Subgroups — C₂×#22562385 #31010567

Order 17152.

526 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863798	C₂	1	1	yes	maximal, characteristic, complemented
2	#22562385	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863220	C₂	1	2	yes	maximal, complemented
4	C₂×#16863217	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863206	C₂	1	2	yes	maximal
6	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14524035	—	2	1	no	—
8	C₂×#14523983	C₂²	1	2	yes	—
9	#16863798	—	2	2	no	—
10	#16863798	C₂²	1	4	yes	complemented
11	C₂×#14523977	—	2	1	no	—
12	C₂×#14523975	C₂²	1	2	yes	—
13	#16863220	C₂²	1	8	yes	complemented
14	#16863217	C₂²	1	4	yes	complemented
15	#16863206	C₂²	1	8	yes	—
16	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#131×C₆₇	—	2	1	no	—
18	#443×C₆₇	—	2	2	no	—
19	#443×C₆₇	C₂²	1	4	yes	—
20	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#104×C₆₇	—	2	1	no	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#14524035	—	2	4	no	—
24	#14524035	—	4	2	no	—
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₂×Dic₆₇⋊C₄	—	4	2	no	—
27	#14523983	—	2	2	no	—
28	#14523983	C₂³	1	4	yes	—
29	C₂×C₆₇⋊C₁₆	—	2	2	no	—
30	#14523977	—	2	4	no	—
31	#14523975	—	2	2	no	—
32	#14523975	C₂³	1	4	yes	—
33	C₂×#41×C₆₇	—	2	1	no	—
34	#135×C₆₇	—	2	1	no	—
35	#135×C₆₇	—	2	1	no	—
36	#135×C₆₇	Q₈	1	2	yes	—
37	#135×C₆₇	C₂³	1	2	yes	—
38	C₂×Q₁₆×C₆₇	Q₈	1	1	yes	characteristic
39	C₂×SD₁₆×C₆₇	—	2	1	no	—
40	C₂×D₈×C₆₇	Q₈	1	1	yes	characteristic
41	#131×C₆₇	—	2	4	no	—
42	#131×C₆₇	—	4	2	no	—
43	C₂×M₁₆×C₆₇	—	2	1	no	—
44	C₂×M₁₆×C₆₇	—	4	1	no	—
45	C₂²×C₈×C₆₇	—	4	1	no	—
46	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
49	#108×C₆₇	—	2	1	no	—
50	#108×C₆₇	C₂³	1	2	yes	—
51	#104×C₆₇	—	2	4	no	—
52	C₄×C₈×C₆₇	D₄	1	2	yes	—
53	C₄×C₈×C₆₇	C₂³	1	2	yes	—
54	C₂²×Dic₆₇	—	4	2	no	—
55	C₄⋊Dic₆₇	—	2	4	no	—
56	C₄⋊Dic₆₇	—	4	2	no	—
57	Dic₆₇⋊C₄	—	4	8	no	—
58	C₆₇⋊C₁₆	—	2	4	no	—
59	#41×C₆₇	—	2	2	no	—
60	#41×C₆₇	—	2	2	no	—
61	#41×C₆₇	—	4	1	no	—
62	#41×C₆₇	—	4	1	no	—
63	C₂×Q₈×C₆₇	—	2	1	no	—
64	C₂×D₄×C₆₇	—	2	1	no	—
65	C₂×D₄×C₆₇	—	4	1	no	—
66	C₂²×C₄×C₆₇	—	2	1	no	—
67	C₂²×C₄×C₆₇	—	4	1	no	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	Q₁₆×C₆₇	—	2	1	no	—
70	Q₁₆×C₆₇	C₂×Q₈	1	2	yes	—
71	SD₁₆×C₆₇	—	2	2	no	—
72	SD₁₆×C₆₇	—	2	2	no	—
73	D₈×C₆₇	—	2	1	no	—
74	D₈×C₆₇	C₂×Q₈	1	2	yes	—
75	M₁₆×C₆₇	—	2	2	no	—
76	M₁₆×C₆₇	—	4	1	no	—
77	M₁₆×C₆₇	—	4	4	no	—
78	C₂×C₈×C₆₇	—	4	1	no	—
79	C₂×C₈×C₆₇	—	4	4	no	—
80	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
81	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
82	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
83	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
88	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
90	C₄²×C₆₇	—	2	1	no	—
91	C₄²×C₆₇	C₂×D₄	1	2	yes	—
92	C₂³×C₆₇	—	4	1	no	—
93	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
94	Q₈×C₆₇	—	2	2	no	—
95	Q₈×C₆₇	—	4	1	no	—
96	D₄×C₆₇	—	2	2	no	—
97	D₄×C₆₇	—	4	2	no	—
98	D₄×C₆₇	—	4	2	no	—
99	D₄×C₆₇	—	4	1	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	4	2	no	—
