Subgroups — C₂×#22574019 #31764015

Order 17152.

6826 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	#22574019	C₂	1	32	yes	maximal, direct product factor
2	C₂×Q₈×Dic₁₃₄	C₂	1	4	yes	maximal, complemented
3	C₂×#16864125	C₂	1	4	yes	maximal, complemented
4	C₂×#16864122	C₂	1	8	yes	maximal, complemented
5	C₂×#16864027	C₂	1	6	yes	maximal
6	C₂×#16863953	C₂	1	6	yes	maximal
7	C₂×#16863874	C₂	1	2	yes	maximal
8	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	Q₈×Dic₁₃₄	C₂²	1	64	yes	complemented
10	#16864125	C₂²	1	64	yes	complemented
11	#16864122	C₂²	1	128	yes	complemented
12	#16864027	C₂²	1	96	yes	—
13	#16863953	C₂²	1	96	yes	—
14	C₂×#14524043	C₂²	1	6	yes	—
15	C₂×#14524043	C₂²	1	6	yes	—
16	C₂×#14524042	C₂²	1	24	yes	—
17	C₂×#14524035	C₂²	1	12	yes	—
18	C₂×#14524034	C₂²	1	24	yes	—
19	C₂×#14524022	C₂²	1	12	yes	—
20	C₂×#14524022	C₂²	1	4	yes	—
21	#16863874	C₂²	1	32	yes	—
22	C₂×Q₈×Dic₆₇	C₂²	1	8	yes	—
23	C₂×Q₈×Dic₆₇	C₂²	1	8	yes	—
24	C₂×C₄×Dic₁₃₄	C₂²	1	12	yes	—
25	C₂×C₄×Dic₁₃₄	C₂²	1	12	yes	—
26	C₂×#14524002	C₂²	1	12	yes	—
27	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₆₇	C₂²	1	16	yes	—
29	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
30	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
31	C₂²×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
32	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
33	C₂²×Dic₁₃₄	C₂³	1	4	yes	—
34	C₂²×Dic₁₃₄	C₂³	1	12	yes	—
35	#14524043	C₂³	1	48	yes	—
36	#14524043	C₂³	1	48	yes	—
37	#14524042	C₂³	1	192	yes	—
38	#14524035	C₂³	1	96	yes	—
39	#14524034	C₂³	1	192	yes	—
40	#14524022	C₂³	1	96	yes	—
41	#14524022	C₂³	1	32	yes	—
42	Q₈×Dic₆₇	C₂³	1	64	yes	—
43	Q₈×Dic₆₇	C₂³	1	64	yes	—
44	C₄×Dic₁₃₄	C₂³	1	96	yes	—
45	C₄×Dic₁₃₄	C₂³	1	96	yes	—
46	#14524002	C₂³	1	96	yes	—
47	C₂×C₄⋊Dic₆₇	C₂³	1	12	yes	—
48	C₂×C₄⋊Dic₆₇	C₂³	1	12	yes	—
49	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
50	C₂×C₄⋊Dic₆₇	C₂³	1	12	yes	—
51	C₂×Dic₆₇⋊C₄	C₂³	1	8	yes	—
52	C₂×Dic₆₇⋊C₄	C₂³	1	24	yes	—
53	C₂×C₂²⋊Dic₆₇	C₂³	1	8	yes	—
54	C₂×C₄×Dic₆₇	C₂³	1	24	yes	—
55	C₂²×Q₈×C₆₇	Q₈	1	16	yes	complemented
56	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
57	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
58	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
60	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
61	C₄×Q₈×C₆₇	C₂³	1	32	yes	—
62	C₄×Q₈×C₆₇	C₂³	1	32	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	24	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
65	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
66	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
