Subgroups — C₂×#22562366 #31010446

Order 17152.

606 conjugacy classes of subgroups.

The subgroup lattice has 292 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	#22562366	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863223	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16863221	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863197	C₂	1	1	yes	maximal, characteristic
7	C₂×#16863197	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14524035	—	2	1	no	—
10	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#16863798	—	2	2	no	—
13	#16863798	C₂²	1	4	yes	complemented
14	C₂×#14523977	C₂²	1	1	yes	characteristic
15	C₂×#14523977	C₂²	1	1	yes	characteristic
16	C₂×#14523975	—	2	1	no	—
17	#16863223	C₂²	1	4	yes	complemented
18	#16863221	C₂²	1	4	yes	complemented
19	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	complemented
20	#16863197	C₂²	1	4	yes	—
21	#16863197	C₂²	1	4	yes	—
22	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₆₇	—	2	1	no	—
24	C₄×D₈×C₆₇	—	2	2	no	—
25	C₄×D₈×C₆₇	C₂²	1	4	yes	—
26	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₆₇	—	2	1	no	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	#14524035	—	2	4	no	—
30	#14524035	—	4	2	no	—
31	C₂×C₄⋊Dic₆₇	—	2	1	no	—
32	C₂×C₄⋊Dic₆₇	—	2	1	no	—
33	C₂×Dic₆₇⋊C₄	—	4	1	no	—
34	C₂×Dic₆₇⋊C₄	—	4	1	no	—
35	C₂×C₆₇⋊C₁₆	—	2	2	no	—
36	C₂×C₆₇⋊C₁₆	—	2	2	no	—
37	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
38	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
39	#14523977	—	2	1	no	—
40	#14523977	—	2	1	no	—
41	#14523977	C₂³	1	2	yes	—
42	#14523977	C₂³	1	2	yes	—
43	#14523975	—	2	4	no	—
44	C₂²×D₄×C₆₇	—	2	1	no	—
45	C₂³×C₄×C₆₇	—	4	1	no	—
46	C₂×D₈×C₆₇	—	2	1	no	—
47	C₂×D₈×C₆₇	—	2	2	no	—
48	C₂×D₈×C₆₇	Q₈	1	4	yes	—
49	C₂×D₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄×D₄×C₆₇	—	2	4	no	—
53	C₄×D₄×C₆₇	—	4	2	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
56	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
57	#107×C₆₇	—	2	1	no	—
58	#107×C₆₇	C₂³	1	2	yes	—
59	#102×C₆₇	—	2	4	no	—
60	C₄×C₈×C₆₇	D₄	1	2	yes	—
61	C₄×C₈×C₆₇	C₂³	1	2	yes	—
62	C₂²×Dic₆₇	—	4	1	no	—
63	C₂²×Dic₆₇	—	4	1	no	—
64	C₄⋊Dic₆₇	—	2	2	no	—
65	C₄⋊Dic₆₇	—	2	2	no	—
66	C₄⋊Dic₆₇	—	4	1	no	—
67	C₄⋊Dic₆₇	—	4	1	no	—
68	Dic₆₇⋊C₄	—	4	4	no	—
69	Dic₆₇⋊C₄	—	4	4	no	—
70	C₆₇⋊C₁₆	—	2	2	no	—
71	C₆₇⋊C₁₆	—	2	2	no	—
72	C₂⁴×C₆₇	—	4	1	no	—
73	C₂×D₄×C₆₇	—	2	4	no	—
74	C₂×D₄×C₆₇	—	2	2	no	—
75	C₂×D₄×C₆₇	—	4	1	no	—
76	C₂×D₄×C₆₇	—	4	2	no	—
77	C₂²×C₄×C₆₇	—	4	1	no	—
78	C₂²×C₄×C₆₇	—	4	4	no	—
79	C₂²×C₄×C₆₇	—	8	4	no	—
80	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	D₈×C₆₇	—	2	4	no	—
84	D₈×C₆₇	—	2	2	no	—
85	D₈×C₆₇	C₂×Q₈	1	4	yes	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₆₇	#41	1	2	yes	—
91	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₄⋊C₄×C₆₇	—	4	1	no	—
94	C₂²⋊C₄×C₆₇	—	4	4	no	—
95	C₄²×C₆₇	D₈	1	2	yes	—
96	C₄²×C₆₇	C₂×D₄	1	2	yes	—
97	C₂³×C₆₇	—	4	2	no	—
98	C₂³×C₆₇	—	4	4	no	—
99	C₂³×C₆₇	—	8	2	no	—
100	C₂³×C₆₇	—	8	2	no	—
101	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
102	D₄×C₆₇	—	2	4	no	—
