Subgroups — C₂×#22562025 #31008958

Order 17152.

538 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562025	C₂	1	8	yes	maximal, direct product factor
2	C₂×#16863217	C₂	1	2	yes	maximal, complemented
3	C₂×#16863178	C₂	1	2	yes	maximal
4	C₂×#16863174	C₂	1	2	yes	maximal, complemented
5	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	4	yes	—
7	#16863217	C₂²	1	8	yes	complemented
8	#16863178	C₂²	1	8	yes	—
9	#16863174	C₂²	1	8	yes	complemented
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₄	1	1	yes	characteristic
12	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
13	#405×C₆₇	C₄	1	4	yes	—
14	#405×C₆₇	C₂²	1	4	yes	—
15	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
16	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#14523983	—	2	4	no	—
20	#14523983	C₂³	1	8	yes	—
21	C₂×C₆₇⋊C₁₆	—	2	4	no	—
22	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₆₇	—	2	2	no	—
24	C₂×M₁₆×C₆₇	—	2	2	no	—
25	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
26	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
27	C₂²×C₈×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	—	2	2	no	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	#117×C₆₇	—	2	2	no	—
35	#117×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	#97×C₆₇	C₂×C₄	1	4	yes	—
41	#97×C₆₇	C₂×C₄	1	2	yes	—
42	#97×C₆₇	C₂³	1	2	yes	—
43	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₆₇	C₂³	1	2	yes	—
46	C₆₇⋊C₁₆	—	2	8	no	—
47	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	—	2	1	no	—
49	C₂²×C₄×C₆₇	—	2	1	no	—
50	C₂²×C₄×C₆₇	—	2	1	no	—
51	C₂²×C₄×C₆₇	—	2	1	no	—
52	C₂²×C₄×C₆₇	—	2	1	no	—
53	C₂²×C₄×C₆₇	—	2	1	no	—
54	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	M₁₆×C₆₇	—	2	4	no	—
59	M₁₆×C₆₇	—	2	4	no	—
60	C₂×C₈×C₆₇	—	2	1	no	—
61	C₂×C₈×C₆₇	—	2	1	no	—
62	C₂×C₈×C₆₇	—	2	1	no	—
63	C₂×C₈×C₆₇	—	2	1	no	—
64	C₂×C₈×C₆₇	—	2	8	no	—
65	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
66	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
67	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₆₇	#41	1	2	yes	—
72	C₂×C₈×C₆₇	#41	1	2	yes	—
73	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₆₇	—	2	4	no	—
75	C₂²⋊C₄×C₆₇	—	2	4	no	—
76	C₄²×C₆₇	—	2	1	no	—
77	C₄²×C₆₇	—	2	1	no	—
78	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂³×C₆₇	—	2	1	no	—
81	C₂³×C₆₇	—	2	1	no	—
82	C₂³×C₆₇	—	4	2	no	—
83	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
84	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	4	2	no	—
94	C₂×C₄×C₆₇	—	4	2	no	—
95	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
102	C₈×C₆₇	—	2	2	no	—
103	C₈×C₆₇	—	2	2	no	—
104	C₈×C₆₇	#117	1	2	yes	—
105	C₈×C₆₇	C₂×M₁₆	1	4	yes	—
106	C₈×C₆₇	C₂×#41	1	2	yes	—
107	C₂²×C₆₇	—	2	2	no	—
108	C₂²×C₆₇	—	2	2	no	—
109	C₂²×C₆₇	—	4	2	no	—
110	C₂²×C₆₇	—	4	2	no	—
111	C₂²×C₆₇	—	4	2	no	—
112	C₂²×C₆₇	#432	1	1	yes	characteristic
113	C₂²×C₆₇	#432	1	1	yes	characteristic
114	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	—	4	2	no	—
117	C₄×C₆₇	—	4	2	no	—
118	C₄×C₆₇	#432	1	1	yes	characteristic
119	C₄×C₆₇	#432	1	1	yes	characteristic
120	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
121	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
122	C₂×#2166	—	67	1	no	maximal, Sylow
123	C₂×C₆₇	—	2	1	no	—
124	C₂×C₆₇	—	2	1	no	—
125	C₂×C₆₇	—	4	2	no	—
126	C₂×C₆₇	#2166	1	2	yes	complemented
127	C₂×C₆₇	C₂×#432	1	1	yes	characteristic
128	C₂×#405	D₆₇	1	1	yes	characteristic
129	#2166	—	67	8	no	—
130	C₂×#364	—	67	2	no	—
131	C₂×#349	—	67	2	no	—
132	C₂×#347	—	67	2	no	—
133	C₆₇	C₂×#2166	1	1	yes	characteristic, complemented, Sylow
134	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
135	C₂³×C₈	Dic₆₇	1	1	yes	characteristic
136	C₂×#117	Dic₆₇	1	1	yes	characteristic
137	C₂×M₃₂	—	67	4	no	—
138	#405	Dic₆₇	1	4	yes	—
139	#405	C₂×D₆₇	1	4	yes	—
140	C₂×#97	Dic₆₇	1	1	yes	characteristic