67	C₂×C₄²×C₆₇	C₂³	1	24	yes	—
68	C₂×Dic₁₃₄	—	2	16	no	—
69	C₂×Dic₁₃₄	—	4	24	no	—
70	C₂×Dic₁₃₄	C₂⁴	1	16	yes	—
71	C₂×Dic₁₃₄	C₂⁴	1	48	yes	—
72	C₂²×Dic₆₇	C₂⁴	1	8	yes	—
73	C₂²×Dic₆₇	C₂⁴	1	8	yes	—
74	C₄⋊Dic₆₇	C₂⁴	1	48	yes	—
75	C₄⋊Dic₆₇	C₂⁴	1	48	yes	—
76	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
77	C₄⋊Dic₆₇	C₂⁴	1	48	yes	—
78	Dic₆₇⋊C₄	C₂⁴	1	32	yes	—
79	Dic₆₇⋊C₄	C₂⁴	1	96	yes	—
80	C₂²⋊Dic₆₇	C₂⁴	1	32	yes	—
81	C₄×Dic₆₇	C₂⁴	1	96	yes	—
82	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₂×Q₈×C₆₇	C₂×Q₈	1	32	yes	—
84	C₂×Q₈×C₆₇	C₂×Q₈	1	64	yes	complemented
85	C₂×Q₈×C₆₇	C₂⁴	1	16	yes	—
86	C₂²×C₄×C₆₇	—	2	4	no	—
87	C₂²×C₄×C₆₇	—	4	6	no	—
88	C₂²×C₄×C₆₇	C₂×Q₈	1	24	yes	—
89	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
91	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
92	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
93	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
94	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
95	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₆₇	C₂⁴	1	48	yes	—
97	C₄²×C₆₇	C₂⁴	1	24	yes	—
98	C₄²×C₆₇	C₂⁴	1	24	yes	—
99	C₂³×C₆₇	—	2	2	no	—
100	C₂³×C₆₇	—	2	2	no	—
101	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
102	C₂³×C₆₇	C₂⁵	1	2	yes	—
103	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
104	Q₈×C₆₇	C₂²×Q₈	1	64	yes	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	12	no	—
107	C₂×C₄×C₆₇	—	2	8	no	—
108	C₂×C₄×C₆₇	—	4	12	no	—
109	C₂×C₄×C₆₇	C₂²×Q₈	1	24	yes	—
110	C₂×C₄×C₆₇	C₂²×Q₈	1	48	yes	—
111	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
112	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
113	C₂×C₄×C₆₇	C₂⁵	1	12	yes	—
114	C₂×C₄×C₆₇	C₂⁵	1	12	yes	—
115	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
116	C₂×Dic₆₇	—	2	16	no	—
117	C₂×Dic₆₇	—	2	16	no	—
118	C₂×Dic₆₇	C₂⁵	1	16	yes	—
119	C₂×Dic₆₇	C₂⁵	1	16	yes	—
120	Dic₁₃₄	—	2	32	no	—
121	Dic₁₃₄	—	4	48	no	—
122	C₂²×C₆₇	—	2	2	no	—
123	C₂²×C₆₇	—	2	2	no	—
124	C₂²×C₆₇	—	2	8	no	—
125	C₂²×C₆₇	C₂³×Q₈	1	4	yes	—
126	C₂²×C₆₇	C₂³×Q₈	1	2	yes	—
127	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
128	C₂²×C₆₇	#585	1	2	yes	—
129	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
130	C₄×C₆₇	—	2	2	no	—
131	C₄×C₆₇	—	2	12	no	—
132	C₄×C₆₇	C₂³×Q₈	1	24	yes	—
133	C₄×C₆₇	C₂×2₋¹⁺⁴	1	4	yes	—
134	Dic₆₇	—	2	16	no	—
135	Dic₆₇	—	2	16	no	—
136	C₂×#3502	—	67	1	no	maximal, Sylow
137	C₂×C₆₇	—	2	4	no	—
138	C₂×C₆₇	#3502	1	4	yes	complemented
139	C₂×C₆₇	C₂⁴×Q₈	1	1	yes	characteristic
140	C₂×C₆₇	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
141	C₂×C₆₇	C₂×#585	1	1	yes	characteristic
142	#3502	—	67	32	no	—
