Subgroups — C₂×#22559690 #30838428

Order 17152.

850 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863593	C₂	1	2	yes	maximal, complemented
2	C₂×#16863526	C₂	1	1	yes	maximal, characteristic, complemented
3	#22559690	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863147	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863140	C₂	1	2	yes	maximal, complemented
6	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761616	—	2	2	no	—
8	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
9	#16863593	—	2	4	no	—
10	#16863593	C₂²	1	8	yes	complemented
11	#16863526	—	2	2	no	—
12	#16863526	C₂²	1	4	yes	complemented
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×Dic₆₇⋊C₈	—	2	1	no	—
16	C₂×#14523964	—	2	2	no	—
17	C₂×#14523953	—	2	1	no	—
18	#16863147	C₂²	1	4	yes	complemented
19	#16863140	C₂²	1	8	yes	complemented
20	C₂²×D₈×C₆₇	—	2	1	no	—
21	C₂²×D₈×C₆₇	—	2	1	no	—
22	C₂×#127×C₆₇	C₂²	1	2	yes	—
23	#493×C₆₇	—	2	2	no	—
24	#493×C₆₇	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
27	C₂×#11761616	—	2	8	no	—
28	C₂×#11761616	—	4	8	no	—
29	C₂²×C₆₇⋊C₈	—	2	1	no	—
30	C₂²×C₆₇⋊C₈	—	2	2	no	—
31	C₄×Dic₁₃₄	—	2	1	no	—
32	C₄×Dic₁₃₄	—	2	1	no	—
33	C₄×Dic₁₃₄	D₄	1	2	yes	—
34	C₄×Dic₁₃₄	C₂³	1	2	yes	—
35	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
36	C₂×Dic₆₇⋊C₄	—	2	1	no	—
37	C₂×C₄×Dic₆₇	—	2	1	no	—
38	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
39	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
40	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
41	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
42	Dic₆₇⋊C₈	—	2	4	no	—
43	#14523964	—	2	8	no	—
44	#14523953	—	2	4	no	—
45	C₂²×D₄×C₆₇	—	2	2	no	—
46	C₂²×D₄×C₆₇	—	2	2	no	—
47	C₂²×D₄×C₆₇	—	2	2	no	—
48	C₂×D₈×C₆₇	—	2	4	no	—
49	C₂×D₈×C₆₇	—	2	4	no	—
50	C₂×D₈×C₆₇	—	4	4	no	—
51	C₂×D₈×C₆₇	—	4	4	no	—
52	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
54	#127×C₆₇	—	2	2	no	—
55	#127×C₆₇	—	2	2	no	—
56	#127×C₆₇	—	2	2	no	—
57	#127×C₆₇	C₂³	1	4	yes	—
58	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₆₇	D₄	1	2	yes	—
60	C₄×C₈×C₆₇	C₂³	1	2	yes	—
61	C₂×Dic₁₃₄	—	2	4	no	—
62	C₂×Dic₁₃₄	—	2	1	no	—
63	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
64	C₂²×Dic₆₇	—	2	1	no	—
65	C₂²×Dic₆₇	—	2	1	no	—
66	#11761616	—	4	16	no	—
67	C₂×C₆₇⋊C₈	—	2	2	no	—
68	C₂×C₆₇⋊C₈	—	2	4	no	—
69	C₂×C₆₇⋊C₈	—	4	2	no	—
70	C₂×C₆₇⋊C₈	—	4	4	no	—
71	C₄⋊Dic₆₇	—	2	1	no	—
72	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
73	Dic₆₇⋊C₄	—	2	2	no	—
74	Dic₆₇⋊C₄	—	2	2	no	—
75	C₄×Dic₆₇	—	2	2	no	—
76	C₄×Dic₆₇	—	2	2	no	—
77	C₂⁴×C₆₇	—	4	2	no	—
78	C₂×D₄×C₆₇	—	2	4	no	—
79	C₂×D₄×C₆₇	—	2	4	no	—
80	C₂×D₄×C₆₇	—	2	4	no	—
81	C₂×D₄×C₆₇	—	4	2	no	—
82	C₂×D₄×C₆₇	—	4	2	no	—
83	C₂×D₄×C₆₇	—	4	2	no	—
84	C₂×D₄×C₆₇	—	4	4	no	—
85	C₂×D₄×C₆₇	—	4	4	no	—
86	C₂×D₄×C₆₇	—	4	4	no	—
87	C₂×D₄×C₆₇	—	4	4	no	—
88	C₂×D₄×C₆₇	—	8	4	no	—
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
92	D₈×C₆₇	—	4	8	no	—
93	D₈×C₆₇	—	4	8	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	SD₁₆	1	4	yes	complemented
96	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
98	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₆₇	#41	1	1	yes	characteristic
101	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
102	C₂³×C₆₇	—	4	4	no	—
103	C₂³×C₆₇	—	8	4	no	—
104	C₂³×C₆₇	—	8	4	no	—
105	C₂³×C₆₇	—	8	4	no	—
