Subgroups — C₂×#22562409 #31010620

Order 17152.

630 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523977	C₂	1	1	yes	maximal, characteristic
2	#22562409	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863221	C₂	1	2	yes	maximal, complemented
4	C₂×#16863200	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
6	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₆₇	—	2	1	no	—
8	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	C₂×#14523977	—	2	2	no	—
10	C₂×#14523977	—	2	1	no	—
11	C₂×#14523977	C₂²	1	4	yes	—
12	C₂×#14523977	C₂²	1	2	yes	—
13	#16863221	C₂²	1	8	yes	complemented
14	#16863200	C₂²	1	8	yes	complemented
15	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
16	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
19	#467×C₆₇	—	2	2	no	—
20	#467×C₆₇	C₂²	1	4	yes	—
21	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#103×C₆₇	—	2	1	no	—
23	C₂³×Dic₆₇	—	4	1	no	—
24	C₂×C₄⋊Dic₆₇	—	2	4	no	—
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	—	4	1	no	—
27	C₂×C₄⋊Dic₆₇	—	4	2	no	—
28	C₂×C₆₇⋊C₁₆	—	2	4	no	—
29	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
30	#14523977	—	2	4	no	—
31	#14523977	—	2	2	no	—
32	#14523977	C₂³	1	4	yes	—
33	C₂²×Q₈×C₆₇	—	2	1	no	—
34	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
35	C₂×Q₁₆×C₆₇	—	2	1	no	—
36	C₂×Q₁₆×C₆₇	—	2	4	no	—
37	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₆₇	—	2	4	no	—
39	C₂²×C₈×C₆₇	D₄	1	2	yes	—
40	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²⋊Q₈×C₆₇	—	2	4	no	—
44	C₂²⋊Q₈×C₆₇	—	4	2	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
46	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
47	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
48	#107×C₆₇	—	2	1	no	—
49	#107×C₆₇	C₂³	1	2	yes	—
50	#103×C₆₇	—	2	4	no	—
51	C₂²×Dic₆₇	—	4	4	no	—
52	C₂²×Dic₆₇	—	4	2	no	—
53	C₂²×Dic₆₇	—	8	4	no	—
54	C₄⋊Dic₆₇	—	2	4	no	—
55	C₄⋊Dic₆₇	—	4	4	no	—
56	C₄⋊Dic₆₇	—	4	2	no	—
57	C₆₇⋊C₁₆	—	2	4	no	—
58	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
59	C₂×Q₈×C₆₇	—	2	2	no	—
60	C₂×Q₈×C₆₇	—	4	1	no	—
61	C₂×Q₈×C₆₇	—	4	2	no	—
62	C₂×Q₈×C₆₇	—	4	2	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	4	1	no	—
66	C₂²×C₄×C₆₇	—	4	1	no	—
67	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	D₈	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	Q₁₆×C₆₇	—	2	4	no	—
72	Q₁₆×C₆₇	—	4	2	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	—	2	4	no	—
76	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	#41	1	4	yes	—
79	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₄⋊C₄×C₆₇	—	4	1	no	—
82	C₄⋊C₄×C₆₇	—	4	4	no	—
83	C₂²⋊C₄×C₆₇	—	4	4	no	—
84	C₂³×C₆₇	—	2	2	no	—
85	C₂³×C₆₇	—	2	2	no	—
86	C₂³×C₆₇	Q₃₂	1	4	yes	—
87	C₂³×C₆₇	C₂×D₈	1	2	yes	—
88	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
89	Q₈×C₆₇	—	4	2	no	—
90	Q₈×C₆₇	—	4	2	no	—
91	Q₈×C₆₇	—	8	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	4	2	no	—
97	C₂×C₄×C₆₇	—	4	2	no	—
98	C₂×C₄×C₆₇	—	8	2	no	—
99	C₂×C₄×C₆₇	—	8	2	no	—
100	C₂×C₄×C₆₇	#123	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#137	1	2	yes	—
104	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×Dic₆₇	—	4	4	no	—
106	C₂×Dic₆₇	—	8	8	no	—
107	C₂×Dic₆₇	—	8	4	no	—
108	C₈×C₆₇	—	2	2	no	—
109	C₈×C₆₇	#123	1	2	yes	—
110	C₈×C₆₇	C₂×#41	1	2	yes	—
111	C₂²×C₆₇	—	2	2	no	—
112	C₂²×C₆₇	—	2	2	no	—
113	C₂²×C₆₇	—	2	8	no	—
114	C₂²×C₆₇	#480	1	2	yes	—
115	C₂²×C₆₇	C₂×Q₃₂	1	4	yes	—
116	C₂²×C₆₇	C₂×Q₃₂	1	2	yes	—
117	C₂²×C₆₇	#509	1	2	yes	—
118	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
119	C₄×C₆₇	—	2	2	no	—
120	C₄×C₆₇	—	8	2	no	—
121	C₄×C₆₇	—	8	2	no	—
122	C₄×C₆₇	#480	1	2	yes	—
123	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
124	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
125	Dic₆₇	—	8	4	no	—
126	C₂×#2236	—	67	1	no	maximal, Sylow
127	C₂×C₆₇	—	2	4	no	—
128	C₂×C₆₇	#2236	1	4	yes	complemented
129	C₂×C₆₇	C₂×#480	1	1	yes	characteristic
130	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
131	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
132	C₂×#467	D₆₇	1	1	yes	characteristic
133	C₂²×#107	—	67	1	no	—
134	#2236	—	67	8	no	—
135	C₂×#366	—	67	2	no	—