143	C₂×Q₈²	—	67	4	no	—
144	C₂×#557	—	67	8	no	—
145	C₂×#557	—	67	4	no	—
146	C₂×#544	—	67	6	no	—
147	C₂×#533	—	67	6	no	—
148	C₂×#519	—	67	2	no	—
149	C₂²×C₄×Q₈	D₆₇	1	1	yes	characteristic
150	C₆₇	C₂×#3502	1	1	yes	characteristic, complemented, Sylow
151	C₂³×Q₈	C₂×D₆₇	1	1	yes	characteristic
152	Q₈²	—	67	64	no	—
153	#557	—	67	128	no	—
154	#557	—	67	64	no	—
155	#544	—	67	96	no	—
156	#533	—	67	96	no	—
157	C₂×C₄⋊Q₈	—	67	24	no	—
158	C₂×C₄⋊Q₈	—	67	6	no	—
159	C₂×C₄⋊Q₈	—	67	6	no	—
160	C₂×#125	—	67	24	no	—
161	C₂×#125	—	67	12	no	—
162	C₂×C₂²⋊Q₈	—	67	4	no	—
163	C₂×C₂²⋊Q₈	—	67	12	no	—
164	#519	—	67	32	no	—
165	C₂×C₄×Q₈	—	67	8	no	—
166	C₂×C₄×Q₈	—	67	8	no	—
167	C₂×C₄×Q₈	—	67	12	no	—
168	C₂×C₄×Q₈	—	67	12	no	—
169	C₂×C₄×Q₈	C₂×D₆₇	1	4	yes	—
170	C₂×C₄×Q₈	C₂×D₆₇	1	4	yes	—
171	C₂×C₄×Q₈	C₂×D₆₇	1	16	yes	—
172	C₂×#117	—	67	12	no	—
173	C₂²×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
174	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
175	C₂²×Q₈	—	67	12	no	—
176	C₂²×Q₈	—	67	4	no	—
177	C₂²×Q₈	Dic₁₃₄	1	16	yes	complemented
178	C₂²×Q₈	C₂²×D₆₇	1	4	yes	—
179	C₂²×Q₈	C₂²×D₆₇	1	8	yes	—
180	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
181	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
182	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
183	C₄⋊Q₈	—	67	192	no	—
184	C₄⋊Q₈	—	67	48	no	—
185	C₄⋊Q₈	—	67	48	no	—
186	#125	—	67	192	no	—
187	#125	—	67	96	no	—
188	C₂²⋊Q₈	—	67	32	no	—
189	C₂²⋊Q₈	—	67	96	no	—
190	C₄×Q₈	—	67	64	no	—
191	C₄×Q₈	—	67	64	no	—
192	C₄×Q₈	—	67	96	no	—
193	C₄×Q₈	—	67	96	no	—
194	C₄×Q₈	C₂²×D₆₇	1	32	yes	—
195	C₄×Q₈	C₂²×D₆₇	1	32	yes	—
196	#117	—	67	96	no	—
197	C₂×C₄⋊C₄	—	67	12	no	—
198	C₂×C₄⋊C₄	—	67	12	no	—
199	C₂×C₄⋊C₄	—	67	8	no	—
200	C₂×C₄⋊C₄	—	67	4	no	—
201	C₂×C₄⋊C₄	—	67	12	no	—
202	C₂×C₄⋊C₄	—	67	24	no	—
203	C₂×C₄⋊C₄	C₂²×D₆₇	1	12	yes	—
204	C₂×C₄⋊C₄	C₂²×D₆₇	1	24	yes	—
205	C₂×C₂²⋊C₄	—	67	8	no	—
206	C₂×C₄²	—	67	24	no	—
207	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
208	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
209	C₂×C₄²	C₂²×D₆₇	1	24	yes	—
210	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
211	C₂×Q₈	—	268	24	no	—
212	C₂×Q₈	—	67	48	no	—
213	C₂×Q₈	—	67	16	no	—
214	C₂×Q₈	—	134	16	no	—
215	C₂×Q₈	C₂×Dic₁₃₄	1	32	yes	—
216	C₂×Q₈	C₂×Dic₁₃₄	1	64	yes	complemented
217	C₂×Q₈	C₂³×D₆₇	1	16	yes	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	6	no	—
220	C₂²×C₄	—	67	8	no	—
221	C₂²×C₄	—	67	8	no	—
222	C₂²×C₄	C₂×Dic₁₃₄	1	24	yes	—
223	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
224	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
226	C₂²×C₄	C₂³×D₆₇	1	12	yes	—
