Subgroups — C₂×#22558184 #30765621

Order 17152.

398 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863514	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863117	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558184	C₂	1	4	yes	maximal, direct product factor
4	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14524065	C₄	1	1	yes	characteristic
6	C₂×#14524049	—	2	1	no	—
7	#16863514	—	2	1	no	—
8	#16863514	—	2	1	no	—
9	#16863514	C₄	1	2	yes	—
10	#16863514	C₂²	1	2	yes	—
11	C₂×#14523979	C₄	1	1	yes	characteristic
12	C₂×#14523971	—	2	1	no	—
13	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
14	#16863117	—	2	1	no	—
15	#16863117	C₂²	1	2	yes	complemented
16	C₂×M₃₂×C₆₇	—	2	1	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#364×C₆₇	—	2	1	no	—
19	#364×C₆₇	C₂²	1	2	yes	complemented
20	C₂×#11761628	—	2	1	no	—
21	#14524065	—	2	1	no	—
22	#14524065	—	2	1	no	—
23	#14524065	C₈	1	2	yes	complemented
24	#14524065	C₂×C₄	1	2	yes	—
25	C₂×Dic₂₆₈	C₈	1	1	yes	characteristic, complemented
26	C₂×#11761617	—	2	1	no	—
27	C₂×D₅₃₆	C₈	1	1	yes	characteristic, complemented
28	#14524049	—	2	2	no	—
29	#14524049	—	2	2	no	—
30	#14524049	—	4	1	no	—
31	#14524049	—	4	1	no	—
32	C₂×#11761612	—	2	1	no	—
33	C₂×#11761611	—	4	1	no	—
34	C₂²×C₆₇⋊C₈	—	2	2	no	—
35	C₂×C₈×D₆₇	—	4	1	no	—
36	#14523979	—	2	1	no	—
37	#14523979	C₂×C₄	1	2	yes	—
38	#14523971	—	2	2	no	—
39	#14523971	—	2	2	no	—
40	C₄×C₆₇⋊C₈	—	2	2	no	—
41	C₄×C₆₇⋊C₈	—	2	2	no	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
45	M₃₂×C₆₇	—	2	2	no	—
46	M₃₂×C₆₇	—	4	1	no	—
47	C₂×C₁₆×C₆₇	—	4	1	no	—
48	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	#11761628	—	2	2	no	—
53	#11761628	—	2	2	no	—
54	#11761628	—	4	1	no	—
55	#11761628	—	4	1	no	—
56	C₂×Dic₁₃₄	—	2	1	no	—
57	C₂×D₂₆₈	—	2	1	no	—
58	C₂×C₂²⋊D₆₇	—	4	1	no	—
59	C₂×C₄×D₆₇	—	4	1	no	—
60	Dic₂₆₈	—	2	1	no	—
61	Dic₂₆₈	C₂×C₈	1	2	yes	complemented
62	#11761617	—	2	2	no	—
63	#11761617	—	2	2	no	—
64	D₅₃₆	—	2	1	no	—
65	D₅₃₆	C₂×C₈	1	2	yes	complemented
66	#11761612	—	2	2	no	—
67	#11761612	—	4	1	no	—
68	#11761611	—	4	2	no	—
69	#11761611	—	4	2	no	—
70	C₂×C₆₇⋊C₈	—	2	4	no	—
71	C₂×C₆₇⋊C₈	—	4	4	no	—
72	C₂×C₆₇⋊C₈	—	4	1	no	—
73	C₈×D₆₇	—	4	2	no	—
74	C₈×D₆₇	—	4	2	no	—
75	C₂²×C₄×C₆₇	—	2	1	no	—
76	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
81	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
82	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
83	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
84	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
85	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
86	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₇	—	2	1	no	—
90	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₁₆×C₆₇	—	4	2	no	—
93	C₂³×C₆₇	#104	1	1	yes	characteristic
94	C₂²×D₆₇	—	4	1	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	2	1	no	—
97	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#104	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂²⋊D₆₇	—	4	2	no	—
105	C₂²⋊D₆₇	—	4	2	no	—
106	C₂×Dic₆₇	—	4	1	no	—
107	D₂₆₈	—	2	2	no	—
108	D₂₆₈	—	4	1	no	—
109	C₄×D₆₇	—	4	2	no	—
110	C₄×D₆₇	—	4	2	no	—
111	Dic₁₃₄	—	2	2	no	—
112	Dic₁₃₄	—	4	1	no	—