103	D₄×C₆₇	—	4	2	no	—
104	D₄×C₆₇	—	4	4	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	4	2	no	—
108	C₂×C₄×C₆₇	—	8	8	no	—
109	C₂×C₄×C₆₇	—	8	2	no	—
110	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#137	1	2	yes	—
115	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×Dic₆₇	—	4	2	no	—
117	C₂×Dic₆₇	—	4	2	no	—
118	C₂×Dic₆₇	—	8	2	no	—
119	C₂×Dic₆₇	—	8	2	no	—
120	C₈×C₆₇	—	2	2	no	—
121	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
122	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
123	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
124	C₂²×C₆₇	—	4	4	no	—
125	C₂²×C₆₇	—	8	2	no	—
126	C₂²×C₆₇	—	8	2	no	—
127	C₂²×C₆₇	—	8	8	no	—
128	C₂²×C₆₇	#474	1	2	yes	—
129	C₂²×C₆₇	#508	1	2	yes	—
130	C₂²×C₆₇	#509	1	2	yes	—
131	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₆₇	—	2	2	no	—
133	C₄×C₆₇	—	2	2	no	—
134	C₄×C₆₇	—	8	2	no	—
135	C₄×C₆₇	#474	1	2	yes	—
136	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
137	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
138	Dic₆₇	—	8	2	no	—
139	Dic₆₇	—	8	2	no	—
140	C₂×#2228	—	67	1	no	maximal, Sylow
141	C₂×C₆₇	—	8	4	no	—
142	C₂×C₆₇	#2228	1	4	yes	complemented
143	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
144	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
145	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
146	C₂×#499	—	67	1	no	—
147	C₂×C₄×D₈	D₆₇	1	1	yes	characteristic
148	#2228	—	67	8	no	—
149	C₂×#367	—	67	1	no	—
150	C₂×#366	—	67	1	no	—
151	C₂×C₄⋊C₁₆	—	67	1	no	—
152	C₂×#357	—	67	1	no	—
153	C₂×#357	—	67	1	no	—
154	C₆₇	C₂×#2228	1	1	yes	characteristic, complemented, Sylow
155	C₂²×D₈	C₂×D₆₇	1	1	yes	characteristic
156	C₂×#125	—	134	1	no	—
157	C₂×C₄×D₄	—	2	1	no	—
158	C₂²×C₁₆	—	67	1	no	—
159	C₂²×C₁₆	—	67	1	no	—
160	#499	—	67	4	no	—
161	#499	—	134	2	no	—
162	C₄×D₈	—	2	2	no	—
163	C₄×D₈	C₂×D₆₇	1	4	yes	—
164	C₂×#107	—	67	1	no	—
165	C₂×#107	—	67	1	no	—
166	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
167	C₂×#106	—	134	1	no	—
168	C₂×#102	—	2	1	no	—
169	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
170	#367	—	67	4	no	—
171	#366	—	67	4	no	—
172	C₄⋊C₁₆	—	67	4	no	—
173	#357	—	67	4	no	—
174	#357	—	67	4	no	—
175	C₂²×D₄	—	2	1	no	—
176	C₂³×C₄	—	4	1	no	—
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	—	2	2	no	—
179	C₂×D₈	Dic₁₃₄	1	4	yes	—
180	C₂×D₈	C₂²×D₆₇	1	2	yes	—
181	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
183	#125	—	268	2	no	—
184	#125	—	134	4	no	—
185	C₄×D₄	—	2	4	no	—
186	C₄×D₄	—	4	2	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	268	1	no	—
189	C₂×C₄⋊C₄	—	268	1	no	—
190	C₂×C₄⋊C₄	—	134	1	no	—
191	C₂×C₄⋊C₄	—	134	1	no	—
192	C₂×C₂²⋊C₄	—	4	1	no	—
193	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
194	C₂×C₁₆	—	67	2	no	—
195	C₂×C₁₆	—	67	2	no	—
196	C₂×C₁₆	—	134	2	no	—
197	C₂×C₁₆	—	134	2	no	—
198	#107	—	2	1	no	—
199	#107	—	67	2	no	—
200	#107	—	67	2	no	—
201	#107	—	134	1	no	—
202	#107	—	134	1	no	—
203	#107	C₂²×D₆₇	1	2	yes	—
204	#106	—	134	4	no	—
205	#102	—	2	4	no	—
206	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
207	C₄×C₈	C₂²×D₆₇	1	2	yes	—