102	C₂×C₄×C₆₇	—	4	1	no	—
103	C₂×C₄×C₆₇	—	4	2	no	—
104	C₂×C₄×C₆₇	—	4	2	no	—
105	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×Dic₆₇	—	4	4	no	—
112	C₂×Dic₆₇	—	8	4	no	—
113	C₈×C₆₇	—	4	2	no	—
114	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
115	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
116	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
117	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
118	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
119	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
120	C₂²×C₆₇	—	2	2	no	—
121	C₂²×C₆₇	—	4	2	no	—
122	C₂²×C₆₇	—	8	2	no	—
123	C₂²×C₆₇	#476	1	1	yes	characteristic
124	C₂²×C₆₇	#476	1	1	yes	characteristic
125	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₄×C₆₇	—	4	2	no	—
127	C₄×C₆₇	—	4	2	no	—
128	C₄×C₆₇	#476	1	1	yes	characteristic
129	C₄×C₆₇	#476	1	1	yes	characteristic
130	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
131	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
132	Dic₆₇	—	8	4	no	—
133	C₂×#2231	—	67	1	no	maximal, Sylow
134	C₂×C₆₇	—	2	1	no	—
135	C₂×C₆₇	—	2	1	no	—
136	C₂×C₆₇	—	8	2	no	—
137	C₂×C₆₇	#2231	1	2	yes	complemented
138	C₂×C₆₇	C₂×#476	1	1	yes	characteristic
139	C₂×#499	—	67	1	no	—
140	C₂×#443	D₆₇	1	1	yes	characteristic
141	#2231	—	67	8	no	—
142	C₂×#365	—	67	2	no	—
143	C₂×#364	—	67	1	no	—
144	C₂×#360	—	67	2	no	—
145	C₆₇	C₂×#2231	1	1	yes	characteristic, complemented, Sylow
146	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
147	C₂×#131	—	2	1	no	—
148	C₂×#125	—	134	1	no	—
149	C₂×M₃₂	—	67	2	no	—
150	#499	—	67	4	no	—
151	#499	—	134	2	no	—
152	#443	—	2	2	no	—
153	#443	C₂×D₆₇	1	4	yes	—
154	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
155	C₂×#107	—	134	1	no	—
156	C₂×#106	—	67	2	no	—
157	C₂×#104	—	2	1	no	—
158	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
159	#365	—	67	8	no	—
160	#364	—	67	4	no	—
161	#360	—	67	8	no	—
162	C₂×#41	—	2	1	no	—
163	#135	—	2	1	no	—
164	#135	—	2	1	no	—
165	#135	Dic₁₃₄	1	2	yes	—
166	#135	C₂²×D₆₇	1	2	yes	—
167	C₂×Q₁₆	Dic₁₃₄	1	1	yes	characteristic
168	C₂×SD₁₆	—	2	1	no	—
169	C₂×D₈	Dic₁₃₄	1	1	yes	characteristic
170	#131	—	2	4	no	—
171	#131	—	4	2	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	4	1	no	—
174	C₂²×C₈	—	4	1	no	—
175	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
177	#125	—	268	2	no	—
178	#125	—	134	4	no	—
179	C₂×C₄⋊C₄	—	268	2	no	—
180	C₂×C₄⋊C₄	—	134	2	no	—
181	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
182	M₃₂	—	67	4	no	—
183	M₃₂	—	134	2	no	—
184	C₂×C₁₆	—	134	2	no	—
185	#108	—	2	1	no	—
186	#108	C₂²×D₆₇	1	2	yes	—
187	#107	—	134	4	no	—
188	#106	—	67	4	no	—
189	#106	—	134	2	no	—
190	#104	—	2	4	no	—
191	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
192	C₄×C₈	C₂²×D₆₇	1	2	yes	—
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	#41	—	4	1	no	—
196	#41	—	4	1	no	—
197	C₂×Q₈	—	2	1	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	—	268	2	no	—
203	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