136	C₂×#358	—	67	2	no	—
137	C₂×C₂²⋊C₁₆	—	67	1	no	—
138	C₆₇	C₂×#2236	1	1	yes	characteristic, complemented, Sylow
139	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
140	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
141	C₂×C₂²⋊Q₈	—	2	1	no	—
142	C₂²×C₄⋊C₄	—	134	1	no	—
143	C₂²×C₁₆	—	67	2	no	—
144	#467	—	2	2	no	—
145	#467	C₂×D₆₇	1	4	yes	—
146	C₂×#107	—	67	2	no	—
147	C₂×#107	—	67	4	no	—
148	C₂×#107	—	134	1	no	—
149	C₂×#107	—	134	2	no	—
150	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
151	C₂×#103	—	2	1	no	—
152	#366	—	67	8	no	—
153	#358	—	67	8	no	—
154	C₂²⋊C₁₆	—	67	4	no	—
155	C₂²×Q₈	—	2	1	no	—
156	C₂³×C₄	—	268	1	no	—
157	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₂×Q₁₆	—	2	1	no	—
159	C₂×Q₁₆	—	2	4	no	—
160	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
161	C₂²×C₈	—	2	4	no	—
162	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
164	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
165	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
166	C₂²⋊Q₈	—	2	4	no	—
167	C₂²⋊Q₈	—	4	2	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	—	4	1	no	—
170	C₂×C₄⋊C₄	—	268	2	no	—
171	C₂×C₄⋊C₄	—	268	1	no	—
172	C₂×C₄⋊C₄	—	134	4	no	—
173	C₂×C₄⋊C₄	—	134	2	no	—
174	C₂×C₂²⋊C₄	—	4	1	no	—
175	C₂×C₁₆	—	67	4	no	—
176	C₂×C₁₆	—	134	4	no	—
177	#107	—	2	1	no	—
178	#107	—	67	4	no	—
179	#107	—	134	4	no	—
180	#107	—	134	2	no	—
181	#107	C₂²×D₆₇	1	2	yes	—
182	#103	—	2	4	no	—
183	C₂⁴	#11761613	1	1	yes	characteristic
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	4	1	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	4	1	no	—
191	C₂²×C₄	—	4	1	no	—
192	C₂²×C₄	—	268	4	no	—
193	C₂²×C₄	—	268	2	no	—
194	C₂²×C₄	—	536	4	no	—
195	C₂²×C₄	#11761613	1	2	yes	—
196	C₂²×C₄	#11761613	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
198	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
199	Q₁₆	—	2	4	no	—
200	Q₁₆	—	4	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
205	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
206	C₂×C₈	#11761627	1	4	yes	—
207	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	268	4	no	—
212	C₄⋊C₄	—	268	2	no	—
213	C₄⋊C₄	—	134	4	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₁₆	—	134	4	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	#14523989	1	4	yes	—
219	C₂³	C₂×#11761613	1	2	yes	—
220	C₂³	C₂×#11761613	1	1	yes	characteristic
221	Q₈	—	4	2	no	—
222	Q₈	—	4	2	no	—
223	Q₈	—	8	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	268	4	no	—
233	C₂×C₄	—	536	8	no	—
234	C₂×C₄	—	536	4	no	—
235	C₂×C₄	#14524024	1	2	yes	—
236	C₂×C₄	C₂×#11761613	1	2	yes	—
237	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
238	C₂×C₄	#14524073	1	2	yes	—
239	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	#14524024	1	2	yes	—
242	C₈	C₂×#11761627	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	8	no	—
246	C₂²	#16863707	1	2	yes	—
247	C₂²	C₂×#14523989	1	4	yes	—
248	C₂²	C₂×#14523989	1	2	yes	—
249	C₂²	#16863831	1	2	yes	—
250	C₂²	C₂²×#11761613	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	8	2	no	—
253	C₄	—	8	2	no	—
254	C₄	—	536	4	no	—
255	C₄	#16863707	1	2	yes	—
256	C₄	C₂×#14524024	1	1	yes	characteristic
257	C₄	C₂×#14524073	1	1	yes	characteristic
258	C₂	—	2	4	no	—
259	C₂	#22562409	1	4	yes	direct product factor
260	C₂	C₂×#16863707	1	1	yes	characteristic
261	C₂	C₂²×#14523989	1	1	yes	characteristic
262	C₂	C₂×#16863831	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#467×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂⁴×C₆₇	D₈
hypercenter	C₂×#467	D₆₇
hypocenter	C₆₇	C₂×#2236
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
2. #22562409	C₂	direct (N × H)
3. C₂×#16863221	C₂	semidirect
4. C₂×#16863200	C₂	semidirect
13. #16863221	C₂²	semidirect
14. #16863200	C₂²	semidirect
128. C₂×C₆₇	#2236	semidirect
138. C₆₇	C₂×#2236	semidirect
259. C₂	#22562409	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523977	2
#22562409	2
C₂×#16863221	2
C₂×#16863200	2
C₂×C₂²⋊(C₆₇⋊C₁₆)	2
C₂×#467×C₆₇	2
C₂×#2236	67

Subgroups — C2×#22562409 #31010620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562409 #31010620