141	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
142	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
144	#364	—	67	8	no	—
145	#349	—	67	8	no	—
146	#347	—	67	8	no	—
147	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
151	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
155	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
156	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
159	#117	—	2	2	no	—
160	#117	C₂×Dic₆₇	1	4	yes	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₂²⋊C₄	—	2	1	no	—
163	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
165	M₃₂	—	67	8	no	—
166	M₃₂	—	134	4	no	—
167	C₂×C₁₆	—	134	4	no	—
168	#97	C₂×Dic₆₇	1	4	yes	—
169	#97	C₂×Dic₆₇	1	2	yes	—
170	#97	C₂²×D₆₇	1	2	yes	—
171	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
172	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
173	C₄×C₈	C₂²×D₆₇	1	2	yes	—
174	C₂⁴	#11761612	1	1	yes	characteristic
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	#11761612	1	1	yes	characteristic
182	C₂²×C₄	#11761612	1	2	yes	—
183	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
184	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
185	M₁₆	—	2	4	no	—
186	M₁₆	—	2	4	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	#11761612	1	2	yes	—
193	C₂×C₈	#11761612	1	2	yes	—
194	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
195	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
196	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
197	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
198	C₂×C₈	#11761628	1	2	yes	—
199	C₂×C₈	#11761628	1	2	yes	—
200	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
201	C₄⋊C₄	—	2	4	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	C₂²×Dic₆₇	1	2	yes	—
206	C₄²	C₂²×Dic₆₇	1	2	yes	—
207	C₁₆	—	134	8	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	4	2	no	—
211	C₂³	C₂×#11761612	1	1	yes	characteristic
212	C₂³	C₂×#11761612	1	2	yes	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	#14524002	1	1	yes	characteristic
224	C₂×C₄	#14524002	1	1	yes	characteristic
225	C₂×C₄	C₂×#11761612	1	2	yes	—
226	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
227	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
228	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
229	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	2	2	no	—
232	C₈	#14524002	1	2	yes	—
233	C₈	C₂×#11761612	1	4	yes	—
234	C₈	C₂×#11761628	1	2	yes	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	#16863453	1	1	yes	characteristic
241	C₂²	#16863453	1	1	yes	characteristic
242	C₂²	C₂²×#11761612	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	#16863453	1	1	yes	characteristic
247	C₄	#16863453	1	1	yes	characteristic
248	C₄	C₂×#14524002	1	1	yes	characteristic
249	C₄	C₂²×#11761612	1	1	yes	characteristic
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	4	2	no	—
253	C₂	#22562025	1	2	yes	direct product factor
254	C₂	C₂×#16863453	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524002
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#405×C₆₇	C₂
socle	C₂²×C₆₇	#432
prime radical	C₂⁴×C₆₇	M₁₆
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#405	D₆₇
hypocenter	C₆₇	C₂×#2166
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. #22562025	C₂	direct (N × H)
2. C₂×#16863217	C₂	semidirect
4. C₂×#16863174	C₂	semidirect
7. #16863217	C₂²	semidirect
9. #16863174	C₂²	semidirect
126. C₂×C₆₇	#2166	semidirect
133. C₆₇	C₂×#2166	semidirect
253. C₂	#22562025	direct (N × H)

Maximal subgroups

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

subgroup	index
#22562025	2
C₂×#16863217	2
C₂×#16863178	2
C₂×#16863174	2
C₂×#405×C₆₇	2
C₂×#2166	67

Subgroups — C2×#22562025 #31008958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562025 #31008958