4	2	no	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	—	4	2	no	—
54	C₄⋊Dic₆₇	—	2	1	no	—
55	C₄⋊Dic₆₇	—	2	1	no	—
56	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
57	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
58	C₂²⋊Dic₆₇	—	2	2	no	—
59	C₂²⋊Dic₆₇	—	2	2	no	—
60	C₄×Dic₆₇	—	2	2	no	—
61	C₄×Dic₆₇	—	2	2	no	—
62	C₆₇⋊C₁₆	—	4	2	no	—
63	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	4	1	no	—
75	C₂×C₈×C₆₇	—	4	1	no	—
76	C₂×C₈×C₆₇	—	4	1	no	—
77	C₂×C₈×C₆₇	—	4	1	no	—
78	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₁₆×C₆₇	—	4	2	no	—
83	C₂³×C₆₇	—	2	2	no	—
84	C₂³×C₆₇	—	2	2	no	—
85	C₂³×C₆₇	—	2	2	no	—
86	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
87	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
88	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	1	no	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	1	no	—
96	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#99	1	2	yes	—
99	C₂×C₄×C₆₇	#101	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
102	C₂×Dic₆₇	—	2	2	no	—
103	C₂×Dic₆₇	—	2	2	no	—
104	C₂×Dic₆₇	—	4	2	no	—
105	C₂×Dic₆₇	—	4	2	no	—
106	C₈×C₆₇	—	2	1	no	—
107	C₈×C₆₇	—	2	1	no	—
108	C₈×C₆₇	—	2	1	no	—
109	C₈×C₆₇	—	2	1	no	—
110	C₆₇⋊C₈	—	4	2	no	—
111	C₂²×C₆₇	—	2	1	no	—
112	C₂²×C₆₇	—	2	1	no	—
113	C₂²×C₆₇	—	2	1	no	—
114	C₂²×C₆₇	—	2	1	no	—
115	C₂²×C₆₇	—	2	1	no	—
116	C₂²×C₆₇	—	2	1	no	—
117	C₂²×C₆₇	—	4	4	no	—
118	C₂²×C₆₇	#324	1	2	yes	—
119	C₂²×C₆₇	#350	1	2	yes	—
120	C₂²×C₆₇	#350	1	2	yes	—
121	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₄×C₆₇	—	2	1	no	—
123	C₄×C₆₇	—	2	1	no	—
124	C₄×C₆₇	#324	1	2	yes	—
125	C₄×C₆₇	C₂×#99	1	1	yes	characteristic
126	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
127	Dic₆₇	—	4	2	no	—
128	Dic₆₇	—	4	2	no	—
129	C₂×#1323	—	67	1	no	maximal, Sylow
130	C₂×C₆₇	—	4	1	no	—
131	C₂×C₆₇	—	4	1	no	—
132	C₂×C₆₇	#1323	1	4	yes	complemented
133	C₂×C₆₇	C₂×#324	1	1	yes	characteristic
134	C₂×C₆₇	C₂×#350	1	1	yes	characteristic
135	C₂×C₆₇	C₂×#350	1	1	yes	characteristic
136	C₂×#424	—	67	1	no	—
137	C₂×C₂²⋊C₁₆	—	67	1	no	—
138	C₂×C₂²⋊C₁₆	D₆₇	1	1	yes	characteristic
139	#1323	—	67	4	no	—
140	C₆₇	C₂×#1323	1	1	yes	characteristic, complemented, Sylow
141	C₂²×M₁₆	—	67	1	no	—
142	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
143	C₂×#117	—	67	1	no	—
144	C₂²×C₁₆	—	2	1	no	—
145	C₂²×C₁₆	—	134	1	no	—
146	#424	—	67	2	no	—
147	#424	—	67	2	no	—
148	#424	—	134	1	no	—
149	#424	—	134	1	no	—
150	C₂×C₄⋊C₈	—	134	1	no	—
151	C₂×C₄⋊C₈	—	134	1	no	—
152	C₂²⋊C₁₆	—	2	1	no	—
153	C₂²⋊C₁₆	—	67	2	no	—
154	C₂²⋊C₁₆	—	134	1	no	—
155	C₂²⋊C₁₆	C₂×D₆₇	1	2	yes	—
156	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
157	C₂×M₁₆	—	268	2	no	—
158	C₂×M₁₆	—	67	2	no	—
159	C₂×M₁₆	—	134	1	no	—
160	C₂²×C₈	—	4	2	no	—
161	C₂²×C₈	—	134	1	no	—
162	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
163	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
164	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
166	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
168	#117	—	67	2	no	—
169	#117	—	67	2	no	—
170	#117	—	134	1	no	—