113	C₈×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
114	C₈×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
115	C₈×C₆₇	#102	1	1	yes	characteristic
116	C₈×C₆₇	#102	1	1	yes	characteristic
117	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
118	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
119	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
120	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
121	C₆₇⋊C₈	—	4	4	no	—
122	C₆₇⋊C₈	—	4	2	no	—
123	C₂²×C₆₇	—	2	2	no	—
124	C₂²×C₆₇	D₄⋊C₈	1	1	yes	characteristic
125	C₂²×C₆₇	D₄⋊C₈	1	1	yes	characteristic
126	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
127	C₂×D₆₇	—	4	2	no	—
128	C₂×D₆₇	—	8	2	no	—
129	C₄×C₆₇	—	2	2	no	—
130	C₄×C₆₇	—	2	2	no	—
131	C₄×C₆₇	D₄⋊C₈	1	1	yes	characteristic
132	C₄×C₆₇	D₄⋊C₈	1	1	yes	characteristic
133	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
135	Dic₆₇	—	4	2	no	—
136	C₂×#1336	—	67	1	no	maximal, Sylow
137	C₂×C₆₇	—	2	1	no	—
138	C₂×C₆₇	—	2	1	no	—
139	C₂×C₆₇	#1336	1	2	yes	complemented
140	C₂×C₆₇	C₂×D₄⋊C₈	1	1	yes	characteristic
141	D₆₇	—	8	2	no	—
142	C₂×#443	—	67	1	no	—
143	C₂×#364	D₆₇	1	1	yes	characteristic, complemented
144	C₂×#322	—	67	1	no	—
145	#1336	—	67	4	no	—
146	C₆₇	C₂×#1336	1	1	yes	characteristic, complemented, Sylow
147	C₂×#135	—	67	1	no	—
148	C₂×#131	—	134	1	no	—
149	C₂×M₃₂	—	2	1	no	—
150	#443	—	67	2	no	—
151	#443	—	67	2	no	—
152	#443	—	134	1	no	—
153	#443	—	134	1	no	—
154	C₂×#108	—	67	1	no	—
155	C₂×#104	—	134	1	no	—
156	C₂×C₄×C₈	—	134	1	no	—
157	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
158	#364	—	2	1	no	—
159	#364	C₂×D₆₇	1	2	yes	complemented
160	#322	—	67	2	no	—
161	#322	—	134	1	no	—
162	C₂×#41	—	134	1	no	—
163	#135	—	67	2	no	—
164	#135	—	67	2	no	—
165	#135	—	134	1	no	—
166	#135	—	134	1	no	—
167	C₂×Q₁₆	—	67	1	no	—
168	C₂×SD₁₆	—	134	1	no	—
169	C₂×D₈	—	67	1	no	—
170	#131	—	268	1	no	—
171	#131	—	268	1	no	—
172	#131	—	134	2	no	—
173	#131	—	134	2	no	—
174	C₂×M₁₆	—	268	1	no	—
175	C₂×M₁₆	—	134	1	no	—
176	C₂²×C₈	—	268	1	no	—
177	C₂²×C₈	—	134	2	no	—
178	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
179	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
180	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
181	M₃₂	—	2	2	no	—
182	M₃₂	—	4	1	no	—
183	C₂×C₁₆	—	4	1	no	—
184	#108	—	67	2	no	—
185	#108	—	134	1	no	—
186	#104	—	134	2	no	—
187	#104	—	134	2	no	—
188	C₄×C₈	—	134	2	no	—
189	C₄×C₈	—	134	2	no	—
190	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
191	C₄×C₈	D₂₆₈	1	1	yes	characteristic
192	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
194	#41	—	268	1	no	—
195	#41	—	268	1	no	—
196	#41	—	134	2	no	—
197	#41	—	134	2	no	—
198	C₂×Q₈	—	134	1	no	—
199	C₂×D₄	—	268	1	no	—
200	C₂×D₄	—	134	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	268	1	no	—
203	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
204	Q₁₆	—	67	2	no	—
205	Q₁₆	—	134	1	no	—
206	SD₁₆	—	134	2	no	—
207	SD₁₆	—	134	2	no	—
208	D₈	—	67	2	no	—
209	D₈	—	134	1	no	—
210	M₁₆	—	268	1	no	—
211	M₁₆	—	268	2	no	—
212	M₁₆	—	268	2	no	—
213	M₁₆	—	134	2	no	—
214	C₂×C₈	—	268	4	no	—
215	C₂×C₈	—	268	1	no	—
216	C₂×C₈	—	268	2	no	—
217	C₂×C₈	—	268	2	no	—
218	C₂×C₈	—	134	4	no	—
219	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