106	C₂³×C₆₇	#121	1	1	yes	characteristic
107	D₄×C₆₇	—	4	4	no	—
108	D₄×C₆₇	—	4	4	no	—
109	D₄×C₆₇	—	4	4	no	—
110	D₄×C₆₇	—	4	4	no	—
111	D₄×C₆₇	—	8	4	no	—
112	D₄×C₆₇	—	8	4	no	—
113	D₄×C₆₇	—	8	4	no	—
114	D₄×C₆₇	—	8	4	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
120	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
121	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₆₇	—	2	2	no	—
126	C₂×Dic₆₇	—	2	2	no	—
127	C₂×Dic₆₇	—	4	2	no	—
128	C₂×Dic₆₇	—	4	2	no	—
129	Dic₁₃₄	—	2	4	no	—
130	Dic₁₃₄	—	4	2	no	—
131	C₈×C₆₇	—	2	2	no	—
132	C₈×C₆₇	C₂×SD₁₆	1	4	yes	complemented
133	C₆₇⋊C₈	—	4	2	no	—
134	C₆₇⋊C₈	—	4	4	no	—
135	C₂²×C₆₇	—	8	4	no	—
136	C₂²×C₆₇	—	8	4	no	—
137	C₂²×C₆₇	—	8	4	no	—
138	C₂²×C₆₇	—	16	8	no	—
139	C₂²×C₆₇	#460	1	2	yes	—
140	C₂²×C₆₇	#469	1	2	yes	—
141	C₂²×C₆₇	#472	1	2	yes	—
142	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
143	C₄×C₆₇	—	2	1	no	—
144	C₄×C₆₇	—	2	1	no	—
145	C₄×C₆₇	#460	1	2	yes	—
146	C₄×C₆₇	#469	1	2	yes	—
147	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
148	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
149	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
150	Dic₆₇	—	4	2	no	—
151	Dic₆₇	—	4	2	no	—
152	C₂×#1688	—	67	1	no	maximal, Sylow
153	C₂×C₆₇	—	16	4	no	—
154	C₂×C₆₇	#1688	1	4	yes	complemented
155	C₂×C₆₇	C₂×#460	1	1	yes	characteristic
156	C₂×C₆₇	C₂×#469	1	1	yes	characteristic
157	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
158	C₂×#493	D₆₇	1	1	yes	characteristic
159	C₂×#460	—	67	2	no	—
160	C₂×#446	—	67	1	no	—
161	#1688	—	67	8	no	—
162	C₂×#334	—	67	1	no	—
163	C₂×#331	—	67	2	no	—
164	C₆₇	C₂×#1688	1	1	yes	characteristic, complemented, Sylow
165	C₂²×SD₁₆	—	134	2	no	—
166	C₂²×D₈	—	2	1	no	—
167	C₂²×D₈	—	2	1	no	—
168	C₂×#127	C₂×D₆₇	1	2	yes	—
169	C₂×C₄×Q₈	—	67	1	no	—
170	#493	—	2	2	no	—
171	#493	C₂×D₆₇	1	4	yes	—
172	#460	—	67	8	no	—
173	#460	—	134	4	no	—
174	#446	—	67	4	no	—
175	#446	—	134	2	no	—
176	C₂×C₄⋊C₈	—	67	1	no	—
177	C₂×C₄⋊C₈	—	67	2	no	—
178	C₂×C₄⋊C₈	—	134	1	no	—
179	C₂×#102	—	134	2	no	—
180	C₂×#97	—	134	1	no	—
181	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
182	#334	—	67	4	no	—
183	#331	—	67	8	no	—
184	C₂²×Q₈	—	67	1	no	—
185	C₂²×D₄	—	2	2	no	—
186	C₂²×D₄	—	2	2	no	—
187	C₂²×D₄	—	2	2	no	—
188	C₂×SD₁₆	—	268	8	no	—
189	C₂×SD₁₆	—	134	8	no	—
190	C₂×D₈	—	2	4	no	—
191	C₂×D₈	—	2	4	no	—
192	C₂×D₈	—	4	4	no	—
193	C₂×D₈	—	4	4	no	—
194	C₂²×C₈	—	134	2	no	—
195	C₂²×C₈	—	134	1	no	—
196	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
198	#127	—	2	2	no	—
199	#127	—	2	2	no	—
200	#127	—	2	2	no	—
201	#127	C₂²×D₆₇	1	4	yes	—
202	C₄×Q₈	—	67	2	no	—
203	C₄×Q₈	—	67	2	no	—
204	C₄×Q₈	—	134	1	no	—
205	C₄×Q₈	—	134	1	no	—
206	C₂×C₄⋊C₄	—	67	1	no	—
207	C₂×C₄⋊C₄	—	134	1	no	—
208	C₂×C₄²	—	134	1	no	—
209	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
210	C₄⋊C₈	—	67	2	no	—
211	C₄⋊C₈	—	67	4	no	—
212	C₄⋊C₈	—	134	4	no	—
213	C₄⋊C₈	—	134	2	no	—
214	C₄⋊C₈	—	134	1	no	—
215	#102	—	134	8	no	—
216	#97	—	134	4	no	—
217	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
218	C₄×C₈	C₂²×D₆₇	1	2	yes	—
219	C₂⁴	—	4	2	no	—
220	C₂×Q₈	—	67	2	no	—
221	C₂×Q₈	—	134	4	no	—
222	C₂×Q₈	—	134	1	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	4	4	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	—	8	4	no	—
234	C₂²×C₄	—	134	1	no	—
235	C₂²×C₄	—	134	1	no	—