171	#117	—	134	1	no	—
172	C₂×C₄⋊C₄	—	67	1	no	—
173	C₂×C₄⋊C₄	—	67	1	no	—
174	C₂×C₂²⋊C₄	—	134	1	no	—
175	C₂×C₄²	—	134	1	no	—
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	4	2	no	—
178	C₂×C₁₆	—	268	2	no	—
179	C₂×C₁₆	—	134	2	no	—
180	C₄⋊C₈	—	134	2	no	—
181	C₄⋊C₈	—	134	2	no	—
182	C₄⋊C₈	—	134	2	no	—
183	C₄⋊C₈	—	134	2	no	—
184	C₂⁴	#11761611	1	1	yes	characteristic
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	134	1	no	—
188	C₂²×C₄	—	134	1	no	—
189	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
192	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
193	C₂²×C₄	#11761611	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
195	M₁₆	—	268	2	no	—
196	M₁₆	—	268	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	268	2	no	—
204	C₂×C₈	—	134	2	no	—
205	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
206	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
207	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
208	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
209	C₄⋊C₄	—	67	2	no	—
210	C₄⋊C₄	—	67	2	no	—
211	C₄⋊C₄	—	134	1	no	—
212	C₄⋊C₄	—	134	1	no	—
213	C₂²⋊C₄	—	134	2	no	—
214	C₂²⋊C₄	—	134	2	no	—
215	C₄²	—	134	2	no	—
216	C₄²	—	134	2	no	—
217	C₁₆	—	4	2	no	—
218	C₁₆	—	268	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
223	C₂³	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
224	C₂³	C₂×#11761611	1	1	yes	characteristic
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	268	2	no	—
233	C₂×C₄	—	268	2	no	—
234	C₂×C₄	—	134	2	no	—
235	C₂×C₄	—	134	2	no	—
236	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
237	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
238	C₂×C₄	#14523957	1	2	yes	—
239	C₂×C₄	#14523961	1	2	yes	—
240	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
241	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	268	2	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	4	no	—
254	C₂²	#16863121	1	2	yes	—
255	C₂²	#16863179	1	2	yes	—
256	C₂²	#16863181	1	2	yes	—
257	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	268	2	no	—
261	C₄	—	268	2	no	—
262	C₄	#16863121	1	2	yes	—
263	C₄	C₂×#14523957	1	1	yes	characteristic
264	C₄	C₂×#14523961	1	1	yes	characteristic
265	C₂	—	4	1	no	—
266	C₂	—	4	1	no	—
267	C₂	#22558150	1	4	yes	direct product factor
268	C₂	C₂×#16863121	1	1	yes	characteristic
269	C₂	C₂×#16863179	1	1	yes	characteristic
270	C₂	C₂×#16863181	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₆₇)⋊C₄
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×C₂²⋊C₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂²⋊C₈
prime radical	C₂⁴×C₆₇	M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₆₇
hypocenter	C₆₇	C₂×#1323
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #22558150	C₂	direct (N × H)
132. C₂×C₆₇	#1323	semidirect
140. C₆₇	C₂×#1323	semidirect
267. C₂	#22558150	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×#16863422	2
C₂×C₂²⋊(C₆₇⋊C₁₆)	2
#22558150	2
C₂×C₂²⋊C₁₆×C₆₇	2
C₂×#1323	67

Subgroups — C2×#22558150 #30764971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558150 #30764971