204	Q₁₆	—	2	1	no	—
205	Q₁₆	C₂×Dic₁₃₄	1	2	yes	—
206	SD₁₆	—	2	2	no	—
207	SD₁₆	—	2	2	no	—
208	D₈	—	2	1	no	—
209	D₈	C₂×Dic₁₃₄	1	2	yes	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	4	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	#11761616	1	1	yes	characteristic
216	C₂×C₈	#11761616	1	1	yes	characteristic
217	C₂×C₈	#11761616	1	2	yes	—
218	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
219	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
220	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
221	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
222	C₂×C₈	#11761628	1	1	yes	characteristic
223	C₂×C₈	#11761628	1	1	yes	characteristic
224	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
225	C₄⋊C₄	—	268	8	no	—
226	C₄⋊C₄	—	268	2	no	—
227	C₄⋊C₄	—	134	4	no	—
228	C₄²	—	2	1	no	—
229	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
230	C₁₆	—	134	4	no	—
231	C₂³	—	4	1	no	—
232	C₂³	#14524068	1	1	yes	characteristic
233	Q₈	—	2	2	no	—
234	Q₈	—	4	1	no	—
235	D₄	—	2	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	2	no	—
238	D₄	—	4	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	268	4	no	—
245	C₂×C₄	—	536	4	no	—
246	C₂×C₄	#14524022	1	1	yes	characteristic
247	C₂×C₄	#14524022	1	1	yes	characteristic
248	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
249	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
250	C₂×C₄	#14524068	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
252	C₈	—	4	2	no	—
253	C₈	#14524022	1	1	yes	characteristic
254	C₈	#14524022	1	1	yes	characteristic
255	C₈	C₂×#11761616	1	2	yes	—
256	C₈	C₂×#11761616	1	2	yes	—
257	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
258	C₈	C₂×#11761628	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	#16863686	1	1	yes	characteristic
263	C₂²	#16863686	1	1	yes	characteristic
264	C₂²	C₂×#14524068	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	536	4	no	—
268	C₄	#16863686	1	1	yes	characteristic
269	C₄	#16863686	1	1	yes	characteristic
270	C₄	C₂×#14524022	1	1	yes	characteristic
271	C₄	C₂²×#11761616	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	8	2	no	—
275	C₂	#22562385	1	2	yes	direct product factor
276	C₂	C₂×#16863686	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863686
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#443×C₆₇	C₂
socle	C₂²×C₆₇	#476
prime radical	C₂×#135×C₆₇	C₂²
hypercenter	C₂×#443	D₆₇
hypocenter	C₆₇	C₂×#2231
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#16863798	C₂	semidirect
2. #22562385	C₂	direct (N × H)
3. C₂×#16863220	C₂	semidirect
4. C₂×#16863217	C₂	semidirect
10. #16863798	C₂²	semidirect
13. #16863220	C₂²	semidirect
14. #16863217	C₂²	semidirect
137. C₂×C₆₇	#2231	semidirect
145. C₆₇	C₂×#2231	semidirect
275. C₂	#22562385	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×#16863798	2
#22562385	2
C₂×#16863220	2
C₂×#16863217	2
C₂×#16863206	2
C₂×#443×C₆₇	2
C₂×#2231	67

Subgroups — C2×#22562385 #31010567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562385 #31010567