236	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
238	C₂²×C₄	#11761627	1	1	yes	characteristic
239	SD₁₆	—	268	16	no	—
240	D₈	—	4	8	no	—
241	D₈	—	4	8	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	268	4	no	—
244	C₂×C₈	—	268	2	no	—
245	C₂×C₈	—	134	4	no	—
246	C₂×C₈	—	134	2	no	—
247	C₂×C₈	#11761616	1	4	yes	complemented
248	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
249	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
250	C₄⋊C₄	—	67	2	no	—
251	C₄⋊C₄	—	134	1	no	—
252	C₄⋊C₄	—	134	2	no	—
253	C₄⋊C₄	—	134	2	no	—
254	C₄²	—	134	2	no	—
255	C₄²	—	134	2	no	—
256	C₄²	D₄×D₆₇	1	1	yes	characteristic
257	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
258	C₄²	#11761627	1	1	yes	characteristic
259	C₄²	C₂³×D₆₇	1	1	yes	characteristic
260	C₂³	—	4	4	no	—
261	C₂³	—	8	4	no	—
262	C₂³	—	8	4	no	—
263	C₂³	—	8	4	no	—
264	C₂³	#14524014	1	1	yes	characteristic
265	Q₈	—	268	2	no	—
266	Q₈	—	134	4	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	4	no	—
270	D₄	—	4	4	no	—
271	D₄	—	8	4	no	—
272	D₄	—	8	4	no	—
273	D₄	—	8	4	no	—
274	D₄	—	8	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	268	2	no	—
277	C₂×C₄	—	268	2	no	—
278	C₂×C₄	—	134	2	no	—
279	C₂×C₄	—	134	2	no	—
280	C₂×C₄	#14524014	1	1	yes	characteristic
281	C₂×C₄	#14524014	1	1	yes	characteristic
282	C₂×C₄	#14524014	1	1	yes	characteristic
283	C₂×C₄	C₂×#11761616	1	2	yes	—
284	C₂×C₄	#14524067	1	4	yes	—
285	C₂×C₄	#14524069	1	2	yes	—
286	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
287	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
288	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	268	4	no	—
291	C₈	—	268	2	no	—
292	C₈	C₂×#11761616	1	4	yes	complemented
293	C₂²	—	8	4	no	—
294	C₂²	—	8	4	no	—
295	C₂²	—	8	4	no	—
296	C₂²	—	16	8	no	—
297	C₂²	#16863593	1	2	yes	—
298	C₂²	#16863646	1	2	yes	—
299	C₂²	#16863663	1	2	yes	—
300	C₂²	C₂×#14524014	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	268	2	no	—
304	C₄	—	268	2	no	—
305	C₄	#16863593	1	2	yes	—
306	C₄	#16863646	1	2	yes	—
307	C₄	C₂²×#11761616	1	1	yes	characteristic
308	C₄	C₂×#14524067	1	2	yes	—
309	C₄	C₂×#14524069	1	1	yes	characteristic
310	C₂	—	16	4	no	—
311	C₂	#22559690	1	4	yes	direct product factor
312	C₂	C₂×#16863593	1	1	yes	characteristic
313	C₂	C₂×#16863646	1	1	yes	characteristic
314	C₂	C₂×#16863663	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524014
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting, prime radical	C₂×#493×C₆₇	C₂
socle	C₂³×C₆₇	#121
hypercenter	C₂×#493	D₆₇
hypocenter	C₆₇	C₂×#1688
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#16863593	C₂	semidirect
2. C₂×#16863526	C₂	semidirect
3. #22559690	C₂	direct (N × H)
4. C₂×#16863147	C₂	semidirect
5. C₂×#16863140	C₂	semidirect
10. #16863593	C₂²	semidirect
12. #16863526	C₂²	semidirect
18. #16863147	C₂²	semidirect
19. #16863140	C₂²	semidirect
95. C₂×C₈×C₆₇	SD₁₆	semidirect
132. C₈×C₆₇	C₂×SD₁₆	semidirect
154. C₂×C₆₇	#1688	semidirect
164. C₆₇	C₂×#1688	semidirect
247. C₂×C₈	#11761616	semidirect
292. C₈	C₂×#11761616	semidirect
311. C₂	#22559690	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×#16863593	2
C₂×#16863526	2
#22559690	2
C₂×#16863147	2
C₂×#16863140	2
C₂×#493×C₆₇	2
C₂×#1688	67

Subgroups — C2×#22559690 #30838428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559690 #30838428