220	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
221	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
222	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
223	C₂×C₈	#11761611	1	1	yes	characteristic
224	C₂×C₈	#11761611	1	1	yes	characteristic
225	C₂×C₈	#11761613	1	1	yes	characteristic
226	C₂×C₈	#11761613	1	1	yes	characteristic
227	C₂×C₈	#11761615	1	1	yes	characteristic
228	C₂×C₈	#11761615	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
231	C₄²	—	2	1	no	—
232	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
233	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
234	C₁₆	—	4	2	no	—
235	C₂³	—	268	1	no	—
236	C₂³	#14523969	1	1	yes	characteristic
237	Q₈	—	268	1	no	—
238	Q₈	—	134	2	no	—
239	D₄	—	268	2	no	—
240	D₄	—	268	2	no	—
241	D₄	—	268	1	no	—
242	D₄	—	134	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	268	1	no	—
247	C₂×C₄	—	268	2	no	—
248	C₂×C₄	—	268	2	no	—
249	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
250	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
251	C₂×C₄	#14523963	1	1	yes	characteristic
252	C₂×C₄	#14523963	1	1	yes	characteristic
253	C₂×C₄	#14523969	1	1	yes	characteristic
254	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
255	C₈	—	268	4	no	—
256	C₈	—	268	2	no	—
257	C₈	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
258	C₈	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
259	C₈	#14523963	1	1	yes	characteristic
260	C₈	#14523963	1	1	yes	characteristic
261	C₈	C₂×C₈×D₆₇	1	1	yes	characteristic
262	C₈	C₂×#11761611	1	1	yes	characteristic
263	C₈	C₂×#11761613	1	1	yes	characteristic
264	C₈	C₂×#11761615	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	268	2	no	—
267	C₂²	—	536	2	no	—
268	C₂²	#16863123	1	1	yes	characteristic
269	C₂²	#16863123	1	1	yes	characteristic
270	C₂²	C₂×#14523969	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	268	2	no	—
274	C₄	#16863123	1	1	yes	characteristic
275	C₄	#16863123	1	1	yes	characteristic
276	C₄	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
277	C₄	C₂×#14523963	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	536	2	no	—
281	C₂	#22558184	1	2	yes	direct product factor
282	C₂	C₂×#16863123	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14523963
commutator	C₈×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×#364×C₆₇	C₂
socle	C₂²×C₆₇	D₄⋊C₈
prime radical	C₂×#14524065	C₄
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×#364	D₆₇
hypocenter	C₆₇	C₂×#1336
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. C₂×#16863117	C₂	semidirect
3. #22558184	C₂	direct (N × H)
4. C₂×#364×C₆₇	C₂	semidirect
15. #16863117	C₂²	semidirect
19. #364×C₆₇	C₂²	semidirect
23. #14524065	C₈	semidirect
25. C₂×Dic₂₆₈	C₈	semidirect
27. C₂×D₅₃₆	C₈	semidirect
61. Dic₂₆₈	C₂×C₈	semidirect
65. D₅₃₆	C₂×C₈	semidirect
139. C₂×C₆₇	#1336	semidirect
143. C₂×#364	D₆₇	semidirect
146. C₆₇	C₂×#1336	semidirect
159. #364	C₂×D₆₇	semidirect
281. C₂	#22558184	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×#16863514	2
C₂×#16863117	2
#22558184	2
C₂×#364×C₆₇	2
C₂×#1336	67

Subgroups — C2×#22558184 #30765621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558184 #30765621