227	C₂²×C₄	C₂³×D₆₇	1	12	yes	—
228	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
229	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
230	C₄⋊C₄	—	67	48	no	—
231	C₄⋊C₄	—	67	48	no	—
232	C₄⋊C₄	—	67	32	no	—
233	C₄⋊C₄	—	67	16	no	—
234	C₄⋊C₄	—	67	48	no	—
235	C₄⋊C₄	—	67	96	no	—
236	C₄⋊C₄	C₂³×D₆₇	1	48	yes	—
237	C₂²⋊C₄	—	67	32	no	—
238	C₄²	—	67	96	no	—
239	C₄²	C₂³×D₆₇	1	24	yes	—
240	C₄²	C₂³×D₆₇	1	24	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂²×Dic₁₃₄	1	4	yes	—
244	C₂³	C₂⁴×D₆₇	1	2	yes	—
245	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
246	Q₈	—	268	48	no	—
247	Q₈	—	134	32	no	—
248	Q₈	C₂²×Dic₁₃₄	1	64	yes	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	12	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	4	12	no	—
253	C₂×C₄	—	67	16	no	—
254	C₂×C₄	—	67	16	no	—
255	C₂×C₄	—	134	16	no	—
256	C₂×C₄	—	134	16	no	—
257	C₂×C₄	C₂²×Dic₁₃₄	1	24	yes	—
258	C₂×C₄	C₂²×Dic₁₃₄	1	48	yes	—
259	C₂×C₄	#14524091	1	4	yes	—
260	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
261	C₂×C₄	C₂⁴×D₆₇	1	12	yes	—
262	C₂×C₄	C₂⁴×D₆₇	1	12	yes	—
263	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	8	no	—
267	C₂²	C₂³×Dic₁₃₄	1	4	yes	—
268	C₂²	C₂³×Dic₁₃₄	1	2	yes	—
269	C₂²	C₂×#14524091	1	2	yes	—
270	C₂²	#16864232	1	2	yes	—
271	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	2	12	no	—
274	C₄	—	134	16	no	—
275	C₄	—	134	16	no	—
276	C₄	C₂³×Dic₁₃₄	1	24	yes	—
277	C₄	C₂×#14524091	1	4	yes	—
278	C₂	—	2	4	no	—
279	C₂	#22574019	1	4	yes	direct product factor
280	C₂	C₂⁴×Dic₁₃₄	1	1	yes	characteristic
281	C₂	C₂²×#14524091	1	1	yes	characteristic
282	C₂	C₂×#16864232	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂²×C₄×Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂⁴×C₆₇	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₆₇
hypocenter	C₆₇	C₂×#3502
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #22574019	C₂	direct (N × H)
2. C₂×Q₈×Dic₁₃₄	C₂	semidirect
3. C₂×#16864125	C₂	semidirect
4. C₂×#16864122	C₂	semidirect
9. Q₈×Dic₁₃₄	C₂²	semidirect
10. #16864125	C₂²	semidirect
11. #16864122	C₂²	semidirect
55. C₂²×Q₈×C₆₇	Q₈	semidirect
84. C₂×Q₈×C₆₇	C₂×Q₈	semidirect
138. C₂×C₆₇	#3502	semidirect
150. C₆₇	C₂×#3502	semidirect
177. C₂²×Q₈	Dic₁₃₄	semidirect
216. C₂×Q₈	C₂×Dic₁₃₄	semidirect
279. C₂	#22574019	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22574019	2
C₂×Q₈×Dic₁₃₄	2
C₂×#16864125	2
C₂×#16864122	2
C₂×#16864027	2
C₂×#16863953	2
C₂×#16863874	2
C₂²×C₄×Q₈×C₆₇	2
C₂×#3502	67

Subgroups — C2×#22574019 #31764015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22574019 #31764015