Subgroups — C₂×#22558313 #30770743

Order 17152.

554 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523979	C₂	1	2	yes	maximal
2	#22558313	C₂	1	4	yes	maximal, direct product factor
3	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11761612	—	2	2	no	—
5	C₂×#14523979	—	2	2	no	—
6	C₂×#14523979	—	2	2	no	—
7	C₂×#14523979	—	2	2	no	—
8	C₂×#14523979	C₄	1	4	yes	—
9	C₂×#14523979	C₄	1	4	yes	—
10	C₂×#14523979	C₂²	1	4	yes	—
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₆₇	—	2	2	no	—
13	C₂×M₃₂×C₆₇	—	2	2	no	—
14	C₂×M₃₂×C₆₇	C₄	1	2	yes	—
15	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₆₇	C₄	1	2	yes	—
17	C₂×#11761612	—	2	4	no	—
18	C₂×#11761612	—	2	4	no	—
19	C₂×#11761612	—	2	4	no	—
20	C₂×#11761612	—	4	2	no	—
21	C₂×#11761612	—	4	2	no	—
22	C₂×#11761612	—	4	2	no	—
23	C₂²×C₆₇⋊C₈	—	4	2	no	—
24	#14523979	—	2	4	no	—
25	#14523979	—	2	4	no	—
26	#14523979	—	2	4	no	—
27	#14523979	C₂×C₄	1	8	yes	—
28	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	2	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	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	D₄	1	2	yes	—
35	C₂²×C₈×C₆₇	D₄	1	2	yes	—
36	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	M₃₂×C₆₇	—	2	4	no	—
40	M₃₂×C₆₇	—	2	4	no	—
41	C₂×C₁₆×C₆₇	—	2	2	no	—
42	C₂×C₁₆×C₆₇	—	2	2	no	—
43	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
44	#11761612	—	2	8	no	—
45	#11761612	—	4	4	no	—
46	#11761612	—	4	4	no	—
47	#11761612	—	4	4	no	—
48	C₂×C₆₇⋊C₈	—	4	4	no	—
49	C₂×C₆₇⋊C₈	—	4	4	no	—
50	C₂×C₆₇⋊C₈	—	4	4	no	—
51	C₂⁴×C₆₇	Q₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
59	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
60	C₂²×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₄²	1	2	yes	—
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
71	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
72	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
73	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
80	C₁₆×C₆₇	—	2	4	no	—
81	C₂³×C₆₇	—	4	2	no	—
82	C₂³×C₆₇	#103	1	2	yes	—
83	C₂³×C₆₇	#103	1	2	yes	—
84	C₂³×C₆₇	#107	1	1	yes	characteristic
85	C₂³×C₆₇	#107	1	1	yes	characteristic
86	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
88	C₂×C₄×C₆₇	#95	1	2	yes	—
89	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#102	1	2	yes	—
91	C₂×C₄×C₆₇	#102	1	2	yes	—
92	C₂×C₄×C₆₇	#102	1	2	yes	—
93	C₂×C₄×C₆₇	#102	1	2	yes	—
94	C₂×C₄×C₆₇	#103	1	2	yes	—
95	C₂×C₄×C₆₇	#103	1	2	yes	—
96	C₂×C₄×C₆₇	#106	1	2	yes	—
97	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
105	C₈×C₆₇	#95	1	1	yes	characteristic
106	C₈×C₆₇	#95	1	1	yes	characteristic
107	C₈×C₆₇	#95	1	2	yes	—
108	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
109	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₆₇⋊C₈	—	4	8	no	—
112	C₂²×C₆₇	—	2	4	no	—
113	C₂²×C₆₇	—	2	2	no	—
114	C₂²×C₆₇	—	4	2	no	—
115	C₂²×C₆₇	—	4	1	no	—
116	C₂²×C₆₇	—	4	1	no	—
117	C₂²×C₆₇	#340	1	2	yes	—
118	C₂²×C₆₇	#340	1	1	yes	characteristic
119	C₂²×C₆₇	#340	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×#103	1	2	yes	—
121	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
122	C₄×C₆₇	#340	1	2	yes	—
123	C₄×C₆₇	#340	1	1	yes	characteristic
124	C₄×C₆₇	#340	1	1	yes	characteristic
125	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
126	C₄×C₆₇	C₂×#102	1	2	yes	—
127	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
128	C₂×#1385	—	67	1	no	maximal, Sylow
129	C₂×C₆₇	—	2	2	no	—
130	C₂×C₆₇	—	2	1	no	—
131	C₂×C₆₇	—	2	2	no	—
132	C₂×C₆₇	—	2	1	no	—
133	C₂×C₆₇	#1385	1	2	yes	complemented
134	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
135	C₂²×M₃₂	D₆₇	1	1	yes	characteristic
136	C₂²×#108	—	67	2	no	—
137	#1385	—	67	4	no	—
138	C₆₇	C₂×#1385	1	1	yes	characteristic, complemented, Sylow
139	C₂²×M₁₆	—	134	2	no	—
140	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
141	C₂×M₃₂	—	2	2	no	—
142	C₂×M₃₂	—	2	2	no	—
143	C₂×M₃₂	Dic₆₇	1	2	yes	—
144	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
145	C₂²×C₁₆	Dic₆₇	1	2	yes	—
146	C₂×#108	—	67	4	no	—
147	C₂×#108	—	67	4	no	—
148	C₂×#108	—	67	4	no	—
149	C₂×#108	—	134	2	no	—
150	C₂×#108	—	134	2	no	—
151	C₂×#108	—	134	2	no	—
152	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
153	C₂×M₁₆	—	268	2	no	—
154	C₂×M₁₆	—	268	2	no	—
155	C₂×M₁₆	—	268	2	no	—
156	C₂×M₁₆	—	134	4	no	—
157	C₂×M₁₆	—	134	4	no	—
158	C₂×M₁₆	—	134	4	no	—
159	C₂²×C₈	—	268	2	no	—
160	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
161	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
162	C₂²×C₈	C₄×D₆₇	1	2	yes	—
163	C₂²×C₈	C₄×D₆₇	1	2	yes	—
164	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