208	C₂⁴	—	4	1	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	—	8	4	no	—
216	C₂²×C₄	—	268	1	no	—
217	C₂²×C₄	—	268	1	no	—
218	C₂²×C₄	#11761613	1	1	yes	characteristic
219	C₂²×C₄	#11761613	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
221	D₈	—	2	4	no	—
222	D₈	—	2	2	no	—
223	D₈	C₂×Dic₁₃₄	1	4	yes	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
226	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
227	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
228	C₂×C₈	#11761628	1	2	yes	—
229	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	268	4	no	—
233	C₄⋊C₄	—	268	4	no	—
234	C₄⋊C₄	—	268	1	no	—
235	C₄⋊C₄	—	268	1	no	—
236	C₄⋊C₄	—	134	2	no	—
237	C₄⋊C₄	—	134	2	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₄²	#11761613	1	2	yes	—
240	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
241	C₁₆	—	134	2	no	—
242	C₁₆	—	134	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	4	4	no	—
245	C₂³	—	8	2	no	—
246	C₂³	—	8	2	no	—
247	C₂³	C₂×#11761613	1	1	yes	characteristic
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	8	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	268	2	no	—
257	C₂×C₄	—	268	2	no	—
258	C₂×C₄	—	536	2	no	—
259	C₂×C₄	—	536	2	no	—
260	C₂×C₄	#14524022	1	2	yes	—
261	C₂×C₄	C₂×#11761613	1	2	yes	—
262	C₂×C₄	C₂×#11761613	1	2	yes	—
263	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
264	C₂×C₄	#14524073	1	2	yes	—
265	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	#14524022	1	2	yes	—
268	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
269	C₈	C₂×#11761628	1	1	yes	characteristic
270	C₂²	—	4	4	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	8	no	—
274	C₂²	#16863676	1	2	yes	—
275	C₂²	#16863826	1	2	yes	—
276	C₂²	#16863832	1	2	yes	—
277	C₂²	C₂²×#11761613	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	536	2	no	—
282	C₄	—	536	2	no	—
283	C₄	#16863676	1	2	yes	—
284	C₄	C₂×#14524022	1	1	yes	characteristic
285	C₄	C₂×#14524073	1	1	yes	characteristic
286	C₂	—	8	4	no	—
287	C₂	#22562366	1	4	yes	direct product factor
288	C₂	C₂×#16863676	1	1	yes	characteristic
289	C₂	C₂×#16863826	1	1	yes	characteristic
290	C₂	C₂×#16863832	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₄×D₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂²×D₈×C₆₇	C₂²
hypercenter	C₂×C₄×D₈	D₆₇
hypocenter	C₆₇	C₂×#2228
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16863798	C₂	semidirect
2. #22562366	C₂	direct (N × H)
3. C₂×#16863223	C₂	semidirect
4. C₂×#16863221	C₂	semidirect
5. C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	semidirect
13. #16863798	C₂²	semidirect
17. #16863223	C₂²	semidirect
18. #16863221	C₂²	semidirect
19. C₄⋊(C₆₇⋊C₁₆)	C₂²	semidirect
142. C₂×C₆₇	#2228	semidirect
154. C₆₇	C₂×#2228	semidirect
287. C₂	#22562366	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
#22562366	2
C₂×#16863223	2
C₂×#16863221	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₂×#16863197	2
C₂×C₄×D₈×C₆₇	2
C₂×#2228	67

Subgroups — C2×#22562366 #31010446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562366 #31010446