165	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
166	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
167	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
168	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
169	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
170	M₃₂	—	2	4	no	—
171	M₃₂	—	2	4	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
175	#108	—	67	8	no	—
176	#108	—	134	4	no	—
177	#108	—	134	4	no	—
178	#108	—	134	4	no	—
179	C₂⁴	Dic₂₆₈	1	1	yes	characteristic
180	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
181	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
182	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
183	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
184	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
185	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
186	C₂²×C₄	#11761617	1	2	yes	—
187	C₂²×C₄	#11761617	1	2	yes	—
188	C₂²×C₄	Dic₂₆₈	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
190	M₁₆	—	268	4	no	—
191	M₁₆	—	268	4	no	—
192	M₁₆	—	268	4	no	—
193	M₁₆	—	134	8	no	—
194	C₂×C₈	—	268	4	no	—
195	C₂×C₈	—	268	4	no	—
196	C₂×C₈	—	268	4	no	—
197	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
198	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
199	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
200	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
201	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
202	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
205	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
206	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
207	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
208	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
209	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
210	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
212	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
213	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
214	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
215	C₁₆	—	2	4	no	—
216	C₂³	—	4	2	no	—
217	C₂³	#14523968	1	2	yes	—
218	C₂³	#14523968	1	2	yes	—
219	C₂³	#14523977	1	1	yes	characteristic
220	C₂³	#14523977	1	1	yes	characteristic
221	C₂³	C₂×Dic₂₆₈	1	1	yes	characteristic
222	C₂×C₄	C₆₇⋊#95	1	2	yes	—
223	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
224	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
225	C₂×C₄	#14523965	1	2	yes	—
226	C₂×C₄	#14523965	1	2	yes	—
227	C₂×C₄	#14523965	1	2	yes	—
228	C₂×C₄	#14523965	1	2	yes	—
229	C₂×C₄	#14523968	1	2	yes	—
230	C₂×C₄	#14523968	1	2	yes	—
231	C₂×C₄	#14523975	1	2	yes	—
232	C₂×C₄	#14523975	1	1	yes	characteristic
233	C₂×C₄	#14523975	1	1	yes	characteristic
234	C₂×C₄	#14523977	1	1	yes	characteristic
235	C₂×C₄	#14523977	1	1	yes	characteristic
236	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
237	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
238	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
239	C₂×C₄	C₂×#11761617	1	2	yes	—
240	C₈	—	268	8	no	—
241	C₈	C₆₇⋊#95	1	1	yes	characteristic
242	C₈	C₆₇⋊#95	1	1	yes	characteristic
243	C₈	C₆₇⋊#95	1	2	yes	—
244	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
245	C₈	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
246	C₈	C₂×Dic₆₇⋊C₄	1	2	yes	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	1	no	—
251	C₂²	—	4	1	no	—
252	C₂²	#16863159	1	2	yes	—
253	C₂²	#16863159	1	1	yes	characteristic
254	C₂²	#16863159	1	1	yes	characteristic
255	C₂²	C₂×#14523968	1	2	yes	—
256	C₂²	C₂×#14523977	1	1	yes	characteristic
257	C₄	#16863159	1	2	yes	—
258	C₄	#16863159	1	1	yes	characteristic
259	C₄	#16863159	1	1	yes	characteristic
260	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
261	C₄	C₂×#14523965	1	2	yes	—
262	C₄	C₂×#14523975	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	2	2	no	—
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	#22558313	1	2	yes	direct product factor
268	C₂	C₂×#16863159	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14523975
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	#340
prime radical	C₂⁴×C₆₇	Q₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×M₃₂	D₆₇
hypocenter	C₆₇	C₂×#1385
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. #22558313	C₂	direct (N × H)
133. C₂×C₆₇	#1385	semidirect
138. C₆₇	C₂×#1385	semidirect
267. C₂	#22558313	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×#14523979	2
#22558313	2
C₂²×M₃₂×C₆₇	2
C₂×#1385	67

Subgroups — C2×#22558313 #30770743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558313 #30770743